ieee antennas and propagation

AUGUST 2004 VOLUME 52 NUMBER 8 IETPAK (ISSN 0018-926X)

EDITORIAL

A Note From the Outgoing Editor-in-Chief . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . A. W. Glisson 1926

PAPERS

Bandwidth Enhancement and Further Size Reduction of a Class of Miniaturized Slot Antennas. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Behdad and K. Sarabandi 1928

Miniature Built-In Multiband Antennas for Mobile Handsets . . . . . . . .. . . . . . . . Y.-X. Guo, M. Y. W. Chia, and Z. N. Chen 1936Miniature Reconfigurable Three-Dimensional Fractal Tree Antennas. . . . . . . . . .. . . . . . . . . . J. S. Petko and D. H. Werner 1945Investigations on Miniaturized Endfire Vertically Polarized Quasi-Fractal Log-Periodic Zigzag Antenna . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. K. Sharma and L. Shafai 1957Compact Wide-Band Multimode Antennas for MIMO and Diversity . . . . . . . . .. . . . . . . . C. Waldschmidt and W. Wiesbeck 1963Ground Influence on the Input Impedance of Transient Dipole and Bow-Tie Antennas . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. A. Lestari, A. G. Yarovoy, and L. P. Ligthart 1970Adaptive Crossed Dipole Antennas Using a Genetic Algorithm. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .R. L. Haupt 1976Modeling and Investigation of a Geometrically Complex UWB GPR Antenna Using FDTD . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .K.-H. Lee, C.-C. Chen, F. L. Teixeira, and R. Lee 1983Radiation Properties of an Arbitrarily Flanged Parallel-Plate Waveguide . . . . .. . . . D. N. Chien, K. Tanaka, and M. Tanaka 1992Scan Blindness Free Phased Array Design Using PBG Materials. . . .. . . .L. Zhang, J. A. Castaneda, and N. G. Alexopoulos 2000Fractile Arrays: A New Class of Tiled Arrays With Fractal Boundaries . . . .. . . D. H. Werner, W. Kuhirun, and P. L. Werner 2008A New Millimeter-Wave Printed Dipole Phased Array Antenna Using Microstrip-Fed Coplanar Stripline Tee Junctions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Y.-H. Suh and K. Chang 2019Physical Limitations of Antennas in a Lossy Medium. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . A. Karlsson 2027Minimum Norm Mutual Coupling Compensation With Applications in Direction of Arrival Estimation . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .C. K. E. Lau, R. S. Adve, and T. K. Sarkar 2034A Phase-Space Beam Summation Formulation for Ultrawide-band Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Shlivinski, E. Heyman, A. Boag, and C. Letrou 2042

(Contents Continued on 925)

(Contents Continued from Front Cover)

Theoretical Considerations in the Optimization of Surface Waves on a Planar Structure . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. F. Mahmoud, Y. M. M. Antar, H. F. Hammad, and A. P. Freundorfer 2057

Generalized System Function Analysis of Exterior and Interior Resonances of Antenna and Scattering Problems . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Li and C.-H. Liang 2064

MIMO Wireless Communication Channel Phenomenology. . . . . . . . .. . . . . . . . .D. W. Bliss, A. M. Chan, and N. B. Chang 2073Service Oriented Statistics of Interruption Time Due to Rainfall in Earth-Space Communication Systems . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Matricciani 2083Full-Wave Analysis of Dielectric Frequency-Selective Surfaces Using a Vectorial Modal Method . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Coves, B. Gimeno, J. Gil, M. V. Andrés, A. A. San Blas, and V. E. Boria 2091On the Interaction Between Electric and Magnetic Currents in Stratified Media . . . .. . . . D. Llorens del Río and J. R. Mosig 2100Scattering by Arbitrarily-Shaped Slots in Thick Conducting Screens: An Approximate Solution . . . . . .. . . . . . . J. R. Mosig 2109Double Higher Order Method of Moments for Surface Integral Equation Modeling of Metallic and Dielectric Antennas

and Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . M. Djordjevic and B. M. Notaros 2118Loop-Tree Implementation of the Adaptive Integral Method (AIM) for Numerically-Stable, Broadband, Fast

Electromagnetic Modeling . . . . . . . . . . . . . . .. . . . . . . . . . . . . . V. I. Okhmatovski, J. D. Morsey, and A. C. Cangellaris 2130A Single-Level Low Rank IE-QR Algorithm for PEC Scattering Problems Using EFIE Formulation . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. M. Seo and J.-F. Lee 2141Accelerated Gradient Based Optimization Using Adjoint Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. K. Nikolova, R. Safian, E. A. Soliman, M. H. Bakr, and J. W. Bandler 2147A Theoretical Study of the Stability Criteria for Hybridized FDTD Algorithms for Multiscale Analysis . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Marrone and R. Mittra 2158

COMMUNICATIONS

Full-Wave Analysis of a Waveguide Printed Slot . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . G. Montisci and G. Mazzarella 2168Dual Polarized Wide-Band Aperture Stacked Patch Antennas . . . . . . . . . .. . . . . . . . . . .K. Ghorbani and R. B. Waterhouse 2171Resonant Frequency of Equilateral Triangular Microstrip Antenna With and Without Air Gap. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .D. Guha and J. Y. Siddiqui 2174Effect of a Cavity Enclosure on the Resonant Frequency of Inverted Microstrip Circular Patch Antenna. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .D. Guha and J. Y. Siddiqui 2177Design and Development of Multiband Coaxial Continuous Transverse Stub (CTS) Antenna Arrays . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Isom, M. F. Iskander, Z. Yun, and Z. Zhang 2180Near-Field, Spherical-Scanning Antenna Measurements With Nonideal Probe Locations . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. C. Wittmann, B. K. Alpert, and M. H. Francis 2184Resonance Series Representation of the Early-Time Field Scattered by a Coated Cylinder. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Vollmer and E. J. Rothwell 2186High Order Symplectic Integration Methods for Finite Element Solutions to Time Dependent Maxwell Equations . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .R. Rieben, D. White, and G. Rodrigue 2190

CORRECTIONS

Corrections to “Phased Arrays Based on Oscillators Coupled on Triangular and Hexagonal Lattices” . . .. . . R. J. Pogorzelski 2196

CALLS FOR PAPERS

Special Issue on Multifunction Antennas and Antenna Systems. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 2197

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1926 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 8, AUGUST 2004

A Note from the Outgoing Editor-in-Chief

I T HAS BEEN an honor and a privilege for me to serve as theEditor-in-Chief of the IEEE TRANSACTIONS ON ANTENNAS

AND PROPAGATION for the past three years. I want to expressmy sincere thanks to the Antennas and Propagation (AP) So-ciety Administrative Committee for giving me the opportunityto serve the Society in this capacity. Although I was initiallyreluctant to accept the great responsibility that goes with thisposition, it has been a unique and rewarding experience. TheTRANSACTIONS is the leading journal in its field and the AP So-ciety is justifiably proud of it. I can only hope that we have beensuccessful during my tenure as Editor-in-Chief (EIC) in main-taining the outstanding quality of the Transactions that APSmembers expect and deserve.

During the last three years the annual number of pagespublished in the TRANSACTIONS has been increased from 1,960pages in 2001 to 3,400 pages in 2003 to decrease the publi-cation backlog. Page budgets that are required to be set far inadvance along with an increasing number of submissions havemade it difficult to reduce the publication backlog. A specialthanks is due to the APS AdCom for authorizing an extraor-dinary expenditure of funds to increase the number of pagespublished in 2003 to help reduce the backlog. Additionally, theIEEE has recently approved more flexible rules with regard topage budgets that will hopefully make it easier to avoid largebacklogs in the future.

The transition to an all-electronic process from submission topublication is also well underway. Galley proof delivery to au-thors has become electronic. Corrections can be noted on thegalley proof files with Adobe Acrobat software and returnedelectronically as well. A new electronic copyright form has re-cently been included in Manuscript Central. The promise of acompletely paper-free process is expected to be achieved soon,possibly within the next year. When you have the chance, pleasethank Wilson Pearson, the previous EIC, and Anthony Martinfor leading the way in implementing the electronic submissionprocess.

As I turn over the reins to the new Editor-in-Chief, Dr. TrevorS. Bird, I want express my deepest thanks to all of you whohave served as reviewers for the TRANSACTIONS over the pastthree years. The review process is critical to maintaining thequality of the Transactions. I also particularly want to thank allthe Associate Editors who have worked so hard. Their namesare still listed on the inside back cover in this issue, but I listthem here again to emphasize their outstanding service:

• Jørgen Bach Andersen• Yahia Antar• Jennifer Bernhard• Trevor Bird• Filippo Capolino


• Lawrence Carin• Christos Chrisodoulou• Cynthia Furse• Stephen Gedney• George Hanson• Michael Jensen• Leo Kempel• Chi Chung Ko• Karl Langenberg• Louis Medgyesi-Mitschang• Kathleen Melde• Krzysztof Michalski• Eric Michielssen• Michal Okoniewski• Hsueh-Yuan Pao• Sembiam Rengarajan• Antoine Roederer• Kamal Sarabandi• Ari Sihvola• Rainee Simons• Parveen Wahid

Thanks are also due to the Editors of the special issuesthat have appeared and are currently in preparation: MagdyIskander and Jim Mink, for the Special Issue on WirelessInformation Technology and Networks; Rick Ziolkowski andNader Engheta, for the Special Issue on Metamaterials; andPer-Simon Kildal, Ahmed Kishk, and Stefano Maci, for theupcoming Special Issue on Artificial Magnetic Conductors,Soft/Hard Surfaces, and other Complex Surfaces.

Special thanks are also extended to former Associate Ed-itor Roena Rabelo Vega and current Associate Editor Dawn L.Menendez at IEEE Headquarters who have worked hard on pro-ducing a quality product. Dawn, in particular, has done a tremen-dous job in getting our publication schedule back on track.

Finally, I want to give a special thanks to Sharon Martinez,who has served admirably as my Editorial Assistant. She haskept us organized, worked to keep Reviewers and Associate Ed-itors on schedule, answered author questions, and generally keptthings running efficiently. Without her help your Editor’s Officewould have dissolved into chaos after the first few months.

In closing, I again thank the AP Society for the opportunityto serve, and I hope that you will all support Trevor as he takeson his new role. He has done a superb job as an Associate Editorand I know he is committed to serving the AP Society and theTRANSACTIONS. There is no doubt that the TRANSACTIONS willbe in good hands.

ALLEN W. GLISSON, Outgoing Editor-in-ChiefThe University of MississippiDepartment of Electrical EngineeringUniversity, MS 38677-1848 USA

0018-926X/04$20.00 © 2004 IEEE

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 8, AUGUST 2004 1927

Allen W. Glisson (S’71–M’78–SM’88–F’02) received the B.S., M.S., and Ph.D. degrees in elec-trical engineering from the University of Mississippi, in 1973, 1975, and 1978, respectively.

In 1978, he joined the faculty of the University of Mississippi, where he is currently aProfessor and Chair of the Department of Electrical Engineering. His current research interestsinclude the development and application of numerical techniques for treating electromagneticradiation and scattering problems, and modeling of dielectric resonators and dielectric resonatorantennas. He has been actively involved in the areas of numerical modeling of arbitrarilyshaped bodies and bodies of revolution with surface integral equation formulations. He hasalso served as a consultant to several different industrial organizations in the area of numericalmodeling in electromagnetics.

Dr. Glisson is a Member of Sigma Xi Research Society and the Tau Beta Pi, Phi Kappa Phi, andEta Kappa Nu Honor Societies. He is a Member of several professional societies within the IEEE,Commission B of the International Union of Radio Science (URSI), and the Applied Computa-

tional Electromagnetics Society. He was a U.S. delegate to the 22nd, 23rd, and 24th General Assemblies of URSI. He was selectedas the Outstanding Engineering Faculty Member in 1986, 1996, and 2004. He received a Ralph R. Teetor Educational Award in1989 and the Faculty Service Award in the School of Engineering in 2002. He received a Best Paper Award from the SUMMAFoundation and twice received a citation for excellence in refereeing from the American Geophysical Union. He is the recipientof the 2004 Microwave Prize awarded by the Microwave Theory and Techniques Society. He has served as a member of the IEEEAntennas and Propagation Society Administrative Committee and is currently a member of the IEEE Press Liaison Committee.He currently serves on the Board of Directors of the Applied Computational Electromagnetics Society and has recently served asCo-Editor-in-Chief of the Applied Computational Electromagnetics Society Journal. He has also served as an Associate Editor forRadio Science and as the Secretary of Commission B of the U.S. National Committee of URSI. From August 2001 to July 2004he was the Editor-in-Chief of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.


Bandwidth Enhancement and Further Size Reductionof a Class of Miniaturized Slot Antennas

Nader Behdad, Student Member, IEEE, and Kamal Sarabandi, Fellow, IEEE

Abstract—In this paper, new methods for further reducingthe size and/or increasing the bandwidth (BW) of a class ofminiaturized slot antennas are presented. This paper examinestechniques such as parasitic coupling and inductive loading toachieve higher BW and further size reduction for this classof miniaturized slot antennas. The overall BW of a proposeddouble resonant antenna is shown to be increased by more than94% compared with a single resonant antenna occupying thesame area. The behavior of miniaturized slot antennas, loadedwith series inductive elements along the radiating section is alsoexamined. The inductive loads are constructed by two balancedshort circuited slot lines placed on opposite sides of the radiatingslot. These inductive loads can considerably reduce the antennasize at its resonance. Prototypes of a double resonant antenna at850 MHz and inductively loaded miniaturized antennas at around1 GHz are designed and tested. Finally the application of bothmethods in a dual band miniaturized antenna is presented. In allcases measured and simulated results show excellent agreement.

Index Terms—Slot antennas, electrically small antennas, para-sitic antennas, multifrequency antennas.

I. INTRODUCTION

CURRENT advancements in communication technologyand significant growth in the wireless communication

market and consumer demands demonstrate the need forsmaller, more reliable and power efficient, integrated wirelesssystems. Integrating entire transceivers on a single chip isthe vision for future wireless systems. This has the benefit ofcost reduction and improving system reliability. Antennas areconsidered to be the largest components of integrated wirelesssystems; therefore antenna miniaturization is a necessary taskin achieving an optimal design for integrated wireless systems.The subject of antenna miniaturization is not new and has beenextensively studied by various authors [1]–[4]. Early studieshave shown that for a resonant antenna, as size decreases,bandwidth (BW) and efficiency will also decrease [1]. This is afundamental limitation which, in general, holds true indepen-dent of antenna architecture. However, research on the designof antenna topologies and architectures must be carried outto achieve maximum possible BW and efficiency for a givenantenna size. Impedance matching for small antennas is alsochallenging and often requires external matching networks;Therefore antenna topologies and structures which inherently

Manuscript received May 22, 2003; revised September 30, 2003. This workwas supported in part by the Engineering Research Centers program of the Na-tional Science Foundation (NSF) under Award EEC-9986866 and by the U.S.Army Research Office under Contract DAA-99-1-01971.

The authors are with The Radiation Laboratory, Department of Electrical En-gineering and Computer Science, The University of Michigan, Ann Arbor, MI48109-2122 USA (e-mail: [email protected]).


allow for impedance matching are highly desirable. The funda-mental limitation introduced by Chu [1] and later re-examinedby McLean [4] relates the radiation Q of a single resonanceantenna with its BW. However, whether such limitation canbe directly extended to multiresonance antenna structures ornot is unclear. In fact, through a comparison with filter theory,designing a relatively wideband antenna may be possible usingmultipole (multiresonance) high Q structures. In this paper weexamine the applicability of multiresonance antenna structuresto enhance the BW of miniaturized slot antennas.

Different techniques have been used for antenna miniaturiza-tion such as: miniaturization using optimal antenna topologies[5]–[7] and miniaturization using magneto-dielectric materials[8], [9]. In pursuit of antenna miniaturization while main-taining ease of impedance matching and attaining relativelyhigh efficiency, a novel miniaturized slot antenna was recentlypresented [6]. Afterwards, a similar architecture in the formof a folded antenna geometry was presented in order to in-crease the BW of the previously mentioned miniaturized slotantenna [7]. Here we re-examine this topology [6] and proposemodifications that can result in further size reduction or BWenhancement without imposing any significant constraint onimpedance matching or cross polarization level. In Section II,a dual-resonant antenna topology is examined for BW en-hancement. This miniaturized antenna shows a BW which is94% larger than that of a single-resonant miniaturized antennawith the same size.

Using series inductive elements distributed along the antennaaperture results in the increase of inductance per unit length ofthe line. Therefore the guided wavelength of the resonant slotline is shortened. Thus, the overall length of the antenna is de-creased. In Section III, this technique is first demonstrated usinga standard resonant slot antenna and then incorporated in theminiaturized antenna topology of [6] to further reduce the res-onant frequency without increasing the area occupied by theantenna.

The aforementioned techniques for BW enhancement andfurther size reduction can be used individually or in combina-tion. The combined application of the techniques of Sections IIand III is presented in Section IV by demonstrating the designof a dual band miniaturized slot antenna.

II. MINIATURIZED SLOT ANTENNA WITH ENHANCED BW

A. Design Procedure

In this section the design of coupled miniaturized slot an-tennas for BW enhancement is studied. The configuration ofthe proposed coupled slot antenna is shown in Fig. 1(b) where

0018-926X/04$20.00 © 2004 IEEE

BEHDAD AND SARABANDI: BW ENHANCEMENT AND FURTHER SIZE REDUCTION 1929

Fig. 1. Geometry of single- and double-element miniaturized slot antennas. (a) Single-element miniaturized slot antenna. (b) Double-element miniaturized slotantenna.

two miniaturized slot antennas are arranged so that they areparasitically coupled. Each antenna occupies an area of about

[Fig. 1(a)] and achieves miniaturization by thevirtue of a special topology described in detail in [6]. However,this antenna demonstrates a small BW (less than 1%). A closeexamination of the antenna topology reveals that the slot-linetrace of the antenna only covers about half of the rectangularprinted-circuit board (PCB) area. Therefore another antenna,with the same geometry, can be placed in the remaining areawithout significantly increasing the overall PCB size. Placingtwo antennas in close proximity of each other creates strongcoupling between the antennas which, if properly controlled,can be employed to increase the total antenna BW.

As seen in Fig. 1(b), only one of the two antennas is fed bya microstrip line. The other antenna is parasitically fed throughcapacitive coupling mostly at the elbow section. The couplingis a mixture of electric and magnetic couplings that counteracteach other. At the elbow section, where the electric field is large,the slots are very close to each other; therefore, it is expected thatthe electric field coupling is the dominant coupling mechanismand the electric fields (magnetic currents) in both antennas willbe in phase and thus the radiated far field is enhanced.

The two coupled antennas are designed to resonate at thesame frequency, , where is the center fre-quency and and are the resonant frequencies of the twoantennas. In this case the spectral response of the coupledantenna will show two nulls, the separation of which is a func-tion of the separation between the two antennas, , and theiroverlap distance . In order to quantify this null separation acoupling coefficient is defined as

(1)

where and are the frequencies of the upper and lower nullsin . Hence can easily be adjusted by varying and[Fig. 1(b)], and decreases as is increased and is decreased.A full-wave electromagnetic simulation tool can be used to ex-tract as a function of and in the design process. BW maxi-mization is accomplished by choosing a coupling coefficient (bychoosing and ) such that remains below dB over the

entire frequency band. Here the resonant frequencies of both an-tennas are fixed at MHz and is used as thetuning parameter. However, it is also possible to change and

slightly, in order to achieve a higher degree of control fortuning the response.

The input impedance of a microstrip-fed slot antenna, for agiven slot width, depends on the location of the microstrip feedrelative to one end of the slot and varies from zero at the shortcircuited end to a high resistance at the center. Therefore anoff-center microstrip feed can be used to easily match a slot an-tenna to a wide range of desired input impedances. The optimumlocation of the feed line can be determined from the full-wavesimulation. In the double antenna example the feed line consistsof a 50 transmission line connected to an open-circuited 75line crossing the slot [Fig. 1(b)]. The 75 line is extended by

beyond the strip-slot crossing to couple the maximumenergy to the slot and also to compensate for the imaginary partof the input impedance. Using this 75 line as the feed, allowsfor compact and localized feed of the antenna and tuning thelocation of the transition from 50 to 75 provides anothertuning parameter for obtaining a good match.

B. Fabrication and Measurement

A double-element antenna (DEA) and two different single-element antennas (SEAs) (SEA 1 and SEA 2) were designed,fabricated, and measured. SEA 1 is the constitutive element ofDEA and SEA 2 is an SEA with the same topology as SEA 1[see Fig. 1(a)] but with the same area as the DEA. SEA 2 is usedto compare the BW of the double resonant miniaturized antennawith that of the single-resonant miniaturized antenna with thesame size. All antennas were simulated using IE3D [12] whichis a full wave simulation software based on method of moments(MoM) and fabricated on a Rogers RO4350B substrate withthickness of 500 m, a dielectric constant of , and aloss tangent of with a copper ground plane of33.5 23 cm . The return losses of the SEAs as well as theDEA are presented in Fig. 2. SEA 1 shows a BW of 8 MHz ornearly 0.9% and SEA 2 shows a BW of 11.7 MHz or 1.31%whereas the BW of the DEA is 21.6 MHz (2.54%) which in-dicates a factor of 1.94 increase over a SEA (SEA) with the


Fig. 2. Return losses of DEA and SEA 2: SEA with the same size as DEA. (a) Return losses of the DEA and SEA of the same size (SEA 2). (b) Return loss ofthe SEA that constitutes the DEA (SEA 1).

same area. Choosing a different substrate with different thick-ness and dielectric constant, can increase the overall BW of bothantennas. However, it is also expected that the BW ratio of theDEA to the SEA remains the same. The overall size of the DEAis which shows a 25% increase in area whencompared to the size of the SEA 1 . The Qof each antenna has also been calculated using the method pre-sented in [10] and compared with the fundamental limit on theQ of small antennas [4] in Table II. Demonstrably the qualityfactors of both SEAs are well above the minimum theoreticallimit. Since Q is only defined for single resonant structures, novalue for Q is reported for the DEA in Table II. In calculating theminimum Q for the slot antennas using the Chu limit, it is nec-essary to find the radius of the smallest sphere than encloses theantenna. At first, it may not be clear whether this sphere shouldonly cover the aperture or, in addition to that, some portion ofthe ground plane too (because of the electric currents that existin the ground plane). This becomes clear by applying the equiv-alence theorem to this problem, which shows that the magneticcurrents responsible for radiation, exist only on the aperture andaccording the the derivation of the Chu limit, the smallest spherethat encloses these radiating magnetic currents should be used.

The gain of the double resonant antenna was mea-sured at three different frequencies and is presented inTable I. Radiation patterns of the antenna were measuredat MHz and found to be similar to eachother. Fig. 3 shows the co- and cross-polarized E- and H-planeradiation patterns at MHz. The E- and H-plane radi-ation patterns of this antenna are expected to be dual of thoseof a short electric dipole. Fig. 3(a) shows the H-Plane radiationpattern which is similar to the E-Plane radiation pattern ofan electric dipole with deeps instead of nulls at .This can be attributed to the finiteness of the ground planewhere some radiation comes from the electric currents on theantenna ground plane at the edges of the substrate. Fig. 3(b),however, does not show a uniform radiation pattern like theH-Plane radiation of a short electric dipole. This is because ofthe 180 difference in phase between the normal component ofthe electric fields at the top and bottom of the antenna groundplane. The H-Plane pattern is expected to have deep nulls atthese angles; therefore, this does not significantly affect the

TABLE ICOMPARISON BETWEEN THE DEA AND ITS CONSTITUTIVE SEAS

SHOWN IN FIG. 1

TABLE IICOMPARISON BETWEEN MEASURED Q AND THE MINIMUM ATTAINABLE Q. �

CALCULATED USING THE FOSTER REACTANCE THEOREM [10]. �� CALCULATED

USING THE CHU-MCLEAN FORMULA FOR A SINGLE-RESONANT ANTENNA [4]

H-Plane pattern. Table I shows the radiation characteristics ofthe DEAs and SEAs. It is seen that the gain-BW product of theproposed double-antenna is significantly higher than that of thesingle antenna.

III. IMPROVED ANTENNA MINIATURIZATION USING

DISTRIBUTED INDUCTIVE LOADING

A. Design Procedure

A microstrip-fed slot antenna has the length of , whereis the wavelength in the slot, at its first resonance. The electric

current distribution can be modeled by the voltage distributionover a transmission line short circuited at both ends. Theresonant length of a transmission line can be made smallerif the inductance per unit length of the line is increased. This canbe accomplished by inserting a number of series inductors in thetransmission line. For slot-lines, insertion of series lumped el-ements is not possible. Besides, series lumped elements have alow Q which adversely affects antenna efficiency (gain). To re-alize a slot line with higher inductance per unit length, an arrayof distributed, short circuited, narrow slot-lines can be placed


Fig. 3. Far field radiation patterns of the double-element miniaturized slot antenna at 852 MHz. (a) H-plane and (b) E-plane.

Fig. 4. Loaded and unloaded straight slot antennas. (a) Geometry of a microstrip-fed straight slot antenna. (b) Geometry of a microstrip-fed straight slot antennaloaded with an array of series inductive elements.

along the radiating segment of the slot antenna as shown inFig. 4(b). The impedance of a short circuited slot line is ob-tained by

(2)

where is the propagation constant, is the characteristicimpedance, and is the length of the short circuited slot-line.The characteristic impedance of a slot-line is inversely propor-tional to its width [11] therefore by using wider series slots,more inductance can be obtained for a fixed length of short cir-cuited transmission line. The best location to put series induc-tors in a slot is near its end where the amplitude of magneticcurrent is small. Putting them at the center of the slot where themagnetic current is at its maximum, strongly degrades radiationefficiency. It can easily be seen that by increasing the numberand value of inductors, the length of transmission line neces-sary to satisfy the boundary conditions at both ends of the slotdecreases.

The size reduction may also be explained by considering theelectric current distribution in the conductor around the slot.There are two components of electric current in the ground planeof the slot, one that circulates around the slot and one that is

Fig. 5. Geometry of a miniaturized slot antenna loaded with series distributedinductors (slits).

perpendicular to it. The latter is described by the continuityof the electric current and displacement current at the slotdiscontinuity. Putting a discontinuity (a slit) normal to thecirculating current path forces the current to circle around


Fig. 6. Simulated and measured return losses of the straight slot antennas and miniaturized slot antennas with and without inductive loading. (see Figs. 4 and 5).(a) Return losses of straight loaded and unloaded slot antennas. (b) Return losses of ordinary and loaded miniaturized slot antennas.

Fig. 7. Far field radiation patterns of loaded straight slot antenna shown in Fig. 4(b). (a) H-plane and (b) E-plane.

the discontinuity. Hence the electric current traverses a longerpath length than the radiating slot length which in turn lowersthe resonant frequency. Fig. 4(b) shows a slot antenna loadedwith a number of narrow slits which act as an array of seriesinductors. These slits are designed to have a length smallerthan and carry a magnetic current with a direction normalto that of the main radiator. Placing them only on one side ofthe radiating slot results in asymmetry in phase and amplitudeof the current along the slot which could create problems inmatching and worsen cross polarization. In order to circumventthis problem, two series slits are placed on the opposite sidesof the main slot. These slits carry magnetic currents with equalamplitudes and opposite directions. Since the lengths of thesenarrow slits are small compared to the wavelength and sincethey are closely spaced, the radiated fields from the oppositeslits cancel each other and they do not contribute to the radiatedfar field. Matching is performed by using an off-centered opencircuited microstrip feed. The optimum location and length

of the microstrip line are found by trial and error, using fullwave simulations. For both straight slots (with and withoutseries inductors) the lengths of the extended microstrip linesare found to be , where is the wavelength in themicrostrip lines at their respective resonance frequencies.

Fig. 5 shows a miniaturized slot antenna (similar to thetopology in [6]) loaded with series inductive slits to furtherreduce its resonant frequency. The antenna without the seriesinductors is already small, and adding series inductive elementsfurther reduces the resonant frequency or equivalently theelectrical dimensions of the antenna. Instead of using identicalinductive elements along the radiating slot, differently sizedinductive slits are used to cover most of the available area onthe PCB in order to maintain the area occupied by the antenna.The antenna is matched to a microstrip transmission line ina manner similar to the straight slots described earlier. Thefeed line is composed of a 75 open-circuited microstrip lineconnected to a 50 feed line. In this case, the open circuited


Fig. 8. Far field radiation patterns of loaded miniaturized slot antenna shown in Fig. 5. (a) H-plane and (b) E-plane.

Fig. 9. Geometry of the dual band inductively loaded miniaturized slot antennaof Section IV.

Fig. 10. Measured and simulated return losses of the miniaturized dual bandslot antenna of Section IV.

microstrip line is extended beyond the slot-strip crossing byand for the miniaturized antenna and the loaded

miniaturized antenna respectively.

B. Fabrication and Measurement

The straight slots with and without series inductors weresimulated using IE3D and fabricated on a 500 m thick RogersRO4350B substrate. Fig. 6(a) shows the simulated and measuredreturn losses for the slot antennas with and without inductiveloading. This figure shows the resonance frequency and dBBW of 2.2 GHz, and 235 MHz (10.7%) for the straight slot.The loaded slot with the same length as that of the unloaded slothas a resonance frequency of 1.24 GHz and a BW of 63 MHz(5%). This result indicates a 44% reduction in the resonantfrequency and a similar reduction in the BW, as expected.The overall size can still be reduced by using longer shortcircuited slits, if they could be designed in a compact fashion.The radiation patterns of the small slot antenna were measuredin the anechoic chamber of the University of Michigan andare presented in Fig. 7. It is seen that the cross polarizationcomponents in the far field region in both E- and H-planes arenegligible, thereby confirming the fact that the radiation fromthe magnetic currents in the inductive loadings with oppositedirections cancel each other in the far field region.

The miniaturized loaded and unloaded slot antennas werealso fabricated using RO4350B substrate. Fig. 6(b) shows thesimulated and measured return losses of the loaded and un-loaded miniaturized antennas. It is shown that, by inserting theseries inductors, the resonant frequency of the antenna shiftsdown from 1116 to 959 MHz (14% reduction). In this design, theoverall PCB size is unchanged. Fig. 8 shows the E- and H-planeco- and cross-polarized radiation patterns of the loaded minia-turized antenna. It is seen that the cross polarization level isnegligible at broadside. The gains of the loaded and unloadedminiaturized slot antennas (antenna in Fig. 5) were also mea-sured in the anechoic chamber using a standard log-periodicreference antenna and were found to be 0.8 and 0.7 dB, respec-tively. Table III shows a comparison between Q of the minia-turized antennas presented in this section and the fundamentallimit on Q of small antennas with the same size [1], [4]. It is ob-served that the Q of these antennas are well above the Chu limit.


TABLE IIICOMPARISON BETWEEN BW, MEASURED Q AND THE MINIMUM ATTAINABLE Q OF THE MINIATURIZED ANTENNAS IN SECTION III

TABLE IVCOMPARISON BETWEEN BW, MEASURED Q AND THE MINIMUM ATTAINABLE Q OF THE DUAL BAND MINIATURIZED ANTENNA

IV. DUAL BAND MINIATURIZED SLOT ANTENNA

In this section the techniques introduced in the previoussections are used in the design of a dual band miniaturizedslot antenna. The geometry of this antenna is shown in Fig. 9.The resonant frequencies of the slot antennas ( and )and the value of the coupling coefficient are used asdesign parameters to achieve the desired response. Increasingthe vertical displacement, , and decreasing the horizontalseparation, , causes to increase or equivalently result ina larger separation between the two frequency bands. Smallchanges in the resonance lengths of the slots result in slightchanges in and which can be used as a means of fine-tuning the response. Note, however, that resonant frequencies

, and should be close to each other so that couplingtakes place. The separation between the two bands is limitedby practical values of . Large values cannot be obtainedeasily, since both electric and magnetic couplings are presentand add destructively. In addition to this problem, matchingthe antenna at the two bands becomes increasingly difficultas the separation increases. A parameter is defined as ameasure of separation between the two frequency bands

(3)

where is the center frequency. In practice by changing ,and , values of up to 10% can easily be obtained. This ar-chitecture is particularly useful for wireless applications that usetwo separate frequency bands (different bands for transmit andreceive for example) that are close to each other but still cannotbe covered with the available BW of these types of miniaturizedantennas.

In order to achieve a higher miniaturization level for the givensize, series inductive elements are also placed along slots to re-duce the resonance frequencies of each element. Fig. 10 showsthe simulated and measured return losses of this dual band an-tenna. The discrepancies between the simulated and measuredresults are due to the finiteness of the ground plane as describedin [6]. The measured results indicate an MHz and

MHz or equivalently a %. A good match atboth bands is obtained by using an off center open-circuited mi-crostrip feed where the microstrip line is extended by 7 cm over

the slot-strip transition. Table IV shows a comparison betweenthe antenna size, dB BW, measured Q, and minimum at-tainable Q for the two bands. It is seen that the Q of both bandsare well above the Chu limit. The overall size of the structureis 5.73 cm 5.94 cm or equivalently at thelowest frequency of operation. Radiation patterns of the antennaat the two bands are measured and found to be similar to thoseof the SEA topology (Fig. 8).

V. CONCLUSION

Two approaches are introduced for increasing the BW andreducing the size of miniaturized slot antennas. Placing twosimilar slot antennas in close proximity of each other creates adouble resonant structure, the response of which is a function ofrelative spacing between the two antennas. The coupled minia-turized antenna can be designed to have a BW which is largerby 94% than the BW of a single resonant antenna with the samearea or to behave as a dual band antenna.

For a fixed resonant frequency, adding series inductive el-ements to a slot antenna reduces its size. The size reductionis a function of number and values of the inserted inductiveelements. Using series inductive elements does not adverselyaffect impedance matching and the cross polarization level.This technique is also used in combination with other miniatur-ization techniques to further decrease the size of the radiatingstructure. The technique is applied to a straight as well as aminiaturized slot antenna and for a given antenna size, sig-nificant reduction in resonant frequencies are observed.

Finally, both techniques are applied to the design of a minia-turized dual band antenna. Series inductors are used to reducethe resonant frequencies of each resonator. A large coupling co-efficient is used to achieve a large separation between the twonulls in the response of the parasitically coupled antenna.The values of , and are used as design parameters inorder to obtain a miniaturized dual band slot antenna with rela-tively good simultaneous matching.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir constructive comments.


REFERENCES

[1] L. J. Chu, “Physical limitations on omni-directional antennas,” J. Appl.Phys., vol. 19, pp. 1163–1175, Dec. 1948.

[2] H. A. Wheeler, “Fundamental limitations of small antennas,” in Proc.IRE., vol. 35, Dec. 1947, pp. 1479–1484.

[3] R. C. Hansen, “Fundamental limitations in antennas,” Proc. IEEE, vol.69, pp. 170–182, Feb. 1981.

[4] J. S. McLean, “A re-examination of the fundamental limits on the radia-tion Q of electrically small antennas,” IEEE Trans. Antennas Propagat.,vol. 44, pp. 672–676, May 1996.

[5] H. K. Kan and R. B. Waterhouse, “Small square dual-spiral printed an-tennas,” Electron. Lett., vol. 37, pp. 478–479, Apr. 2001.

[6] K. Sarabandi and R. Azadegan, “Design of an efficient miniaturizedUHF planar antenna,” in Proc. IEEE Int. Antennas Propagat. & URSISymp., Boston, MA, July 8–13, 2001.

[7] R. Azadegan and K. Sarabandi, “Miniaturized folded-slot: An approachto increase the bandwidth and efficiency of miniaturized slot antennas,”in Proc. IEEE Int. Antennas Propagat. & URSI Symp., San Antonio, TX,June 16–21, 2002.

[8] K. Sarabandi and H. Mosallaie, “Antenna miniaturization with enhancedbandwidth and radiation characteristics: A novel design utilizing peri-odic magneto-dielectric,” in Proc. IEEE Int. Antennas Propagat. & URSISymp., San Antonio, TX, June 16–21, 2002.

[9] T. Ozdemir, P. Frantzis, K. Sabet, L. Katehi, K. Sarabandi, and J. Harvey,“Compact wireless antennas using a superstrate dielectric lens,” in Proc.IEEE Trans. Antennas Propagat. & URSI Symp., Salt Lake City, Utah,July 2000.

[10] W. Geyi, P. Jarmuszewski, and Y. Qi, “The foster reactance theorem forantennas and radiation Q,” IEEE Trans. Antennas Propagat., vol. 48, pp.401–408, Mar. 2000.

[11] J. J. Lee, “Slotline impedance,” IEEE Trans. Microwave Theory andTechniques, vol. 39, pp. 666–672, Apr. 1991.

[12] Electromagnetic Simulation and Optimization Software. IE3D.

Nader Behdad (S’97) was born in Mashhad, Iran,in 1977. He received the Bachelor of Science degreefrom Sharif University of Technology, Tehran,Iran, and the Master of Science degree from theUniversity of Michigan, Ann Arbor, in 2000 and2003, respectively, where he is currently workingtoward the Ph.D. degree on bandwidth enhancementand miniaturization of printed antennas in the De-partment of Electrical Engineering and ComputerScience.

From 2000 to 2001, he was with the ElectronicsResearch Center, Sharif Unviersity of Technology, as an antenna design En-gineer working on design of antennas for wireless local loop (WLL) systems.Since January 2002, he has been working as a Research Assistant in the Radia-tion Laboratory, University of Michigan.

Mr. Behdad is the recipient of the Best Student Paper Award in the AntennaApplications Symposium held in Monticelo, IL, in September 2003 and winnerof the Second Prize in the student paper competition of the USNC/URSI Na-tional Radio Science Meeting, Boulder, CO, in January 2004.

Kamal Sarabandi (S’87–M’90–SM’92–F’00)received the B.S. degree in electrical engineeringfrom Sharif University of Technology, Tehran,Iran, in 1980, the M.S. degree in electrical engi-neering/mathematics, and the Ph.D. degree in elec-trical engineering from The University of Michigan,Ann Arbor, in 1986 and 1989, respectively.

He is Director of the Radiation Laboratory anda Professor in the Department of Electrical Engi-neering and Computer Science, The University ofMichigan. He has 20 years of experience with wave

propagation in random media, communication channel modeling, microwavesensors, and radar systems and is leading a large research group consistingof four research scientists, ten Ph.D. students, and two M.S. students. Overthe past ten years he has graduated 15 Ph.D. students. He has served as thePrincipal Investigator on many projects sponsored by NASA, JPL, ARO, ONR,ARL, NSF, DARPA, and numerous industries. He has published many bookchapters and more than 105 papers in refereed journals on electromagneticscattering, random media modeling, wave propagation, antennas, microwavemeasurement techniques, radar calibration, inverse scattering problems, andmicrowave sensors. He has had more than 220 papers and invited presentationsin national and international conferences and symposia on similar subjects. Hisresearch areas of interest include microwave and millimeter-wave radar remotesensing, electromagnetic wave propagation, and antenna miniaturization.

Dr. Sarabandi is a Member of the International Scientific Radio Union (URSI)Commission F and of The Electromagnetic Academy. He received the HenryRussel Award from the Regent of The University of Michigan (the highest honorthe University of Michigan bestows on a faculty member at the assistant orassociate level). In 1999, he received a GAAC Distinguished Lecturer Awardfrom the German Federal Ministry for Education, Science, and Technology. Healso received a 1996 Teaching Excellence Award from the Department of Elec-trical Engineering and Computer Science, and the 2003/2004 College of Engi-neering Research Excellence Award, The University of Michigan. He is a VicePresident of the IEEE Geoscience and Remote Sensing Society (GRSS), a pastChairman of the Awards Committee of the IEEE GRSS from 1998 to 2002, and aMember of the IEEE Technical Activities Board Awards Committee from 2000to 2002. He is an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS

AND PROPAGATION and the IEEE SENSORS JOURNAL. He is listed in AmericanMen & Women of Science, Who’s Who in America, and Who’s Who in Electro-magnetics.


Miniature Built-In Multiband Antennas forMobile Handsets

Yong-Xin Guo, Member, IEEE, Michael Yan Wah Chia, Member, IEEE, and Zhi Ning Chen, Member, IEEE

Abstract—In this paper, we propose a new design for built-inhandset antennas in that metal strips as additional resonators aredirectly connected with a feed strip. With the new design scheme,a quad-band antenna for covering GSM900, DCS1800, PCS1900,and UMTS2000 bands and a five-band antenna for coveringGSM900, DCS1800, PCS1900, UMTS2000, and ISM2450 bandsfor use in mobile built-in handsets are experimentally carried out.Compared with the parasitic form with a shorted strip placedaway from the main radiator in the open literature, the size of theproposed antennas can be reduced by an order of 10 20%, whichis desirable since the size of mobile phones is becoming smalleraccording to consumer preferences. Moreover, the impedancematching for each band of the new antennas becomes easy. Thenew quad-band and five-band built-in handset antennas aredeveloped within the limits of a 36 16 8 mm

3 volume. Theantennas are also analyzed using the finite-difference time-domaintechnique. A good agreement is achieved between measurementand simulation.

Index Terms—Antennas, built-in antennas, handset antennas,planar inverted-F antennas (PIFAs) antennas, small antennas.

I. INTRODUCTION

THE SIZES AND weights of mobile handsets have rapidlybeen reduced due to the development of modern inte-

grated circuit technology and the requirements of the users.Conventional monopole-like antennas have remained relativelylarge compared to the handset itself. Thus, built-in antennasare becoming very promising candidates for applications inmobile handsets. Most built-in antennas currently used inmobile phones include microstrip antennas, inverted-F shapedwire-form antennas (IFAs), and planar inverted-F antennas(PIFAs). Microstrip antennas are small in size and light inweight. However, at the lower band for mobile applicationssuch as GSM900, half-wavelength microstrip antennas aretoo large to be incorporated into a mobile handset. Basic IFAand PIFA elements, which have a length equal to a quarterwavelength of the center frequency in the operating band, arenarrow in bandwidth. In addition to reduced antenna sizes, it isenvisaged that next generation mobile phones will require thecapability to tune to a number of frequency bands for cellularapplications, wireless local area networks (WLAN) and otherwireless communications. The trend in the development ofwireless personal communication systems has been in thepursuit of a single system that can accommodate the needs ofall users. To develop compact, highly efficient and broadband

Manuscript received February 24, 2003; revised November 2, 2003.The authors are with the Institute for Infocomm Research, Singapore 117674,

Singapore (e-mail: [email protected]).Digital Object Identifier 10.1109/TAP.2004.832375

antennas, which are well suited for integration with multibandmultifunctional personal communication transceiver systems,is a big challenge for antenna engineers.

Currently, many mobile telephones use one or more of the fol-lowing frequency bands: the GSM band centered at 900 MHz,the DCS band centered at 1800 MHz, and the PCS band centeredat 1900 MHz. Many interesting designs based on the IFA andPIFA concepts for achieving dual-band operations have beenavailable in open literatures [1]–[11]. Triple-band built-in an-tennas to operate at GSM900, DCS1800, and PCS1900 bandswere demonstrated [12]–[14]. These tri-band antennas consistof a main radiator operating at a low frequency band and afirst high band and a shorted parasitic radiator operating at asecond high band. The parasitic radiator lies in a plane par-allel to and away from the main radiator and therefore occupiesvaluable space in mobile phones that are constantly shrinkingin size. Moreover, the parasitic-feed technique used for intro-ducing one more mode may have problems in tuning of the an-tenna. More recently, some customers may need the designedmobile antennas can also include the UMTS2000 band for 3Gmobile applications or 2450 MHz ISM band for indoor cord-less phones, WLAN and Bluetooth applications. Antenna de-signs for covering GSM900, DCS1800, and ISM2450 bands canbe found in the literatures [15], [16]. Furthermore, a quad-bandbuilt-in antenna for covering GSM900, DCS1800, GSM1900,and UMTS2000 was reported in [17].

In this paper, we propose a new design in that metal strips asadditional resonators are directly connected with a feed strip.With this direct-feed scheme, the forgoing problems relatingto the parasitic-feed technique for an additional resonance ina conventional multiple-band antenna can be alleviated. Asan example, a quad-band antenna for covering the GSM900,DCS1800, PCS1900, and UMTS2000 was achieved, whichwas initially presented in [18], [19]. Herein, we would like toreport extensive results on this quad-band antenna. Further,by the addition of a second metal strip, a five-frequency bandoperation to cover GSM900, DCS1800, PCS1900, UMTS2000,and ISM2450 can be implemented. In the wireless industry,there are several ways to fabricate handset antennas, suchas: 1) molded intrusion design (MID) technology; 2) using aconductive pattern, screen-printed on an adhesive flexible film;3) thin film technology; and 4) metal cutting. It is possible tofabricate our newly designed antennas using MID technology ormetal cutting. The simulations were performed using Remcomsoftware XFDTD5.3, which is based on the FDTD method.

This paper is organized as follows. Section II presents asimple and efficient measurement setup, which is very impor-tant for measuring small handset antennas. In Section III, a

0018-926X/04$20.00 © 2004 IEEE

GUO et al.: MINIATURE BUILT-IN MULTIBAND ANTENNAS FOR MOBILE HANDSETS 1937

Fig. 1. Geometry of the test bed.

starting point for the quad-band and five-band antennas is pro-vided. After that, the built-in quad-band handset antenna designand parametric study are extensively described in Section IV.Then, Section V shows the built-in five-band handset antenna.Finally, the entire work is summarized in Section VI.

II. MEASUREMENT SETUP

The measurement methods of mobile handset antennas areof much concern by many investigators [20]–[23]. During thedevelopment of such a handset antenna, the antenna under test(DUT) is connected to a network analyzer via a coaxial cable.Errors in the measured resonant frequency, bandwidth, radia-tion pattern, and antenna gain can be expected owing to thefeed cable placed in the near field of the antenna and the coaxialcable acting as a secondary radiator driven by the surface cur-rents flowing on its outer surface of the shield. Ferrite chokes onthe exterior of the cables can reduce the cable-related effect sig-nificantly [20]. However, ferrites typically work well as chokesup to 1 GHz. The use of sleeve-like baluns on the cable, locatednear the handset, can reduce the effect of the RF cable on the an-tenna measurement as well [21], [22]. As shown in [22], the de-sign of such a balun in multiband operation is very complicated.Moreover, it is commonly known to us in wireless industry thata semi-rigid coaxial transmission line is placed on the groundplane with its centre conductor connected to the antenna whileits shield soldered onto the ground place as in [23]. With thisarrangement, the effects of the feeding coaxial cable on the an-tenna can be reduced to an acceptable level. Thus this feedingscheme can best model the real mobile phone where a separateRF transceiver, residing inside a metal enclosure, is employedto drive the antenna that is mounted very close to the RF trans-ceiver.

In this paper, the test printed circuit board (PCB) with theDUT for use in a mobile telephone is shown in Fig. 1. The PCBis in a rectangular shape. It has a ground plane and a microstripline etched on the back. In practical use the PCB will have anumber of electronic components mounted thereon, which arenecessary for the operation of the mobile telephone, but whichare omitted here for brevity. The rectangular ground plane has a

Fig. 2. (a) Existing dual-band internal handset antenna and (b) dual-bandinternal handset antenna with a fine-tuning stub.

length of and a width of . The sub-strate is Duroid RO4003 with a thickness of 1.5 mm and dielec-tric constant of 3.38. The width of the microstrip line is 3.5 mmto keep its characteristic impedance at 50 . The DUT is placedat the top of the ground plane. A coaxial cable is connected tonetwork analyzer HP8753E at the bottom of the board. The gainand radiation patterns are measured using Orbit/FR system in ananechoic chamber. The above mentioned measurement setup forhandset antenna design was validated and confirmed in terms ofmeasured and simulated input reflection, near-field currents onthe ground plane and far-field radiation patterns in house.

III. STARTING POINT

The antenna shown in Fig. 2(a) is the one as in [12], whichwas used as the starting point of this work. The antenna com-prises a folded radiating patch in the first layer, a ground planein the second layer, a supporting foam in-between, a short-cir-cuited strip, and a feed strip. The patch is connected to theground plane via a vertical short-circuited strip and is fed viaa feed strip connected to a 50- transmission line etched on theback of the ground plane. The folded PIFA is spaced from theground plane by a dielectric substrate of foam. At the first layer,the long bent portion of the antenna is tuned to have a relativelylow band resonance frequency, such as 900 MHz, and the short


Fig. 3. (a) Measured and simulated return losses of the antenna in Fig. 2(a)and (b) measured and simulated return losses of the antenna in Fig. 2(b).

part of the antenna is tuned to have a high band resonance fre-quency, such as 1800 MHz. The ground plane has dimensionsof length 80 mm, and width 36 mm. The dielectric constant offoam is around 1.07. The dimensions of the dual-band antennaare , , , ,

, ,, , ,

and .The measured and simulated return losses of the dual-band

antenna in Fig. 2(a) is shown in Fig. 3(a). The measured band-widths, defined for 6 dB return loss (SWR 3), are 82 MHz(972–1054 MHz) for the lower band and 263 MHz (1769–2032MHz) for the upper band, respectively. The corresponding sim-ulated bandwidths are 120 MHz (965–1085 MHz) and 270 MHz(1830–2100 MHz). Reasonable agreement between measure-ment and simulation is achieved. The wide bandwidth at theupper band in this design may also be due to one resonancegenerated by the ground plane, which has a half-wave lengthwith the center frequency being around 1.8 GHz. If the influ-ence of the plastic casing as in real situation is considered witha rough 5% reduction of the resonant frequency [12], it is ob-served that the corresponding frequencies are still a little out ofthe GSM900 and DCS1800 bands.

Fig. 4. Proposed quad-band internal handset antenna.

To reduce the resonant frequency further, we may extend thearm length or bend at the open end of the folded antenna inFig. 2(a). However, the bending at the open end in Fig. 2(a) maybe a little difficult to manufacture. Fig. 2(b) shows one possiblevariation of the folded antenna in Fig. 2(a) with an additionalstrip stub bent perpendicularly toward the ground plane. Thestub has dimensions of length and width . All otherparameters are kept same as those in Fig. 2(a). The measuredand simulated return losses of the antenna in Fig. 2(b) is shownin Fig. 3(b). The size of the stub for the results shown in Fig. 3(b)is and . The measured bandwidthsfor 6 dB return loss are 76 MHz (942–1018 MHz) for thelower band and 239 MHz (1752–1991 MHz) for the upper band,respectively. The corresponding simulated results are 122 MHz(928–1050 MHz) and 242 MHz (1808–2050 MHz). In this case,the resonant frequencies can meet the requirement of GSM900and DCS1800 with a plastic cover being added.

IV. QUAD-BAND ANTENNA

A. Antenna Structure

The proposed antenna in this work is shown in Fig. 4.Compared with the dual-band folded patch in Fig. 2(b), anew radiating strip is added. The new radiating strip as anadditional resonator is directly connected to the feed strip andpositioned at a plane perpendicular to the ground plane andthe original folded patch, whereas in the previous triple-banddesigns [12]–[14], the additional parasitic radiator has a smalldistance from the original folded patch. Due to that, the designsin [12]–[14] may occupy valuable space in mobile phones thatare constantly shrinking in size. Compared with the designsin [12]–[14], the size of the newly proposed antenna can bereduced by an order of 10 20%, which is desirable sincethe size of mobile phones is becoming smaller accordingto consumer preferences. The new additional strip is like aPIFA antenna and is tuned to have a second high resonancefrequency, such as 2100 MHz. The new quad-band antenna wasdeveloped within the limits of a volume. Therectangular ground plane has a length of and awidth of . The dimensions of the new antennaare , , , ,

, , , ,,


Fig. 5. Measured and simulated return losses of the antenna in Fig. 4.

Fig. 6. Measured gains of the dual-band antenna in Fig. 2(b) and the quad-bandantenna in Fig. 4.

, , , and.

B. Measured and Simulated Results

The return losses and radiation properties of the new antennaas shown in Fig. 4 were investigated using measurement andsimulation. The simulation was performed using the commer-cial software XFDTD5.3, which is based on the FDTD method.

Fig. 5 shows the measured and simulated return losses ofthe new antenna presented in Fig. 4. In the actual design, weneed to consider around 5% frequency-shifting due to the ef-fect of the plastic cover [7]. Thus, the simulated result with theplastic cover is also provided. In the simulation, 2-mm thickdielectric sheet with dielectric constant and 1-mmspacing between the cover and the antenna is used to simulatethe actual effect of the plastic cover [7], [24]. The measuredbandwidths without the plastic cover according to 6 dB re-turn loss matching are 78 MHz (933–1010 MHz) at the lowerband and 516 MHz (1772–2288 MHz) at the upper band, respec-tively. The corresponding simulated results without the plasticcover are 130 MHz (924–1054 MHz) at the lower band and 486MHz (1824–2310 MHz) at the upper band, respectively. A good

agreement between measurement and simulation is obtained.Referring to Fig. 5, it is observed that there are some differ-ences for the null depth in the simulated and measured returnlosses of the upper band, which may come from that the antennasize cannot be modeled very accurately by the FDTD methoddue to its meshing scheme. The simulated bandwidths with theplastic cover as in real case are 126 MHz (883–1009 MHz) atthe lower band and 573 MHz (1659–2232 MHz) at the upperband, respectively. The antenna has a capacity for covering theGSM900, DCS1800, PCS1900, and UMTS2000 bands. With re-gard to Fig. 5, the return loss has one distinct minimum at a lowfrequency band and two minima at two high frequency bandsrelatively close to each other. It is very clear to observe thatthe wide bandwidth of the higher band of the new antenna isdue to the introduced strip connected to the feed. Note that thewide bandwidth at the upper band in this design may also comefrom one resonance generated by the ground plane, which hasa half-wave length with the center frequency being around 1.8GHz.

The -plane far-field radiation patterns of the new quad-band antenna at 935, 1795, 1935, and 2100 MHz are depictedin Fig. 7(a)–(d), respectively. They are similar to those of otherintegrated antennas for mobile handsets [1]. Referring to Fig. 7,the overall shape of the radiation patterns can be suitable formobile communications terminals. The measured values of thegain of the quad-band antenna are shown in Fig. 6. Also, themeasured gains for the dual-band antenna as in Fig. 2(b) are in-cluded for comparison. The measured gains are varying from0.4 to 3.6 dBi.

C. Parametric Study

In this section, effects of varying key antenna parameters, i.e.the substrate thickness, the ground plane size, and the additionalstrip position and width are considered on the antenna band-width. Again, for all the simulations in this section, 2-mm thickdielectric sheet with dielectric constant and 1-mmspacing between the cover and the antenna are used to simu-late the actual effect of the plastic cover as before [7], [24].

1) Effects of the Substrate Thickness: The first variation isperformed by varying the height of the substrate. Other param-eters of the antenna are as follows: , ,

, , , ,, , ,

, ,, , and . It can

be seen from Fig. 8, that this variation has a large impact onthe impedance matching of the upper band. With the substrateheight increasing, the bandwidths for the upper band increasesignificantly, while the bandwidths of the lower band increaseslightly. The lower cutoff frequency (f1L), upper cutoff fre-quency (f1U) and the absolute bandwidth (BW1) for the lowerband and the lower cutoff frequency (f2L), upper cutoff fre-quency (f2U) and the absolute bandwidth (BW2) for the upperband are tabulated in Table I for reference, respectively. Addi-tionally, the resonant frequencies of the lower band shift up withthe substrate height being increased, which may be due to thedecreased capacitance between the folded radiating patch andthe ground plane.


Fig. 7. Radiation patterns of the antenna in Fig. 4 at xz plane: (a) 925 MHz, (b) 1795 MHz, (c) 1935 MHz, and (d) 2100 MHz.

Fig. 8. Simulated return losses with variation of H for the antenna in Fig. 4.

2) Effects of the Ground Plane Size: For small PIFA-like an-tennas, the finite ground plane can be considered as a radiator.Therefore, it is necessary to study the effects of the ground planesize on the impedance characteristics of the new antenna. Withregard to Fig. 9 it is observed that the ground plane length hasa large effect on the upper band with the length varying from60 to 100 mm, while the lower band almost keeps unchanged.

Only small variations are seen for the bandwidths of both thelower and upper bands with the ground plane width varyingfrom 36 to 44 mm. Other parameters of the antenna are as fol-lows: , , , ,

, , , ,,

, , , and.

3) Effects of the Additional Strip Position and ItsSize: Fig. 10 shows the effects of the additional strip position

and its width on the new antenna. Other parame-ters of the antenna are as follows: , ,

, , , ,, , , ,

,, and . Referring to the Fig. 10,

it can be seen that the additional strip position and its widthmainly affect the impedance matching of the upper band.The matching will become deteriorated when the additionalstrip bottom approaches very near the ground plane as highcapacitance may be introduced in this case. The bandwidthsof the lower band almost keep constant with the additionalstrip position and its size varying. The antennas with otheradditional strip positions and the strip sizes were also simulated


TABLE ISIMULATED BANDWIDTHS WITH DIFFERENT SUBSTRATE THICKNESS H OF THE QUAD-BAND ANTENNA

Fig. 9. Simulated return losses with variation of L for the antenna in Fig. 4.

Fig. 10. Simulated return losses with variation of W9 for the antenna in Fig. 4,d = 2 mm.

and tabulated in Table II. Again, the lower cutoff frequency(f1L), upper cutoff frequency (f1U) and the absolute bandwidth(BW1) for the lower band and the lower cutoff frequency(f2L), upper cutoff frequency (f2U) and the absolute bandwidth(BW2) for the upper band are listed in Table II, respectively.With regard to Table II, the bandwidth is defined accordingto return loss of 6 dB. Moreover, it is easy to realize thatthe length of the additional strip mainly affects its resonant

frequency of the introduced second high band as it acts like aquarter-wavelength IFA antenna.

V. FIVE-BAND ANTENNA

A. Antenna Structure

Fig. 11 depicts a five-band antenna for covering the GSM900,DCS1800, PCS1900, UMTS2000, and ISM2450 bands byadding a second additional strip and connecting it to the feedstrip. The second metal strip is parallel to the ground planeand the original dual-band PIFA patch but orthogonal to thefirst additional radiating strip with a horizontal separationd3 and a height h1. The additional separation and positionarrangement between the first radiating strip and the secondadditional strip reduces the mutual coupling therebetween. Thenew five-band antenna was still developed within the limits ofa volume. The rectangular ground plane hasa length of and a width of .The dimensions of this antenna are ,

, , , ,, , , ,

,, , , ,

, , and .

B. Measured and Simulated Results

Fig. 12 shows the measured and simulated return losses of theproposed five-band antenna presented in Fig. 11. The measuredbandwidths for 6 dB return loss are 78 MHz (932–1010 MHz)at the GSM900 band, 456 MHz (1818–2274 MHz), and115 MHz (2523–2638 MHz), respectively. The correspondingsimulated results are 130 MHz (920–1050 MHz) at the lowerband, 552 MHz (1760–2312 MHz) at the first high band, and80 MHz (2480–2560 MHz), respectively. A reasonable agree-ment between measurement and simulation is obtained. Theantenna has a capacity for covering the GSM900, DCS1800,PCS1900, UMTS2000, and ISM2450 bands after the frequencyshifting is considered due to the plastic cover. Again, it isobserved that there are some differences for the null depthin the simulated and measured return losses of the first highband, which may come from that the antenna size cannotbe modeled very accurately by the FDTD method due to its


TABLE IISIMULATED BANDWIDTHS WITH DIFFERENT W9 AND d2 OF THE QUAD-BAND ANTENNA

Fig. 11. Geometry of proposed five-band internal handset antenna.

meshing scheme. It is obvious that the band around 2450 MHzis due to the introduced second strip shown in Fig. 11. Theradiation patterns were also measured and similar to those forthe antenna presented in Fig. 4, thus are not shown for brevity.The measured gain at 2.45 GHz is around 1.5 dBi.

VI. CONCLUSION

In this paper, we have proposed a new design in that a newmetal strip as an additional resonator is directly connected witha feed strip and positioned at a plane perpendicular to a groundplane. With the new design scheme, a quad-band antenna

Fig. 12. Measured and simulated return losses for the antenna in Fig. 11.

for covering GSM900, DCS1800, PCS1900, and UMTS2000bands and a five-band antenna for covering GSM900, DCS1800,PCS1900, UMTS2000, and ISM2450 bands for use in mobilehandsets have been experimentally carried out. Compared withthe parasitic form, the size of the proposed antennas can bereduced by an order of 10 20%. Moreover, the impedancematching for each band becomes easy. The new quad-bandand five-band antennas have been developed within the limitsof a volume. The antennas have also been


analyzed using the FDTD technique. A good agreement hasbeen achieved between measurement and simulation.

ACKNOWLEDGMENT

The authors would like to thank the two anonymous reviewersfor their comments which improved this paper.

REFERENCES

[1] K. Hirasawa and M. Haneishi, Eds., Analysis, Design, and Measurementof Small and Low-Profile Antennas. Norwood, MA: Artech House,1992.

[2] Z. D. Liu, P. S. Hall, and D. Wake, “Dual-frequency planar inverted-Fantenna,” IEEE Trans. Antennas Propagat., vol. 45, pp. 1451–1457, Oct.1997.

[3] C. R. Rowell and R. D. Murch, “A compact PIFA suitable for dual-frequency 900/1800-MHz operation,” IEEE Trans. Antennas Propagat.,vol. 46, pp. 596–598, Apr. 1998.

[4] M. Sanad and N. Hassas, “Compact wide-band microstrip antennas forPCS and cellular phones,” in Proc. IEEE Conf. Antennas Propagationfor Wireless Commun., Nov. 1998, pp. 152–155.

[5] R. Mittra and S. Dey, “Challenges in PCS antenna design,” in Proc. IEEEAntennas Propagation Symp. Dig., Orlando, FL, July 1999, pp. 544–547.

[6] S. Tarvas and A. Isohatala, “An internal dual-band mobile phone an-tenna,” in Proc. IEEE Antennas Propagation Symp. Dig., Salt Lake City,UT, July 2000, pp. 266–269.

[7] M. Yang and Y. Chen, “A novel U-shaped planar microstrip antennafor dual-frequency mobile telephone communications,” IEEE Trans. An-tennas Propagat., vol. 49, pp. 1002–1004, June 2001.

[8] A. Taflove and L. Vasilyeva, “Elongate Radiator Conformal Antenna forPortable Communication Devices,” U.S. patent 6 292 144, Sept. 2001.

[9] M. Martinez-Vazquez, M. Geissler, D. Heberling, A. Martinez-Gon-zalez, and D. Sanchez-Hernandez, “Compact dual-band antenna formobile handsets,” Microwave and Opt. Technol. Lett., vol. 32, no. 2, pp.87–88, Jan. 2002.

[10] R. Chair, K. M. Luk, and K. F. Lee, “Measurement and analysis of minia-ture multilayer patch antenna,” IEEE Trans. Antennas Propagat., vol. 50,pp. 244–250, Feb. 2002.

[11] K. L. Wong, “A short course note on planar antennas for wirelesscommunications,” in Proc. IEEE Antennas and Propagation SocietyInt. Symp., San Antonio, TX, June 2002, pp. 6–14.

[12] D. Manteuffel, A. Bahr, D. Heberling, and I. Wolff, “Design con-sideration for integrated mobile phone antennas,” in Proc. 11th Int.Conf. Antennas and Propagation, Manchest, U.K., April 2001, pp.252–256.

[13] Z. Ying, Multi Frequency-Band Antenna PCT application WO01/91233,May 2001.

[14] I. Egorov, “Antenna,” U.S. patent application 09/908 817, July 2001.[15] W. P. Dou and Y. M. W. Chia, “Novel meandered planar inverted-F an-

tenna for triple-frequency operation,” Microwave Opt. Technol. Lett.,vol. 27, pp. 58–60, Oct. 2000.

[16] C. T. P. Song, P. S. Hall, H. Ghafouri-Shiraz, and D. Wake, “Triple bandplanar inverted F antennas for handheld devices,” Electron Lett., vol. 36,pp. 112–114, Jan. 2000.

[17] M. Martinez-Vazquez and O. Litschke, “Design considerations forquadband antennas integrated in personal communications devices,” inProc. Int. Symp. Antennas (JINA), vol. 1, Nice, France, Nov. 2002, pp.195–198.

[18] Y. X. Guo, M. Y. W. Chia, and Z. N. Chen, “Compact multi-band an-tennas for wireless communications,” in Proc. Progress in Electromag-netics Research Symp., Singapore, Jan. 2003, p. 130.

[19] , “Miniature built-in quad-band antennas for mobile handsets,”IEEE Antennas Wireless Propagat. Lett., vol. 2, pp. 30–32, 2003.

[20] S. Saario, D. V. Thiel, J. W. Lu, and S. G. O’Keefe, “An assessmentof cable radiation effects on mobile communications antenna measure-ments,” in Proc. IEEE Antennas Propagt. Symp. Dig., Montreal, Canada,July 1997, pp. 550–553.

[21] C. Icheln, J. Ollikainen, and P. Vainikainen, “Reducing the influence offeed cables on small antenna measurements,” Electron. Lett, vol. 35, no.15, pp. 1212–1214, July 1999.

[22] C. Icheln and P. Vainikainen, “Dual-frequency balun to decrease influ-ence of RF feed calbes in small antenna measurements,” Electron. Lett.,vol. 36, no. 21, pp. 1760–1761, Oct. 2000.

[23] J. Haley, T. Moore, and J. T. Bernhard, “Experimental investigationof antenna-handset-feed interaction during wireless product testing,”Microwave and Opt. Technol. Lett., vol. 34, no. 3, pp. 169–172,Aug. 2002.

[24] H. S. Hwang, “private communication,” unpublished.

Yong-Xin Guo (M’01) received the B.Eng. andM.Eng. degrees from Nanjing University of Scienceand Technology, Nanjing, China, and the Ph.D.degree from City University of Hong Kong, all inelectronic engineering, in 1992, 1995, and 2001,respectively.

From 1995 to 1997, he was a Teaching andResearch Assistant and then a Lecturer in theDepartment of Electronic Engineering, NanjingUniversity of Science and Technology. From January1998 to August 1998, he was a Research Associate

in the Department of Electronic Engineering, City University of Hong Kong.Since September 2001, he has been with the Institute for Infocomm Research,Singapore, as a Scientist. He also holds an appointment of Adjunct AssistantProfessor at the National University of Singapore. He has published over50 technical papers in international journals and conferences. He holds oneChinese Patent and one pending U.S. patent. His current research interestsinclude design and modeling of microstrip and dielectric resonator antennas forhandsets and other wireless communications, UWB antennas, LTCC filters andbaluns and LTCC optoelectronic transceiver for radio-over-fiber application,and numerical methods in electromagnetics.

Dr. Guo has served as a reviewer for IEEE TRANSACTIONS ON ANTENNAS AND

PROPAGATION since 2002. He also served as one session chair for 2003 Asia andPacific Microwave Conference (APMC2003), Korea.

Michael Yan Wah Chia (M’94) was born in Singa-pore. He received the B.Sc. (1st Class Honors) andPh.D. degrees from Loughborough University, U.K.

He joined the Center for Wireless Communica-tions (CWC), Singapore, in 1994 as a Member ofTechnical Staff (MTS), was promoted to SeniorMTS, then Principal MTS, and finally Senior Prin-cipal MTS. He was also holding the appointmentas Division Director of Radio Division. In 1999, hestarted UWB work at CWC, which was later mergedto I R. Currently, he is the Division Director of

Communications and Devices Division (which consists of five Departments),Institute for Infocomm Research (I R) of ASTAR (Agency of Science Tech-nology And Research). Concurrently, he has been appointed as an AdjunctAssociate Professor at the National University of Singapore. Externally, he hasalso served in the Public Service Funding Panel of ASTAR-2002, Radio Stan-dard Committee and UWB Task Force Committee of Infocomm DevelopmentAuthority(IDA) of Singapore. He was also on the Technical Program Com-mittee International Workshop on UWB System (IWUWBS) 2003, Finland,and has been listed in Marquis’s Who’s Who in Engineering since 2002. Hehas published 28 international journal papers and 50 international conferencepapers. He has 10 patents both filed and granted. Some of the patents hasbeen commercialized and licensed to companies. His main research interestare ultraeide-band(UWB) system, antenna, transceiver, radio over fiber, RFIC,linearization and communication, and radar system architecture.

Dr. Chia was awarded Overseas Research Studentship (ORS) and BritishAerospace Studentship from the U.K.


Zhi Ning Chen (M’99) was born in China andreceived the B.Eng., M.Eng., and Ph.D. degrees inelectrical engineering from the Institute of Commu-nications Engineering (ICE), China, and the Ph.D.degree from the University of Tsukuba, Japan.

From 1988 to 1995, he was with the ICE and wasappointed Teaching Assistant, Lecturer, and then pro-moted to Associate Professor. After that, he was withthe Southeast University (SEU), Nanjing, China, asa Postdoctoral Fellow and then appointed AssociateProfessor. From 1995 to 1997, he undertook his re-

search in the City University of Hong Kong, China, as a Research Assistant, Re-search Associate, Senior Research Associate, and then Research Fellow. From1997 to 1999, he pursued research at the University of Tsukuba, Japan, withthe Fellowship awarded by Japan Society for Promotion of Science (JSPS).He visited SEU in 2000 and 2001, as a Visiting Scholar awarded by the Min-istry of Education, China. In 2001, he visited the University of Tsukuba, Japan,again under the Invitation Fellowship Program (senior level) of JSPS. In 1999,he joined the Centre for Wireless Communications (CWC) (later known theInstitute for Communications Research (ICR) and now the Institute for Info-comm Research (I R)) as a Member of Technical Staff (MTS), Senior MTS,and then promoted Principal MTS. Currently, he is working as a Lead Scientistand a Manager for Department of Radio Systems. He is concurrently teachingand supervising postgraduate students at the National University of Singapore(NUS), as an Adjunct Associate Professor. Since 1990, he has authored andcoauthored over 110 technical papers published in international journals andpresented at international conferences. One of his patents has been granted andfour are pending. His main research interests include applied computationalelectromagnetics, and antenna theory and designs. Currently, he is focused onsmall, broadband, lightweight antennas for wireless systems and ultrawide-band(UWB) radio systems, and metamaterials and their applications. He managed aresearch project on Small and Multiband Antennas during 2000 to2003.

Dr. Chen was a Member of the technical program and organizer/chair of UWBtechnology workshop at IEEE Radio and Wireless Conference (RAWCON),2003. He also organized and chaired special session on Antennas for UWBWireless Communication Systems at the IEEE International Symposium on An-tennas and Propagation (AP-S), 2003 and IEEE Asia and Pacific MicrowaveConference (APMC), Korea, 2003.


Miniature Reconfigurable Three-Dimensional FractalTree Antennas

Joshua S. Petko, Student Member, IEEE, and Douglas H. Werner, Senior Member, IEEE

Abstract—This paper introduces a design methodology forminiature multiband as well as reconfigurable (i.e., tunable)antennas that exploits the self-similar branching structure ofthree-dimensional (3-D) fractal trees. Several fundamental rela-tionships, useful for design purposes, are established between thegeometrical structure of the fractal tree antenna and its corre-sponding radiation characteristics. In particular, it will be shownthat the density and elevation angle of the branches play a keyrole in the effective design of miniature 3-D fractal tree antennas.Several design examples are considered where fractal trees areused as end-loads in order to miniaturize conventional dipole ormonopole antennas. Multiband and reconfigurable versions ofthese miniature antennas are also proposed, where either reactiveLC traps or RF switches are strategically placed throughout thebranches and/or along the trunk of the trees. Included amongthese designs is a miniature reconfigurable dipole antenna thatachieves a 57% size reduction for the center frequency of thelowest intended band of operation and has a tunable bandwidthof nearly 70%.

Index Terms—Fractal antennas, fractal tree antennas, miniatureanennas, reconfigurable antennas.

I. INTRODUCTION

AS PART OF AN effort to further improve modern commu-nication system technology, researchers are now studying

many different approaches for creating new and innovativeantennas. One technique that has received a lot of recent at-tention involves combining aspects of the modern theory offractal geometry with antenna design. This rapidly growingarea of research is known as fractal antenna engineering [1],[2]. One particular class of antenna configurations that havebeen studied recently is based on fractal trees [2]–[10]. Fractaltree structures can be exploited in antenna designs to producemultiband characteristics [1], [3]–[5], [9] or to achieve minia-turization [6]–[10]. A set of dipole antennas that use fractal treestructures as end loads to achieve a resonant frequency lowerthan a standard dipole of comparable length have been recentlystudied in [6]–[10].

This paper begins by discussing ways to improve antennaminiaturization techniques that employ fractal tree geometriesas end loads by increasing the density of branches (i.e., by usingtrees with a higher fractal dimension). Several miniaturizationschemes for fractal tree antennas are introduced, which arebased on various combinations of different branch lengths orangles. The addition of a center stub is also considered as ameans for improving existing designs for miniature fractal tree

Manuscript received February 10, 2003; revised September 26, 2003.The authors are with the The Pennsylvania State University, Department of

Electrical Engineering, University Park, PA 16802 USA ([email protected]).Digital Object Identifier 10.1109/TAP.2004.832491

Fig. 1. First 4 iterations of the four-branch class of fractal tree antennas [7].Also included is a pictorial representation of the nomenclature used to describefractal trees.

TABLE IPARAMETERS FOR GENERATING FOUR-BRANCH, 30 FRACTAL TREES

antennas. Finally, the unique self-similar wire branch structureof the three-dimensional (3-D) fractal tree is exploited todevelop new design methodologies for reconfigurable minia-ture dipole and monopole antennas. Several design examplesare considered where these miniature fractal tree antennasare made reconfigurable by the introduction of strategicallyplaced reactive loads or RF switches. Among these designs isa reconfigurable miniature dipole antenna that achieves a 57%size reduction with respect to its conventional counterpart (i.e.,a half-wave dipole designed for the center frequency of thelowest operating band) and has a tunable bandwidth of nearly70%.

II. FRACTAL TREE RADIATION STUDIES

A. Dense Fractal Tree Generators

Fractals are objects which have a self-similar structure re-peated throughout their geometry [11], [12]. This self-similarstructure may be produced by the repeated application of a gen-erator, and in the case of fractal trees, the generator is definedas a junction from which several smaller branches, called childbranches, split from a parent branch. Every branch, with the ex-ception of the first and final branches, has a generator connected

0018-926X/04$20.00 © 2004 IEEE


TABLE IIPARAMETERS FOR GENERATING SIX- AND EIGHT-BRANCH FRACTAL TREES

Fig. 2. First two iterations of the six-branch and eight-branch classes of fractaltree antennas.

to it at each end: one from which it is a child and the other towhich it is the parent.

Three different families of 3-D fractal trees are first evalu-ated for their suitability as miniature dipole antennas. Theseclasses of antennas will be referred to throughout the paperas four-branch, six-branch, and eight-branch fractal trees. Thefour-branch class of antennas, shown in Fig. 1, has been adaptedfrom [7] and is used as a benchmark of comparison for thesix-branch and the eight-branch classes introduced in this paper.Table I shows the generation parameters for the four-branchfractal trees shown in Fig. 1.

The fractal generators of these antennas have several prop-erties in common. First, the child branches are half the lengthof the parent branches from which they separate. In addition,child branches bend 30 from the direction the parent branchis aimed. Also, all child branches have equal angles separatingthem (i.e., for the four-branch class, there are 90 between eachchild branch, for the six-branch class, there are 60 betweeneach branch, and for the eight-branch class, there are 45 be-tween each branch). Finally, one of these child branches mustcontinue the arc path of its parent and the parent branch beforethat. The generation parameters are listed in Table II for boththe six- and eight-branch cases. The similarities between four-,six-, and eight-branch antenna classes are evident from Figs. 1and 2.

Several of the other fractal tree design parameters are alsoheld constant for the sake of comparison. First, the distance fromtip to tip for each fractal tree dipole antenna is fixed at 7.5 cm (or3.75 cm from tip to source). More specifically, the base or trunklength of the first stage is 2.5 cm, the second stage is 2.143 cm,

the third stage is 2 cm, and the fourth stage is 1.936 cm. Inaddition, the diameter of the wire assumed for each antenna is

. Finally, all of the fractal tree dipole antennasare assumed to be center-fed.

The resulting radiation characteristics of these fractal treeantennas were evaluated using a numerically rigorous approachbased on the method of moments (MoM). Fig. 3 plots the forthe first two stages of the six- and eight-branch antenna classesand compares them to the first four stages for the four-branchclass. The results show that the six-branch antennas have aresonant frequency 100 MHz lower than the four-branch antennaat the same fractal stage of growth, whereas the eight-branchantenna has a resonant frequency approximately 150 MHzlower. The six- and eight-branch classes follow the same trendas the four-branch in the sense that the resonant frequencydecreases with an increase in the fractal stage of growth;however, there is no valid geometry for the six- and eight-branchantenna classes beyond stage 3, because wire segments willintersect. Nevertheless, at these lower stages of growth, thesix- and eight-branch antenna classes are more effective thanthe four-branch and are easier to fabricate than higher orderfour-branch antennas because the six- and eight-branch classeshave less junctions. Despite the complicated geometry of thesefractal tree structures, the radiation patterns exhibited by thesetypes of antennas are very similar to those of typical dipoleantennas. Fig. 4 shows the radiation patterns produced by thefour-, six-, and eight- branch fractal tree dipoles comparedto the radiation pattern of a conventional half-wave dipole.The radiation patterns for these fractal tree antennas havenegligible cross-polarization components (i.e., 150 dB)and are nearly identical to the radiation pattern of the half-wavedipole antenna.

B. Fractal Tree Generators of Varying Angle

In this section, several different 3-D fractal trees are evalu-ated for their suitability as miniature dipole antennas at theirsecond and third stages of growth. These antennas are relatedto the four-branch class of antennas adapted from [7]; how-ever, the elevation angle that the child branches bend from theparent branch is not held constant at 30 but is varied over arange of angles from 10 to 90 . All other independent designparameters are assumed to be the same for each antenna. Forinstance, the antennas are all center fed dipoles with fractaltree loads placed on both ends, each generator has only fourequally spaced branches, child branches are half the length of

PETKO AND WERNER: MINIATURE RECONFIGURABLE 3-D FRACTAL TREE ANTENNAS 1947

Fig. 3. Comparison of S versus frequency for four-branch, six-branch, and eight-branch fractal trees. The S was calculated with respect to 50 line.

Fig. 4. Radiation patterns of dense fractal tree antennas compared to those ofa conventional half-wave dipole.

the parent branches, and the arc length of wire from tip to sourceis 3.75 cm. Fig. 5 shows how modifying this angle can dramat-

Fig. 5. Third iteration of four-branch fractal tree antennas with the elevationangle at 10 , 30 , 45 , 60 and 90 respectively.

TABLE IIIPARAMETERS FOR GENERATING FOUR-BRANCH FRACTAL TREES WITH

VARYING ELEVATION ANGLE

ically change the shape of the antenna. The generation parame-ters for each case considered in Fig. 5 are listed in Table III.

Next, we consider variations of a second stage and a thirdstage four-branch fractal tree antenna. In both cases, the eleva-tion angle is varied from 10 to 90 and each antenna that resultsis individually simulated. From this data, the value of voltagestanding wave ratio (VSWR) that corresponds to the resonantfrequency of each fractal tree antenna is obtained and plottedas shown in Fig. 6. The results for both stages show that fractaltree antennas with small elevation angles have a lower VSWRbut a higher resonant frequency than those with large eleva-tion angles. The resonant frequency continues to move loweras the elevation angle increases until it reaches approximately50 . From that point, the resonant frequency begins to increaseagain. Throughout the entire range of elevation angles that wereconsidered for each stage, the VSWR increases as the elevationangle increases. Trends uncovered by the elevation angle studycan be used to design effective miniature fractal tree dipoles that


Fig. 6. Fractal tree dipole VSWR versus resonant frequency for elevation angles ranging from 10 to 90 . The VSWR was calculated with respect to 50 line.

also possess good VSWR performance characteristics. The ra-diation patterns, not shown in this case, have very similar char-acteristics to the patterns illustrated in Fig. 4.

III. FRACTAL TREE ANTENNA DESIGNS

A. Hybrid Fractal Tree Antenna

While the eight-branch class of fractal tree antennas consid-ered in Section II does not have a valid third stage of growth,modifications can be performed on the generator in order tomake a valid self-avoiding third stage possible. As shown bythe generation parameters listed in Table IV, the child branchesat 0 , 90 , 180 , and 270 are 50% as long as the parent branch,which is the same as the previous antenna classes; however,the child branches at 45 , 135 , 225 , and 315 are only 40%as long as the parent branch. This difference in scale creates ahybrid pattern of short and long wires, which also impacts thelength of wires at later stages of growth (see Fig. 7). All otherfactors in the design remain unchanged, and the distance of thelongest possible path from the source to a wire end is 3.75 cm(7.5 cm end-to-end). This class of antenna is not as dense as theeight-branch class for stages one and two; however, its advan-tage is its self-avoiding geometry at stage three.

The results (see Fig. 8) show a reduction in the resonantfrequencies for the hybrid class of antennas when compared tothe four-branch class of antennas for the same stage of growth:100 MHz difference for the first stage, 80 MHz difference for thesecond, and 60 MHz difference for the third. Fig. 8 demonstratesthat the resonant frequency of the third stage hybrid antenna isabout the same as the resonance of the fourth stage four-branchantenna, illustrating that denser fractal tree structures can ef-fectively reduce resonant frequency in a similar manner as lessdense fractal tree structures at higher stages of growth. As inthe previous cases considered, the radiation patterns (not shown)are nearly indistinguishable from the pattern of a conventionalhalf-wave dipole antenna and possess essentially no cross-po-larized field components.

TABLE IVPARAMETERS FOR GENERATING EIGHT-BRANCH HYBRID SCALED

SELF-AVOIDING FRACTAL TREES

Fig. 7. First three iterations for the eight-branch hybrid scaled self-avoidingclass of fractal tree antennas.

B. Center-Stubbed Fractal Tree Antennas

It will be demonstrated here that fractal tree antennas canbe designed to have a reduced resonant frequency and lowreflection properties by incorporating a center stub. As shownpreviously in Section II-B, fractal tree dipole antennas can bedesigned to have low values of reflection by using generatorbranching schemes with sufficiently small elevation angles.Also, the resonant frequency of fractal tree dipole antennas maybe reduced by considering geometries which have a more denseconfiguration of branches. In this manner a center stubbedfractal tree antenna can take into account both of these design


Fig. 8. Comparison of S for eight-branch hybrid fractal tree dipoles with standard four-branch and eight-branch fractal trees. The S was calculated withrespect to 50 line.

considerations by increasing the density of the fractal branchingstructure while, at the same time, keeping the elevation anglessmall with respect to the generator. Here we consider two gen-erators, one with and one without a center stub. Each generatorhas four branches that bend out with elevation angles of 45that are 50% as long as the previous branch. The branches ofeach generator are equally spaced in the phi direction; however,the generator is rotated an additional 45 so that they are offsetfrom the path of the previous two branches. For this reason, therotational angles are offset from 0 , 90 , 180 and 270 to 45 ,135 , 225 , and 315 respectively. A fractal tree antenna with agenerator having only these 4 branches is compared to a fractaltree antenna with a generator having these 4 branches and anadditional branch as a center stub. This center stub continuesas an extension of the previous branch, but is rotated by 45 . Inthis way the child branches of the center stub are offset by 45from the branches of the level below it. All other parametersin the design remain unchanged. The distance of the longestpossible path from the source to an end of a wire is 3.75 cm(7.5 cm end-to-end). These structures are shown in Fig. 9 andthe corresponding generation parameters are listed in Table V.

The MoM simulation results indicate that fractal tree an-tennas with center stubs, when compared to fractal trees withouta center stub, exhibit a downward shift in the resonant frequencywith only a minimal increase in the reflection at resonance. The

characteristics (see Fig. 10) show that for each of the firstthree stages of growth there is a 60 MHz reduction in resonantfrequency for fractal tree antennas with a center stub comparedto those without a center stub. The results plotted in Fig. 10 alsoshow that for each of the first three stages of growth there is onlya minimal increase in the level of (between 3 dB and 1 dB)for tree antennas with a center stub as opposed to those withouta center stub. Thus, fractal tree antennas with center stubs canbe very effective designs because of the similar efficiency andthe lower resonant frequency. Finally, the radiation patternsfor miniature fractal tree antennas with center stubs were also

Fig. 9. An example of a four-branch, center-stubbed fractal tree antenna.

found to be comparable to the radiation pattern produced by aconventional half-wave dipole antenna.

C. Six Branch 50 –30 Fractal Tree Antenna

In this section we consider an approach for designing minia-ture fractal tree dipole antennas that have relatively low valuesof . This is achieved by using a combination of differentbranches with small elevation angles to increase the density orspace-filling property of the end-loads. One particular designconsidered here has 6 branches, with a 60 rotational angle be-tween each branch of the generator. The elevation angles of thegenerator alternate between 30 and 50 , and the 50 branch isaligned with the 0 rotational angle. Fig. 11 shows these struc-tures in detail and Table VI lists the corresponding generationparameters. Finally, all other design parameters are kept thesame as before and the length of the antenna is 3.75 cm fromtip to source.

The resulting plots shown in Fig. 12 indicate that thereis a significant reduction in the resonant frequency when com-pared to the four-branch 30 fractal tree antenna evaluated in[7]. The second iteration six-branch 50 –30 antenna has a res-onant frequency of 920 MHz, which is the same as the third


TABLE VPARAMETERS FOR GENERATING FOUR-BRANCH, 45 FRACTAL TREES WITH AND WITHOUT CENTER STUB

Fig. 10. S versus frequency for a four-branch, 45 fractal tree dipole antenna (stage 1, stage 2, and stage 3) with and without a center stub. The S wascalculated with respect to 50 line.

Fig. 11. First three stages of a six-branch 50 –30 fractal tree dipole antenna.

TABLE VIPARAMETERS FOR GENERATING SIX BRANCH 50 –30 FRACTAL TREES

iteration for the four-branch fractal tree antenna. For the thirditeration six-branch 50 –30 antenna, the resonant frequencyis 790 MHz, 70 MHz lower than the fourth iteration of the

four-branch fractal tree antenna. The radiation patterns for astage 1, stage 2, and stage 3 six-branch 50 –30 fractal treedipole antenna are again similar to those illustrated in Fig. 4with negligible cross-polarization.

IV. RECONFIGURABLE FRACTAL TREE ANTENNAS

A. Self-Reconfigurable Reactive Loaded Fractal TreeMonopoles

It has been shown that reactive loads acting as traps can beused to make a typical monopole antenna resonant at more thanone frequency [13], [14]. These reactive loads behave as anopen circuit at some frequencies and a short circuit at others, ef-fectively making an antenna self-reconfigurable (i.e., reconfig-urable without the need for RF switches). This concept can alsobe applied to fractal tree monopoles to produce an antenna thatnot only is resonant at more than one frequency but also is minia-ture in size due to the presence of the space-filling end-loadstructure.

The first case study evaluates a monopole version of the thirdstage four-branch fractal tree, this time including parallel LCreactive load elements on the base (i.e., trunk) of the antenna.By placing the reactive loads below the fractal tree end-load,the antenna can be designed so that the feed effectively sees theentire antenna at the lowest resonant frequency and only partsof the base at higher resonant frequencies. Both a dual-bandand a tri-band version of the third stage four-branch fractal tree


Fig. 12. S versus frequency for a stage 1, stage 2, and stage 3, six-branch 50 –30 fractal tree antenna. Also shown for comparison is the S for the first fourstages of a standard four-branch fractal tree antenna. The S was calculated with respect to 50 line.

TABLE VIILOAD COMPONENT VALUES FOR THE DUAL- AND TRI-BAND FOUR-BRANCH FRACTAL TREE MONOPOLE ANTENNAS

Fig. 13. Load locations on dual- and tri-band four-branch fractal treemonopole antennas.

antenna are considered here. For the dual-band antenna, oneresonant load is placed near the top of the fractal tree’s base,1.95 cm above the ground plane along the 2-cm tall trunk. Forthe tri-band antenna, an additional load is placed on the baseat a height of 1.65 cm above the ground plane. Fig. 13 illus-trates the load locations on the dual- and tri-band monopoleantennas and Table VII lists the corresponding load compo-nent values. Plots of the versus frequency are shown inFig. 14 for the single-band unloaded fractal tree monopole, thedual-band fractal tree monopole with one reactive load, and the

tri-band fractal tree monopole with two reactive loads. The re-active loads also have another advantage in further reducing theminimum resonant frequencies of the antenna structure. Theunloaded fractal tree structure has one primary resonance at910 MHz, the dual-band antenna has two primary resonances at800 and 2460 MHz, and the tri-band antenna has three primaryresonances at 550, 2300, and 5240 MHz. Also, fundamentalor primary resonant frequencies can be distinguished from sec-ondary resonant frequencies in that there is only one main lobepresent in the radiation pattern at a primary resonant frequency.Fig. 15 shows the radiation patterns at each resonant frequencywithin the 400 to 6400 MHz band. In the plot for the unloadedfractal tree, we see that there is a primary resonance at 910 MHz,characterized by a single lobe in the radiation pattern, and a sec-ondary resonance at 6100 MHz, characterized by two lobes inthe radiation pattern. For the fractal tree with one load, the pri-mary resonances are confirmed to be at 800 MHz and 2460 MHzwhile there is a secondary resonance at 6230 MHz. Finally forthe fractal tree with two loads, primary resonances occur at 550,2300, and 5240 MHz.

The next case study considers the third stage of a four-branch,45 fractal tree monopole with center stub. In this instance, se-ries LC traps are placed inside the fractal tree end-load structureinstead of, as in the previous instance, parallel LC traps beingplaced on the trunk or base of the tree. The capability of placing


Fig. 14. S versus frequency for self-reconfigurable four-branch fractal tree monopoles. The response of the single-band unloaded fractal tree monopole iscompared to a dual-band and a tri-band version of the same antenna with one and two reactive loads respectively. The S was calculated with respect to 50 line.

Fig. 15. Radiation patterns for self-reconfigurable four-branch fractal treemonopoles.

Fig. 16. Load locations on a tri-band four-branch, center stubbed fractal treemonopole.

LC traps in the end-load structure is an important advantage ofthe fractal tree antenna. Resonant traps can be placed on thebase of any end-loaded monopole antenna; however, by placing

Fig. 17. Matching network for the tri-band four-branch, center stubbed fractaltree monopole antenna.

the traps only on the base, the antennas will exhibit a charac-teristically large separation between the lowest operating bandand any higher frequency bands. Having the ability to place thetraps in the branches of the fractal tree structure provides moreflexibility in the design of miniature multiband monopole an-tennas by allowing the resonances to be placed closer together.Using five distinct reactive LC traps, a tri-band version of thethird stage four-branch, 45 fractal tree monopole with centerstub is presented and discussed here. Four traps are placed nearthe top of the first stage of the outer four branches. A fifth trap isplaced near the bottom of the first stage of the center stub. Fig. 16shows the position of the traps and Table VIII provides the cor-responding component values. The antenna has three primaryresonances at 330 MHz, 800 MHz, and 2220 MHz. The sepa-ration between the first and third resonant frequencies is only1890 MHz, 2800 MHz less than the separation between the firstand third resonant frequencies of the tri-band four-branch fractaltree shown in Fig. 13. In this case, a five-element matching net-work can be used to allow the antenna to perform efficiently ateach resonant frequency. The topology of a candidate matchingnetwork is shown in Fig. 17 along with the required componentvalues. The candidate matching network was designed using atrial and error approach in combination with a random variable


TABLE VIIILOAD COMPONENT VALUES FOR THE TRI-BAND FOUR-BRANCH, CENTER STUBBED FRACTAL TREE MONOPOLE ANTENNA SHOWN IN FIG. 16

Fig. 18. S versus frequency comparison of matched and unmatched tri-band four-branch, center stubbed fractal tree monopole antennas. TheS was calculatedwith respect to 50 line.

Fig. 19. Radiation patterns for each band of the tri-band four-branch, centerstubbed fractal tree monopole antenna shown in Fig. 16.

optimizer [15]. The plots for the matched and unmatchedtri-band center-stubbed fractal tree monopoles are comparedin Fig. 18. These plots demonstrate that by placing a reactivematching network at the feed of the antenna, all three resonancescan be matched to operate at a VSWR under 2:1. Finally, Fig. 19shows the radiation patterns at each of the resonances of thetri-band center stub fractal tree monopole antenna. The radia-

tion patterns at 300 MHz, 800 MHz, and 2220 MHz each onlyhave a single lobe, indicating that all three cases represent pri-mary resonant frequencies.

B. Miniature Reconfigurable/Tunable Fractal Tree AntennasUsing Electronic Switches

Recently there has been a considerable amount of interestin design concepts for reconfigurable antennas capable ofoperating over a broad range of frequencies. In this section areconfigurable antenna design approach will be introduced thatexploits the branching structure of fractal tree dipoles. Becausethe current is distributed over the entire end-load, the effectiveremoval or switching off of even relatively large sections of thefractal tree structure will reduce the resonant frequency only bysmall amounts. This property of reconfigurable fractal tree an-tennas can be exploited to allow the resonances of the antennato be spaced close enough together to cover all of the frequen-cies between adjacent bands. Second, because the fractal treestructure has many parallel paths for the current to flow, thenonly a relatively small number of switches may be required toachieve tunability over a desired range of frequencies.

A design is presented here where RF switches are strate-gically placed on the third stage, six-branch 50 –30 fractal


Fig. 20. Switch layout for the reconfigurable six-branch, 50 –30 fractal treedipole antenna.

tree dipole antenna to make it reconfigurable (i.e., tunable) overa bandwidth of 68%. The antenna uses 204 separate switchesplaced throughout both tree structures (a total of 102 on eachend) to produce 20 reconfigurable states. The locations of theseRF switches are illustrated in Fig. 20. The switches are placed atevery junction inside the tree structure with the exception of thejunction joining the base to the first stage. The junctions joiningthe first and second stages all have six switches associated witheach of the six branches. In addition the 36 junctions betweenthe second and third stage of the fractal structure have switchesassociated with each of them. The six junctions between thesecond and third stages that are nearest to the center axis of theantenna (i.e., those that are closest to being vertical) also havesix switches associated with each of the six branches. The re-mainder of the junctions between the second and third stage ofthe fractal tree have only one switch, which is placed near theend of the branch below the junction. For this particular design,there are 20 different combinations of switch settings which cor-respond to 20 different resonant frequencies at which the recon-figurable antenna is capable of operating.

A rigorous moment method simulation is used to individuallymodel each of the 20 different states of the reconfigurable fractaltree antenna. The switches in the fractal tree structure are mod-eled as ideal. Also, the MoM model for each state is generatedfrom what remains of the fractal tree antenna after removing theappropriate sections that have been switched off. This is justifiedsince coupling effects between these removed sections, whichare significantly less than a half-wavelength long, and the re-mainder of the antenna still being directly fed are expected tobe minimal.

In an effort to maintain the omni-directional radiation patternof the fractal tree antenna in the azimuthal plane, the order inwhich the switches are turned off is performed in a mannerthat preserves the symmetry of the tree about its base as muchas possible. Fig. 21 illustrates several different states of thereconfigurable fractal tree antenna, visualized by removingportions of the fractal tree above switches that are turned off.The resulting antenna can be reconfigurable from 770 to 1570MHz for a bandwidth of 800 MHz with a VSWR under 3:1

Fig. 21. Several states for the reconfigurable six-branch, 50 –30 fractal treedipole antenna. For illustrative purposes, the branches that are switched off foreach of the five different bands have been removed.

Fig. 22. S versus frequency for the reconfigurable six-branch, 50 –30fractal tree antenna. The light gray curves represent each of the 20 statesthe antenna can be configured to operate at. The dark gray curves representreconfigured states which operate as stage 1, stage 2, and stage 3 fractal treeantennas. The black line represents the overall minimum S the antenna canoperate over the entire band. The S was calculated with respect to 50 line.

and is reconfigurable from 970 to 1570 MHz for a bandwidthof 560 MHz with a VSWR below 2:1. In Fig. 22 each ofthe 20 reconfigurable states is represented by a separatecurve (indicated by light gray lines) with the lowest resonantfrequency representing the state with all the switches closedand the highest resonant frequency representing the state withall the switches open. The remaining states are achieved byopening the switches progressively from the top to the bottom.In addition, for three of the reconfigurable states the antennaeffectively operate as a 50 –30 fractal tree dipole with fractalstages 3, 2, or 1. The curves for these three special casesare indicated on the graph by thick dark gray lines. Finally thesolid black line represents the overall minimum the antennacan be configured to for a particular frequency over the entireoperating range of the antenna. Fig. 23 presents an overlay of theco-polarized and cross-polarized radiation patterns plotted at theresonance for each state of the reconfigurable 50 –30 fractaltree dipole. Because of the asymmetry in the tree geometry forseveral of the reconfigurable states, the cross-polarized patternscorresponding to these particular states are somewhat higher( 30 dB), but still remain within acceptable limits. Theco-polarized radiation pattern plots are again seen to closelyresemble those of a conventional linear half-wave dipole. Finally,


Fig. 23. Co-polarized and cross-polarized radiation patterns for each state of the reconfigurable six-branch, 50 –30 fractal tree dipole antenna.

we note that a monopole version of this antenna can also becreated with half the number of switches. In this case a broadbandor tunable matching network would be also required.

V. CONCLUSION

This paper begins by investigating the relationship betweenthe geometrical structure of fractal tree antennas and their cor-responding radiation characteristics. It was found, through aseries of systematic MoM simulations, that the two most crit-ical factors influencing the successful design of miniature 3-Dfractal tree antennas appear to be the density and elevation angleof their branches. These observations subsequently led to thedevelopment of several new design configurations that employfractal tree end-loads as a means of miniaturizing conventionaldipole or monopole antennas. Hybrid fractal trees and center-stubbed fractal trees represent two types of end-load structuresthat have proven to be particularly effective in achieving a sig-nificant amount of size reduction. Next, multiband and recon-figurable versions of these miniature antennas are introduced,where either reactive LC traps or RF switches are strategicallyplaced throughout the branches and/or along the trunk of the

trees. A prime advantage of placing the traps or RF switchesat critical locations on branches of the fractal tree end-loadsis that the resonant frequencies of the antenna can be spacedmuch closer together than if they were applied only to the treetrunks. Among the designs considered was a miniature reac-tively loaded tri-band center stubbed fractal tree monopole withassociated matching network. Also considered was a design fora miniature reconfigurable dipole antenna that achieves a 57%size reduction for the lowest intended band of operation and hasa tunable bandwidth of nearly 70%.

REFERENCES

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[2] D. H. Werner and R. Mittra, Frontiers in Electromagnetics. Piscat-away, NJ: IEEE Press, 2000.

[3] M. Sindou, G. Ablart, and C. Sourdois, “Multiband and wideband prop-erties of printed fractal branched antennas,” IEE Electron. Lett., vol. 35,no. 3, pp. 181–182, Feb. 1999.

[4] C. Puente, J. Claret, F. Sagues, J. Romeu, M. Q. Lopez-Salvans, andR. Pous, “Multiband properties of a fractal tree antenna generated byelectrochemical deposition,” IEE Electron. Lett., vol. 32, no. 25, pp.2298–2299, Dec. 1996.


[5] D. H. Werner, A. R. Bretones, and B. R. Long, “Radiation characteristicsof thin-wire ternary fractal trees,” IEE Electron. Lett., vol. 35, no. 8, pp.609–610, Apr. 1999.

[6] J. P. Gianvittorio and Y. Rahmat-Samii, “Fractal element antennas: acompilation of configurations with novel characteristics,” in Proc. IEEEAntennas and Propagation Society Int. Symp., vol. 3, Salt Lake City, UT,July 2000, pp. 1688–1691.

[7] J. P. Gianvittorio, “Fractal Antennas: Design, Characterization, and Ap-plications,” M.S. thesis, Dept. Elect. Eng., University of California LosAngeles, 2000.

[8] J. P. Gianvittorio and Y. Rahmat-Samii, “Fractal antennas: a novelantenna miniaturization technique, and applications,” IEEE AntennasPropagat. Mag., vol. 44, Feb. 2002.

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[10] J. S. Petko and D. H. Werner, “Dense 3-D fractal tree structures as minia-ture end-loaded dipole antennas,” in Proc. IEEE Antennas and Propaga-tion Society Int. Symp., vol. 4, San Antonio, TX, June 2002, pp. 94–97.

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[15] Microwave Office, Applied Wave Research Inc., 2000.

Joshua S. Petko (S’02) was born in Brownsville,PA, in 1979. He received the B.S. degree in electricalengineering from The Pennsylvania State University,University Park, in 2002, where he is currentlyworking toward the M.S. degree.

Currently, he is a Research Assistant for theCommunications and Space Sciences Laboratoryand the Applied Research Laboratory, PennsylvaniaState University. His research interests includeantenna theory, computational electromagnetics,fractal electrodynamics, and evolutionary algorithms

with a focus on fractal antenna elements and arrays.Mr. Petko is a Member of Eta Kappa Nu. He has been awarded the 2002

James A. Barnak Outstanding Senior Award from the Pennsylvania State Uni-versity Eta Kappa Nu chapter and was a finalist and an honorable mention for theAlton B. Zerby and Carl T. Koerner National Outstanding Senior Award fromEta Kappa Nu. He also has been awarded second place in both the 2002 PennState Undergraduate Poster Competition and the 2002 IEEE Region 2 StudentPaper Competition.

Douglas H. Werner (S’81–M’89–SM’94) receivedthe B.S., M.S., and Ph.D. degrees in electrical en-gineering and the M.A. degree in mathematics fromThe Pennsylvania State University (Penn State),University Park, in 1983, 1985, 1989, and 1986,respectively.

He is an Associate Professor in the Department ofElectrical Engineering, Penn State. He is a member ofthe Communications and Space Sciences Lab (CSSL)and is affiliated with the Electromagnetic Communi-cation Research Lab. He is also a Senior Research

Associate in the Electromagnetics and Environmental Effects Department ofthe Applied Research Laboratory at Penn State. He is a former Associate Editorof Radio Science. He has published numerous technical papers and proceedingsarticles and is the author of nine book chapters. He is an Editor of Frontiersin Electromagnetics (Piscataway, NJ: IEEE Press, 2000). He also contributeda chapter for Electromagnetic Optimization by Genetic Algorithms (New York:Wiley Interscience, 1999). His research interests include theoretical and com-putational electromagnetics with applications to antenna theory and design, mi-crowaves, wireless and personal communication systems, electromagnetic waveinteractions with complex media, meta-materials, fractal and knot electrody-namics, and genetic algorithms.

Dr. Werner is a Member of the American Geophysical Union (AGU), Inter-national Scientific Radio Union (URSI) Commissions B and G, the AppliedComputational Electromagnetics Society (ACES), Eta Kappa Nu, Tau Beta Pi,and Sigma Xi. He received the 1993 Applied Computational ElectromagneticsSociety (ACES) Best Paper Award and a 1993 URSI Young Scientist Award.In 1994, he received the Pennsylvania State University Applied Research Lab-oratory Outstanding Publication Award. He received a College of EngineeringPSES Outstanding Research Award and Outstanding Teaching Award in March2000 and March 2002, respectively. He recently received an IEEE Central Penn-sylvania Section Millennium Medal. He has also received several Letters ofCommendation from Penn State’s Department of Electrical Engineering foroutstanding teaching and research. He is an Editor of IEEE ANTENNAS AND

PROPAGATION MAGAZINE.


Investigations on Miniaturized Endfire VerticallyPolarized Quasi-Fractal Log-Periodic

Zigzag AntennaSatish K. Sharma, Member, IEEE, and Lotfollah Shafai, Fellow, IEEE

Abstract—This paper presents the investigations on a miniatur-ized vertically polarized traveling wave antenna for operation inthe high frequency band (3–6 MHz), with a specific requirementof keeping its height near 1 8th of a wavelength. The antenna isdesired to have a good endfire gain and front to back ratio, andsmall radiation levels in the vertical direction at broadside angle. Alog-periodic Zigzag antenna (LPZA) has acceptable performancein both gain and polarization. Its height however is large, at aboutone wavelength (1 ). The concept of fractal antenna is employedin this antenna to achieve the necessary height reduction to 8,while keeping its radiation characteristics nearly constant. Bothsingle and dual arm quasifractal log-periodic zigzag antenna(QFLPZA) configurations are investigated, with a maximumantenna height of only 1 8th of a wavelength, showing the desiredradiation characteristics, and a wide impedance bandwidth of67%. This type of antenna may find applications in surveillanceradar.

Index Terms—Miniaturized wire antenna, vertically polarized,log-periodic (LP), zigzag (Z), backfire radiation, quasi-fractal(QF).

I. INTRODUCTION

THE antennas whose current and voltage distributions canbe represented by one or more traveling waves, usually in

the same direction, are referred to as traveling wave antennas.This antenna radiates from a continuous source. There arevarious examples of traveling wave antennas, such as dielectricrod, helix, and various log periodic antennas (LPA) [1], mostof which are suitable for microwave frequencies. In the presentstudy, different variations of traveling wave antennas wereconsidered, i.e., traveling wave linear [2] and V-antennas [3],sandwich wire antennas [4]–[6], meander line planar arrayantennas [7], [8], and most importantly log-periodic zigzagantennas (LPZA) [9], [10]. Another antenna of interest was theelectrically short umbrella top-loaded antenna [11]. However,among these options, the log periodic zigzag antenna seemedto provide the desired antenna radiation characteristics hencewas selected for further study. Some investigation results onthis study were presented by the authors in [12].

Zigzag antennas are classified as periodic structures [9]. Thebasic zigzag antenna is shown in Fig. 1(a), in which is thepitch angle. Assuming that the current along the wire travelswith free space phase velocity, the near fields of the zigzag

Manuscript received July 16, 2003; revised November 5, 2003.The authors are with the Department of Electrical and Computer En-

gineering, The University of Manitoba, Winnipeg, MB R3T 5V6, Canada(e-mail: [email protected]).


Fig. 1. Configurations of (a) basic zigzag traveling wave antenna, (b) singlearm log-periodic zigzag traveling wave antenna, and (c) balanced LPZA [9],[10].

wire will have a fundamental-wave phase constant given by:, where is the free space wave number. This

monofilar zigzag antenna produces a backfire radiation below itsresonance frequency, and a broadside beam at its resonance. Avariation of this zigzag antenna as a single arm LPZA is shownin Fig. 1(b). It shows the small cells termed as transmissionline region, which correspond to the low-frequency condition,and causes little radiation to occur as the fundamental and allspace harmonics are slow waves. The region where the first re-verse traveling space harmonics approaches the backfire condi-tion produces appreciable radiation and corresponds

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TABLE I

to the active region. If the radiation is sufficient, the larger cellswill not be excited, which is desirable to avoid radiation in di-rections away from backfire [9]. Fig. 1(c) shows the geometryof the double arm thin wire LPZA [10]. The angle refers tothe angle from tip to tip, and the angle between the planes of thezigzag elements is called . By controlling the angles and ,the E- and H-plane beamwidths and antenna directivity can becontrolled. On the other hand, the control of input impedance,beamwidth and other radiation characteristics may be achievedusing different log-periodic parameters like , and , where

is the geometric factor defined as [Fig. 1(c)].Hence, for the present investigation, the LPZA is selected as apossible candidate, and will be studied next. The computationaltool used for this study is NEC [16], which is a full wave anal-ysis software based on the method of moments (MOM). Thewire radius is 4.00 mm for the entire investigation. The antennadesign requirements are shown in Table I, where andare the gains along vertical (broadside) and horizontal (endfire)directions.

II. DOUBLE ARM LPZA

In this section, a LPZA is investigated [Fig. 1(c)] based on [9],[10], which will be the basis of comparison for the miniaturizedantenna presented next. The most important design parametersfor this LPZA are the geometrical factor, which is the ratioof two adjacent similar dimensions on the antenna ,and angles and . As shown in Fig. 1(c), determines the an-tenna lengths measured from the apex and is the property of afrequency independent log-periodic antenna. For this antenna,the selected parameters were: , itsmaximum element height , and length[Fig. 1(c)] along the principal axis of its single arm, whereis the free space wavelength at the lower end of the frequencyband, i.e., , where MHz. Thisantenna height of is much larger than the desired height of

. It was assumed that the maximum length of the antenna isnot a problem. Hence, the study was devoted to the reduction ofthe antenna height from to near , without sacrificingsignificantly the antenna’s radiation performance. This requiredan extensive study. Design details for this antenna are not givenhere, as it was known in literature [9], [10]. For the purposeof computations, the antenna is placed over an infinite groundplane at a height of and is fed at the input end with avoltage source w.r.t. the ground plane. Its impedance character-istics and radiation gain patterns for 67% frequency bandwidthare shown in Fig. 2(a) and (b), respectively.

An examination of Fig. 2(a) reveals that, with frequency vari-ation the resistance and reactance oscillate around 700 and 0

, respectively. Their maximum span, for both resistance andreactance, is around 400 and shows a fairly constant trend.The radiation gain characteristic is shown in Fig. 2(b) remaining

Fig. 2. Characteristics of a double arm LPZA (Fig. 1(c)), (a) Resistance andreactance variation with frequency, and (b) Radiation patterns (� = 0 plane) atdifferent frequencies within 67% bandwidth. The antenna is lifted from groundplane by 0:002� .

between 10.50 to 12.50 dBi. The gain level at broadside angleis always less than dBi. The front-to-back (F/B) ratio isvarying between 6.50 dB and 9.00 dB. Since, the radiation pat-tern peak is near it is also referred as the backfire antenna.Next, height miniaturization of this antenna is presented, em-ploying similar to the Koch type of fractal element [13] conceptfor a single arm LPZA.

III. ANTENNA MINIATURIZATION TECHNIQUES FOR

SINGLE ARM

In this section, the height miniaturization of a single armLPZA is discussed, while keeping its radiation characteristicsalmost unaltered. The techniques used to reduce the antennaheight are as follows.

1) Fold the lower half of the antenna along its axis, i.e.,axis, by 90 , keeping the upper half vertical and making thelower half horizontal, i.e., plane (Fig. 3, Step II). Theantenna length along its axis remains constant and equal to

. Its height reduces to .2) Fold back the antenna triangular arms, when their heightexceeds (Fig. 3, Step III). The antenna length along itsaxis remains constant. Its height decreases to .3) Fold back the arms again, when their length exceeds(Fig. 3, Step IV). The antenna length along its axis remainsconstant. The antenna height decreases to .4) The entire antenna is lifted (20 cm) aboveground plane to prevent shorting the horizontal arms toground plane during its operation.

SHARMA AND SHAFAI: INVESTIGATIONS ON MINIATURIZED ENDFIRE VERTICALLY POLARIZED QFLPZA 1959

Fig. 3. Step by step antenna miniaturization technique. (a) Step I: original antenna height of H , (b) Step II: reduced height, H=2, (c) Step III: reduced height,H=4, and (d) Step IV: final reduced height, H=8. In all cases, the antenna is lifted from ground plane by 0:002� .

Fig. 4. Effect of antenna height reduction on vertically polarized radiationpattern (� = 0 plane) at different frequencies for single arm QFLPZA. (a)Step I and (b) Step IV.

In the above process, the elements exceeding a specifiedheight limit are reduced by the quasifractal elements. Thequasifractal element concept employed is similar to the Istiteration of the Koch curve [13]. The condition that geometricalfactor, remains unaltered makes sure that after reducing theheight the antenna is still log-periodic. The horizontal wiresparallel to the ground plane will not add much to the radiation,though they will cause the necessary phase delay to occur [14],[15], behaving as a transmission lines. Contrary to this, thevertical wires will directly contribute to the vertically polarizedradiation due to vertical component of current present in them.Furthermore, after braking the original height of wires, forboth horizontal and vertical wires and converting them into

Fig. 5. Configuration of the reduced length single arm quasifractal LPZA(QFLPZA), L = 1:96 � , and H = � =8. The antenna is lifted from groundplane by 0:002� .

fractal form, the resultant path lengths for propagation remainsalmost the same as the original antenna. Breaking the wires,however, modifies the direction of the radiating currents andthus, will influence the antenna radiation characteristics. Thisis investigated and the resulting radiation patterns are shown inFig. 4(a) and (b), respectively, for the original antenna and thefinal reduced height model.

An examination of Fig. 4(a) reveals that for the original an-tenna in Step I of Fig. 3, the backfire gain varies between 9.50to 11.40 dBi, and the F/B ratio changes from 11.20 to 14.60 dB.The radiating gain in the vertical direction is below

6 dBi, about 15 dB below the main beam. Its performanceafter the full height reduction to (Step IV of Fig. 3) asshown in Fig. 4(b), remains satisfactory. The backfire gain isvarying between 6.00 to 9.00 dBi, and the radiation in the ver-tical direction is below 2.50 dBi. The back radiation, however,seems sensitive to the operating frequency. An interesting phe-nomenon is the null filling in the main beam and elimination ofthe first sidelobe. As shown inc Fig. 4(b) the resulting gain pat-terns become smoother, and the gain drops gradually away from


Fig. 6. Radiation pattern (� = 0 plane) for the reduced length(L = 1:96� ) single arm QFLPZA at different frequencies.

Fig. 7. Configuration of double arm reduced length QFLPZA shown in xyzviews. L = 1:96� ;H = � =8; � = 25 . The antenna is lifted from groundplane by 0:002� .

the main beam. This antenna is termed as the quasi-fractal logperiodic zigzag antenna (QFLPZA), and studied further for itslength reduction and fractal length variations.

A. Length Reduction

In this subsection, the Step IV antenna is further investigatedfor reduction in its length along the axis, which originallywas . Extensive simulations were run, by reducing thewire elements near the transmission line region, which do notcontribute significantly to the radiation. For the sake of brevity,these are not presented here, instead the results for an antennawith length reduced to along the -coordinate axis ispresented, which eliminates of length, and still per-forms similar to the Step IV antenna. The antenna configurationis shown in Fig. 5. It is clear that the angle as defined in Fig. 1is now irrelevant, except at the antenna input end. The geomet-rical factor , still defines the configuration. To prevent shortingof horizontal arms to ground, the antenna is placed over an infi-nite ground plane at a height of and is fed at the inputend with a voltage source w.r.t. the ground plane. The computeddata for radiation patterns at is shown in Fig. 6. The gainvaries within 3.00 dB range between 9.00 and 6.00 dBi for dif-ferent frequencies within the band.

B. Effect of Fractal Element Lengths

It is known that the introduction of bends in a line reducesits effective electrical length due to the corner effects. The

Fig. 8. Effect of varying the arm-to-arm angle � on double arm reduced length(L = 1:96� )QFLPZA. (a) Gain and (b) F/B ratio.

amount of the length reduction depends on the corner angleand increases as the corner angle reduces. In the proposedheight reduction method the number of fractal and thus cornersincreases progressively along the antennas. However, since eachfractal has a single corner, this can easily be compensated for byincreasing the length of each fractal element. This phenomenonwas studied at different frequencies within 2 % length variationof the fractals. The results showed a rapid degradation of theperformance, at all frequencies, with the length reduction.The improvement in the performance, due to length increase,however, was very slow. Consequently, and the fact that theantenna height reduction was desired, the length increase wasnot incorporated. Hence, the fractal element lengths were keptas in the original antenna (Section III: Step IV, and III-A).

IV. DOUBLE ARM QFLPZA

The double arm antenna is a more practical one, in which theH-plane beamwidth can be controlled by the angle betweenthe arms. Thus, in this part a double arm QFLPZA is investi-gated using the single arm design of QFLPZA presented in theprevious Section III-A. The geometry of the antenna is shownin Fig. 7. A parametric study was carried out to find the effectof the arm-to-arm angle as defined in Figs. 1(c) and 7 on thegain and F/B ratio. The results are shown in Fig. 8(a) and (b).

Fig. 8(a) reveals that by increasing the angle the gain dropsslowly at the low frequency end of 0.75 , whereas for it isnearly constant. For 1.25 the gain increases with frequency.Similarly, Fig. 8(b) shows that by increasing the F/B ratiodrops rapidly for 0.75 f , whereas for and 1.25 , it is nearlyconstant. Thus, based on Fig. 8(a) and (b), the arm-to-arm angle

is an important parameter. Near a tradeoff between

SHARMA AND SHAFAI: INVESTIGATIONS ON MINIATURIZED ENDFIRE VERTICALLY POLARIZED QFLPZA 1961

Fig. 9. Performance of reduced length (L = 1:96� ) double arm quasifractalLPZA (QFLPZA). (a) Resistance and reactance, (b) input impedance on SmithChart with 575 normalization impedance, (c) return loss, and (d) verticallypolarized radiation pattern at � = 0 plane.

the gain and bandwidth can be achieved, and therefore isselected for the double arm QFLPZA. The results for itsimpedance, return loss, and radiation patterns are shown inFig. 9(a)–(d), respectively.

From Fig. 9(a) the resistance undergoes significant variationbetween 200 and 1000 in the frequency range, but the reac-tance varies only between , except at 0.825 , where itis around 680 . The input impedance was also plotted on SmithChart with 575 of normalization characteristic impedance,and is shown in Fig. 9(b). From Fig. 9(b) it is evident thatimpedance is balanced and can be matched to a 50 circuitusing a commercial RF impedance transformer of impedanceratio 50 /575 . The impedance plot lies at the center ofthe Smith chart. Only a few points lie outside 2:1 circle. Thisis further clear from Fig. 9(c), which shows the return loss,

(dB), computed using a reference impedance of 575 . Itreveals a very good impedance match ( dB) over thecomplete frequency band, except at 0.775 and 1.00 whereit becomes poor. Fig. 9(d) shows the radiation patterns withinthe 67% frequency band. The antenna shows good backfiregain at all frequencies, except at the lower end frequencies,which is attributed to the reduced effective antenna height atthese frequencies. The gain variation is within 3.60 dB rangingbetween 11.60 and 8.00 dBi within the band. The F/B ratio isbetween 6.00 and 9.00 dB. The gain in the vertical directionis less than 5 dBi, except at the center frequency. Its perfor-mance is superior to the single arm QFLPZA, and its gain hasimproved by at least 2.00 dB throughout the frequency range.Further effort may be needed to improve the F/B ratio.

The antenna was intended to be transportable and easy toinstall. The entire antenna was, therefore, to be mounted on aflexible net like material, using flexible wires. To install, non-conducting rigid poles of proper heights, were to be installedat location of the vertical arms, with predetermined mountinghooks. The deployed antenna was to be mounted on the poles,in a trellis like structure. For transportation, the antenna to be re-moved from the poles, folded along bends, and simply rolled forstorage as a bundle. The mounting and removal of the antennarequires negligible time. Only the installation of poles needscareful planning, to insure its survival during operation.

V. CONCLUSION

Vertically polarized antennas are an important class of an-tennas that are used efficiently over conducting ground planes,and monopoles are the most popular configurations. Theirheight of , however, becomes large at low frequencies.They are also narrow band. In this study, we have investigatedthe possibility of using LPZA to achieve wide impedance bandsand modified their geometry to reduce the height to . Theresulting antenna is a quasifractal antenna. Steps leading to theheight reduction are described and their effects on the antennaperformance are studied. While the height reduction processhas deteriorated the front-to-back ratio at some frequencies, theoverall antenna performance has remained remarkably good.The gain pattern becomes smoother and decreases graduallywith the angle away from the main beam.

REFERENCES

[1] C. A. Balanis, Antenna Theory and Design, 2nd ed, New York: Wiley.[2] D. P. Nyquist and K. Chen, “The traveling wave linear antenna with

nondissipative loading,” IEEE Trans. Antennas Propagat., vol. AP-16,pp. 21–31, Jan. 1968.


[3] K. Iizuka, “The traveling wave V-antenna and related antennas,” IEEETrans. Antennas Propagat., vol. AP-15, pp. 236–243, Mar. 1967.

[4] W. Rotman and N. Karas, “Sandwich wire antenna: A new type ofmicrowave line source radiator,” Proc. IRE Nat. Conf. Rec., pt. 1, pp.166–172, 1957.

[5] K. Chen, “Sandwich wire antenna,” IRE Trans. Antennas Propagat., vol.AP-10, pp. 159–164, Mar. 1962.

[6] H. E. Green and J. L. Whitrow, “The new analysis of the sandwich wireantenna,” IEEE Trans. Antennas Propagat., vol. AP-19, pp. 600–605,Jan. 1971.

[7] G. A. Hockham and R. I. Wolfson, “Broadband meander-line planararray antenna,” IEEE Trans. Antennas Propagat., vol. AP-27, pp.645–648, 1979.

[8] O. Aboul-Atta and L. Shafai, “Hemispherically radiating meander-lineplanar array antenna,” in Proc. Int. Conf. Antennas and Propagation,1987, pp. 141–144.

[9] Antenna Handbook, Theory, Applications, and Design, Y. T. Lo and S.W. Lee, Eds., Van Nostrand Reinhold, New York, 1988.

[10] S. H. Lee and K. K. Mei, “Analysis of zigzag antennas,” IEEE Trans.Antennas Propagation, vol. AP-18, pp. 760–764, Nov. 1970.

[11] A. F. Gangi, S. Sensiper, and G. R. Dunn, “The characteristics of electri-cally short, umbrella top-loaded antennas,” IEEE Trans. Antennas Prop-agat., vol. AP-13, pp. 864–871, Nov. 1965.

[12] S. K. Sharma and L. Shafai, “Investigations of a compact vertically po-larized backfire high frequency traveling wave antenna,” in Proc. IEEEAntennas Propagation Int. Symp., vol. 1, Columbus, OH, June 22–27,2003, pp. 253–256.

[13] J. P. Gianvittorio and Y. Rahmat-Samii, “Fractal antennas: A novelantenna miniaturization technique, and applications,” IEEE AntennasPropagat. Mag., vol. 44, pp. 20–36, Feb. 2002.

[14] E. C. Jordan and K. G. Balmain, “Ch. 15,” in Electromagnetic Wavesand Radiating Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-HallInc., 1968, pp. 620–621.

[15] V. H. Rumsey, Frequency Independent Antennas, New York: Academic,1966, pp. 105–110.

[16] Numerical Electromagnetic Code.

Satish Kumar Sharma (M’00) was born in Sul-tanpur (U.P.), India, in April 1970. He received theB. Tech. degree from Kamla Nehru Institute of Tech-nology, Sultanpur affiliated with Avadh University,Faizabad, and the Ph.D. degree from Institute ofTechnology, Banaras Hindu University, Varanasi,both in India, in 1991 and 1997, respectively, bothin electronics engineering.

He was a Lecturer and Project Officer with theKamla Nehru Institute of Technology, Sultanpur,and Institute of Engineering and Rural Technology,

Allahabad, from February 1992 to December 1993, respectively, both in UttarPradesh. There, he taught courses in electromagnetics, antennas and propaga-tion, electronics instrumentation and electronic communication, etc. DuringDecember 1993 to February 1999, he was a Research Scholar, and Junior andSenior Research Fellow of the Council of Scientific and Industrial Research(CSIR), Government of India, in the Department of Electronics Engineering,Institute of Technology, Banaras Hindu University. He was a PostdoctoralFellow with the Department of Electrical and Computer Engineering, TheUniversity of Manitoba, with Professor L. Shafai from March 1999 to April2001. Since May 2001, he has been a Senior Antenna Researcher/Engineer withInfoMagnetics Technologies Corporation, Winnipeg, MB, Canada. Since June2001, he has also been a part-time Research Associate with the Department ofElectrical and Computer Engineering, The University of Manitoba. Here, he hasbeen involved in the design and development of several antennas for wirelessand satellite communications as feed for reflectors, polarizers, and MEMSphase shifters. His main research interests are in applied electromagnetics,antennas, and RF MEMS.

He is a registered Professional Engineer (P. Eng.) of the Province of Man-itoba, Canada. He is reviewer of IEEE TRANSACTIONS ON ANTENNAS AND

PROPAGATION and Indian Journal of Radio and Space Physics (IJRSP).

Lotfollah Shafai (F’87) received the B.Sc. degreefrom the University of Tehran, Iran, in 1963 and theM.Sc. and Ph.D. degrees from the Faculty of AppliedSciences and Engineering, University of Toronto,ON, Canada, in 1966 and 1969, respectively, all inelectrical engineering.

In November 1969, he joined the Department ofElectrical and Computer Engineering, University ofManitoba, Canada, as a Sessional Lecturer, then as anAssistant Professor in 1970, an Associate Professorin 1973, and as Professor in 1979. Since 1975, he

has made a special effort to link the University research to the industrial de-velopment by assisting industries in the development of new products or es-tablishing new technologies. To enhance the University of Manitoba’s contactwith industry, in 1985 he assisted in establishing The Institute for TechnologyDevelopment, and was its Director until 1987, when he became the Head of theElectrical Engineering Department. His assistance to industry was instrumentalin establishing an Industrial Research Chair in Applied Electromagnetics at theUniversity of Manitoba in 1989, which he held until July 1994.

Dr. Shafai is a Member of the International Scientific Radio Union (URSI)Commission B, was its Chairman during 1985 to 1988, and is currently theVice-Chairman. He was elected a Fellow of The Royal Society of Canada in1998. He has been the recipient of numerous awards. In 1978, his contributionto the design of a small ground station for the Hermus satellite was selected asthe 3rd Meritorious Industrial Design. In 1984, he received the Professional En-gineers Merit Award and in 1985, “The Thinker” Award from Canadian Patentsand Development Corporation. From the University of Manitoba, he receivedthe “Research Awards” in 1983, 1987, and 1989, the Outreach Award in 1987and the Sigma Xi, Senior Scientist Award in 1989. In 1990, he received theMaxwell Premium Award from IEE (London) and in 1993 and 1994, the Dis-tinguished Achievement Awards from Corporate Higher Education Forum. In1998, he received the Winnipeg RH Institute Foundation Medal for Excellencein Research. In 1999 and 2000, he received the University of Manitoba, FacultyAssociation Research Award. He was a recipient of the IEEE Third MilleniumMedal in 2000 and in 2002 was elected a Fellow of The Canadian Academyof Engineering and Distinguished Professor at The University of Manitoba. Heholds a Canada Research Chair in Applied Electromagnetics. He has been a par-ticipant in nearly all Antennas and Propagation symposia and participates in thereview committees. In 1986, he established the symposium on Antenna Tech-nology and Applied Electromagnetics, ANTEM, at the University of Manitoba,which is held every two years.


Compact Wide-Band Multimode Antennas forMIMO and Diversity

Christian Waldschmidt, Student Member, IEEE, and Werner Wiesbeck, Fellow, IEEE

Abstract—This paper presents broadband multimode antennasfor multiple-input multiple-output (MIMO) and diversity applica-tions. The antenna system is not based on spatial diversity, as usualMIMO systems, but on a combination of pattern and polarizationdiversity. Different modes of self-complementary, thus extremelybroadband, spiral and sinuous antennas are used to decorrelatethe signals. It is shown that only one antenna is necessary to re-ceive three uncorrelated signals, thus the space required to placethe MIMO antenna is very small. Simulation results and measure-ments of a typical indoor scenario are given.

Index Terms—Multimode diversity, multiple-input multiple-output (MIMO), sinuous antenna, spiral antenna.

I. INTRODUCTION

FUTURE communication systems have to fulfill the require-ments of high data rates and flexible interfaces for dif-

ferent communication system standards. Multistandard radios,offering the demanded flexibility to use different standards, re-quire very broadband antennas. multiple-input multiple-output(MIMO) and diversity systems allow exploitation of the spatialchannel properties. If the signals received by different antennasare uncorrelated, very high data rates may be reached as recentstudies have shown, first in [1] and later in [2], [3]. Usually un-correlated signals are obtained by spatial diversity, which re-quires large antenna spacings.

This paper presents new broadband antenna solutions, thatare small enough to fit into laptops or organizers, but that stillyield uncorrelated signals for MIMO or diversity applications.The compactness of the broadband MIMO antenna system is notachieved by using different antennas, but by one antenna withdifferent, independently fed, modes. This results in multimodediversity, a combination of pattern- and polarization diversity toobtain uncorrelated channel impulse responses for the MIMO ordiversity system. As far as the authors are aware multimode di-versity has first been suggested in [4], where orthogonal azimuthpatterns were used. In [5] a multimode patch antenna with dif-ferent modes for diversity was presented. Multimode diversityfor MIMO has been suggested in [6], but this paper presentsa new and practical antenna concept, based on spiral and sin-uous antennas. In [7] the ability of logarithmic spiral antennasto radiate in different polarizations is discussed and a possibleapplication for diversity is mentioned, but not explicated.

Besides uncorrelated signals at the antennas, which are ob-tained by orthogonal patterns the mean signal to noise ratio

Manuscript received February 4, 2003; revised August 25, 2003.The authors are with the Institut für Höchstfrequenztechnik und Elektronik

(IHE), Universität Karlsruhe (TH), Karlsruhe D-76128 Germany (e-mail: [email protected]).


Fig. 1. Geometry of a spiral antenna with voltage sources between the singlearms of the spiral.

(SNR) of all the signals has to be “similar,” see [8], to obtaina diversity gain or capable MIMO systems. “Similar” in thiscontext means, e.g., less than 10 dB difference for two branchmaximum ratio combining, [8]. In this paper it is shown, thatthe mean effective gain (MEG), which is linked to the SNR, ofthe single modes differs by only 1 to 2 dB, thus a high diver-sity gain is obtained. For MIMO the total received power or themean SNR respectively is an important quality measure for anantenna array. By a comparison with a dipole array with largeantenna spacings, which is generally considered as a capablearray for MIMO, the ability of multimode antennas for MIMOis shown.

This paper is organized as follows. In the first sectionfour-arm spiral and sinuous antennas and the different exci-tations for the modes are presented. Second, the correlationproperties of signals received by different modes of the antennaand the mean effective gains are given as a function of theincident field and its spatial distribution. In the last sectionMIMO capacity calculations and measurements with spiralantennas are given.

II. SPIRAL AND SINUOUS ANTENNAS

The self-complementary, archimedian, four-arm spiral an-tenna and sinuous antennas are well described in the literature,see, e.g., [9]–[11], thus only the properties crucial for multi-mode diversity are given here. The spiral antenna consists offour arms, that are rotated around the center of the antenna,see Fig. 1. The antenna can basically radiate three differentmodes depending on the excitation. For this application mode1 and mode 2 are used. Mode 1 is characterized by a phaseshift of 90 between adjacent sources at the single arms of thespiral, see Fig. 1. Mode 2 has a phase shift of 180 . Both modesare circularly polarized in the direction of the main radiation

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Fig. 2. Geometry of a four-arm sinuous antenna. � describes the lengths ofthe teeth and is therefore a determining antenna parameter.

Fig. 3. Pattern of mode 1 of the spiral antenna with a radius of 10 cm at 2 GHzseparated into left (lhc) and right hand circular (rhc) polarization. If the spiral isfed at the outer end of the arms, the polarization is orthogonal to the one obtainedby exciting at the center of the spiral.

and elliptically polarized otherwise. Due to the self-comple-mentarily the antennas are frequency-independent or, in otherwords, extremely broadband. Since the geometrical structureof the spiral antenna is finite, there exists a lower frequencybound. This bound is

(1)

where is the speed of light, the outer radius of the spiraland the effective substrate permittivity. has to be de-termined by simulations of the spiral antenna or experimentally.According to experience, is close to one for etched spirals,also for a high of the substrate. For all simulations presentedin this paper the antennas were simulated with FEKO [12], astandard software tool based on method of moments. Equation(1) is explicable by the current distribution within the activezones of the single modes [10]. The active zone is a circular arealocated around the center of the antenna. The energy is radiatedfrom the antenna in the active zone. This zone is characterizedby a certain ratio of its circumference to the wavelength. For

Fig. 4. Pattern of mode 2 of the spiral antenna with a radius of 10 cm at 2 GHz.The pattern hardly changes versus frequency for frequencies above 1.2 GHz.

Fig. 5. Phase of the pattern of mode 1 and mode 2, shown in Figs. 3 and 4.The phase of mode 1 changes 360 per circulation around the antenna, mode 2changes 720 .

mode 1 the circumference is one wavelength, for mode 2 it istwo wavelengths. Thus, the current distribution on the arms ofthe spiral in the active zone has two maxima for mode 1 andfour for mode 2. Above this lower frequency bound all antennaproperties are almost stable and change only slightly with fre-quency. The pattern in elevation of mode 1 and 2 are givenin Fig. 3 and Fig. 4. The azimuth patterns are omnidirec-tional. The phase of the complex radiation pattern, which amongother parameters determines the correlation among the receivesignals, is shown in Fig. 5. The phase of mode 1 changes 360and the one of mode 2 720 for each circulation around the an-tenna, which is explicable by the current distribution within theactive zones. The modes can be excited in two ways: first byfeeding the spiral arms at the inner ends that is at the centerof the spiral and second at the outer ends of the arms. Thosemodes are orthogonally polarized left-hand circular (lhc) andright-hand circular (rhc). The third mode of the spiral antenna(270 phase shift between adjacent arms at the excitation) hasa pattern, whose amplitude is equal to mode 1, but the polar-ization changes from lhc to rhc. Thu,s mode 1 and mode 3 areorthogonally polarized. The unwrapped phase of the pattern ofmode 3 changes 1080 per circulation around the antenna.

WALDSCHMIDT AND WIESBECK: COMPACT WIDE-BAND MULTIMODE ANTENNAS FOR MIMO AND DIVERSITY 1965

Fig. 6. Pattern of mode 2 of the sinuous antenna with a radius of 10 cm at2 GHz. The pattern hardly changes versus frequency for frequencies above1.7 GHz.

Fig. 7. Pattern of mode 1 of the sinuous antenna with a radius of 10 cm at2 GHz. � is 50 . Mode 3 is orthogonally polarized, but has the same pattern.

The geometry of a four-arm sinuous antenna is given in Fig. 2and described in detail in [11]. The antenna is self-complemen-tary and used as a multimode antenna. The modes are excited thesame way as for the spiral antenna. The lower frequency boundsof the modes are a function of different geometry parameters,thus are not as easy accessible as for the spiral. In general thelower frequency bounds are higher than the ones for the spiralantenna for a given outer radius of the antennas. They decreasewith increasing (for see Fig. 2), since the antenna resemblesin sections a spiral antenna for large . The patterns of mode1 and 2 are given in Figs. 6 and 7. The shapes of the patternschange only slightly with frequency or , but the polarizationchanges. The pattern is alternately left and right hand ellipti-cally polarized versus frequency, see Fig. 8. The axial ratio ofthe sinuous antenna depends on . For large the antenna actsin sections like a spiral, thus the axial ratio is almost 0 dB. Forsmall the antenna is rather linearly polarized. Both modes maybe excited at the center or at the outer ends of the arms, but incontrast to the spiral antenna, orthogonal polarizations are onlyobtained for large ( , spiral-like behavior). The shape

Fig. 8. Gain of the rhc and lhc polarized field of the sinuous antenna with� = 50 and an outer radius of 0.1 m.

of the pattern of mode 3 is equivalent to mode 1, but both modesare orthogonally polarized.

III. MULTIMODE-DIVERSITY

MIMO transmission channels are characterized by thechannel matrix , which contains the channel impulse re-sponses or the channel coefficient in the flat fading casebetween the different sets of transmit and receive antennaports. For broadband systems the spectrum can be divided intonarrowband sections with flat fading. The diversity gain orMIMO capacity depends on the correlation coefficients amongthose channel coefficients of , see [3], and the SNR. Thecorrelation is influenced by the statistical properties of thewave propagation and the antenna properties, in this case theproperties of the single modes. In the following the correlationcoefficient among two receive signals as a function of theincident field is calculated. This is equivalent to the correlationamong the channel coefficients of for one transmit and tworeceive antennas in a MIMO system.

The spatial wave propagation properties are describable bythe power azimuth and elevation profile andfor both polarizations and . To allow for analytical calcula-tions typical statistical functions to model the wave propagationare chosen. Measurements have shown, that the power azimuthspectrum is best modeled by a Laplacian function [13]for both polarizations. For the power elevation profile aGaussian function is assumed. The total power angle spectrum isgiven by the product of the Laplacian function for the azimuthand a Gaussian function for the elevation, normalized so that

. With [14] (earlier shown in [8] in asimilar way) it can easily be shown that the complex correlationcoefficient among two signals received by different antennas, inthis case different modes, is given by

(2)


Fig. 9. Correlation coefficient among mode 1 (a) and mode 2 (c), excited atthe center of the spiral, and mode 1 (b), excited at the outer edge of the spiral togenerate orthogonal polarizations. The incident field has an elevation spread of5 and azimuth spread of 60 . The lower frequency bound of mode 2 is 1.2 GHz,thus the spiral does not work correctly for lower frequencies.

with the covariance

(3)

where is constant and the variance

(4)

where is the ratio of the power in -polarization to thepower in polarization at the receiver. Note, that is a func-tion of the polarimetric radiation pattern, thus disappears fororthogonally polarized antennas in this case lhc and rhc polar-ized modes. The power correlation coefficient is obtained by

, according to [15].Basically it is possible to use spiral or sinuous antennas with

any different modes and polarizations for multimode-diversity.In the following first a spiral and second a sinuous antenna areused to calculate the correlation coefficients among receive sig-nals. The orientation of the both antenna planes is vertical. Aspiral antenna with mode 1 and mode 2, excited at the center ofthe antenna, and a third mode (mode 1) with orthogonal polar-ization excited at the outer edge of the antenna is used. Figs. 9and 10 show the power correlation coefficient betweendifferent modes for a large azimuth angular spread of 60 anda small spread of 20 for a mean direction of 10 in azimuthand 0 in elevation of the incident waves. The third mode is or-thogonally polarized to the other modes, thus the correlation isalmost zero. The other modes are more strongly correlated asthe pattern of mode 1 and 2 partly overlap. On the other hand,

Fig. 10. Correlation coefficient among the same modes as in Fig. 9, but theincident field has an elevation spread of 5 and an azimuth spread of 20 .Due to slight changes in the pattern for different frequencies the correlationcoefficient changes. But it is over the whole frequency range low enough toobtain a diversity gain.

Fig. 11. Correlation coefficient among mode 1 (1), mode 2 (2) and mode 3(3) of the sinuous antenna. The incident field has an elevation spread of 5 andazimuth spread of 60 . The lower frequency bound of mode 2 is 1.7 GHz, thusthe antenna does not work correctly for lower frequencies.

the different phases of the patterns of mode 1 and 2 (see Fig. 5)decorrelate the received signals, since the single plane wavesfrom different directions superpose differently for each mode.The influence of the feed network on the pattern of the modesis neglected.

The sinuous antenna is used with three modes, allexcited at the center of the antenna. Figs. 11 and 12 show thecorrelation coefficient versus frequency for the scenarios men-tioned above. Mode 3 is orthogonally polarized to mode 1 and2, thus the correlation is low. Mode 1 and 2 hardly overlap, thusdifferent signals are received.

In order to fulfill the requirement of an equal or “similar”SNR of the signals received by different modes to obtain a diver-sity gain the MEG may be used, see [16]. The MEG is defined asthe ratio of the mean received power of one antenna under testto the mean received power of a reference antenna, when both


Fig. 12. Correlation coefficient among the same modes as in Fig. 11, but theincident field has an elevation spread of 5 and an azimuth spread of 20 . Withdecreasing angular spread the correlation increases.

TABLE IMEG OF DIFFERENT MODES IN DECIBELS (ELEVATION ANGULAR SPREAD 5 )

antennas are used in the same channel with the same transmitantenna, see [17], [18]. For the assumptions on the wave propa-gation made above the MEG can be calculated analytically foran isotropic reference antenna.

(5)

where are the gain patterns for both polarizations. Table Ishows the MEGs for different antennas and modes for a cross-polarization coupling of 8 dB. The MEGs of mode 3 of bothantennas are equal to the one of mode 1. Since the requirementof orthogonal patterns, i.e., uncorrelated signals, and similarMEGs are fulfilled, a diversity gain over a large bandwidth withboth antenna types, used as multimode antennas, is obtained.

IV. MIMO SYSTEMS BASED ON MULTIMODE-DIVERSITY

In order to show the potential of multimode antennas inMIMO systems, simulations and measurements of the capacityof a MIMO system with one multimode spiral antenna on eachside of the link, were performed. Additionally a comparisonwith dipole antennas, arranged in parallel, is drawn. For thesimulations a sophisticated channel model is used. This modeldoes not allow for analytical calculations like in Section III,but it allows to assess the MIMO performance in very realisticenvironments.

A. Simulations of the Capacity

The channel model used to calculate the capacity of MIMOsystems consisting of one spiral antenna at the transmitter andone at the receiver is an extended version of the model describedin [19]. This stochastic channel model is based on ray-tracingsimulations and measurement campaigns in indoor scenarios.It is a three dimensional double-directional channel model, inother words provides the angle of departure and arrival of eachpath. The channel model takes only non line-of-sight (NLOS)connections into account. The power azimuth spectrumis modeled by multiple Laplacian functions, each modeling acluster of scatterers. The elevation profile is modeled by a sinefunction. The cross polarization coupling is 8 dB. The an-tennas used for the simulation are one spiral antenna at the trans-mitter and one at the receiver. Mode 1 and 2 are excited at thecenter of the spiral, and mode 1 with orthogonal polarization isexcited at the outer edge of the spiral. Thus, the same modesas in Section III at both transmitter and receiver are used. Theorientation of the spiral plane is again vertical. The result ofthe simulations with this channel model are channel matrices(obtained the same way as in [20]). Therefore, the capacity ofa MIMO system with no channel state information at the trans-mitter can be calculated [2]

SNR(6)

where SNR denotes SNR conjugate complex transpose andis the number of transmit antennas, in this case the number

of different modes. The channel matrices in (6) are normalizedwith

(7)

to obtain a constant mean gain of each channel matrix, see [21].The SNR in (7) is the average SNR at the receiver. This normal-ization allows to show the influence on the capacity of the cor-relation properties and the distribution of the mean gains of thechannel coefficients. This distribution influences the capacity.The channel coefficients between co-polarized modes have alarger mean gain than those between cross-polarized modes.Thus, the mean gains of the channel coefficients are not equal.Equality is considered to be optimal, [2]. Fig. 14 shows the ca-pacity distribution for a constant mean SNR at the receiver of10 dB for 1000 channel realizations at 2 GHz. The 10% outagecapacity is approximately 7.3 bit/s/Hz.

B. Power Considerations

When the normalization in (7) is used in other words whenthe gain of each channel matrix is normalized, the informationabout it is lost. But to assess arrays for MIMO completely, thisinformation needs to be taken into account to assure a high effi-ciency of the complete MIMO channel. Fig. 15 shows the cumu-lative distribution function of the gain of the channel matrices

of the simulations. The comparison with a MIMOsystem with arrays consisting of three vertical half-wavelength


Fig. 13. Scenarios for the measurements. For the LOS scenarios transmitterand receiver are placed in the same room. For the NLOS scenarios thetransmitter is placed in the corridor.

Fig. 14. Measured cumulative distribution functions of the capacity fordifferent antenna scenarios at 2 GHz. The three dipoles have spacings of �=4.The capacity of the NLOS measurements reaches higher capacities as in theLOS scenarios for a constant mean SNR of 10 dB.

dipoles (also simulated with FEKO) with spacings on eachside of the link shows, that the gain of the channel matrices ofthe multimode MIMO system is not worse than with the dipolearrays. Additionally this distribution function is given for dipolearrays with spacings on each side of the link.

C. Measurements

The measurements were performed with two spiral antennas.The antenna were designed for a frequency range from 1.2 GHzup to 2.5 GHz, limited by the feeding network. Mode 1 and 2 areexcited with the feeding network given in [9]. At the outer endsof the arms a hybrid mode with orthogonal polarization com-pared to the other modes is excited. The coupling between thesingle modes is below 20 dB. The measurement system con-sists of a two channel network analyzer, amplifiers and coaxialswitches. The channel coefficients were measured one by one.All measurements were done during night, in order to reducethe time variance of the channel. The measurements were per-formed in an office building, with concrete ceilings and concreteand wood covered walls. The average office size is 4 5 m, seeFig. 13. The receive antennas were placed at the same position

Fig. 15. Transmission gain of different MIMO systems for the path basedchannel model. The SISO system has one transmit and one receive dipole.

for all measurements. The transmitter was moved along two dif-ferent routes, shown in Fig. 13. During the first route a strongLOS component is present, whereas the other route is alwaysNLOS. Along each route measurements at 801 discrete frequen-cies in the frequency range from 1.5 to 2.5 GHz at 210 differentpositions were performed. The measured data are normalized,according to (7), to obtain a constant mean SNR of each channelmatrix of 10 dB. Fig. 14 shows the cumulative capacity distribu-tion for both routes at 2 GHz. The capacity distribution changesnegligibly with frequency. Due to the higher multipath richnessof the NLOS route, it outperforms the LOS route.

For comparison two dipole arrays, consisting of three dipoleseach, were used, one at each side of the link. The dipoles werearranged in parallel with spacings of and vertical polariza-tion. The array covers approximately the same area as the spiralantenna with dimensions, so that the resonance frequency of thedipoles equals the lower frequency bound of the spiral. Fig. 14shows, that the dipoles perform worse than the spiral, since nei-ther polarization nor pattern diversity is exploited. The spacediversity is very limited due to the small antenna spacings.

V. CONCLUSION

This paper shows that four-arm spiral and sinuous antennasallow to exploit multimode diversity, which is a combinationof pattern and polarization diversity. The antennas are ex-tremely broadband, thus allow applications for multistandardradios. The space required for the antennas is relatively small.If placing dipoles on the same space required by the spiral,the dipoles do not reach the capacity of multimode-basedMIMO-systems.

REFERENCES

[1] J. H. Winters, “On the capacity of radio communication systems withdiversity in a rayleigh fading environment,” IEEE J. Select. AreasCommun., vol. 5, pp. 871–877, May 1987.

[2] G. J. Foschini and M. J. Gans, “On limits of wireless communications ina fading environment when using multiple antennas,” Wireless PersonalCommun., vol. 6, pp. 311–335, 1998.


[3] C. Chuah, D. N. C. Tse, J. M. Kahn, and R. A. Valenzuela, “Capacityscaling in MIMO wireless systems under correlated fading,” IEEETrans. Inform. Theory, vol. 48, pp. 637–650, 2002.

[4] E. N. Gilbert, “Energy reception for mobile radio,” Bell Syst. Tech. J.,vol. 44, pp. 1779–1803, 1965.

[5] R. G. Vaughan and J. B. Andersen, “A multiport patch antenna for mo-bile communications,” in Proc. 14th Eur. Microwave Conf., 1984, pp.607–612.

[6] T. Svantesson, “An antenna solution for mimo channels: the multimodeantenna,” in Conf. Record 34th Asilomar Conf., vol. 2, 2000, pp.1617–1621.

[7] O. K. Kim and J. D. Dyson, “A log-spiral antenna with selectable po-larization,” IEEE Trans. Antennas Propagat., vol. AP-19, pp. 675–677,Apr. 1971.

[8] R. G. Vaughan and J. B. Andersen, “Antenna diversity in mobile com-munications,” IEEE Trans. Veh. Technol., vol. 36, no. 4, pp. 149–172,July 1987.

[9] E. Gschwendtner and W. Wiesbeck, “Multi-service dual-mode spiralantenna for conformal integration into vehicle roofs,” in Proc. IEEEInt. Symp. Antennas and Propagation AP-S, vol. 3, Davos, Switzerland,2000, pp. 1532–1535.

[10] R. G. Corzine and J. A. Mosko, Four-Arm Spiral Antennas. Norwood,MA: Artech House, 1990.

[11] T. T. Chu and H. G. Oltman, “The sinuous antenna,” Microwave Syst.,News and Commun. Technol., vol. 18, pp. 40–48, 1988.

[12] www.emss.de [Online][13] K. I. Pedersen, P. M. Mogensen, and B. H. Fleury, “Spatial channel

characteristics in outdoor environments and their impact on BS antennasystem performance,” in Proc. IEEE Vehicular Technology Conf., 1998,pp. 719–724.

[14] K. Fujimoto and J. R. James, Mobile Antenna Systems Handbook. Nor-wood, MA: Artech House, 1994.

[15] J. R. Pierce and S. Stein, “Multiple diversity with nonindependentfading,” in Proce. IRE, vol. 48, 1960, pp. 89–104.

[16] M. G. Douglas, M. Okoniewski, and M. A. Stuchly, “Performance ofpcs handset antennas in mobile environments,” in Proc. IEEE MTT-SInt. Microwave Symp. Dig., vol. 3, 1997, pp. 1759–1762.

[17] J. B. Andersen and F. Hansen, “Antennas for VHF/UHF personal radio: atheoretical and experimental study of characteristics and performance,”IEEE Trans. Veh. Technol., vol. AP-26, pp. 349–357, 1977.

[18] T. Taga, “Analysis for mean effective gain in mobile antennas in landmobile radio environments,” IEEE Trans. Veh. Technol., vol. 39, pp.117–131, 1990.

[19] T. Zwick, C. Fischer, and W. Wiesbeck, “A stochastic multipath channelmodel including path directions for indoor environments,” IEEE J. Se-lect. Areas Commun., vol. 20, pp. 1178–1192, 2002.

[20] C. Waldschmidt, T. Fügen, and W. Wiesbeck, “Spiral and dipole an-tennas for indoor MIMO-systems,” Antennas Wireless Propagat. Lett.,vol. 1, no. 9, pp. 176–178, 2002.

[21] J. W. Wallace and M. A. Jensen, “Characteristics of measured 4� 4 and10� 10 MIMO wireless channel data at 2.4 GHz,” in Proc. IEEE Symp.Antennas and Propagation, vol. 3, 2001, pp. 96–99.

Christian Waldschmidt (S’01) was born in Basel,Switzerland, in 1976. He received the Dipl.-Ing.(M.S.E.E.) degree in electrical engineering from theUniversität Karlsruhe, Karlsruhe, Germany, in 2000,where he is currently working toward the Ph.D.degree.

From 2001 to 2003, he was with the Institut fürHöchstfrequenztechnik und Elektronik (IHE), Uni-versität Karlsruhe (TH), as a Research Assistant. Heserves as a Lecturer for smart antennas and radar an-tenna systems for the Carl Cranz Series for scientific

education. His research activities mainly focus on multiple input multiple outputsystems, smart antennas, small antennas, integration of antennas and vehicularantennas for radar and mobile communications applications.

Werner Wiesbeck (SM’87–F’94) received the Dipl.-Ing. (M.S.E.E.) and the Dr.-Ing. (Ph.D.E.E.) degreesfrom the Technical University of Munich, Munich,Germany, in 1969 and 1972, respectively.

From 1972 to 1983, he was with AEG-Tele-funken in various positions including the Head ofResearch and Development, Microwave Division,Flensburg, Germany, and Marketing Director inthe Receiver and Direction Finder Division, Ulm.During this period he had product responsibility formillimeter-wave radars, receivers, direction finders

and electronic warfare systems. Since 1983, he has been Director of the Institutfür Höchstfrequenztechnik und Elektronik (IHE), University of Karlsruhe,Karlsruhe, Germany, where he is presently Dean of the Faculty of ElectricalEngineering. In 1989 and 1994, respectively, he spent a six month sabbaticalat the Jet Propulsion Laboratory, Pasadena. He serves as a Permanent Lecturerfor radar system engineering and for wave propagation For the Carl CranzSeries for Scientific Education. He is a Member of an Advisory Committee ofthe EU-Joint Research Centre (Ispra/Italy), and he is an advisor to the GermanResearch Council (DFG), to the Federal German Ministry for Research and toindustry in Germany. His research topics include radar, remote sensing, wavepropagation and antennas.

Dr. Wiesbeck has received a number of awards including the IEEE Mil-lennium Medal. Since 2002, he has been a Member of the “HeidelbergerAkademie der Wissenschaften.” He was a Member of the IEEE GRS-S AdComfrom 1992–2000, Chairman of the GRS-S Awards Committee from 1994 to1998, Executive Vice President IEEE GRS-S from 1998 to 1999, PresidentIEEE GRS-S from 2000 to 2002, Associate Editor IEEE TRANSACTIONS ON

ANTENNAS AND PROPAGATION from 1996 to 1999, past Treasurer of the IEEEGerman Section. He has been General Chairman of the 1988 Heinrich HertzCentennial Symposium, the 1993 Conference on Microwaves and Optics(MIOP ’93) and he has been a member of scientific committees of manyconferences.


Ground Influence on the Input Impedance of TransientDipole and Bow-Tie Antennas

Andrian Andaya Lestari, Alexander G. Yarovoy, Member, IEEE, and Leo P. Ligthart, Fellow, IEEE

Abstract—In this paper, the influence of a lossy ground on theinput impedance of dipole and bow-tie antennas excited by a shortpulse is investigated. It is shown that the ground influence on theinput impedance of transient dipole and bow-tie antennas is signif-icant only for elevations smaller than 1 5 of the wavelength thatcorresponds to the central frequency of the exciting pulse. Further-more, a principal difference between the input impedance due totraveling-wave and standing-wave current distributions is pointedout.

Index Terms—Bow-tie antenna, dipole antenna, inputimpedance, transient antenna.

I. INTRODUCTION

D IPOLE and bow-tie antennas are employed in manytransient applications such as impulse ground penetrating

radar (GPR) for transmitting short transient pulses. The largeantenna bandwidth required to transmit such pulses with min-imal distortion (e.g., antenna ringing) is usually obtained bythe application of resistive loading [1], [2]. As resistive loadingsubstantially reduces radiation efficiency [1], it is essential toachieve maximum power transfer from the generator to theantenna, for which the input impedance of the antenna shouldbe known.

The input impedances of time-harmonic and transient an-tennas are principally different since the former is due tostanding-wave current distribution, while the latter is due totraveling-wave current distribution. Publications with regard tothe input impedance of time-harmonic dipole and bow-tie an-tennas near the ground are abundantly available in the literature.On the contrary, not much of the input impedance of transientdipole and bow-tie antennas near the ground has been reported.In the free-space case, significant contributions were given byWu [3] and Carrel [4] who presented analytical expressions ofthe input impedance of transient dipole and bow-tie antennas,respectively. In this paper we analyze the input impedance oftransient dipole and bow-tie antennas near a lossy ground.

A numerical method to predict the input impedance ofarbitrary metallic transient antennas in free space using thetime-domain integral equation (TDIE) method has beendemonstrated by Booker, et al. [5]. In their work the TDIE is

Manuscript received March 7, 2003; revised August 20, 2003. This workwas supported by the Dutch Technology Foundation (STW) under the projects“Improved Ground Penetrating Radar Technology” (1999–2000) and “Ad-vanced Re-Locatable Multisensor System for Buried Landmine Detection”(2001–2002).

The authors are with the International Research Centre for Telecommunica-tions-Transmission and Radar (IRCTR), Delft University of Technology, Delft,The Netherlands (e-mail: [email protected]).


numerically solved by the method of moments (MoM) usingthe marching-on-in time approach. By neglecting end reflec-tions in the time-domain solution, the input impedance is givenby the high-frequency limit of the frequency-domain solutionobtained by Fourier transforming the mentioned time-domainsolution. Unfortunately, when it comes to layered-mediumproblems, well-suited Green’s functions in space-time domainare not yet well documented. One of few developments of suchGreen’s functions has recently been reported for analyzing theresponse of a transient dipole in stratified media [6]. However,layered-medium Green’s function formulations in space-timedomain which are directly applicable to surface-patch MoMmethodologies, are not yet widely reported.

The numerical analysis carried out in this work is based onthe frequency-domain integral equation (FDIE) method, as themethods of solution for problems with layered media in fre-quency domain are already well established. The FDIE incor-porates a layered-medium Green’s function and is numericallysolved by a surface-patch MoM scheme for metallic nonwirestructures, whereas wire structures are approximated by narrowstrips. The input impedance of the transient antennas is ob-tained by the Fourier transformation and a time-window tech-nique for excluding end reflections. An experimental analysis isperformed to verify the computed results.

II. NUMERICAL METHOD

The work reported in this paper is based on a frequency-domain mixed-potential integral equation (MPIE) formula-tion. To account for the presence of the ground the dyadicGreen’s function formulation C for layered-medium problemsby Michalski and Zheng [7] is incorporated into the MPIE,which is numerically solved by MoM according to the trian-gular surface-patch methodology introduced by Rao, et al. [8].Computation time is minimized by employing the efficientnumerical implementation introduced in [9].

In this work the antennas are excited by a monocycle with0.8-ns duration shown in Fig. 1(a). In Fig. 1(b) its normalizedspectrum is given. The central frequency of this pulse is about1 GHz and the dB levels are found at frequencies 424 MHzand 1.670 GHz. It can be seen in Fig. 1(b) that the spectrum ofthe exciting pulse is essentially contained within the 0–5 GHzrange.

In this paper, we develop a nonstraightforward numericalmethod to predict the input impedance of a transient antenna infour steps as follows.

1) Antenna feed current is computed in the frequencydomain with 1 Volt input voltage by means of the MoM

0018-926X/04$20.00 © 2004 IEEE

LESTARI et al.: GROUND INFLUENCE ON THE INPUT IMPEDANCE OF TRANSIENT DIPOLE AND BOW-TIE ANTENNAS 1971

Fig. 1. Exciting pulse used in this work: (a) a monocycle with 0.8-ns durationand (b) its normalized spectrum.

scheme mentioned above. For obtaining time-domain so-lutions, the computations are performed at 100 frequencypoints from 50 MHz to 5 GHz with 50-MHz steps.2) Exciting pulse function is synthesized directly from

measurement of the 0.8-ns monocycle in Fig. 1(a), and usedas the excitation model in the computations.3) The feed current of the same antenna with infinite length

(thus, no end reflections) is computed by means of the Fouriertransformation and a time-window technique for removal ofend reflections, which can concisely be written as

F W F (1)

where is the normalized F is the discreteFourier transformation operator, F is the discrete inverseFourier transformation operator, and W is the time-windowoperator with smooth truncation process given by

for the left end (2a)

for the right end (2b)

Fig. 2. Setup for input impedance measurements.

in which is the value of the real or imaginary part of theargument, and is the decay rate of the truncation processthat assumes values in the range . We have foundthat typically the performance of (2) is optimal with

. Note that when (2) reduces to a rectangular timewindow.4) Finally, the input impedance of the transient antenna isobtained as

(3)

III. MEASUREMENT TECHNIQUE

We perform input impedance measurements in frequency do-main using a standard network analyzer. The antennas are situ-ated horizontally over a lossy ground, which in this case is drysand. The measurements are carried out without an anechoicchamber, and consequently the results are heavily disturbed byreflections from surrounding objects. To deal with this, the mea-surement results are inverse-Fourier transformed into time do-main, after which use of time gating is made to remove thoseunwanted reflections. The actual impedance of the antenna canbe extracted afterwards by performing the Fourier transforma-tion of the results back to frequency domain.

Furthermore, to properly measure balanced loads such as theantennas discussed here, a balun is required. However, sinceultra-wideband baluns are difficult to produce and commer-cially available ones are expensive, baluns are not used in themeasurements. A technique to measure the input impedanceof balanced antennas without baluns introduced in [10] is heresimplified. The antenna under test is fed using two identical50-Ohm semi-rigid cables soldered together over their length,except for a small part near the ends where SMA connectorsare attached. Each of the inner conductors of the other endsis soldered to one of the antenna terminals in the way shownin Fig. 2. The semi-rigid cables are connected by 50-OhmSucoflex coaxial cables to the ports of the network analyzer,which has been previously calibrated at the SMA connectors.Hence, the reflection coefficient at the SMA connectors isgiven by

(4)

where and are the -parameters measured by the net-work analyzer. Disturbances caused by unwanted reflections can


be removed from (4) by means of a time-gating operation, whichcan be described by

F W F (5)

in which the time-window operator W implements the trunca-tion process in (2). Using the dielectric constant and theinsertion loss of the semi-rigid cables provided by the man-ufacturer, the reflection coefficient at the antenna terminal canthen be written as

(6)

where is the phase constant which depends on is twotimes the length of the semi-rigid cables and is given in dB/m.Several available values of for different frequencies are inter-polated to obtain the values of over the whole frequency range.Finally, the input impedance of the antenna is given by

(7)

where is the characteristic impedance of the semi-rigidcables, which has a value of 100 Ohms due to the double-lineconfiguration shown in Fig. 2.

It is worth noting that antenna length is actually not importantfor the input impedance of transient antennas. However, in theabove technique antenna length is important for computationand measurement procedures, i.e.,

— to ensure separation in time between feed-point andend reflections for time window application,

— to obtain adequate duration of the time window sincea finite time window limits the lowest frequency atwhich input impedance can be determined.

In this work we use antennas with 50-cm length, which allowsapplication of a time window with 2-ns duration for excitationwith the 0.8-ns monocycle.

IV. ANALYSIS

As the input impedance of a transient dipole is frequency de-pendent [3], the input impedance is determined with respect tothe exciting pulse using the averaging given by [11]

(8)

in which and are the frequencies which correspond to thelower and upper limits of the pulse spectrum, respectively, and

is the antenna input impedance. It has been indicated in[11] that it is adequate to assume over the range

, and this is followed here with and cor-responding to the dB limits of the exciting pulse in Fig. 1.Hence, in (8) can be interpreted as the input impedancewith respect to the exciting pulse (the 0.8-ns monocycle). In thiswork in (8) is replaced with in (3). In this way,(8) is improved because prior to using the averaging in (8) we

Fig. 3. Experimental dipole over dry sand. Length= 50 cm and wire diameter= 2 mm.

perform time windowing in (1), which greatly reduces the oscil-lation of the impedance curve. In effect, this approach improvesthe averaging process.

The computed and measured input impedances of the hori-zontal transient dipole in Fig. 3 with respect to the 0.8-ns mono-cycle as function of elevation above the sand are presented inFig. 4(a). The wire is modeled as a thin strip using the equiva-lent radius formula [12]. The result computed by the NumericalElectromagnetics Code (NEC-2) is included for comparison. Itcan be seen that in general the computations agree with the mea-surement. At the highest elevation the computed and measuredvalues of the reactance are about 200 , in accordance with theresult obtained using the expression given in [3]. For very smalldistances from the interface we observe that the result computedby the triangular-patch MoM suffers from a slight discrepancywith the measurement. We notice that this discrepancy might becaused by the variation of the electrical length of the feed gap ofthe experimental antenna when it approaches the ground. Sucha phenomenon is however not experienced by the delta-functiongenerator assumed in the triangular-patch MoM. The delta-gapmodel used by NEC, on the other hand, accommodates this phe-nomenon as it uses a feed segment with the same length as thefeed gap width of the experimental antenna. This leads to betteragreement with the measurement for small distances from theinterface as shown in the figure. The result computed using thecommercial MoM code FEKO, which is based on the same tri-angular-patch MoM methodology, is also included. It can beseen that generally agreement between our code and FEKO isachieved. To further test the accuracy of our results, computa-tions are carried out using our code and FEKO for a 1-mm wirediameter and the results are presented in Fig. 4(b), in which theresults for a 2-mm diameter in Fig. 4(a) are also shown. It isdemonstrated in Fig. 4(b) that agreement is generally achieved.The observed slight discrepancy in reactance may be attributedto different densities of the mesh generated by the codes, es-pecially in the feed region of the antenna. Moreover, we notethat agreement between the results for small elevations indicatesthe accuracy of numerical evaluation of the layered-mediumGreen’s function.


Fig. 4. (a) Input impedance of the transient dipole in Fig. 3 with respect tothe 0.8-ns monocycle as function of elevation above the sand. (b) Comparisonbetween this work and FEKO for two different wire diameters (1 and 2 mm).

Additional proof of the accuracy of the numerical results isgiven in Fig. 5 by comparison with analytical results, obtainedusing the theory in [3]. The input impedance of a transient dipolein free space is plotted for two different wire diameters, 1 and 2mm. It can be seen that for sufficiently high frequencies gen-erally agreement between numerical and analytical results isachieved. The observed slight discrepancy at high frequenciesmay be explained by difference in the excitation models of theantenna. The theory in [3] assumes excitation from a coaxialfeed system, while for excitation we use a delta-function gen-erator. The large discrepancy at low frequencies is due to thefinite length of the time window, which imposes limitation onthe lowest frequency at which accuracy of the result is ensured.To obtain improved accuracy at lower frequencies one shoulduse a longer antenna for allowing a longer time window.

It is advisable to mention the advantages and disadvantagesof the used codes. The advantage of NEC-2 is the efficiencyfor handling wire structures as it employs thin-wire approx-imation, which reduces formulation of the problem into a

Fig. 5. Input impedance of transient dipoles in free space for two different wirediameters: analytical against numerical results. Analytical results are obtainedusing [3]; numerical results are computed by our code.

one-dimensional integral equation. However, to this work themain drawback of NEC-2 is that it renders inaccurate whenmodeling antennas very close to the ground. In Fig. 4(a) NEC-2computation is interrupted at 5-mm elevation because for lowerelevations the results become inaccurate. One of the advan-tages of our triangular-patch MoM code is its capability ofmodeling antennas touching the interface. In addition, it offersflexibility for modeling metallic antennas of arbitrary shape.In comparison with NEC-2 the obvious disadvantage of thecode when treating wire structures is its lower computationefficiency since it employs a surface integral formulation.

To analyze the influence of a lossy ground on the inputimpedance of the transient dipole we compute the inputimpedance with respect to the 0.8-ns monocycle as func-tions of antenna elevation for different ground types.In particular, we assume the ground to be sandy soil( S/m), dry clay ( S/m),wet clay ( S/m), and muddy soil( S/m). Furthermore, two different wirediameters of 1 and 2 mm are assumed to investigate the influ-ence of the wire thickness on the results. The computed inputresistance and reactance with respect to the 0.8-ns monocycleare plotted in Fig. 6, where it is evident that the presence ofthe ground significantly affects the impedance only for verysmall distances from the interface. In Fig. 6 it is shown that,the ground influence is already very small at elevations higherthan 6 cm for a wide range of ground types. Noting that thecentral frequency of the 0.8-ns monocycle is about 1 GHz(corresponding to a wavelength of 30 cm), as a generalizationof the results one may state that the ground essentially affectsthe input impedance of a transient dipole only when the antennaelevation is smaller than of the wavelength that correspondsto the central frequency of the exciting pulse.

Evidently, wire diameter exhibits a considerable influenceon the resistance as demonstrated in Fig. 6, which indicatesthat doubling the wire diameter from 1 to 2 mm reduces theresistance by about 19%. It is also worth noting that for a very


Fig. 6. Computed input impedance of a transient dipole with respect to the0.8-ns monocycle as functions of elevation for different ground types.

close proximity to the interface, the input resistance decreases asthe dipole approaches the interface. We note that this behavioris the opposite of the time-harmonic case, in which for verysmall distances from the interface the input resistance increasesas the dipole is lowered [13], [14]. This result indicates aprincipal difference between traveling-wave and standing-wavecurrent distributions of transient and time-harmonic antennas,respectively.

In this paper the input impedance of a transient bow-tieantenna is computed as functions of elevation above the groundfor flare angles of 30 , 50 , and 70 . The computed inputimpedance of a transient bow tie above sandy soil ( and

S/m) with respect to the 0.8-ns monocycle is pre-sented in Fig. 7. We notice in the figure that for the three flareangles the value of the reactance at a 6-cm elevation is alreadyclose to zero, which is the free-space value of the characteristicreactance of a transient bow tie [4]. Moreover, by inspection ofthe result given in [4] it is found that that at a 6-cm elevation theresistance nearly assumes its free-space value. Hence, similarto the case of the transient dipole discussed previously it isshown that the ground essentially affects the input impedanceof a transient bow-tie antenna only when the antenna elevation

Fig. 7. Computed input impedance of a transient bow tie as functions ofelevation above sandy soil for different flare angles.

is smaller than of the wavelength that corresponds to thecentral frequency of the exciting pulse.

V. CONCLUSION

The influence of a lossy ground on the input impedanceof dipole and bow-tie antennas excited by a short pulse isinvestigated. It is shown that the ground influence on the inputimpedance of a transient dipole and bow-tie antennas is signif-icant only for elevations smaller than of the wavelengththat corresponds to the central frequency of the exciting pulse.Furthermore, it is shown that for a very close proximity to theinterface, the input resistance of a transient dipole decreasesas the dipole approaches the interface. This behavior is theopposite of the time-harmonic case, in which for very smalldistances from the interface the input resistance increases asthe dipole is lowered.

ACKNOWLEDGMENT

The authors thank P. Hakkaart for his assistance in the con-struction of the experimental antenna and J. Zijderveld for hisassistance in the measurements.

REFERENCES

[1] T. P. Montoya and G. S. Smith, “A study of pulse radiation from severalbroad-band loaded monopoles,” IEEE Trans. Antennas Propagat., vol.44, pp. 1172–1182, Aug. 1996.

[2] K. L. Shlager, G. S. Smith, and J. G. Maloney, “Optimization of bow-tieantennas for pulse radiation,” IEEE Trans. Antennas Propagat., vol. 42,pp. 975–982, July 1994.

[3] T. T. Wu, “Input admittance of infinitely long dipole antennas drivenfrom coaxial lines,” J. Math. Phys., vol. 3, pp. 1298–1301, 1962.

[4] R. L. Carrel, “The characteristic impedance of two infinite cones of ar-bitrary cross section,” IRE Trans. Antennas Propagat., vol. AP-6, pp.197–201, Apr. 1958.

[5] S. M. Booker, A. P. Lambert, and P. D. Smith, “A numerical calculationof transient antenna impedance,” in Proc. 2nd Int. Conf. Computation inElectromagnetics, 1994, pp. 359–362.

[6] A. G. Tijhuis and A. Rubio Bretones, “Transient excitation of a layereddielectric medium by a pulsed electric dipole,” IEEE Trans. AntennasPropagat., vol. 48, pp. 1673–1684, Oct. 2000.


[7] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radia-tion by surfaces of arbitrary shape in layered media, part I: Theory, partII: Implementation and results for contiguous half-spaces,” IEEE Trans.Antennas Propagat., vol. 38, pp. 335–352, Mar. 1990.

[8] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scatteringby surfaces of arbitrary shape,” IEEE Trans. Antennas Propagat., vol.AP-30, pp. 409–418, May 1982.

[9] A. A. Lestari, A. G. Yarovoy, and L. P. Ligthart, “Numerical and ex-perimental analysis of circular-end wire bow-tie antennas over a lossyground,” IEEE Trans. Antennas Propagat., vol. 52, pp. 26–35, Jan. 2004.

[10] K. D. Palmer and M. W. van Rooyen, “Simple broadband measurementof balanced loads using a network analyzer,” in CD-ROM Proc. Mil-lenium Conf. Antennas Propagat. (AP-2000), Davos, Switzerland, Apr.2000.

[11] R. W. P. King and H. J. Schmitt, “The transient response of linear an-tennas and loops,” IRE Trans. Antennas Propagat., vol. 10, pp. 222–228,May 1962.

[12] C. M. Butler, “The equivalent radius of a narrow conducting strip,” IEEETrans. Antennas Propagat., vol. AP-30, pp. 755–758, July 1982.

[13] G. Turner, “The Influence of subsurface properties on ground pene-trating radar pulses,” Ph.D. dissertation, Macquarie University, Sydney,NSW, Australia, 1993.

[14] C. J. Leat, N. V. Shuley, and G. F. Stickley, “Complex image model forground-penetrating radar antennas,” IEEE Trans. Antennas Propagat.,vol. 46, pp. 1483–1488, Oct. 1998.

Andrian Andaya Lestari was born in Bogor,Indonesia. He received the Ingenieur and Ph.D. de-grees in electrical engineering from Delft Universityof Technology, Delft, The Netherlands, in 1993 and2003, respectively.

From 1993 to 1998, he was with a governmentresearch agency in Jakarta, Indonesia. He joinedthe International Research Centre for Telecommu-nications-transmission and Radar (IRCTR), DelftUniversity of Technology, as a Researcher in 1998.His work at IRCTR has resulted in over 20 publica-

tions, which include national and international patents, journal and conferencepapers, and scientific reports. Currently he works on ultrawide-band antennasand numerical tools for transient antenna analysis.

Alexander G. Yarovoy (M’96) received the Diploma(with honors) in radiophysics and electronics andthe Cand. Phys. & Math. Sci. and Dr. Phys. &Math. Sci. degrees in radiophysics, from KharkovState University, Kharkov, Ukraine, in 1984, 1987,and 1994, respectively.

In 1987, he joined the Department of Radio-physics, Kharkov State University, as a Researcherand became a Professor in 1997. From September1994 through 1996, he was with the TechnicalUniversity of Ilmenau, Germany, as a Visiting

Researcher. Since 1999, he has been with the International Research Centrefor Telecommunications-Transmission and Radar (IRCTR), Delft Universityof Technology, Delft, The Netherlands, where he coordinates all GPR-relatedprojects. His main research interests are in ultrawide-band electromagnetics,wave scattering from statistically rough surfaces and penetrable obstacles andcomputational methods in electromagnetics.

Leo P. Ligthart (M’94–SM’95–F’02) was born inRotterdam, the Netherlands, on September 15, 1946.He received the Engineer’s degree (cum laude)and the Doctor of Technology degree from DelftUniversity of Technology, Delft, The Netherlands, in1969 and 1985, respectively, the Doctorates (honoriscausa) from Moscow State Technical Universityof Civil Aviation, Moscow, Russia, in 1999, andthe Doctorates (honoris causa) from Tomsk StateUniversity of Control Systems and Radioelectronics,Tomsk, Russia, in 2001.

Since 1992, he has held the Chair of Microwave Transmission, Radar andRemote Sensing in the Department of Information Technology and Systems,Delft University of Technology, where in 1994, he became Director of the In-ternational Research Centre for Telecommunications-Transmission and Radar.His principal areas of specialization include antennas and propagation, radarand remote sensing, but he has also been active in satellite, mobile, and radiocommunications.


Adaptive Crossed Dipole Antennas Using aGenetic Algorithm

Randy L. Haupt, Fellow, IEEE

Abstract—Antenna misalignment in a mobile wireless commu-nications system results in a signal loss due to a decrease in an-tenna directivity and a polarization mismatch. A genetic algorithm(GA) is used to adaptively alter the polarization and directivity of acrossed dipole receive antenna in order to increase the link budget.The three orthogonal dipole configuration works better than onlytwo crossed dipoles, but both improved the link loss as the angularpointing errors increased. A GA with a high mutation rate worksbest for a noiseless open loop adaptation, while a GA with a lowmutation rate works best for noisy fully adaptive system.

Index Terms—Adaptive antenna, crossed dipole, genetic algo-rithm (GA), polarization, smart antenna.

I. INTRODUCTION

I F ONE OR MORE of the antennas in a wireless commu-nication system is mobile, then as the antennas move, the

direction of the peak gains and the polarization of the antennaschange. As a result, the power received goes down. For instance,a wireless system that transmits vertical polarization has someof its power converted to horizontal polarization as the signal re-flects from the environment. Unless the receive antenna can de-tect both polarizations, the received power decreases. Anotherexample is when a spacecraft orbits the earth; the antennas inthe communications system no longer align for optimum powertransfer. Antenna engineers design the antennas for maximumdirectivity and polarization match when the antennas point ateach other. Both the directivity and polarization of an antennachange with angle. If the antennas do not point at each other,then the product of the receive and transmit antenna directivi-ties goes down. The directivity loss coupled with the polariza-tion mismatch reduces the received power.

An obvious solution to this problem is to keep the antennaspointing at each other. Constantly maneuvering a spacecraft re-quires an unacceptable expenditure of precious fuel, though.Steering the ground antenna is another option but only solveshalf of the problem, since the spacecraft might still be out ofalignment. If the antenna is a phased array, then steering themain beam maximizes the directivity but does not improve thepolarization mismatch.

One way to improve the link budget is to adaptively changethe antenna polarization and directivity to maximize the powertransfer. In order to improve the link budget, the polarization

Manuscript received February 10, 2003; revised May 27, 2003.The author was with the Utah State University, Electrical and Computer

Engineering, Logan, UT 84322-4120 USA. He is now with the AppliedResearch Laboratory, Pennsylvania State University, State College, PA 16804USA (e-mail: [email protected]).


and directivity of the receive and/or transmit antennas must beadaptively changed if the positions of the two antennas change.Adaptive antennas usually place a null in the antenna patternto reject interference or steer a beam toward a desired signal.Phased arrays have more than one antenna, so they are perfectfor adapting their patterns. Adapting the polarization, however,requires an antenna that can modify the major and minor axesof its polarization ellipse. Crossed dipoles are perfect for thisapplication.

The crossed dipole antenna has found use before in systemsrequiring antennas that change their polarization. A wirelesscommunications system in a high multipath environment canimprove the link better through polarization diversity in theform of crossed dipoles than through spatial diversity (antennaseparation) [1]. A crossed dipole consists of two or threeorthogonal dipoles. When linearly polarized signals becomedepolarized due to reflections, each dipole can receive theelectric field component parallel to it. Using three orthogonalcrossed dipoles has been experimentally shown to significantlyincrease channel capacity of a wireless communication systeminside a building [2].

The polarization and directivity of a crossed dipole antennaare easy to control. One dipole controls the electric field parallelto it and the orthogonal dipoles control the electric fields par-allel to them. Each dipole has an independent complex weight.Controlling the amplitude and phase of the signal at each dipolemodifies the electric field amplitude and phase in orthogonaldirections resulting in any polarization from linear through el-liptical to circular. Modifying the amplitude and phase of thesignal at the dipoles also modifies the directivity of the antennaas well.

Adaptive crossed dipoles alter their polarization based uponenvironmental conditions. When a transmitted circularly polar-ized millimeter wave passes through rain, it becomes ellipticallypolarized. The depolarization can be calculated if the rainfallrate is known. Reference [3] proposed an open loop adaptivetransmit antenna that adjusted its polarization based upon themeasured rainfall in the propagation path. In [4], a least meansquare (LMS) algorithm adapted the polarization and pattern ofa two element array of crossed dipoles to improve the signal tointerference plus noise ratio (SINR). As long as the desired andinterference signals are not at the same angles and have the samepolarizations, the SINR was improved. In another paper, theLMS algorithm was used to find amplitude and phase weightsfor three orthogonal dipole antennas in order to improve theSINR. This arrangement provided some rejection for interfer-ence signals for most angles of arrival and polarizations [5]. Aprevious paper presented results from adaptively adjusting the

0018-926X/04$20.00 © 2004 IEEE

HAUPT: ADAPTIVE CROSSED DIPOLE ANTENNAS USING A GA 1977

Fig. 1. Coordinate system for the crossed dipole transmit and receive antennas.

amplitude and phase of the current fed to one dipole while theother dipole had an amplitude of one and phase of zero. The Nu-merical Electromagnetic Code generated the electromagneticresponse of the dipole antennas and a local optimizer performedthe optimization [6]. It was found that optimizing only for cir-cular polarization produces losses in radiated power that offsetthe polarization correction. An improvement in the power trans-ferred increased up to a maximum of 2.0 dB at . Usingadaptive crossed dipoles at the transmitter and receiver was alsoconsidered and further improved the model.

This paper expands upon a recent presentation that introducedthe application of a genetic algorithm (GA) to adaptively changethe current fed to crossed dipole antennas in order to improvethe link budget [7]. The dipole model consists of three orthog-onal short dipoles with variable control of the phase and am-plitude fed to each element. A GA is used to maximize thereceived signal by improving the directivity and polarizationmatch through weighting the currents at each dipole. Improve-ments in the link budget of up to 6 dB are possible.

II. CROSSED DIPOLE MODEL

Satellite communications systems use circularly polarizedantennas for the satellite and the ground antennas. In thispaper, the circularly polarized antennas are modeled as crosseddipoles. Consequently, controlling the amplitude and phase ofthe signals at the dipoles of the transmit and receive antennas,modifies the directivity and polarization of both antennas.In this case, the crossed dipole has three orthogonal dipoles.The receive antenna is located at an angle of fromthe transmit antenna (Fig. 1). Similarly, the transmit antennais located at an angle of from the receive antenna.Maximum power transfer occurs when and .

In order to determine the directivity and polarization of theantennas, the electric fields can be found from the currents onthe dipoles. If the dipoles are assumed to be short ,

the crossed dipole current is the sum of the constant currents oneach short dipole

(1)

Substituting this current into the equation for the magneticvector potential for a short dipole yields

(2)

wheredistance from the origin to the field point at ;dipole length in the and directions;radial frequency;wave number;permeability;constant current in or direction.

In the far field, the electric field in rectangular coordinates isfound from the magnetic vector potential by

(3)

The transmitted electric field is given by

Converting this rectangular form of the electric field into spher-ical coordinates produces the far field components

(4)

(5)

The directivity is given by

(6)and the polarization loss factor is

(7)

where with a perfect match. Theand subscripts represent transmit and receive, respectively.

Equations (10) and (11) are key ingredients to the link budget.The examples presented here assume the earth station con-

sists of a pair of orthogonal crossed dipoles in the planetransmitting a circularly polarized field in the -direction. In-creasing toward the horizon transitions from circular polar-ization through elliptical until linear polarization results at thehorizon. In this paper, the transmit antenna has the following


Fig. 2. Circularly polarized crossed dipoles along the x and y axes. This is ahemispherical plot of the inverse axial ratio. Light color indicates high with avalue of 1 in the z-direction and a value of 0 in the x � y plane.

Fig. 3. Plot of the inverse axial ratio versus the elevation angle.

currents: , , and . Fig. 2 shows a hemispher-ical plot and Fig. 3 a linear plot of the inverse axial ratio as afunction of . White (at the poles) represents an inverse axialratio of one (circular polarization), while black (at the equator)is an inverse axial ratio of zero (linear polarization). Increasing

also changes the antenna directivity as shown in Figs. 4 and5 from a maximum at 0 (white) to a minimum at 90 (black).Compensating for the loss in directivity and polarization matchcan improve the link budget by several decibels.

III. GENETIC ALGORITHM OPTIMIZATION

In a wireless communications system, the goal is to maximizethe power transfer between the transmit and receive antennas.The following fitness function calculates the portion of the linkbudget relating to polarization and directivity from the ampli-tude and phase of the currents for each dipole:

(8)

where is the directivity of the transmit crossed dipoles in thedirection of the receive crossed dipoles, and is the directivityof the receive crossed dipoles in the direction of the transmit

Fig. 4. Circularly polarized crossed dipoles along the x and y axes. This is ahemispherical plot of the directivity. Light color indicates high with a value inthe y-direction and a low value in the x � y plane.

Fig. 5. Plot of the directivity versus the elevation angle.

crossed dipoles. Since the crossed dipoles have a maximum di-rectivity close to one, the directivities are not normalized in theobjective function. The maximum possible value of this fitnessfunction is approximately 2.25 or 3.5 dB.

There are several approaches to performing the adaptation.With current technology, the most likely approach is to do anopen loop adaptation. The satellite uses various sensors to makeit aware of its orientation. Once its orientation is known, then thedipole currents can be found from a lookup table of optimizedvalues, or the optimization could be done at that time. Noise andorientation errors limit the improvement possible. The other ap-proach is a fully adaptive system capable of correcting for noiseand system inaccuracies. This approach would be necessary forusing the adaptive dipoles on another mobile system, such as anairplane, where the orientation and environment of the dipolesquickly changes and cannot be predicted ahead of time.

A continuous parameter GA was used to find the values of theamplitude and phase of the receive dipole currents that max-imize (12). The GA has a population size of 8, mutation rateof .2, single point crossover, and 50% replacement. This smallpopulation size and high mutation rate results in a very fast con-vergence as will be shown in the following section. The goal of


Fig. 6. Average number of function calls needed to get the fitness above 3 forvarious population sizes and mutation rates.

Fig. 7. Link is optimal when the two antennas face each other or � = � = 0.

the optimization process is to quickly improve the communica-tions link, not necessarily find the global minimum. Fig. 6 showsthe results of optimizing (8) using a GA for population sizes be-tween 8 and 32 and mutation rate between 0.1 and 0.2. No noisewas used in these runs. The plot is of the mean number of func-tion calls to get (8) above 3 dB averaged over 50 independentruns when the GA begins with a random population. A smallpopulation size and large mutation rate produce the fastest con-vergence on average for the open loop adaptation.

IV. RESULTS

In all the examples presented here, the orientation of theground and satellite antennas are assumed to change with timeunless otherwise specified Fig. 7. Even though the distancesbetween the antennas would also change, this variation isignored. As the orientations of the antennas vary with time, sodo their directivity and polarizations in the directions of eachother.

Assume that the transmit antenna (ground station) tracks thesatellite and the receive antenna (satellite) points atthe ground ( varies). The transmit antenna continues to de-liver a circularly polarized signal at maximum directivity tothe moving receive antenna. If the receive antenna consists oftwo crossed dipoles, then the maximum receive power transferoccurs when the receive antenna is directly overhead of thetransmit antenna. If the receive antenna remains circularly po-larized as it moves, then the power received drops off at the rateshown by the dashed line in Fig. 8. The loss in power transfer

Fig. 8. In this case, � varies with time and � = 0. The receive antennaconsists of two crossed dipoles. The solid line results from adaptation and thedashed line has no adaptation.

Fig. 9. In this case, � varies with time and � = 0. The receive antennaconsists of three crossed dipoles. The solid line results from adaptation and thedashed line has no adaptation.

is due to the reduction in the directivity and the PLF. If the cur-rents at each dipole are optimally weighted using the GA, thenthe power loss follows the solid line in Fig. 8. This curve re-sults from running the GA to find the optimum weights for arange of angles. The difference between the two curves is thelink improvement. The link improvement is as much as 3 dB at

. In this case, all the improvement is due to increasingthe directivity of the receive antenna.

Adding a third orthogonal dipole to the receive antenna pro-vides another degree of freedom. Now, adapting the receive an-tenna to the tracking transmit antenna results in no change in thelink budget as a function of (solid line in Fig. 9). The threeorthogonal dipoles can compensate for the change in directivityand polarization of the receive antenna as it moves. This sce-nario produces up to 6 dB improvement in the link budget at

.Another scenario has both two dipole antennas pointing

straight ahead (no tracking) while the satellite moves (and change with time). Fig. 10 shows the improvement (solidline) possible through adaptation compared to the link loss withno adaptation (dashed line). The link improvement is as muchas 3 dB at . Adding a third dipole produces even better


Fig. 10. In this case, � = � vary with time. The receive antenna consists of two crossed dipoles. The solid line results from adaptation and the dashed line hasno adaptation.

Fig. 11. In this case, � = � vary with time. The receive antenna consists of three crossed dipoles. The solid line results from adaptation and the dashed line hasno adaptation.

link improvement than the two dipole case, particularly atsmaller angles (Fig. 11). The maximum improvement is 3 dBat .

How fast can a GA adapt? If the receive antenna continuouslyadapts as it moves, then only small perturbations are necessaryat each angle and the adaptation is very fast. As an example,consider maximizing the link budget of the receive antenna withthree orthogonal dipoles at . If the adaptation startswith the optimal weightings at , then the solid curvein Fig. 12 results. In order to reach the steady state solution at

generation 21, the number of fitness function evaluations madeby this GA run is

(9)

A fitness function evaluation equates to a power measurementin a real system. If the adaptation starts with the optimal weight-ings at , then the dashed curve in results. Continuously


Fig. 12. Typical link improvement versus generation for a three orthogonal dipole receive antenna at an angle of 50 . If the adaptation process is started with thereceive antenna at circular polarization (0 ), then the GA finds an optimum in 21 generations or about 100 power measurements (solid line). If the process startswith the optimum dipole weights at 45 , then minimal adaptation is necessary (dashed line).

Fig. 13. Performance of the GA when normally distributed noise is added tothe amplitude and phase of the currents fed to the dipoles (� = 0 and � = 0:1).The GA has a population size of 8 and mutation rate of 0.2.

adapting the signal results in constant incremental improvementof the link. Even if the antenna must be adapted from circularpolarization, the GA quickly finds an acceptable solution. Theresults are not dependent upon .

Noise was added to the currents of the dipoles of the transmitantennas to see how well the GA performs in a noisy envi-ronment. Using a population size of 8 and mutation rate of0.2, a plot of the link improvement versus generation showsups and downs due to the random variations (Fig. 13). Thetransmit dipole current amplitude and phase errors are normallydistributed with a mean and standard deviation given byand . Note that the mean of the population has highvariations due to the large mutation rate. Using a small mutationrate of 0.02, results in a much lower variations in the meanof the population (Fig. 14). Even though the higher mutationrate finds an optimal solution faster than the lower mutationrate, the average power measurements associated with the high

TABLE IAVERAGE MAXIMUM AND MEAN OF THE POPULATION OVER 100

GENERATIONS FOR VARIOUS MUTATION RATES AND NOISE VARIANCES

Fig. 14. Performance of the GA when normally distributed noise is added tothe amplitude and phase of the currents fed to the dipoles (� = 0 and � = 0:1).The GA has a population size of 8 and mutation rate of 0.02.

mutation rate are lower. Consequently, when noise is included,a lower mutation rate is more desirable. Table I shows that theaverage for the population mean over 100 generations is betterwhen the mutation rate is smaller.


Fig. 15. Performance of the GA when normally distributed noise is added tothe amplitude and phase of the currents fed to the dipoles (� = 0 and � = 0:1)and the satellite is moving 1 per generation. The GA has a population size of8 and mutation rate of 0.2.

Fig. 16. Performance of the GA when normally distributed noise is added tothe amplitude and phase of the currents fed to the dipoles (� = 0 and � = 0:1)and the satellite is moving 1 per generation. The GA has a population size of8 and mutation rate of 0.02.

In a fully adaptive system, the GA would also have to adaptwhile the satellite moves. The next set of simulations used thesame error statistics as before and had the satellite move 1 pergeneration. Fig. 15 shows the convergence curve when the mu-tation rate is 0.2. Again, the mean of the population has veryhigh variations. Fig. 16 shows the convergence curve when themutation rate is 0.02. The smaller mutation rate is more desir-able, because the variations in the population mean are small. Ahigh mutation rate works best for a no noise environment, and alow mutation rate works best in the presence of noise.

V. CONCLUSION

The received signal in a mobile communications systemloses strength due to a decrease in antenna directivity and polar-ization mismatch. The current fed to a set of crossed dipoles canbe modified to increase the directivity and polarization matchbetween the transmit and receive antennas. Two orthogonaldipoles can compensate for the loss in gain but not polarization.Three adaptive orthogonal dipoles can fully restore the loss dueto loss in directivity and polarization mismatch if tracked by thetransmit antenna. A GA quickly adapts the receive antenna tothe transmitted signal. Three orthogonal dipoles provide moreimprovement than just two orthogonal dipoles.

REFERENCES

[1] A. Singer, “Space versus polarization diversity,” Wireless Review, pp.164–168, Feb. 15, 1998.

[2] M. R. Andrews, P. P. Mitra, and R. deCarvalho, “Tripling the capacity ofwireless communications using electromagnetic polarization,” Nature,vol. 409, pp. 316–318, Jan 18, 2001.

[3] R. E. Marshall and C. W. Bostian, “An adaptive polarization correctionscheme using circular polarization,” in Proc. IEEE Int. Antennas andPropagation Society Symp., Atlanta, GA, June 1974, pp. 395–397.

[4] R. T. Compton, “On the performance of a polarization sensitive adaptivearray,” IEEE Trans. Antennas Propagat., vol. AP-29, pp. 718–725, Sept.1981.

[5] , “The tripole antenna: an adaptive array with full polarization flexi-bility,” IEEE Trans. Antennas Propagat., vol. AP-29, pp. 944–952, Nov.1981.

[6] B. D. Griffin, R. Haupt, and Y. C. Chung, “Adaptive polarizationfor spacecraft communications system,” presented at the Proc. IEEEAerospace Conf., Big Sky, MT, Mar. 2002.

[7] R. Haupt, “Adaptive crossed dipole antennas,” in URSI General As-sembly, Maastricht, Netherlands, Aug. 2002.

Randy L. Haupt (M’82–SM’90–F’00) received theB.S. degree in electrical engineering from the U.S.Air Force Academy, U.S. Academy, CO, the M.S.degree in engineering management from WesternNew England College, Springfield, MA, in 1981,the M.S. degree in electrical engineering fromNortheastern University, Boston, MA, in 1983, andthe Ph.D. degree in electrical engineering from theUniversity of Michigan, Ann Arbor, in 1987.

He was a Professor of electrical engineering at theU.S. Air Force Academy and Professor and Chair of

Electrical Engineering at the University of Nevada - Reno. In 1997, he retired asa Lt. Col. in the U.S. Air Force. He was a Project Engineer for the OTH-B radarand a Research Antenna Engineer for Rome Air Development Center. From1999 to 2003, he was Professor and Department Head of Electrical and Com-puter Engineering at Utah State University, Logan. He is currently a Senior Sci-entist at the Applied Research Laboratory, Pennsylvania State University, StateCollege. He has many journal articles, conference publications, and book chap-ters on antennas, radar cross section and numerical methods and is coauthorof the book Practical Genetic Algorithms, 2nd edition (New York: Wiley, May2004). He has eight patents in antenna technology.

Dr. Haupt is a Member of Tau Beta Pi, Eta Kappa Nu, International ScientificRadio Union (URSI) Commission B, and the Electromagnetics Academy. Hewas the Federal Engineer of the Year in 1993.


Modeling and Investigation of a GeometricallyComplex UWB GPR Antenna Using FDTD

Kwan-Ho Lee, Student Member, IEEE, Chi-Chih Chen, Member, IEEE, Fernando L. Teixeira, Member, IEEE,and Robert Lee, Member, IEEE

Abstract—A detailed analysis of ultrawide-band (UWB),dual-polarized, dielectric-loaded horn-fed bow-tie (HFB) an-tennas is carried out using the finite-difference time-domain(FDTD) method. The FDTD model includes realistic features ofthe antenna structure such as the feeding cables, wave launchers,dielectric loading, and resistive-film loading. Important antennacharacteristics that are usually difficult to obtain via measure-ments can be obtained more directly from this FDTD model. Sincethe HFB antennas under consideration are intended for groundpenetrating radar (GPR) applications, the effects of the half-spacemedium are also investigated. The simulated results serve to verifythe performance of the HFB antenna design, and to optimizevarious antenna parameters.

Index Terms—Bow-tie antenna, coaxial cable, dielectric loading,finite-difference time-domain (FDTD), ground penetrating radar(GPR), impedance, resistive, ultrawide-band (UWB).

I. INTRODUCTION

GROUND PENETRATING RADAR (GPR) find appli-cations in many areas such as geophysical prospecting,

archeology, civil engineering, environmental engineering, anddefense technologies as a noninvasive sensing tool [1], [2].One key component in any GPR system is the receiver/trans-mitter antenna(s). Desirable features for GPR antennas includebroadband operation, good impedance matching, and smallsize. The frequency range of a GPR antenna is determinedby the particular application and its relation to the natureof the target, soil constitution, desired depth of penetration,and inversion/classification method being used. For example,the frequency of operation for detection and classification ofanti-tank and anti-personnel landmines is usually from 0.1 to1 GHz [3] and from 1 to 6 GHz [4], [5], respectively. A goodfrequency range for detecting 6-inch drainage pipes is foundto be from 100 to 400 MHz. For unexploded ordnance (UXO)detection, the 10 800 MHz frequency range is often used [6],[7].

For the detection of shallow objects where high sensitivityis not an issue, elevated antennas are often used for easier scan-ning and better antenna calibration. In particular, many antennasused for detection of shallow landmines [5], [8], evaluation ofintegrity of concrete [9] and soil hardness [10] are all elevated

Manuscript received December 2, 2002; revised November 3, 2003. Thiswork was supported in part by the Department of Defense (DoD) Strategic En-vironmental Research and Development Program (SERDP) Project 1122 by theNational Science Foundation (NSF) under Grant ECS-0347502.

The authors are with ElectroScience Laboratory, Department of ElectricalEngineering, The Ohio State University, Columbus OH, 43210 USA (e-mail:[email protected]).


systems that exhibits low antenna-ground interaction. On theother hand, most GPR antennas used for the detection of deeptargets are operated very close to the ground so that most of theenergy is radiated into the ground to improve sensitivity. Thisconfiguration also minimizes radiation into the air to complywith the FCC regulations.

The characteristics of such GPR antennas while in fieldoperation are usually difficult to determine a priori because ofthe large coupling with the environment. For instance, the inputimpedance of the commonly used dipoles or flat bowtie dipolesare directly affected by the electrical property of the particularground for antennas operated close to the surface. Moreover,the amount of energy coupled into the ground changes as thepermittivity increases and hence the radiation patterns alsodepend of the soil permittivity [11]–[13]. Hence, one majordisadvantage is that the antenna characteristics in the fieldbecome dependent on the electrical properties of the groundand surroundings. This also makes calibration more difficult. Inorder to make antenna characteristics less susceptible to groundcharacteristics, a new dielectric-loaded horn-fed bowtie (HFB)antenna design was introduced in [7]. The HFB antenna wasdesigned to minimize the antenna ringing by: 1) employing astable and well matched surge impedance and 2) using speciallydesigned tapered resistive loadings. Unlike most conventionalantennas, the surge impedance was designed to be less depen-dent on the ground property because the feed point is elevatedoff the ground. Low loss dielectric material was then used tofill the space between the feed front and the ground surface toreduce ground-surface reflections and increase the electricalheight of the feed. Both single-polarized and dual-polarizedHFB antenna prototypes have been built and employed in actualapplications.

Due to its flexibility, the finite-difference time-domain(FDTD) method has been widely used in recent years for thenumerical simulation of GPR systems [14]–[17]. Some of theprevious studies have modeled GPR antennas as a series ofpoint sources or short dipoles with or without the presence ofconducting shields [18], [19]. In order to better characterizeHFB antennas and to provide a more convenient tool for theirdesign and optimization, a full-scale detailed three-dimensional(3-D) FDTD model of a dual-polarized HFB prototype wasdeveloped in this work and simulated for GPR applications.To reduce the computational cost, a special partition scheme[20] is adopted for the 3-D FDTD domain. This scheme dividesthe whole inhomogeneous region into several small homo-geneous regions. In each homogeneous region, volumetricmaterial property matrices are replaced by constants to save the

0018-926X/04$20.00 © 2004 IEEE


Fig. 1. Prototype of HFB antenna.

memory. This partition scheme for modeling the electricallylarge HFB antenna in the presence of ground also allows forfaster simulations on a personal computer. An anisotropicperfectly matched layer (APML) especially formulated for thedielectric or lossy half spaces [16], [21], [22] is implemented.

This paper is organized as follows. The HFB antenna designis discussed in Section II. Section III describes the constructionof the FDTD model for the dual-polarized HFB design and theperformance of the resistive-film loading which is optimizedfor a given length. Section IV presents various HFB antennacharacteristics obtained from the FDTD simulations.

II. BASIC DUAL-POLARIZED HFB ANTENNA DESIGN

Fig. 1 illustrates the basic structure of the dual-polarizedUWB HFB antenna design. This is somewhat similar to aplanar bowtie dipole with the feed point being raised off theground. The feed section resembles that of a small transverseelectromagnetic (TEM) horn except that it is filled with lowloss dielectric material. Each antenna arm is smoothly curved inthe transition from the horn section to the planar bowtie dipolesection. The ends of the dipoles are terminated with taperedresistive cards (R-card) to reduce antenna ringing.

A. Resistive Taper Section

Tapered R-cards have many useful applications for radia-tion and scattering control [23]–[25], but commercial taperedR-cards are often expensive and have very limited choicesof tapering profile and taper length. The R-card used in the

HFB prototype was constructed in-house using multiple layersof commercial window films [23]. These have various sheetresistance for different percentage of light transmission. Whenmultiple films are overlaid properly together, one can obtaina desired resistivity profile with desired taper length. Fig. 1illustrates how the tapered R-card was constructed for the HFBprototype.

The objective of the resistive card is to reduce reflections bygradually dissipating the currents propagating toward the end ofeach antenna arm. This requires the resistivity on the R-card tobe tapered from a small value to a large value along the antennaarm. An exponential taper of the resistivity was adopted in theHFB prototype with a tapering shown as follows:

(1)

where is the initial sheet resistance of theR-card at the perfect electric conducting (PEC)/R-card inter-face, and is the sheet resistance at the farend of the R-card, is the length of the R-card, andis the distance along the R-card from the PEC arm.

B. Feed Section

The feed section of HFB resembles a dual polarized TEMhorn except that the end of each antenna arm is curved outwardgradually to be connected to the flat bowtie section, and theinternal space of the horn was filled with low loss dielectricmaterial. The geometry of the horn and the antenna arms was

LEE et al.: MODELING AND INVESTIGATION OF A GEOMETRICALLY COMPLEX UWB GPR ANTENNA USING FDTD 1985

Fig. 2. Dimension and computation domain partitioning of the fully polarimetric dielectric filled HFB antenna.

chosen based upon the tradeoff among the dielectric constant,size, weight, and cost. The objective was to obtain a surgeimpedance of 100 to match to the characteristic impedanceof the feeding twin-coaxial cables shown at the bottom ofFig. 1 (each cable has a characteristic impedance of 50 ohms).Although tabulated characteristic (or surge) impedances for aninfinite TEM horn with arbitrary geometry are available [26],[27], the exact impedance of a dual-polarization TEM hornwith dielectric filling is complicated to obtain analytically.The experimental data obtained from [28] was used duringthe construction of HFB prototype. Note that the center ofeach coaxial cable was connected to one antenna arm andeach pair of the 50 coaxial cable feed one polarization.A 0–180 broadband hybrid was used as a balun for eachpair of cables. Accurate FDTD models recently constructed tocalculate the surge impedance for such an antenna geometryare employed here [29]. The prototype to be analyzed herehas a dielectric constant of 5. The plate angle of each antennaarm is 11.5 . The horn angle itself is approximately 150 .

III. FDTD MODEL DESCRIPTION

A full scale model of the UWB HFB antenna prototype re-quires a minimum of space. A spatialcell size of 6.3 mm was chosen to accurately model the geomet-rical details of the antenna and cable structure [30]. This yieldsapproximately 96 million unknowns. The FDTD grid is shownin Fig. 2. All dimensions in the model were chosen to be as closeto the actual prototype as possible. The four antenna arms weremodeled as PEC plates, and the curved edges and surfaces wereapproximated by staircases. Each tapered R-card attached to the

end of the PEC arm is 63 cm in length and is implemented via aconductive sheet. The ground was assumed to be a lossless halfspace with relative permittivity of 5.

A. Heterogeneous FDTD Domain Partition

The antenna geometry under study is very complicated andresides in a complex environment. A traditional FDTD approachto represent the geometry would require either the storage of thematerial properties for each cell or else a data organization sim-ilar to what is used in the finite element method, which wouldalso require a significant amount of memory overhead. To mini-mize the memory usage, we have adopted a partitioning scheme[20]. The FDTD domain is divided into blocks. The size andnumber of the blocks are judiciously chosen, so that the materialproperties within most of the blocks are homogeneous. Withinthe code, the FDTD algorithm is computed in different ways,and based on the properties of the block, the appropriate FDTDalgorithm will be chosen. If the block is a perfect conductor, theFDTD code will recognize this, and not perform any computa-tions for that block. Thus, there is no need to store either thefields or the material properties for that particular block. If theblock is an homogeneous dielectric, the material properties arenot treated as a function of the grid points within the block butinstead represented just as a constant parameter. Thus only thefield values need to be stored for each cell within that block. Ifthe block is an inhomogeneous dielectric, then the FDTD algo-rithm used will assume a constant permeability and no conduc-tivity. Thus, only the fields and permittivity must be stored foreach cell. In our case, we divide the geometry into 196 blockswith only five of the blocks being heterogeneous as demon-strated in Fig. 2.


Fig. 3. Coaxial cable modeling in rectangular FDTD grid and the TEM current excitation scheme. (a) Top view, (b) side view, and (c) J(t).

B. Feed Cable Modeling

In the discretized FDTD model, each coaxial cable has asquare cross sectional area with a single-cell PEC wire sur-rounded by four PEC walls. As shown in Fig. 3(a), a relative di-electric constant of 1.5 is specified between the center wire andthe PEC walls. Each cable is terminated with perfectly matchedlayer at one end and connected to the tip of an antenna arm atthe other end. A balanced excitation is introduced to the oppo-site pair of cables to excite one antenna polarization as shownin Fig. 3(a) and (b). The time history of the response is alsorecorded at the excitation position to obtain reflection and trans-mission data. The reflection data is obtained with the ex-citation and observation points co-located in the same cable andcross-coupling data is obtained with the observation pointlocated at the second cable. A differential Gaussian pulse ischosen as the time-domain excitation current

(2)

where , and .These parameters for the Gaussian pulse are determined so asto provide significant spectral energy in the frequency range of10 to 800 MHz. Fig. 3(c) illustrates the pulse.

C. Resistive Card Modeling

In the FDTD model, the R-card is modeled as a single-celllayer with a tapered conductivity corresponding to the de-sired sheet resistance . Conductivity along the directionis calculated by where is the thickness of

Fig. 4. Resistive card overlay configurations for the PEC launcher section(R = 300 = , R = 3 = ).

single layer. This assumption is valid when is much greaterthan the penetration depth but much smaller than the free spacewavelength [31].

In addition to the exponential taper described in Section II-A,a linear taper with the following taper function was also inves-tigated using the FDTD model as a comparison

(3)

where is the initial sheet resistance of the R-cardat the PEC/R-card junction, is the end sheetresistance. The taper length is equal to that of the previous


Fig. 5. S , S , and surge impedance of the HFB antenna. (a) Reflection coefficient S and S and (b) antenna surge impedance.

exponential taper, i.e., 0.63 m. As it will be shown shortly, alinear taper provides a better performance, i.e., lower reflectionat low frequency end, due to relatively short taper length withrespect to wavelength. A more detailed analysis on this aspectcan be found in [29]. Fig. 4 plots the linear resistive taper as wellas its position relationship with respect to the antenna arm. Thelateral edges of the R-card were kept aligned to the edges of thePEC arms to avoid undesired diffractions (see Fig. 1).

IV. CHARACTERISTICS OF DUAL-POLARIZED HFBANTENNA DESIGN

A. & and Input Impedance

The simulated and measured reflection and transmission co-efficients, and , of the HFB design are compared inFig. 5(a). Note that the antenna is located on the surface of a halfspace with a dielectric of 5, corresponding to the dry sand in re-ality. The is similar to since the both antenna arms have

the same design. and provide the co-polarized backscat-tering data. provides the cross-polarized kscattering data.A calibration procedure was carried out in a similar manner asdone in real measurement using “short” and “matched” (PML)reference loads at the end of the feed cables

(4)In the above, is the response obtained with the coaxialcables connected to the antenna, is the response obtainedwith the coaxial cables connected to matched load, is theresponse obtained when the coaxial cables are shorted at the endwith a conducting wire, is the response obtained at coaxialcable 2 with antenna connected when the excitation is appliedto cable 1, and is the incident wave.

It is observed that the both linear and exponential taper havesimilar performance at frequency above 0.3 GHz where the


Fig. 6. Comparison of reflected electric field difference with various groundprofiles.

taper length becomes comparable or longer than one wave-length (considering dielectric constant of 5). It is also observedthat the linear taper produces lower reflection level than theexponential taper at frequencies below 0.1 GHz. Overall, the re-flection level is less than 10 dB above 0.05 GHz. This verifiesbroadband characteristic of the HFB design. The measureddata is found to be on average 10 dB higher than that predictedfrom the simulation. This difference is most likely caused bythe asymmetry of the construction of prototype antenna armsand the feed structure. Good agreement between the measure-ment and simulation is the result of the geometrical fidelitybetween of the FDTD numerical model and the prototype,including the R-card geometry, conductive plates, dielectricfilling and coaxial cable feed modeling. However, the prototypemeasurement introduces additional environmental variablesmore difficult to control such as ground loss, slight asymmetryof the antenna arm design due to hand-made fabrication, anddiscrepancies between the equivalent conductive single layerR-card used in the FDTD model and the thin film R-cardconductivity value.bac

The surge impedance can also be calculated from the asshown in (5), often applying a time gate to keep only the firstpeak associated with the feed point near 0 ns position as shownin Fig. 6

(5)

where is the characteristic impedance of the twin-coaxialcable. The resultant surge impedance is shown in Fig. 5(b). Formost of the band , the surgeimpedance is found to be within range, as desired.

B. Ground Effect

In order to see how the ground properties affect the surgeimpedance of the HFB design, four different ground dielectricconstants: 5, 7, 9, and 11 are simulated. Fig. 6 shows the re-flected field from 0 to 3 ns. The height of the antenna feed abovethe ground is equal to 0.1 m. This causes the reflection fromthe ground surface to be delayed by approximately 1.5 ns sincethe antenna dielectric filler has a relative permittivity of 5. This

Fig. 7. Reflected (E ) field in time domain for HFB antennas with differentresistive card overlay configurations and using same conductivity profile.

Fig. 8. Comparison of co-polarized (E ) reflected field in time domain fromHFB antenna with the different resistive cards (R ).

agrees with the significant variations shown in the data near 1.5ns position. Most importantly, the first reflection peak arisingfrom the feed point remain unaffected by the ground property,as desired.

C. R-Card Performance Investigation

We investigate two parameters that play an important role inminimizing reflections from the truncated antenna arms. Thefirst parameter is the overlay distance between the PEC andR-card. In the actual HFB prototype, a 5 cm overlay was used toallow the electromagnetic energy to be coupled into the R-cardsection because the R-card was coated with a protective insu-lator and could not have a direct electrical contact with the an-tenna arm. The second parameter is the far-end resistance valuethat affects the tapering rate of the R-card. If the taper is done toorapidly, undesired diffractions would be produced by the R-card.On the other hand, if the taper is too slow, the far-end reflectionmay still be too strong.

To investigate the effect of PEC and R-card overlay distances,the following three cases were simulated as shown in Fig. 4. Incase 1 through 3, the overlay distances are 11.3, 5, and 0 cm,respectively. The simulated reflection responses are plotted in


Fig. 9. Snap shots from FDTD simulation for E field strength in dB scale where R = 300 = . (a) t = 7:4539 ns with R-card attached-(dB) scale;(b) t = 7:4539 ns without R-card-(dB) scale; (c) t = 13:0954 ns with R-card attached-(dB) scale; (d) t = 13:0954 ns without R-card-(dB) scale.

Fig. 7. As expected, the overlay distance of the tapered R-cardaffects the reflection at the PEC end. Note that the R-card in theoverlay section is shorted out by the PEC, this section wouldhave an effective resistance of zero regardless of the R-cardvalue. The larger geometric discontinuity in Case 3 provides thestronger junction reflection observed in the figure. Case 1 and 2provide a smoother transition and result in a 35 dB reflectedfield at the end of the R-card. Based on the simulations, we con-cluded that a linear-tapered R-card with either a 11.3 or 5 cmoverlay at the PEC/R-card does the best job of suppressing thereflections.

To optimize the choice of , values of 100, 200, 300, and400 were implemented and simulated separately. From thereflected field observed at the feed point, the amount of end-reflection suppression was compared as shown in Fig. 8, wherelate time (after 20 ns) antenna reflections can be observed. Theseresults indicate that provides the maximalsuppression of the arm end reflections.

D. Antenna Ringing

Fig. 9 compares snapshots of the instantaneous field dis-tribution in the vertical (or ) plane with and without the

R-card attached to the HFB antenna arms. Without the R-card,significant diffraction and reflection at the end of the PEC armsare observed. The reflected fields later propagate back to theobservation point inside the cables as shown in Fig. 9(b). Onthe other hand, the R-card extension significantly reduces thediffraction and reflection at the ends as depicted in Fig. 9(a) andlowers the antenna ringing by approximately 20 dB. Note thatthe signals that propagate back to the feed point are partially re-flected due to the imperfect matching. This reflected fields gen-erate the secondary reflection. This process repeats and becomesthe well known “antenna ringing” effect, a major clutter sourcein GPR measurements.

E. Radiated Field Distribution & Polarization

The near-field radiation characteristics are investigated next.Fig. 10 depicts the simulated horizontal co-polarized and cross-polarized field distributions at a plane 40 cm below the antennaaperture, (corresponding to the ground surface plane), at thecenter frequency of 400 MHz. The cases with and without theR-card are also plotted for comparison. The fields are nearlylinearly polarized in the principal planes. The results with theR-card clearly show a more uniform distribution, because the


Fig. 10. Comparison of co- and cross-polarized aperture field distributions at f = 400 MHz, depth z = 40 cm or 0.53 � , R = 300 = in R-card.(a) Co-polarized field with R-card, (b) cross-polarized field with R-card, (c) co-polarized field without R-card, and (d) cross-polarized field without R-card.

diffracted fields from the antenna arm ends modify the radiatedfields that otherwise would have been close to simple sphericalwavefronts. The more uniform field distributions simplify thesubsequent signal processing and inverse problem and improvethe overall detection/classification capabilities of a GPR system.

As the observation point moves away from the principalplanes, the level of depolarization increases and reaches amaximum of approximately 12 dB between the two antennapolarizations. This is, of course, due to the spherical natureof the wavefront. We note that the cross-polarized field levelswith the R-card present are a little bit higher. This again maybe caused by distributed diffractions along the resistive cards.

V. CONCLUSION

In this work, a detailed FDTD model was used to incorporaterealistic features of UWB HFB antennas such as feedingcables, dielectric loading and tapered resistive terminations. TheFDTD model is flexible enough to model different geometries,structures, and materials for both the antenna and the groundmedium. Fully-polarimetric simulations were performed toobtain the radiation characteristics of HFB antennas over a broadfrequency range. A parametric study on the effect of the resistivetaper of the R-card termination was also performed. It wasfound that a linear taper performs better than the commonlyused exponential taper for short taper length. It was alsofound that a proper overlapping between the PEC and R-cardimproves the transition and reduces the diffraction at the endof PEC. The R-card termination also significantly reducesthe undesired antenna ringing. The surge impedance of the

HFB antenna was calculated from the reflection coefficientsand was found to be approximately 100 ohms over the entirefrequency band of interest. This result confirms the broadbandcharacteristic of the HFB design. The FDTD model also provideduseful visualization of dynamic field distributions that canhelp identify undesired radiations and reflections sources. Thenear-field distributions of the co-polarized and cross-polarizedfields were examined. This information is particularly usefulin GPR applications where the depth of the target is unknown.Overall, the simulated results confirm that the optimized HFBantenna design is a very attractive choice for broadband, fullypolarimetric GPR applications.

ACKNOWLEDGMENT

The authors acknowledge the reviewers for their helpfulcomments.

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[31] T. B. A. Senior, “Approximate boundary conditions,” IEEE Trans. An-tennas Propagat., vol. 29, pp. 826–829, Sept. 1981.

Kwan-Ho Lee (M’02) received the B.S. degree fromthe Department of Radio Science and Engineering,Kwangwoon University, Seoul, Korea, in 1997 andthe M.S. degree in electrical engineering from TheOhio State University, Columbus, in 1999, where heis currently working toward the Ph.D. degree.

Since 1997, he has been a Graduate ResearchAssociate at the ElectroScience Laboratory, De-partment of Electrical Engineering, The Ohio StateUniversity. His research interests include com-putational electromagnetics, ultrawide-bandwidth

antenna development, subsurface target detections and classifications, RFcircuits and object oriented programming.

Chi-Chih Chen (S’92–M’97) was born in Taiwan,R.O.C., in 1966. He received the B.S.E.E. degreefrom the National Taiwan University, Taiwan,R.O.C., in 1988 and the M.S.E.E. and Ph.D. degreesfrom The Ohio State University, Columbus, in 1993and 1997, respectively.

He joined the ElectroScience Laboratory, TheOhio State University, as a Postdoctoral Researcherin 1997 and became a Senior Research Associate in1999. His main research interests include the groundpenetrating radar, UWB antenna designs, radar target

detection and classification methods, automobile radar systems. In recent years,his research activities have been focused on the detection and classification ofburied landmines, unexploded ordnance and underground pipes.

Dr. Chen is a Member of Sigma Xi and Phi Kappa Phi.

Fernando L. Teixeira (S’89–M’93) received theB.S. and M.S. degree in electrical engineering fromthe Pontifical Catholic University of Rio de Janeiro(PUC-Rio), Brazil, in 1991 and 1995, respectively,and the Ph.D. degree in electrical engineering fromthe University of Illinois at Urbana-Champaign, in1999.

From 1999 to 2000, he was a Postdoctoral Re-search Associate with the Research Laboratory ofElectronics, Massachusetts Institute of Technology(MIT), Cambridge. Since 2000, he has been an

Assistant Professor at the ElectroScience Laboratory (ESL) and the Depart-ment of Electrical Engineering, The Ohio State University, Columbus. Hiscurrent research interests include analytical and numerical techniques for wavepropagation and scattering problems in communication, sensing, and devicesapplications. He has edited one book Geometric Methods for ComputationalElectromagnetics (PIER 32, EMW: Cambridge, MA, 2001), and has publishedover 30 journal articles and 50 conference papers in those areas.

Dr. Teixeira is a Member of Phi Kappa Phi. He was awarded the Raj MittraOutstanding Research Award from the University of Illinois, and a 1998 MTT-SGraduate Fellowship Award. He received paper awards at 1999 USNC/URSINational Symposium (Orlando, FL), and received a Young Scientist Award atthe 2002 URSI General Assembly. He was the Technical Program Coordinatorof the Progress in Electromagnetics Research Symposium (PIERS), Cambridge,MA, in 2000.

Robert Lee (M’92) received the B.S.E.E. degree in1983 from Lehigh University, Bethlehem, PA, and theM.S.E.E. and Ph.D. degree in 1988 and 1990, respec-tively, from the University of Arizona, Tucson.

From 1983 to 1984, he worked for MicrowaveSemiconductor Corporation, Somerset, NJ, as aMicrowave Engineer. From 1984 to 1986, he wasa Member of the Technical Staff, Hughes AircraftCompany, Tucson, AZ. From 1986 to 1990, he wasa Research Assistant at the University of Arizona.During summer 1987 through 1989, he worked at

Sandia National Laboratories, Albuquerque, NM. Since 1990, he has beenat The Ohio State University, where he is currently a Professor. His majorresearch interests are in the development and application of numerical methodsfor electromagnetics.


Radiation Properties of an Arbitrarily FlangedParallel-Plate Waveguide

Dao Ngoc Chien, Student Member, IEEE, Kazuo Tanaka, Member, IEEE, and Masahiro Tanaka, Member, IEEE

Abstract—The radiation properties of an arbitrarily flanged par-allel-plate waveguide are investigated by means of the boundaryintegral equations that are called guided-mode extracted integralequations. The boundary integral equations derived in this papercan be solved by the conventional boundary-element method. Nu-merical results are presented for a number of cases of flanged par-allel-plate waveguide. Reflection coefficient, reflected and radiatedpowers as well as radiation patterns are numerically calculated forthe incidence of transverse electric guided-mode wave.

Index Terms—Boundary-element method (BEM), boundaryintegral equations (BIE), electromagnetic radiation, numericalanalysis, parallel-plate waveguides (PPW).

I. INTRODUCTION

THERE has been remarkable progress in the developmentof communication systems over the last decade. Signif-

icant improvements in noise figure, gain, output power, andefficiency have been achieved at millimeter-wave frequencies.However, the demand of the wireless broadband communi-cation at millimeter-wave frequency recently increases withactivities of digital multimedia-contents circulation. One ofthe problems of millimeter-wave communication is the largetransmission loss in free space. For instance, the transmissionloss of the signal at 60 GHz frequency for 5 m distance be-tween transmitter and receiver is about 82 dB [1]. Therefore,the antenna with high output radiation power is required tocompensate the large transmission loss.

The flange-shaped parallel-plate waveguide (PPW) is knownwell as a fundamental structure extensively used for electro-magnetic wave radiation (as, e.g., in feed horns, flush-mountedantennas, etc.). So far, even though a closed-form solution tothe problem of the flanged PPW radiation is unavailable, thewaveguide-radiation behavior has been well understood usinga number of numerical techniques and approximate theories[2]–[13]. However, since most of the studies have based on theapproximation solution, the presented results have restricted tothe problem of perpendicularly flanged PPW. As far as we know,no one has reported to the problem of an arbitrarily flanged PPWthat expects to give high output radiation power.

In this paper, the radiation properties of an arbitrarilyflanged PPW are investigated by the boundary-element method(BEM) based on the guided-mode extracted integral equations

Manuscript received July 9, 2003. The work of D. N. Chien was supportedby the Rotary Yoneyama Memorial Foundation, Inc., Japan, under a YoneyamaScholarship.

The authors are with the Department of Electronics and Computer Engi-neering, Gifu University, Gifu 501-1193, Japan (e-mail: [email protected]).


(GMEIEs). We derive GMEIEs for the problems of dielectricfilled and unfilled PPW having an arbitrarily flanged surface.By treating these problems, we can easily understand the ad-vantages of GMEIEs compared with other techniques proposedbefore. Since the method in this paper does not employ anyapproximation, the results are accurate in principle. The numer-ical results of computer simulations are presented, in which,the reflection coefficient, the reflected and radiated powers aswell as the radiation pattern are calculated numerically for theincidence of TE guided-mode wave. The results are comparedwith those reported in the literature, and are confirmed by thelaw of energy conservation.

II. PPW WITH AN ARBITRARILY FLANGED SURFACE

A. Formulation of the Reflection Coefficient in Terms ofGMEIEs

Consider a dielectric filled PPW of width having a tiltedflange surface radiating into a free space as shown in Fig. 1(a).The dielectric is with refractive index of . The waveguide isassumed to be satisfied the single-mode condition.

Referring to Fig. 1(d), we denote the actual boundaries (solidlines) of the waveguide by – . The boundary (dottedline) does not express an actual boundary, but rather express avirtual boundary. The whole space is assumed to be magneti-cally homogeneous with a magnetic permeability

H/m. In the following analysis, a harmonic time depen-dence is supposed and suppressed for the electromag-netic field quantities, the free-space wave number is denoted by

, where is the velocity of light in a vacuum. Sincethe waveguide is assumed to be infinite-extended in the -direc-tion, all field quantities are independent of (i.e., )and thus the electromagnetic field can be decomposed in termsof TE mode.

To derive GMEIEs, we assume that a dominant guided-mode wave is incident upon the aperture in the tilted flange sur-face from inside of the waveguide. Since the electric fields haveonly a -component under the above-mentioned condition, wedenote the electric fields of the -component by

(1)

in the coordinate systems and , as shown inFig. 1(a). The incident guided-mode wave , the reflectedguided-mode wave , and the radiated wave areused to express electric field quantities.

We first consider the case in which an observation pointis in the region surrounded by the boundary

. From Maxwell’s equations and Green’s theorem,

0018-926X/04$20.00 © 2004 IEEE

CHIEN et al.: RADIATION PROPERTIES OF AN ARBITRARILY FLANGED PPW 1993

Fig. 1. (a)–(c) Models of arbitrarily flanged PPW. (b) Location of theboundaries on integral equations.

the well-known boundary integral equation (BIE) for the totalelectric field is given by

(2)

where denotes the derivative with respect to the unitnormal vector to boundary C as shown in Fig. 1(d). Theboundary condition of perfect electric conductor, on

, is enforced in the process of deriving(2). In (2), represents Green’s function in free space,whose refractive index is given by , and it is expressed as

(3)

with denotes the zeroth-order Hankel function ofthe second kind. As can be seen, it is difficult to solve theboundary integral (2) by use of the conventional BEM ormethod of moments (MoM) directly because of that the (2)has an infinite-length integral boundary . To overcomethis difficulty, we use the previously proposed idea [14]–[18]that: Even though the total electric fields near the aperture arevery complicated, only the reflected guided-mode wave cansurvive at points far away from the aperture. Therefore wedecompose the total electric fields on the boundaryinto the field components as

(4)

and we call the field the disturbed field. In (4), is thereflection coefficient. We also express the total electric fieldson the boundary by the same notation with thedisturbed field as follows:

(5)

In (4), it is possible to consider that the disturbed field willvanish at points far away from the aperture.

Substituting (4) and (5) into (2), we obtain an integral equa-tion that includes the semi-infinite line integrals of the guided-mode wave along the boundary as follows:

(6)

with

(7)

Here the Green’s theorem for the guided-mode waves inthe region surrounded by the boundary is appliedas

(8)


to the process of deriving (6). Since the boundary is a virtualboundary, theoretically, we can obtain the (8) with arbitrary po-sition of the boundary .

To derive the expression of the reflection coefficient interms of GMEIE, we put the observation point to far awayfrom the aperture. Under this condition, it is possible to approx-imate Green’s function by the asymptotic expression as

(9)

with

(10)

(11)

Substituting (9) into (6) and dividing both sides of the resultantequation by , we can obtain the relation

(12)

with

(13)

Since it is impossible for a reflected radiation field to exist atpoints in the waveguide far away from the aperture, we can set

(14)

So if we use (14) in (12), we can find that the reflection coeffi-cient can be expressed in terms of GMEIE as

(15)

Physically, the reflection coefficient is an invariable value for aspecific structure of the waveguide and thus we can use (15) toverify the independence of the numerical results on the locationof the virtual boundary .

Substitution of (15) into (6) yields

(16)

where

(17)

(18)

Since will vanish at points far away from the aperture,the integral boundary , which has infinite length, canbe regarded as finite length in (16).

When the observation point is in the free space region thatsurrounded by the boundary , as shown inFig. 1(d), the well-known BIE for the total electric fields is givenby

(19)

It can be seen that the (19) has the integral boundaryalso with semi-infinite length. However, it is easy to truncatethe boundary in the numerical solution procedure at afinite length where the total electric fields are enough small tobe regarded as vanished.

The BIEs (16) and (19) are equations to be solved numeri-cally by using the conventional BEM or MM for the problem ofan arbitrarily flanged PPW as shown in Fig. 1(a)–(c). Once thefields on all the boundaries have been obtained. The reflectioncoefficient can be obtained by the use of (15). And fields atany point can also be calculated by the boundary integral repre-sentations similar to (16) and (19).

B. Radiation Fields in the Free Space

The radiation field in the free-space region can beexpressed by using the asymptotic form of Green’s function infree-space with the refractive index of as follows:

(20)

with

(21)

So far, we have discussed to the case in which a dielectric withthe refractive index of is filled inside the waveguide. For thecase of dielectric unfilled PPW, only one GMEIE is required.Because it is easy to derive by using the same procedure as thatused in the above derivation of (16), it is not necessary to showhere for saving space.


TABLE ICOMPARISON BETWEEN THE VARIOUS METHODS USED TO CALCULATE THE

REFLECTION COEFFICIENT R OF A DIELECTRIC UNFILLED PPW HAVING A

PERPENDICULAR FLANGE SURFACE FOR d=� = 0:5001

TABLE IIREFLECTED POWER � , RADIATED POWER � , AND THEIR TOTAL

� OF A DIELECTRIC FILLED PPW HAVING A TILTED

FLANGE SURFACE FOR d=� = 0:5001, AND n = 1:6

Fig. 2. Reflection coefficient R of a dielectric filled PPW having a tiltedflange surface as a function of location of the virtual boundary C ford=� = 0:5001; n = 1:6, and ' = 10 .

III. NUMERICAL RESULTS AND DISCUSSION

The BIEs derived in this paper were solved with using theconventional BEM. The quadratic functions are used as basicfunctions, and the delta functions are used as testing functions.

A. Accuracy and Convergence Tests

We first consider the problem of a dielectric unfilled PPWhaving a perpendicular flange surface. Many papers have re-ported to this problem before and thus we can compare ourresults with those obtained by the methods appearing in pre-viously published papers. In Table I the results of comparisonfor reflection coefficient , including amplitude and phase, ofthe incident guided-mode wave are presented. It can beseen that our results are in good agreement with the resultsreported in the literature. Notice that owing to the different

time convention used, there is a minus sign differ-ence in the phase of reflection coefficient in the literature.

Fig. 3. (a) Distribution of the disturbed field j@E =@nj on the boundaryC . (b) Distribution of the total field j@E =@nj on the boundary C . Theparameters used in calculations are the same as for Fig. 2, the virtual boundaryC is located at k a = �2.

Fig. 4. Radiated power � as a function of refractive index n of a dielectricfilled PPW having a perpendicular flange surface of width d=� = 0:5001.

These numerical results show the validity of the method inthis paper.


Fig. 5. Numerical results of a dielectric filled PPW having a tilted flangesurface for d=� = 0:5001 and n = 1:6. (a) Dependence of radiated power� on tilting angle '. (b) Typical radiation patterns.

Fig. 6. Relationship between the angle of beam center and the tilting angle ofa dielectric filled PPW having a tilted flange surface for d=� = 0:5001 andn = 1:6.

To verify the feasibility of the method in this paper, we nextapply the method to the problem of a dielectric filled PPWhaving a tilted flange surface as shown in Fig. 1(a). Becauseit seems to be difficult to solve using the methods based onapproximate theories, to our knowledge, no one has reported to

Fig. 7. Numerical results of a dielectric filled PPW having a tapered flangesurface for d=� = 0:5001 and n = 1:6. (a) Dependence of radiated power� on tapering angle '. (b) Typical radiation patterns.

this kind of problem before. In Table II the results of reflectedpower , radiated power , and their total arepresented for the case of , and . As canbe seen, the results satisfy the energy conservation law withinan accuracy of 1% well.

In Section II, mathematically and physically, it has beenshown that the reflection coefficient is independent of loca-tion of the virtual boundary . For numerical demonstration,the reflection coefficient of a dielectric filled PPW having atilted flange surface as a function of location of the virtualboundary is plotted in Fig. 2 for ,and .

It is observed that the reflection coefficient is independent oflocation of the virtual boundary , except at . Thiserror is caused by the numerical method used, because whenthe virtual boundary approaches the aperture the segments ofdiscretized boundary approach zero.

The validity of truncation of the infinite-length boundariesin the numerical solution procedure is proved numerically inFig. 3(a) and (b). Where Fig. 3(a) shows distribution of the dis-turbed field on boundary , and Fig. 3(b) showsdistribution of the total field on boundary . Theparameters used in calculations are the same as for Fig. 2, thevirtual boundary is located at .


Fig. 8. Numerical results of a dielectric filled PPW having an up-tapered aperture for d=� = 0:5001 and n = 1:6. (a) Dependence of radiated power �on tapering width w=d for h=d = 0:5. (b) Typical radiation patterns of (a)-case. (c) Dependence of radiated power � on tapering height h=d for w=d = 2.(d) Typical radiation patterns of (c)-case.

From Fig. 3, it is found that the use of BEM based on GMEIEsis possible to treat waveguide discontinuity problems as an iso-lated object of finite size. So that it is suitable for the basictheory of computer-aided design (CAD) software for waveguidecircuits.

B. Examples

In the first sequence of examples we consider the conven-tional problem of a dielectric filled PPW having a perpendicularflange surface. The result of radiated power as a function ofrefractive index is shown in Fig. 4 for .

From Fig. 4, it is evident improvement of radiated power of adielectric filled PPW compared with a dielectric unfilled PPW.Since most of the solid dielectrics have the index larger thanapproximate 1.4, we choose the dielectric with index of 1.6 forthe next investigations.

In subsequent examples we apply the method to a number ofcases of arbitrarily flanged PPW as shown in Fig. 1(a)–(c). Theresults of computer simulations are shown below.

1) PPW Having a Tilted Flange Surface [Fig. 1(a)]: For adielectric filled PPW of width having a tiltedflange surface, the dependence of radiated power on thetilting angle is shown in Fig. 5(a), and the typical radiationpatterns are plotted in Fig. 5(b).

The results in Fig. 5 show that the radiated power of a dielec-tric filled PPW can be improved by the use of a tilted flange sur-face, and the symmetry of radiation pattern is maintained eventhough changing the tilting angle. In particular, from Fig. 5(b)it is found that the angle of beam center (i.e., the angle of centerof radiation pattern) with respect to the -axis is changed withchanging the tilting angle. In order to see the relationship be-tween the angle of beam center and the tilting angle , wenumerically plot that relationship in Fig. 6.

It is observed that for the tilting angle less than 15 theabove-mentioned relationship is linear, i.e., the center axis ofbeam is perpendicular to the flange surface, but for the tiltingangle larger than 15 that relationship is nonlinear. This effectmay be important in the prediction of radiation properties fromantennas.

2) PPW Having a Tapered Flange Surface [Fig. 1(b)]: Forthe case of a dielectric filled PPW having a tapered flange sur-face with and , the dependence of ra-diated power on the tapering angle is shown in Fig. 7(a),and the typical radiation patterns are plotted in Fig. 7(b).

In Fig. 7(a), the weak effect of change of tapering angle onradiated power is observed. But on the contrary, the strongeffect of that on radiation pattern is found from Fig. 7(b). Itseems obvious that the beam width decreases and the far-fieldintensity increases with up-tapering the flange surface. This is an


interesting and important result for millimeter-wave free-spacecommunication systems.

3) PPW Having an Up-Tapered Aperture [Fig. 1(c)]: Theradiated power as functions of tapering width and ta-pering height , as shown in Fig. 1(c), are respectively shownin Fig. 8(a) and (c) for . Typically, thecorresponding radiation patterns are shown in Fig. 8(b) and (d).

Notice that in Fig. 8(a) and (b) the tapering height isgiven by , and in Fig. 8(c) and (d) the tapering width

is given by . From Fig. 8, it is found that the radiatedpower of a dielectric filled PPW can be improved significantlyby using an up-tapered aperture. In particular, as shown inFig. 8(c), the radiated power is improved to approximate 0.99by a tapering height , i.e., only 1% ofpower is reflected. The strong effect on radiation patterns isalso found by changing the tapering parameters. However, itis observed from Fig. 8(d) that the number of lobe of radiationpattern is more than one, and the far-field intensity fluctuateswith increasing the size of aperture. Let us note that theseresults, which are very interesting and potentially important indesign of antennas, have not shown by any researcher so far.

IV. CONCLUSION

The radiation properties of a dielectric filled and unfilledPPW having an arbitrarily flanged surface have been studiedby the BEM based on GMEIEs. Based on the theory developedin Section II, the typical numerical evaluations have been per-formed for the case of incident guided-mode wave. Thenumerical results were confirmed by using the law of energyconservation. It has been found that the numerical results arein good agreement with previous results and physical consid-eration.

It is apparent that the method in this paper is suitable forthe basic theory of CAD)software for the antennas systems.Since we do not employ any approximations, such as simpleend-shape, in the formulation of GMEIEs used in this paper, sothat it is easy to extend the GMEIEs to more complicated wave-guide circuits that have more than one port, etc.

REFERENCES

[1] H. Shiomi and S. Yamamoto, “Numerical simulation of fat dielectricloaded waveguide antenna using FDTD method,” in IEICE Proc. Int.Symp. Antennas and Propagation ISAP i-02, Nov. 2002, pp. 520–523.

[2] R. C. Rudduck and D. C. F. Wu, “Slope diffraction analysis of TEM par-allel-plate guide radiation patterns,” IEEE Trans. Antennas Propagat.,vol. AP-17, pp. 797–799, Nov. 1969.

[3] D. C. F. Wu, R. C. Rudduck, and E. L. Pelton, “Application of a surfaceintegration technique to parallel-plate waveguide radiation-pattern anal-ysis,” IEEE Trans. Antennas a Propagat., vol. AP-17, pp. 280–285, May1969.

[4] S. W. Lee, “Ray theory of diffraction by open-ended waveguide, I, fieldin waveguides,” J. Math. Phys., vol. 11, pp. 2830–2850, 1970.

[5] K. Hongo, “Diffraction by a flanged parallel-plate waveguide,” RadioSci., vol. 7, pp. 955–963, Oct. 1972.

[6] T. Itoh and R. Mittra, “TEM reflection from a flanged and dielectric-filled parallel-plate waveguide,” Radio Sci., vol. 9, pp. 849–855, Oct.1974.

[7] K. Hongo, Y. Ogawa, T. Itoh, and K. Ogusu, “Field distribution in aflanged parallel-plate waveguide,” IEEE Trans. Antennas Propagat., vol.AP-23, pp. 558–560, July 1975.

[8] S. Lee and L. Grun, “Radiation from flanged waveguide: Comparison ofsolutions,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 147–148,Jan. 1982.

[9] M. S. Leong, P. S. Kooi, and XQXQXQ Chandra, “Radiation from aflanged parallel-plate waveguide: Solution by moment method with in-clusion of edge condition,” in Proc. Inst. Elect. Eng. Microwaves, An-tenna and Propagation, vol. 135, Aug. 1988, pp. 249–255.

[10] C. M. Butler, C. C. Courtney, P. D. Mannikko, and J. W. Silvestro,“Flanged parallel-plate waveguide coupled to a conducting cylinder,”in Proc. Inst. Elect. Eng. Microwaves, Antenna and Propagation, vol.138, Dec. 1991, pp. 549–558.

[11] C. H. Kim, H. J. Eom, and T. J. Park, “A series solution for TM-mode ra-diation from a flanged parallel-plate waveguide,” IEEE Trans. AntennasPropagat., vol. 41, pp. 1469–1471, Oct. 1993.

[12] T. J. Park and H. J. Eom, “Analytic solution for TE-mode radiationfrom a flanged parallel-plate waveguide,” in Proc. Inst. Elect. Eng. Mi-crowaves, Antenna and Propagation, vol. 140, Oct. 1993, pp. 387–389.

[13] J. W. Lee, H. J. Eom, and J. H. Lee, “TM-wave radiation from flangedparallel plate into dielectric slab,” in Proc. Inst. Elect. Eng. Microwaves,Antenna and Propagation, vol. 143, June 1996, pp. 207–210.

[14] M. Tanaka and K. Tanaka, “Computer simulation for two-dimensionalnear-field optics with use of a metal-coated dielectric probe,” J. Opt. Soc.Amer. A, Opt. Image Sci., vol. 18, pp. 919–925, Apr. 2001.

[15] D. N. Chien, M. Tanaka, and K. Tanaka, “Numerical simulation of an ar-bitrarily ended asymmetrical slab waveguide by guided-mode extractedintegral equations,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 19, pp.1649–1657, Aug. 2002.

[16] D. N. Chien, K. Tanaka, and M. Tanaka, “Accurate analysis of powercoupling between two arbitrarily ended dielectric slab waveguides byboundary-element method,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol.20, pp. 1608–1616, Aug. 2003.

[17] , “Optimum design of power coupling between two dielectric slabwaveguides by the boundary-element method based on guided-mode ex-tracted integration equation,” IEICE Trans. Electron., vol. E86-C, Nov.2003.

[18] , “Guided wave equivalents of Snell’s and Brewster’s Laws,” Opt.Commun., vol. 225, pp. 319–329, Oct. 2003.

Dao Ngoc Chien (S’03) received the B.E. de-gree from the Department of TelecommunicationSystems, Faculty of Electronics and Telecommu-nications, Hanoi University of Technology, Hanoi,Vietnam, in 1997 and the M.S. degree from the De-partment of Electronics and Computer Engineering,Gifu University, Gifu, Japan, in 2002, where he iscurrently working toward the Ph.D. degree.

In 1997, he became a Teaching Assistant in the De-partment of Telecommunication Systems, Faculty ofElectronics and Telecommunications, Hanoi Univer-

sity of Technology. He is currently on leave from Hanoi University of Tech-nology and is a Visiting Researcher in the Department of Electronics and Com-puter Engineering, Gifu University. His current research interests are the CADof optical waveguide circuits, and waveguide technology for antennas and feeds.

Mr. Chien is a Student Member of the Optical Society of America (OSA),Washington, DC, and the Institute of Electrical, Information and Communica-tion Engineers (IEICE), Japan. He was awarded a Yoneyama Scholarship by theRotary Yoneyama Memorial Foundation, Inc., Japan.

Kazuo Tanaka (M’75) received the B.E., M.S., andPh.D., degrees from the Department of Communica-tions Engineering, Osaka University, Osaka, Japan,in 1970, 1972, and 1975, respectively.

In 1975, he became a Research Associate in theDepartment of Electrical Engineering, Gifu Univer-sity, Gifu, Japan, where he became an Associate Pro-fessor in 1985 and a Professor in 1990. His researchsince 1970 has been a general-relativistic electromag-netic theory and application, radiographic image pro-cessing and computational electromagnetic and he is

currently interested in the CAD of integral optical circuits, near-field optical cir-cuits and simulation of Anderson localization hypothesis of ball-lightning. Hewas a Visiting Professor at the University of Toronto, ON, Canada, in 1994.

Dr. Tanaka was awarded the Uchida Paper Award by the Japan Society ofMedical Imaging and Information Science. He was a Chair of the TechnicalGroup of Electromagnetic Theory of the Institute of Electrical, Information andCommunication Engineers (IEICE), Japan.


Masahiro Tanaka (M’00) received the B.E. andM.S. degrees from the Department of Electricaland Computer Engineering, Gifu University, Gifu,Japan, in 1992 and 1994, respectively, and the Ph.D.degree from the Department of CommunicationEngineering, Osaka University, Osaka, Japan, in2002.

He was a Research Associate at Tokoha-GakuenHamamatsu University, Japan, from 1994 to 1996.He joined the Department of Electrical and ComputerEngineering, Gifu University, as a Research Assistant

in 1996. He was a Visiting Researcher at the Department of Electrical and Com-puter Engineering, The University of Arizona, Tempe, from 1997 to 1998. Hisresearch interests are the CAD of optical waveguide circuits and near-field op-tical circuits.


Scan Blindness Free Phased Array Design UsingPBG Materials

Lijun Zhang, Jesus A. Castaneda, and Nicolaos G. Alexopoulos, Fellow, IEEE

Abstract—Scan blindness occurs for phased arrays when propa-gation constants of Floquet modes (space harmonics) coincide withthose of surface waves supported by the array structure. In thispaper, we studied the possibility of using photonic band-gap (PBG)substrate to eliminate scan blindness. A specially designed printedPBG substrate can suppress surface wave propagation inside itsbandgap range, therefore it can be used to eliminate scan blind-ness. In this paper, we presented a method of moments (MoM) anal-ysis of the scan properties of dipole arrays on PBG substrates withominidirectional bandgap(s). We found that scan blindness is com-pletely eliminated. The elimination of scan blindness makes PBGmaterials very attractive in phased array design.

Index Terms—Phased array, photonic band-gap (PBG) sub-strate, scan blindness.

I. INTRODUCTION

SCAN blindness for phased arrays can be traced to the forcedsurface waves by phase matching with those of the Floquet

modes (space harmonics). This is common for printed arrays ondielectric substrates, phased arrays with radomes, etc. [1]–[3].Scan blindness limits the scan range and lowers the antenna ef-ficiency, therefore it must be considered in phased array design.

Many efforts have been devoted to eliminate scan blindness,for example, the subarray technique is used to suppress scanblindness but at the expense of a larger unit cell size, whichcauses an increase in power loss to the grating lobes [3]. Butthe idea of using subarray to perturb the phase progressionof surface waves in the substrate is quite an inspring idea.In [4] the authors studied the scan properties of antennas onperturbed inhomogeneous substrates, and they found that theinhomogeneous substrate can mitigate the scan blindness forproper designed substrate media. Scan properties of phasedarrays on ferrite substrates have been investigated in [5], thescan properties of dipole arrays in a two-layer structure havebeen studied in [6], [7]. It is found that in a two layer structureit is possible to select the parameters to prevent the excitationof any surface waves, therefore to eliminate scan blindness.This however may be a very narrow band operation.

Manuscript received May 4, 2001; revised March 26, 2003. This work wassupported in part by MURI.

L. Zhang and J. A. Castaneda were with the Department of Electrical En-gineering, University of California, Los Angeles, CA 90095-1544 USA. Theyare now with Broadcom Corporation, Irvine, CA 92619-7013 USA (e-mail:[email protected]; [email protected]).

N. G. Alexopoulos is with the Department of Electrical and Computer En-gineering, University of California, Irvine, CA 92695-2625 USA (e-mail: [email protected]).


In this paper photonic band-gap (PBG) materials are usedas antenna substrates to treat the scan blindness problem. PBGmaterials are essentially periodic structures whose dispersiveproperties may be controlled by the periodicity and the electro-magnetic properties of lattice elements [8]. For PBG materialsrealized on dielectric substrates, surface waves can be con-trolled. For example, they can be suppressed along certaindirection for PBG materials with partial bandgaps or alongany direction for those with ominidirectional (or complete)bandgaps [9]–[12] inside a certain frequency range. Such prop-erties of PBG could be used to eliminate scan blindness. In[13] the authors studied the active array pattern of phasedarrays on a PBG substrate which is composed of air holesinside a dielectric substrate, but since that kind of material withfinite thickness does not have an ominidirectional bandgap,the scan blindness cannot be eliminated completely.

Recently, a novel printed PBG substrate with ominidirec-tional bandgap was presented in [9], [10]. In this PBG material,periodic metallic patches are printed on a substrate and eachpatch is connected to the ground plane through a via. It is bothexperimentally and numerically verified that this PBG materialhas a complete surface wave bandgap. In [11] a planar PBGwithout any via was also fabricated and an ominidirectionalbandgap was reported. In this paper our analysis is based onthe first kind of PBG substrate in [9], [10], however we believethat any PBG substrate with complete bandgap can eliminatescan blindness.

In this paper, we first present the theorectical method ofmoments (MoM) formulation for the array analysis, then wediscuss the surface-wave bandgap properties of the PBG ma-terials. After that we present detailed theoretical case studiesfor phased arrays on PBG substrates, from low permittivityto high permittivity substrates, from thin to thick substrates.Finally, waveguide simulator experimental results are presentedto validate the theory and the code we developed.

II. NUMERICAL ANALYSIS: THE METHOD OF MOMENTS

In Fig. 1, we show the PBG material [9] and a dipole phasedarray printed on it. The unit cell of the PBG is around .For phased arrays the radiating dipole is printed in every by

PBG unit cell, here in the figure the dipole is printed in every2-by-2 PBG unit cells as an example.

Printed arrays on uniform substrate and their scan blindnessphenomenon have been extensively analyzed using a MoM in[1], [2]. The MoM is a very fast and accurate full-wave analysismethod for the analysis of phased array and therefore is used in

0018-926X/04$20.00 © 2004 IEEE

ZHANG et al.: SCAN BLINDNESS FREE PHASED ARRAY DESIGN USING PBG MATERIALS 2001

Fig. 1. Structure of the phased dipole array on a PBG substrate. The topfigure shows 3-by-3 unit cells of the PBG substrate. The bottom figure givesthe top view of a unit cell of the infinite phased dipole array printed on thePBG susbstrate. Some key parameters are: patch size L by W , gap betweenpatches g, via size4x by4y, dipole sizeD (length) by4w (width), substratethickness h and substrate permittivity � .

our analysis. The Dyadic Green’s function, which is the electricfield caused by an infinite array of such dipoles is given by

(1)

where and are the Floquet’s propagation constants deter-mined by the scan angle, and are the array period, is thefree space impedance. The dyadic quantity is defined in thefollowing [15]

(2)

On metallic surfaces tangential electric fields should be zero

(3)

where is the tangential dyadic and is the current expansionwhich will be discussed in following parts.

A. Current Expansion From Attachment Modes

For the specific PBG structure, it is more advantageous touse the entire domain basis (EDB) function and the attachmentmode expansion technique in the MoM solution to achieve fastconvergence [16]. Since in this paper we are also interested inthick substrates, we used more than one attachment mode. Thecurrent expansion for each attachment mode is obtained throughthe solving of a Sturm-Liouville problem with a line currentsource excitation inside an equivalent cavity. Current distribu-tions for attachment modes on each patch-via in the unit cell ofPBG is given in the following equations,

(4)

where

(5)

(6)

(7)

where stands for the number of attachment modes,and are the center coordinates of each patch

(8)

(9)

(10)

(also see (11) shown at the bottom of the page), and similarlyfor .

B. Entire Domain Current Expansion on the Dipole and Patch

Current expansions on each patch are simply entire domainbasis functions given as follows:

(12)

for

for .

(11)


(13)

(14)

Current expansion on the dipole follows:

(15)where is the number of entire domain basis functionsused in the dipole current expansion.

C. Final Matrix Equation

After weighting on (3), the following matrix equation isobtained:

(16)

The total matrix size is , withthe number of PBG patches in each unite cell of the array.

Each submatrix entry results from the current basis functionfrom structure type and the weighting function from

structure type , with , from , , standing for attachmentmode, entire domain basis (EDB) function on patch, or EDB ondipole, respectively. The matrix entries are given as

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

Notice that , are local indexes for submatrices. For detailedformulation of the matrix entries please refer to [18], they areomitted here for brevity.

The input impedance of the dipole is calculated as

(26)

(27)

where is the width of voltage gap source, and is thewidth of the dipole.

III. SURFACE WAVE BANDGAP OF PRINTED PBG SUBSTRATES

The MoM procedure discussed in the previous section canalso be used to calculate the bandgap properties of printed PBG

Fig. 2. Surface wave bandgap of the PBG substrate. h = 2:3855mm, lenx =

leny = 2:88 mm, W = L = 2:061 mm, g = 0:819 mm, � = 4:4.

Fig. 3. Convergence of the input impedance versus the number of Floquetmodes and the number of EDB functions. Impedances are for broadside scan.

materials. Here we present some simulation data of the printedPBG material calculated using three different methods whichare the MoM, the finite-element-integral-equation method (FE-IEM) [17] and the FE perfect-matched-layer method (FE-PML)[14].

Fig. 2, shows the eigenmodes of the PBG material with pa-rameters given in the caption of the Figure. , , and arevertices of a reduced Brillouin zone. From to , the wavesare propagating along the direction, varies from 0 to .From to , the waves are propagating at an angle between0 and 45 with respect to the direction, where and

varies from 0 to . From to , waves are propagatingalong the 45 and both and varies from or to 0.

The MoM and FE based codes produce consistent results.The MoM code only searches for bounded modes, therefore, itstops when it intercepts the light line in the air (straight line inthis figure) for the second mode. Between the first and secondmode is the surface wave bandgap. The higher band edge isdetermined by the interception point of the second mode andthe light line. Simulation suggests that the second mode is


Fig. 4. Comparison of the scanned impedances for the dipole array on PBG and uniform substrates. FREQ = 13:0 GHz. � = 2:2. For PBG case dipole sizeis 8.982 mm� 0.06 mm, broadside impedance is Z = 71:083 � j0:032 . Array unit cell is 0:4992� . For uniform case the dipole size is tuned to have aresonance at broadside.

radiating inside the bandgap. In the FE-PML calculation, theeigen frequencies are complex inside the bandgap region forgiven real propagation constants, which implies that modesinside this bandgap region are damping in the lateral direction.The leakage properties of these modes need further investigation,and some similar work has been done in [12] on dielectricPBGs. The good agreement between the MoM and FEM basedcodes verifies the validity of the MoM analysis.

IV. NUMERICAL RESULTS ON PHASED ARRAY

In the first example of phased array, one unit cell is made ofone dipole in the middle of two by two PBG unit cells. Dipolesin the rectangular array are separated by mmin and directions, respectively. The substrate thickness is

mm, , mm,mm, mm. The surface wave bandgap is

between 9.55 and 14.10 GHz.Fig. 3 shows convergence plot of the input impedance versus

the number of Floquet modes and the number of entire domainbasis (EDB) functions for patch current. In the axis, isthe highest index of the Floquet mode, which means that thetotal number of Floquet modes is . It is seen that theimpedance converges as the number of the Floquet modes andthe EDB functions are increased. In our following calculation,we use for the Floquet modes and 16 EDB functions.

The reflection of the array at two different frequencies areplotted in Figs. 4 and 5. In Fig. 4, the array operates at 13.0 GHz.For the case with uniform substrate scan blindness occurs at69.0 in the -plane, there is no blindness in the -plane be-cause of polarization mismatch. For the PBG case, there is noblindness spot. In Fig. 5 the operating frequency is at 13.5 GHz.Since the array unit cell is greater than , a grating lobe

occurs for both the uniform and PBG substrate. For the arrayon the uniform substrate, the scan blindness spot moves towardbroadside comparing to the 13.0 GHz case. There is no blind-ness spot for the array on the PBG substrate.

In the next example, a substrate with higher dielectric con-stant is picked. The PBG lattice sizes are

, patch width mm, mm,. One array unit is made of one dipole in the middle of

four-by-four PBG units. The surface wave bandgap is from 9.25to 12.85 GHz. Fig. 6 shows the scanned impedance at 12.8 GHz.For the case on uniform substrate, the blindness spot moves to-ward the broadside when the permittivity is higher. Again, noblindness spot is observed for the PBG case.

Compact arrays can be realized on high permittivity sub-strates, however strong surface waves excited in the substratecause problems such as small scan range, strong mutualcoupling between elements, low efficiency, etc. Use of PBGsubstrate can avoid the formentioned drawbacks.

In the third example, an array on a thick substrate is analyzed.Thicker substrate provides wider bandwidth. A unit of the arrayis made of one dipole in the middle of four by four PBG unitcells. The substrate thickness is mm, ,with the PBG unit period of mm,

mm and mm. The bandgap is from 9.7 to15.1 GHz. Fig. 7 shows the scanned impedance. Again no scanblindness exists for the array on the PBG substrate.

V. PBG SUBSTRATE DESIGN PROCEDURES

In practical phased array design, the most important issue ishow to design the PBG substrate. For example, given the sub-strate thickness ( ), permittivity ( ), array unit cell size ( ),


Fig. 5. Scan impedance of the dipole array on PBG substrate. FREQ = 13:5 GHz, dipole 8.668 mm� 0.06 mm, broadside impedance is Z = 62:635 +j0:085 . Array unit cell is 0:5184� .

Fig. 6. Scan impedance of the dipole array on PBG and uniform substrates. � = 4:4.FREQ = 12:8GHz. For PBG case, dipole length is 7.14 mm, broadsideimpedance is Z = 50:30� j0:08 . Array unit cell is 0:49152� .

how to design the PBG substrate to meet the bandgap and band-width requirement ( ). The suggested steps are listed inthe following:

(1) Design a reference PBG substrate, with

given h, �r, a; b, the bandgap is denoted as

fref ;4fref,

(2) Scale the unit cell size a; b to make the

new bandgap to be ~f; ~4f, which is close to the

aiming frequency f; 4f,

(3) Fine tune the gap between the patches,

(4) Check the loading effect. Go back to step

3 if necessary.

In the array design, one also needs to consider the effects ofthe feeding and the radiating element loading.

VI. EXPERIMENTS

A large phased array is always an extremely costly pieceof hardware, and unfortunately some phenomena are related tosize specifically. A commonly used method to examine the scanproperties of phased arrays is called waveguide simulator, whichprovides a compact and inexpensive test piece for phased arrays.For references on the theory of waveguide simulator, pleaserefer to [19], [20].


Fig. 7. Scanned impedance of a dipole array on thick PBG substrate. FREQ = 13:0 GHz, Dipole length is 9.766 mm, broadside impedance is Z = 82:43�j0:15 .

The experimental setup is explained in Fig. 8[18]. An S-bandWR284 rectangular waveguide is used, the inner dimensions are2.84 inches by 1.34 inches. The single mode ( ) band isbetween 2.60 and 3.95 GHz. The PBG substrate is fabricatedand then press fitted into the waveguide, a shorting plate is addedat the bottom of the substrate ground plane. The coax is fedthrough two holes drilled from the waveguide broadside wall.The substrate is put into a shallow waveguide section so thatwe can access the monopole to solder it to the inner conductorof the coax excitation. Another long waveguide section is putabove the shallow section to allow the excited high order modesbe attenuated, and finally a matched load is put at the end tosimulate the wave propagation in free space.

The waveguide simulator models the phased array with an-plane scan angle governed by the following equation:

(28)

with which is the broadside dimension of the wave-guide and the free space wavelength. For frequencies from2.6 to 3.9 GHz, the simulated scan range is from 51.22 to31.32 .

In the experiment, a power divider was used and the twooutputs are connected to two coax cables used to excite themonopoles. The return loss is measured at the input of the powerdivider, and then de-embedded to obtain the reflection coeffi-cient at the feeding point of the monopole. The substrate is aRT 5870 Duriod board, with , the thickness is390 mil.

In the first case we fabricated a PBG substrate using onelayer of this board, the metal patch size is 8.9 mm by 10.5 mm,with a period of 17.018 mm by 18.034 mm. By mirroring, theequivalent dipole is between every two by four PBG unit cells.The monopole length is 16.32 mm. The experimental and MoM

Fig. 8. Waveguide simulator experimental setup.

simulation data are plotted in Fig. 9, where consistency can beobserved between the two.

In the second case, we fabricated a PBG material with doublesubstrate thickness, the surface wave bandgap is between 1.9and 3.9 GHz. The patch size is 6.5 mm by 7.5 mm. Themonopole length is 21.0 mm. Results are shown in Fig. 10.For the uniform substrate, scan blindness occurs at 3.7 GHz,which is corresponding to an -plane scan angle of 34 fromthe broadside. This agrees with the scan angle predicted by thefollowing [1]:

(29)

where is the surface wave propagation constant,mm is the period of the PBG in broadside. The nonunit


Fig. 9. Waveguide simulator results for dipole array on PBG substrate.

Fig. 10. Waveguide simulator results for dipole array on PBG andcorresponding uniform substrate with substrate thickness twice those inFig. 9.

reflection at scan blindness point is due to the loss associatedwith the waveguide simulator set up and possibly loss in thesusbtrate. Mismatch between the simulation and measurementfor the PBG case may be due to the nonlongitudinal currentsexcited on the patches and vias which affects the waveguidesimulator accuracy. For a resonance frequency at 3.25 GHz, thesubstrate thickness is . The good agreement verifies thatthe MoM analysis is also valid for thick substrates.

VII. CONCLUSION

Scan properties of phased arrays on PBG substrates have beeninvestigated. Through the example of printed dipoles on PBGsubstrates, it is found in both simulation and experiment thatscan blindness can be completely eliminated, due to the sup-pression of surface wave propagation inside PBG substrates.PBG substrates, especially those with complete surface wavebandgaps, will find extensive applications in printed antennasand arrays.

ACKNOWLEDGMENT

The authors would like to acknowledge Prof. F. De Flaviisand Mr. R. Ramirez from the University of California at Irvinefor their kind help in the waveguide experiments. The authorswould like to thank Prof. D. Yang from the University of Illi-nois at Chicago for helpful suggestions on the use of attachmentmodes in the MoM solution.

REFERENCES

[1] D. M. Pozar and D. H. Schaubert, “Scan blindness in infinite phasedarrays of printed dipoles,” IEEE Trans. Antennas Propagat., vol. 32, pp.602–610, June 1984.

[2] , “Analysis of an infinite array of rectangular microstrip patcheswith idealized probe feeds,” IEEE Trans. Antennas Propagat., vol. 32,pp. 1101–1107, Oct. 1984.

[3] D. M. Pozar, “Scanning characteristics of infinite arrays of printedantenna subarrays,” IEEE Trans. Antennas Propagat., vol. 40, pp.666–674, June 1992.

[4] W. J. Tsay and D. M. Pozar, “Radiation and scattering from infinite pe-riodic printed antennas with inhomogeneous media,” IEEE Trans. An-tennas Propagat., vol. 46, pp. 1641–1650, Nov. 1998.

[5] H. Y. Yang and J. A. Castaneda, “Infinite phased arrays of microstripantennas on generalized anisotropic substrates,” Electromagn., vol. 11,no. 1, pp. 107–124, Jan.-Mar. 1991.

[6] J. A. Castaneda, “Infinite phased array of microstrip dipoles in twolayers,” Ph.D. dissertation, University of California, Los Angeles, 1988.

[7] J. Castaneda and N. G. Alexopoulos, “Infinite arrays of microstripdipoles with a superstrate (cover) layer,” in Proc. Antennas and Prop-agation Int. Symp., vol. 2, 1985, pp. 713–717.

[8] E. Yablonovitch, “Photonic band-gap structures,” J. Opt. Soc. Amer. B,vol. 10, no. 2, pp. 283–295, Feb. 1993.

[9] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, andE. Yablonovitch, “High-impedance electromagnetic surfaces witha forbidden frequency band,” IEEE Trans. Microwave Theory andTechniques, vol. 47, pp. 2059–2074, Nov. 1999.

[10] D. Sievenpiper, “High-impedance electromagnetic surfaces,” Ph.D. dis-sertation, Univ. California, Los Angeles, 1999.

[11] F. R. Yang, K. P. Ma, Y. Qian, and T. Itoh, “A novel TEM waveguideusing uniplanar compact photonic-bandgap (UC-PBG) structure,” IEEETrans. Microwave Theory and Techniques, vol. 47, pp. 2092–2098, Nov.1999.

[12] H. Y. Yang, “Characteristics of guided and leaky waves on multilayerthin-film structures with planar material gratings,” IEEE Trans. Mi-crowave Theory and Techniques, vol. 45, pp. 428–435, Mar. 1997.

[13] P. K. Kelly, L. Diaz, M. Piket-May, and I. Rumsey, “Investigation ofscan blindness mitigation using photonic bandgap structure in phasedarrays,” in Proc. SPIE, vol. 3464, July 1999, pp. 239–248.

[14] L. Zhang, N. G. Alexopoulos, D. Sievenpiper, and E. Yablonovitch, “Anefficient finite-element method for the analysis of photonic band-gapmaterials,” in Proc. IEEE Int. Microwave Symp. Dig., vol. 4, 1999, pp.1703–1706.

[15] I. E. Rana and N. G. Alexopoulos, “Current distribution and inputimpedance of printed dipoles,” IEEE Trans. Antennas Propagat., vol.29, pp. 99–105, 1981.

[16] J. T. Aberle and D. M. Pozar, “Analysis of infinite arrays of probe-fedrectangular microstrip patches using a rigorous feed model,” IEE Proc.,pt. H, vol. 136, no. 2, pp. 109–119, Apr. 1989.

[17] L. Zhang and N. G. Alexopoulos, “Finite-element based techniquesfor the modeling of PBG materials,” Electromagn., Special Issue onTheory and Applications of Photonic Band-Gap Materials, vol. 19, pp.225–239, May-June 1999.

[18] L. Zhang, “Numerical characterization of electromagnetic band-gap ma-terials and applications in printed antennas and arrays,” Ph.D. disserta-tion, Univ. Calif., Los Angeles, 2000.

[19] P. W. Hannan and M. A. Balfour, “Simulation of a phase-array antennain waveguide,” IEEE Trans. Antennas Propagat., vol. 13, pp. 342–353,May 1965.

[20] N. Amitay, V. Galindo, and C. P. Wu, Theory and Analysis of PhasedArray Antennas. New York: Wiley-Interscience, 1972, pp. 59–63.


Lijun Zhang received the B.S. and M.S. degrees in electrical engineering fromUniversity of Science and Technology of China, in 1993 and 1996, respectively,and the Ph.D. degree in electrical engineering from the Department of ElectricalEngineering, University of California, Los Angeles, in 2000.

From June 2000 to December 2000, he was with Agilent Technology in WestLake Village, CA, working on RF-CMOS CAD. Since December 2000, he hasbeen working for Broadcom Corporation, Irvine, CA, in the areas of on chippassives modeling and wireless radio transceiver design.

Jesus A. Castaneda received the B.S. degree in physics from Saint Mary’s Col-lege, Moraga, CA, in 1970, and the M.S.E.E. and Ph.D.E.E. degrees from theDepartment of Electrical Engineering, University of California, Los Angeles(UCLA), in 1978, 1981, and 1988, respectively.

From 1978 to 1985, he was with the Antenna Department, Radar SystemsGroup, Hughes Aircraft Company, working in the area of microwave antennadesign and analysis, including electronically scanned antenna arrays. Otherwork areas included planar arrays, frequency selective surfaces, radomes,and adaptive arrays. From 1986 to 1997, he was with Phraxos Research andDevelopment, Inc. as Senior Research Engineer and Engineering Managerwith responsibility for the technical management of projects in the area ofelectromagnetic modeling for microwave and millimeter wave applications.From 1995 to 2000, he was a Senior Lecturer at the School of Engineeringand Applied Science, UCLA. Since 2000, he has been with BroadcomCorporation, Irvine, CA, as a Senior Principal Scientist working in theareas of antennas for wireless systems and on-chip passives design.

Nicoloas G. Alexopoulos (S’68–M’69–SM’82–F’87) received the B.S.E.E.,M.S.E.E., and Ph.D.E.E. degrees from the University of Michigan, Ann Arbor,in 1965, 1967, and 1968, respectively.

He was a member of the faculty in the Department of Electrical Engineering,University of California, Los Angeles (UCLA), from 1969 to 1996. While atUCLA, he served as Associate Dean for Faculty Affairs from 1986 to 1987,and Chair of the Electrical Engineering Department from 1987 to 1992. Underhis leadership and tenure as Chair the department doubled in size, created ahighly successful Corporate Affiliates Program, raised more than $30 millionin gifts and endowments and established the High Frequency ElectronicsLaboratory. In 1997, he joined the Electrical Engineering and ComputerScience Department, University of California, Irvine, and has been Deanof The Henry Samueli School of Engineering. As Dean he led the effortsto establish The Integrated Nanosystems Research Facility, The BiomedicalEngineering Department, The Center for Pervasive Communications andComputing, The California Institute for Telecommunications and InformationTechnology, supported the establishment of The National Fuel Cell ResearchCenter and initiated The Arts, Computing and Engineering Program. Inaddition, he is the Principal Investigator of the University of CaliforniaIrvine Mathematics Engineering Science Achievement Program (MESA)and Co-PI of The National Science Foundation UC Systemwide CaliforniaAlliance for Minority Participation (CAMP). His research contributions overtime include the first contributions in the interaction of electromagneticwaves with active surfaces and particles in the early 1970s. He was thefirst to define and publish on Active and Passive Magnetic Walls (PBGand EBG structures) and their realization with artificial periodic structuresand specifically arrays of antennas terminated at variable load impedances.He demonstrated how such surfaces can be used for beam scanning, andradar cross section elimination or enhancement. Subsequently he focused ondeveloping, with his students, a single full wave theory for the simultaneousdesign of microstrip circuits and printed antennas, thus taking into accountall wave phenomena and mutual interactions. This work also led to thestudy of substrate-superstrate effects and anisotropic and gyrotropic substratematerials. This body of research contributed significant progress in the useof the MoMs for the development of useful design algorithms for microstripantennas and circuits. More recently, he and his students focused in thedevelopment of percolation theory and its applications in materials andwave propagation in complex media, as well as the design of artificialmaterials. Presently, he is working on the integration of the above mentionedexperience in the research of electromagnetically metamorphic objects andinterfaces. He has more than 250 publications and lectures on a varietyof subjects including a popular lecture on “The Genesis and Destructionof The First Research University; The Library/Museum of Alexandria.”

Dr. Alexopoulos was the corecipient (with his students) of the SchelkunoffAward in 1985 and 1998.


Fractile Arrays: A New Class of Tiled Arrays WithFractal Boundaries

Douglas H. Werner, Senior Member, IEEE, Waroth Kuhirun, and Pingjuan L. Werner, Senior Member, IEEE

Abstract—In this paper, a new class of antenna arrays are in-troduced, which we call fractile arrays. A fractile array is definedas any array with a fractal boundary contour that tiles the planewithout gaps or overlaps. It will be shown that the unique geo-metrical features of fractiles may be exploited in order to makeavailable a family of deterministic arrays that offer several highlydesirable performance advantages over their conventional peri-odic planar array counterparts. Most notably, fractile arrays haveno grating lobes even when the minimum spacing between ele-ments is increased to at least one-wavelength. This has led to thedevelopment of a new design methodology for modular broadbandlow-sidelobe arrays that is based on fractal tilings. Several exam-ples of fractile arrays will be considered including Peano–Gosper,terdragon, 6-terdragon, and fudgeflake arrays. Efficient iterativeprocedures for calculating the radiation patterns of these fractilearrays to arbitrary stage of growth are also introduced in thispaper.

Index Terms—Fractal antennas, fractal arrays, broad-bandarrays, grating lobes, low-sidelobe arrays.

I. INTRODUCTION

SEVERAL book chapters and review articles have been pub-lished recently that deal with the subject of fractal antenna

engineering [1]–[5]. A considerable amount of this literature isdevoted to new concepts for antenna arrays that employ fractalgeometries in their design. The first application of fractal geom-etry to antenna array theory was proposed by Kim and Jaggard[3], [5], [6], where properties of random fractals were used todevelop a design methodology for quasirandom arrays. Thesequasirandom arrays were shown to possess radiation character-istics capable of bridging the gap between those produced bycompletely ordered (i.e., periodic) arrays and completely disor-dered (i.e., random) arrays.

The design of multiband and low-sidelobe linear arrays basedon a Cantor fractal distribution of elements was considered in[1], [3], [5], [7]. Other properties of Cantor fractal linear arrayshave been studied more recently in [2], [3], [5]. The electro-magnetic radiation produced by planar concentric-ring Cantorarrays was investigated in [3], [5], [8]. These arrays were gener-ated using polyadic Cantor bars, which are described by theirsimilarity fractal dimension, number of gaps, and lacunarity

Manuscript received June 16, 2003; revised August 18, 2003.The authors are with the The Pennsylvania State University, Depart-

ment of Electrical Engineering, University Park, PA 16802 USA (e-mail:[email protected]).


parameter. Planar fractal array configurations, based on Sier-pinski carpets, were also considered in [2], [3], [5], [9]. Thefact that Sierpinski carpet and related arrays can be generatedrecursively (i.e., via successive stages of growth starting with asimple generating array) has been exploited in order to developrapid algorithms for use in efficient radiation pattern computa-tions and adaptive beamforming, especially for arrays with mul-tiple stages of growth that contain a relatively large number ofelements [2], [3], [5]. The Cantor linear and Sierpinski carpetplanar fractal arrays have also been shown to be examples ofdeterministically thinned arrays [2], [3], [5].

More recently, a new type of deterministic fractal array wasintroduced in [10]–[12] that is based on the Peano–Gosperfamily of space-filling curves. The elements of the array areuniformly distributed along a Peano–Gosper curve, whichleads to a planar array configuration with parallelogram cellsthat is bounded by a closed Koch-type fractal curve. Theseunique properties were exploited in [10]–[12] to develop adesign methodology for deterministic arrays that have nograting lobes even when the minimum spacing between ele-ments is increased to at least one wavelength. Hence, thesePeano–Gosper arrays are relatively broadband when comparedto more conventional periodic planar arrays with square orrectangular cells and regular boundary contours. This type offractal array differs fundamentally from other types of fractalarray configurations that have been studied previously, such asthose reported in [1]–[9], which have regular boundaries withelements distributed in a fractal pattern on the interior of thearray. However, in direct contrast to this, the boundary contourof the Peano–Gosper array is fractal but the elements on theinterior of the array do not follow a fractal distribution.

A new category of fractal arrays, which we call fractile ar-rays, will be introduced in this paper. A fractile array is definedto be any array which has a fractal boundary contour that tilesthe plane. Tilings of the plane using fractal shaped tiles havebeen considered in [13]–[15]. These fractal tiles, or fractiles,represent a unique subset of all possible tile geometries that canbe used to cover the plane without gaps or overlaps. Here weexploit the unique geometrical properties of fractiles to developa new design methodology for modular broadband low-sidelobeantenna arrays.

In Section II-A we demonstrate that the Peano–Gosper arraysrecently considered in [10]–[12] may be classified as fractilearrays. The radiation characteristics of other types of fractilearrays will also be investigated in this paper. These include thetredragon, 6-terdragon, and fudgeflake fractile arrays discussedin Section II-B, Section II-C, and Section II-D, respectively.

0018-926X/04$20.00 © 2004 IEEE

WERNER et al.: FRACTILE ARRAYS: A NEW CLASS OF TILED ARRAYS WITH FRACTAL BOUNDARIES 2009

(a)

(b)

(c)

Fig. 1. Gosper island fractiles and their corresponding Peano–Gosper curvesfor (a) stage 1, (b) stage 2, and (c) stage 4.

II. SOME EXAMPLES OF FRACTILE ARRAYS

A. The Peano–Gosper Fractile Array

The radiation properties of Peano–Gosper arrays have beenrecently investigated in [10]–[12]. These arrays derive theirname from the fact that the elements are uniformly distributedalong a space-filling Peano–Gosper curve. This results in adeterministic planar array configuration composed of a uniquearrangement of parallelogram cells that is bounded by a variantof an irregular closed Koch fractal curve. It was shown in[10]–[12] that these arrays exhibit relatively broadband lowside-lobe performance when compared to their conventionalcounterparts.

Here, we show that Peano–Gosper arrays are in actuality atype of fractile array. In order to see this, it is convenient tostart by considering the sequence of Gosper islands illustrated inFig. 1. Also shown in Fig. 1 are the Peano–Gosper curves that fillthe interior of the associated Gosper islands. The array elementsare assumed to be equally spaced along these Peano–Gospercurves [10]–[12]. For example, the generating array represented

Fig. 2. First three stages in the construction of a terdragon curve. The initiatoris shown in (a) as a dashed line superimposed on the stage 1 generator. Thegenerator (unscaled) is shown again in (b) as the dashed curve superimposed onthe stage 2 terdragon curve. The stage 3 terdragon curve is shown in (c).

by the stage 1 Peano–Gosper curve shown in Fig. 1(a) con-tains a total of eight elements, while the stage 2 Peano–Gosperarray represented by the curve shown in Fig. 1(b) contains 50elements.

Fig. 1(b) indicates that seven stage 1 Gosper islands can betiled together to form a stage 2 Gosper island. Likewise, sevenstage 2 Gosper islands can be tiled together in a similar wayto form a stage 3 Gosper island, and so on. Fig. 1(c) shows astage 4 Gosper island (which consists of seven stage 3 Gosperislands tiled together) as well as the corresponding stage 4Peano–Gosper curve that fills its interior. The tiling processdepicted in Fig. 1 can be repeated to produce Gosper islandshaving any desired stage of growth. This implies that Gosperisland tiles are self-similar since they may be divided intoseven equal tiles that are similar to the whole [13]. Moreover, itfollows that the boundary of these Gosper island tiles is repre-sented by a type of Koch fractal curve. It is also obvious fromFig. 1 that these Gosper islands are examples of fractiles sincethey can be used to tile the plane. Finally, because each Gosperisland has a corresponding Peano–Gosper curve that fills itsinterior, then we are led to the conclusion that Peano–Gosperarrays do in fact belong to the family of fractile arrays.

B. The Terdragon Fractile Array

In this section, we will introduce the terdragon fractile arrayas well as derive a useful compact product representation for thecorresponding array factor. The terdragon is a member of thefamily of space-filling dragon curves [14]. The first three stagesin the construction of a terdragon curve are shown in Fig. 2. Theinitiator for the terdragon curve is indicated in Fig. 2(a) by thedashed line segment of unit length. The generator for the ter-dragon curve is obtained from the initiator by replacing it witha three-sided polygon as shown in Fig. 2(a), where each sidehas a length of . Now in order to obtain the stageconstruction of the terdragon curve shown in Fig. 2(b), each ofthe three sides of the generator polygon at stage (shown


Fig. 3. Element locations and associated current distributions for the (a) stage 1, (b) stage 3, and (c) stage 6 terdragon fractile arrays. The spacing d betweenconsecutive array elements uniformly distributed along the terdragon curve is assumed to be the same for each stage.

reproduced in Fig. 2(b) as the dashed curve) are replaced by anappropriately scaled, rotated, and translated copy of the entiregenerator. This iterative process may be repeated to generateterdragon curves up to an arbitrary stage of growth . For in-stance, the stage construction of the terdragon curve isshown in Fig. 2(c) superimposed on a copy of the stagecurve from Fig. 2(b). The geometry for a stage 1, a stage 3, anda stage 6 fractile array based on the terdragon curve are shownin Fig. 3. Fig. 3 also indicates the location of the elements inthe plane and their corresponding values of current am-plitude excitation. For this example, the minimum spacing be-tween array elements is held fixed at a value of for eachstage of growth. The nonuniform current amplitude distribu-tions arise from the fact that the initiator consists of a uniformlyexcited two-element linear array with spacing between the ele-ments denoted by . Hence, we can consider the generatorarray shown in Fig. 3(a) to be composed of three copies of thetwo-element initiator array appropriately rotated and translated.In this case there are two instances where two of the array ele-ments will share a common location. From a physical point ofview the two colocated elements can be interpreted as a singleelement having twice the value of current amplitude excitation.With this in mind the mathematical six-element uniformly ex-cited array model can be replaced by a physically equivalentfour-element array model that has a nonuniform current dis-tribution of 1:2:2:1. This process is then repeated to generatehigher-order versions of the fractile array.

The array factor for a stage terdragon fractile array may beconveniently expressed in terms of a product of matrices

TABLE IEXPRESSIONS OF x AND y IN TERMS OF THE PARAMETERS d AND �

Fig. 4. First stage in the construction of a 6-terdragon fractile. The initiator isshown as the dashed curve superimposed on the stage 1 generator shown as thesolid curve.


Fig. 5. Element locations and associated current distributions for the (a) stage1, (b) stage 2 and (c) stage 3 6-terdragon fractile arrays. The spacing d

between consecutive array elements uniformly distributed along the 6-terdragoncurve is assumed to be the same for each stage.

which are pre-multiplied by a vector and postmultiplied bya vector such that

(1)

where

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

Fig. 6. 6-terdragon fractile arrays for (a) stage 4 and (b) stage 6.

Fig. 7. The first three stages in the construction of a fudgeflake fractile.The initiator is shown in (a) as the dashed curve superimposed on the stage 1generator. The generator (unscaled) is shown again in (b) as the dashed curvesuperimposed on the stage 2 fudgeflake. The stage 3 fudgeflake curve is shownin (c).

(12)

(13)


Fig. 8. Element locations and associated current distributions for the (a) stage 1, (b) stage 3, and (c) stage 5 fudgeflake fractile arrays. The spacing d betweenconsecutive array elements uniformly distributed along the fudgeflake curve is assumed to be the same for each stage.

Note that the parameter represents the scale factor used togenerate the terdragon fractile arrays shown in Fig. 3. The valuesof required in (8) are found from

(14)

where represents the empty set. Note that if thenfor the corresponding values of and . The values of

and for – required to evaluate (9) and (10) arelisted in Table I.

C. The 6-Terdragon Fractile Array

Fig. 4 shows the first stage in the construction of a fractilearray that is based on six copies of the stage 1 terdragon array,


Fig. 9. Fudgeflake fractile array aperture divided into three self-similarsubarray apertures.

Fig. 10. Plot of the normalized stage 6 terdragon fractile array factor versus �for ' = 90 . The dashed curve represents the case where d = �=2 and thesolid curve represents the case where d = �.

shown in Fig. 3, arranged in the plane around the point atthe origin. Therefore, we introduce the terminology 6-terdragonarray to denote a fractile array generated from the curve shownin Fig. 4. The first three stages (i.e., , and )in the construction of a 6-terdragon fractile array are shown inFig. 5. Also indicated in Fig. 5 are the locations of the elementsand their corresponding values of current amplitude excitation.The minimum spacing between array elements is held fixed at avalue of for each stage of growth. Fig. 6 shows the geom-etry for a stage 4 and a stage 6 6-terdragon fractile array. Thisfigure also clearly illustrates how these arrays can be consideredas being composed of six associated terdragon subarrays tiledtogether around a common central point. The array factor for astage 6-terdragon fractile array may be expressed in terms ofthe product of matrices which are premultiplied by avector and postmultiplied by a vector such that

(15)

(16)

(17)

Fig. 11. Plot of the normalized stage 6 terdragon fractile array factor versus 'for � = 90 and d = �.

Fig. 12. Plot of the normalized array factor versus � with ' = 90 for auniformly excited 18� 18 periodic square array. The dashed curve representsthe case where d = �=2 and the solid curve represents the case whered = �.

where the matrices and are defined in (2) and (6),respectively.

D. The Fudgeflake Fractile Array

In this section, another type of fractile, known as the fudge-flake, is investigated for its potential utility in the design ofbroadband low-sidelobe antenna arrays. The first three stages inthe construction of a fudgeflake fractile are illustrated in Fig. 7[14]. The initiator appears as the dashed curve (i.e., the triangle)in Fig. 7(a) superimposed on the stage 1 generator. This gener-ator is shown again in Fig. 7(b) as the dashed curve superim-posed on the stage 2 fudgeflake, while Fig. 7(c) shows the stage3 fudgeflake with the associated generator from stage 2 super-imposed. The geometry and current distributions for a stage 1,stage 3, and stage 5 fudgeflake fractile array located in theplane are depicted in Fig. 8. Finally, an example is presented inFig. 9 that illustrates how a fudgeflake fractile array can be di-vided into three self-similar subarray apertures.


TABLE IIMAXIMUM DIRECTIVITY FOR SEVERAL DIFFERENT TERDRAGON FRACTILE ARRAYS

TABLE IIICOMPARISON OF MAXIMUM DIRECTIVITY FOR A STAGE 6 TERDRAGON FRACTILE ARRAY WITH 308 ELEMENTS AND AN 18� 18 SQUARE ARRAY OF

COMPARABLE SIZE WITH 324 ELEMENTS

The array factor for a stage fudgeflake fractile array maybe expressed as

(18)

where the matrices and have been defined in (2) and (6),respectively, and

(19)

(20)

Finally, we point out that the self-similar and associated itera-tive properties of fractile arrays could be exploited to developfast algorithms for calculating their driving point impedances.This could be accomplished by following a similar procedure tothat introduced in [5] and [16] for the more conventional Cantorlinear and Sierpinski carpet planar fractal arrays.

III. RESULTS

Fig. 10 contains a plot of the normalized array factor (in deci-bels) versus with for the stage 6 terdragon fractilearray shown in Fig. 3(c). The dashed curve represents the ra-diation pattern slice for a terdragon fractile array with elementspacings of while the solid curve represents thecorresponding radiation pattern slice for the same array with

. Fig. 11 shows a plot of the normalized array factorfor the case where , and .This plot demonstrates that there are no grating lobes presentanywhere in the azimuthal plane of the terdragon fractile array,even with elements spaced one-wavelength apart. For compar-ison purposes, we consider a uniformly excited periodic 18 18square array of comparable size to the stage 6 terdragon fractilearray, which contains a total of 308 elements. Plots of the nor-malized array factor for the 18 18 periodic square array areshown in Fig. 12 for element spacings of(dashed curve) and (solid curve). A grating lobeis clearly visible for the case in which the elements are periodi-cally spaced one wavelength apart.


Fig. 13. Plots of the normalized array factor versus � for ' = 0 withmainbeam steered to � = 45 and ' = 0 . The solid curve represents theradiation pattern of a stage 6 terdragon fractile array with d = �=2 and thedashed curve represents the radiation pattern of a uniformly excited 18� 18square array with d = d = �=2. Note that terdragon arrays are examplesof almost uniformly excited arrays.

Fig. 14. Plot of the normalized stage 4 6-terdragon fractile array factor versus� for ' = 90 . The dashed curve represents the case where d = �=2 andthe solid curve represents the case where d = �.

The array factor of any stage planar fractile array withelements may be expressed in the general form:

(21)

where and represent the excitation current amplitude andphase of the th element respectively, is the horizontal posi-tion vector for the th element with magnitude and angle ,and is the unit vector in the direction of the far-field observa-tion point. Therefore, an expression for the maximum directivityof a broadside stage planar fractile array of isotropic sources

Fig. 15. Plot of the normalized stage 5 fudgeflake fractile array factor versus� for ' = 90 . The dashed curve represents the case where d = �=2 andthe solid curve represents the case where d = �.

may be readily obtained by setting in (21) and substi-tuting the result into

(22)

This leads to the following expression for the maximum direc-tivity given by [10]:

(23)

Table II lists the values of maximum directivity, calcu-lated using (23), for several terdragon fractile arrays withdifferent minimum element spacings and stages of growth

. Table III provides a comparison between the maximumdirectivity of a stage 6 terdragon fractile array and that of a con-ventional uniformly excited 18 18 planar square array. Thesedirectivity comparisons are made for three different valuesof array element spacings (i.e., ,and ). In the first case, where the element spacingis assumed to be , we find that the maximumdirectivity of the stage 6 terdragon array and the 18 18 squarearray are comparable. This is also found to be the case whenthe element spacing is increased to (see Table III).However, in the third case where the element spacing is in-creased to , we see that the maximum directivity forthe stage 6 terdragon fractile array is about 9 dB higher than itsconventional 18 18 square array counterpart. This is becausethe maximum directivity for the stage 6 terdragon fractile arrayincreases from 25.6 to 29.8 dB when the element spacing ischanged from a half-wavelength to one-wavelength, respec-tively, while on the other hand, the maximum directivity for the18 18 square array drops from 26.9 dB down to 20.9 dB. Thedrop in value of maximum directivity for the 18 18 square


TABLE IVMAXIMUM DIRECTIVITY FOR SEVERAL DIFFERENT 6-TERDRAGON FRACTILE ARRAYS

array may be attributed to the appearance of grating lobes inthe radiation pattern.

Next, we consider the case where the mainbeam of the ter-dragon fractile array is steered in the direction corresponding to

and . In order to accomplish this, the elementphases for the terdragon fractile array are chosen according to

(24)

Fig. 13 shows normalized array factor plots with the mainbeamsteered to and , where the solid curve re-sults from a stage 6 terdragon fractile array and the dashed curveresults from a conventional uniformly excited 18 18 squarearray. The minimum spacing between elements for both arraysis assumed to be a half-wavelength. This comparison demon-strates that the terdragon fractile array is superior to the 18 18square array in terms of its overall sidelobe characteristics.

At this point the radiation characteristics of the 6-terdragonfractile array illustrated in Fig. 6(a) will be investigated. A plotof the normalizad array factor as a function of for the stage 46-terdragon fractile array is shown in Fig. 14 for . Thedashed curve represents the case where , whereasthe solid curve represents the case where . Again,we see from Fig. 14 that there are no grating lobes present forthis array when the minimum spacing between elements is asmuch as one-wavelength. The values of maximum directivityfor several 6-terdragon fractile arrays with different minimumelement spacings and stages of growth are listed in Table IV.Table V provides a comparison between the maximum direc-tivity of a stage 4 6-terdragon fractile array and a uniformly ex-

cited 15 15 planar square array for three different values ofelement spacings.

Finally, the last example to be considered will be the stage5 fudgeflake fractile array illustrated in Fig. 8(c). A plot ofthe normalized array factor for this array is shown in Fig. 15,where the dashed curve and solid curve represent the caseswhere and , respectively. Table VIlists the values of maximum directivity for several fudgeflakefractile arrays with different minimum element spacings andstages of growth , while Table VII provides a comparisonbetween the maximum directivity of a stage 5 fudgeflakefractile array and a uniformly excited 18 18 planar squarearray for element spacings of , and . Therefore,this example provides yet another illustration of the uniquefeature characteristic of fractile arrays; namely, the fact thatthey possess very low sidelobes and no grating lobes willappear in the radiation patterns when the minimum spacing ischanged from a half-wavelength to at least a full-wavelength.It is also interesting to note that terdragon, 6-terdragon, andfudgeflake fractile arrays are all deterministic examples ofalmost uniformly excited arrays [17].

IV. CONCLUSION

A new class of antenna arrays, which we call fractile arrays,has been introduced in this paper. These fractile arrays arecharacterized by having a fractal boundary contour that tilesthe plane without gaps or overlaps. The unique geometricalproperties of fractiles have been exploited in order to developa deterministic design methodology for modular broadbandlow-sidelobe arrays. The radiation properties of several different


TABLE VCOMPARISON OF MAXIMUM DIRECTIVITY FOR A STAGE 4 6-TERDRAGON FRACTILE ARRAY WITH 211 ELEMENTS AND A 15� 15 SQUARE ARRAY OF

COMPARABLE SIZE WITH 225 ELEMENTS

TABLE VIMAXIMUM DIRECTIVITY FOR SEVERAL DIFFERENT FUDGEFLAKE FRACTILE ARRAYS

TABLE VIICOMPARISON OF MAXIMUM DIRECTIVITY FOR A STAGE 5 FUDGEFLAKE FRACTILE ARRAY WITH 292 ELEMENTS AND AN 18� 18 SQUARE ARRAY

WITH 324 ELEMENTS

fractile arrays have been investigated including Peano–Gosper,terdragon, 6-terdragon, and fudgeflake arrays. Efficient iterativeprocedures for calculating the radiation patterns of these fractilearrays to arbitrary stage of growth have also been developedin this paper.

REFERENCES

[1] J. L. Vehl, E. Lutton, and C. Tricot, Eds., Fractals in Engineering. NewYork: Springer-Verlag, 1997.

[2] D. H. Werner, R. L. Haupt, and P. L. Werner, “Fractal antenna engi-neering: The theory and design of fractal antenna arrays,” IEEE An-tennas Propagat. Mag., vol. 41, no. 5, pp. 37–59, Oct. 1999.


[3] D. H. Werner and R. Mittra, Eds., Frontiers in Electromagnetics. Pis-cataway, NJ: IEEE Press, 2000.

[4] J. P. Gianvittorio and Y. Rahmat-Samii, “Fractal antennas: A novelantenna miniaturization technique, and applications,” IEEE AntennasPropagat. Mag., vol. 44, pp. 20–36, Feb. 2002.

[5] D. H. Werner and S. Ganguly, “An overview of fractal antenna engi-neering research,” IEEE Antennas Propagat. Mag., vol. 45, pp. 38–57,Feb. 2003.

[6] Y. Kim and D. L. Jaggard, “The fractal random array,” Proc. IEEE, vol.74, no. 9, pp. 1278–1280, 1986.

[7] C. P. Baliarda and R. Pous, “Fractal design of multiband and low side-lobe arrays,” IEEE Trans. Antennas Propagat., vol. 44, pp. 730–739,May 1996.

[8] D. L. Jaggard and A. D. Jaggard, “Cantor ring arrays,” Microwave andOpt. Technol. Lett., vol. 19, pp. 121–125, 1998.

[9] D. H. Werner, K. C. Anushko, and P. L. Werner, “The generation of sumand difference patterns using fractal subarrays,” Microwave and Opt.Technol. Lett., vol. 22, no. 1, pp. 54–57, July 1999.

[10] D. H. Werner, W. Kuhirun, and P. L. Werner, “The Peano–Gosper fractalarray,” IEEE Trans. Antennas Propagat., vol. 51, pp. 2063–2072, Aug.2003.

[11] , “A new class of modular broadband arrays based on gosper islandsand associated Peano–Gosper curves,” in Proc. IEEE Int. Antennas andPropagation Symp. and URSI North American Radio Science Meeting,vol. 4, Columbus, OH, June 22–27, 2003, pp. 250–253.

[12] , “A new design methodology for modular broadband arrays basedon fractal tilings,” in Proc. IEEE Topical Conf. Wireless CommunicationTechnology, Honolulu, HI, Oct. 15–17, 2003.

[13] B. B. Mandelbrot, The Fractal Geometry of Nature. New York:Freeman, 1983.

[14] G. A. Edgar, Measure, Topology, and Fractal Geometry. New York:Springer-Verlag, 1990.

[15] B. Grunbaum and G. C. Shephard, Tilings and Patterns. New York: W.H. Freeman and Company, 1987.

[16] D. H. Werner, D. Baldacci, and P. L. Werner, “An efficient recursive pro-cedure for evaluating the impedance matrix of linear and planar fractalarrays,” IEEE Trans. Antennas Propagat., vol. 52, pp. 380–387, Feb.2004.

[17] P. Lopez, J. A. Rodríguez, F. Ares, and E. Moreno, “Low sidelobe levelin almost uniformly excited array,” Inst. Elect. Eng. Electron. Lett., vol.36, no. 24, pp. 1991–1993, Nov. 2000.

Douglas H. Werner (S’81–M’89–SM’94) receivedthe B.S., M.S., and Ph.D. degrees in electrical en-gineering and the M.A. degree in mathematics fromThe Pennsylvania State University (Penn State),University Park, in 1983, 1985, 1989, and 1986,respectively.

He is an Associate Professor in the Department ofElectrical Engineering, Penn State. He is a member ofthe Communications and Space Sciences Lab (CSSL)and is affiliated with the Electromagnetic Communi-cation Research Lab. He is also a Senior Research

Associate in the Electromagnetics and Environmental Effects Department ofthe Applied Research Laboratory at Penn State. He is a former Associate Editorof Radio Science. He has published numerous technical papers and proceedingsarticles and is the author of nine book chapters. He is an Editor of Frontiersin Electromagnetics (Piscataway, NJ: IEEE Press, 2000). He also contributeda chapter for Electromagnetic Optimization by Genetic Algorithms (New York:Wiley Interscience, 1999). His research interests include theoretical and com-putational electromagnetics with applications to antenna theory and design, mi-crowaves, wireless and personal communication systems, electromagnetic waveinteractions with complex media, meta-materials, fractal and knot electrody-namics, and genetic algorithms.

Dr. Werner is a Member of the American Geophysical Union (AGU), Inter-national Scientific Radio Union (URSI) Commissions B and G, the AppliedComputational Electromagnetics Society (ACES), Eta Kappa Nu, Tau Beta Pi,and Sigma Xi. He received the 1993 Applied Computational ElectromagneticsSociety (ACES) Best Paper Award and a 1993 URSI Young Scientist Award.In 1994, he received the Pennsylvania State University Applied Research Lab-oratory Outstanding Publication Award. He received a College of EngineeringPSES Outstanding Research Award and Outstanding Teaching Award in March2000 and March 2002, respectively. He recently received an IEEE Central Penn-sylvania Section Millennium Medal. He has also received several Letters ofCommendation from Penn State’s Department of Electrical Engineering foroutstanding teaching and research. He is an Editor of IEEE ANTENNAS AND


Waroth Kuhirun received the B. Eng. degree from Chulalongkorn University,Thailand, in 1994 and the M.S. and Ph.D. degrees in electrical engineering fromPennsylvania State University, University Park, in 1998 and 2003, respectively.

From 1994 to 1995, he worked at Kasetsart University, Thailand. His researchinterest is in the area of fractal and fractile antenna arrays.

Dr. Kuhirun received a scholarship from the Thai Government for his M.S.and Ph.D. studies.

Pingjuan L. Werner (SM’02) is an Associate Pro-fessor with the Pennsylvania State University Col-lege of Engineering. Her primary research focusesare in the area of electromagnetics, including fractalantenna engineering and the application of genetic al-gorithms in electromagnetics.

Prof. Werner is a Fellow of the Leonhard Center,College of Engineering, The Pennsylvania State Uni-versity, and a Member of Tau Beta Pi National Egi-neering Honor Society, Eta Kappa Nu National Elec-trical Engineering Honor Society, Sigma Xi National

Research Honor Society. She received The Best Paper Award from the AppliedComputational Electromagnetics Society in 1993.


A New Millimeter-Wave Printed Dipole Phased ArrayAntenna Using Microstrip-Fed Coplanar Stripline

Tee JunctionsYoung-Ho Suh, Member, IEEE, and Kai Chang, Fellow, IEEE

Abstract—A new millimeter-wave printed twin dipole phasedarray antenna is developed at Ka band using a new microstrip-fedCPS Tee junction, which does not require any bonding wires,air bridges, or via holes. The phased array used a piezoelectrictransducer (PET) controlled tunable multitransmission line phaseshifter to accomplish a progressive phase shift. A progressivephase shift of 88.8 is achieved with the 5 mm of perturber lengthwhen the PET has full deflection. Measured return loss of thetwin dipole antenna is better than 10 dB from 29.5 to 30.35 GHz.Measured return loss of better than 15 dB is achieved from 30 to31.5 GHz for a 1 8 phased array. The phased array antenna hasa measured antenna gain of 14.4 dBi with 42 beam scanning andhas more than 11 dB side lobe suppression across the scan.

Index Terms—Coplanar stripline (CPS), CPS Tee junction,coplanar transmission lines, dipole antenna, microstrip-to-CPStransition, phase shifter, phased array antenna, piezoelectrictransducer phase shifter, twin dipole antenna.

I. INTRODUCTION

PHASED array antenna systems usually associated withlarge and complex active device networks for phase

shifters, which occupies large portion of the system expenses.Phased array used in military radar system requires low profilefor invisibility against opponents. It also needs to be lightweight especially in the applications of satellite communica-tions. Correspondingly, the demands for low cost, low profile,small size, light weight, and less complicated phased arrayantenna systems are increasing nowadays for both commercialand military applications.

A printed dipole antenna satisfies the benefits of low profile,light weight, low cost and compact size, which is suitable forbuilding phased arrays if proper phase shifters are provided.To construct a printed dipole array, several configurations havebeen proposed. Nesic et al. [1] reported a one-dimensionalprinted dipole antenna array fed by microstrip at 5.2 GHz.Scott [2] introduced a microstrip-fed printed dipole array usinga microstrip-to-coplanar stripline (CPS) balun. In [1] and [2],the balun designs were not easy to match the impedanceand the structures were too big and complicated to build an

Manuscript received August 1, 2002; revised June 17, 2003. This work wassupported in part by the National Science Foundation and NASA Glenn Re-search Center.

Y.-H. Suh was with the Department of Electrical Engineering, Texas A&MUniversity, College Station, TX 77840 USA. He is now with Mimix BroadbandInc., Houston, TX 77099 USA (e-mail: [email protected]).

K. Chang is with Department of Electrical Engineering, Texas A&M Univer-sity, College Station, TX 77840 USA (e-mail: chang@ee. tamu.edu).


Fig. 1. CCPS structure (a) original CPS, (b) CCPS, (c) cross-sectional view atA-A’ with fields distributions of the CCPS for different layers of metallization.

array. In 1998, a wideband microstrip-fed twin dipole antennawas introduced with double-sided structure operating at thefrequency range from 0.61 to 0.96 GHz [3]. Zhu and Wu [4]developed a 3.5 GHz twin dipole antenna fed by a hybridfinite ground coplanar waveguide (FGCPW)/CPS Tee junction.An X-band monolithic integrated twin dipole antenna mixerwas reported in [5] with devices directly integrated into theantenna, so no feeding network was necessary.

In this paper, a new planar printed dipole phased arrayantenna using a tunable phase shifter controlled by PET ispresented at 30 GHz. The phased array antenna uses a new twindipole antenna excited by a microstrip-fed CPS Tee junction[8]. The piezoelectric transducer (PET) controlled phase shifterdoes not require any solid-state devices and their associateddriving circuits. The 1 8 twin dipole phased array antennahas compact size, low loss, low cost, light weight and reducedcomplexity as well as good beam scanning with low side lobelevels.

The PET controlled phase shifter was adopted for the lowcost phased array antenna systems for the first time in [6] and[7]. In this structure, a dielectric perturber controlled by PETis used to introduce a progressive phase shift. The deflectiontakes place at the PET when the proper voltages are applied.Using this property of the PET, a dielectric perturber can haveupward and downward movement according to the appliedvoltages. Consequently, if a transmission line is perturbed bya PET actuated dielectric perturber, its propagation constant

0018-926X/04$20.00 © 2004 IEEE


Fig. 2. Simulated performances comparison at Ka band between conventional and coupled CPS.

will be changed. This phenomenon induces a variable phaseshift along the transmission line controlled by PET. In [6] and[7], an end-fire Vivaldi antenna was used for covering a widebandwidth, and a transition was required to feed antennas.Consequently, the system was large and bulky.

The new printed twin dipole phased array using a mi-crostrip-fed CPS combined with a PET phase shifter provideslow-cost, low-loss, low-profile, compact-size and low-com-plexity with simple antenna feeding.

II. A MICROSTRIP-FED CPS TEE JUNCTION

The twin dipole antenna is fed by a CPS. Since conventionalplanar transmission line is microstrip line, a microstrip-to-CPStransition is needed to feed the dipole. A microstrip-fed CPSTee junction without using bonding wires or air bridges was in-troduced in [8]. In [8], the operating frequency is centered near3.5 GHz with 0.7 dB insertion loss ranged from 2 to 4.15 GHz.The Tee junction utilized novel coupled CPS (CCPS). Thistransmission line can have a physical discontinuity while fieldsare continuous over the whole transmission line using CCPS.

The structure of original CPS and CCPS at 30 GHz is shownin Fig. 1(a) and (b). For the CPS, CCPS, Tee junction andantenna design, IE3D software [9], which uses the method ofmoments, is employed for full wave electromagnetic simulation.A 31 mil RT/Duroid 5870 substrate with a dielectric constantof 2.33 is used for the antenna and feeding network fabrication.

The width ( ) of CPS strip is 0.65 mm and gap ( ) betweenthe strips is 0.5 mm, which has the characteristic impedanceof 202 . This impedance is chosen to match a dipole an-tenna input impedance which will be shown later. As shownin Fig. 1(b), one of the CPS strips is discontinued and isterminated with radial stubs with a rotation angle of 30 anda radius of 0.65 mm for coupling to the bottom layer met-allization. The bottom layer metallization, which is coupledfrom the top layer’s radial stubs, works as a CPS strip shownin Fig. 1(c). The radial stub is used to accomplish the smoothfield transition. The wideband coupling performance of radialstubs has been reported in the microstrip-to CPS-to-microstripback-to-back transition for lower frequency operation [10]. Theback-to-back transition has a measured 3 dB insertion loss overa frequency range from 1.3 to 13.3 GHz (1:10.2) and return loss

is better than 10 dB. The radial stub provides virtual short tothe bottom layer metallization, which depends on the radius ofthe radial stub. Hence, smaller radius of radial stub gives higheroperating frequency with minimal insertion loss and return lossdeteriorations compared to the original CPS configuration.

Performances of CCPS are simulated with IE3D and com-pared with those of conventional CPS as shown in Fig. 2. Thesimulated transmission line length is about 5 mm and the con-ventional CPS has almost zero insertion loss with that shortlength transmission line. Fig. 2 shows that the insertion loss ofCCPS is deteriorated by about 1 dB as compared with that ofconventional CPS for the frequency range from 29.2 GHz to 35GHz and the return loss is better than 10 dB. Insertion loss de-terioration of less than 2 dB covers the wider frequency rangefrom 26.4 GHz to 35 GHz. From the above results, CCPS showsthat fields are continuous all over the transmission line with theaid of radial stub, though a discontinuity is introduced at one ofthe CPS strips.

The structure of microstrip-fed CPS Tee junction at 30 GHzis shown in Fig. 3. The Tee junction has the characteristicimpedance of 202 at each output port 1 and 2. The inputimpedance to the microstrip feed at port 3 is about 101 , whichis half of 202 . Radial stubs effectively rotate the electricfields from parallel to the normal to the substrate to have agood coupling to the bottom metallization, which provides theground of microstrip line.

The Tee junction is simulated with IE3D to verify the per-formance at 30 GHz. Simulated performance of the Tee junc-tion is shown in Fig. 4. The simulated performance shows thatthe Tee junction equally splits the power to each CPS port with1.2 dB insertion loss at 30 GHz. Simulated 2 dB insertion lossbandwidth of the Tee junction is from 27.2 to 34.8 GHz, and thereturn loss is better than 20 dB. Because of high frequency oper-ation bandwidth restriction of the microstrip-to-CPS transitionin [8], the Tee junction is not measured but simulation resultsquite verifies its performances.

III. TWIN DIPOLE ANTENNA USING MICROSTRIP-FED CPSTEE JUNCTION

The structure of the twin dipole antenna is illustrated inFig. 5. The twin dipole antenna utilizes the microstrip-fed CPS

SUH AND CHANG: A NEW MILLIMETER-WAVE PRINTED DIPOLE PHASED ARRAY ANTENNA 2021

Fig. 3. Structure of Ka-band microstrip-fed CPS Tee junction for twin dipole antenna feeding near 30 GHz.

Fig. 4. Simulated performance of the Tee junction near 30 GHz.

Fig. 5. Structure of printed twin dipole antenna using a microstrip-fed CPSTee junction.

Tee junction as discussed in Section II. The antenna is placedin front of a reflector for uni-directional radiation. The reflectoris spaced from the antenna at the distance of 1.5 mm (60 mil),which is about 0.15 . The length of dipole is 5.3 mm or0.53 . The spacing between dipoles was optimized to be0.36 because of an insertion loss increase in CCPS in theTee-junction with a long coupled line such as 0.5 , causing again drop. Mutual coupling normally takes place when antennaspacing is less than a half wavelength. Twin dipole antenna’sinput impedance is supposed to have some reactance due tothis coupling effect. By adjusting the reflector’s spacing, thisreactance can be minimized with a small change in inputimpedance.

The input impedance of a single dipole antenna is around202 . The strip width ( ) and gap ( ) between strips of CCPSat the CPS Tee junction in Section II are determined to have aCCPS characteristic impedance identical to the dipole antennainput impedance for good impedance matching.

Measured return loss of the twin dipole antenna is better than10 dB from 29.5 to 30.35 GHz as shown in Fig. 6. Measuredand simulated return losses have good agreements. For mea-surements, a quarter-wavelength transformer with limited band-width is used and causes small discrepancies between simulateddata and measurements. Radiation patterns of the antenna aremeasured in an anechoic chamber. The measured radiation pat-terns are shown in Fig. 7. and -plane radiation patterns arequite similar to each other for the twin dipole antenna as dis-cussed in [4]. Measured and -plane gains are about 7.6 and7.7 dBi with the 3 dB beamwidths of 32 and 34 , respectively.The measured cross-polarizations at broadside are about 47.7and 42.4 dB down compared with the copolarization levels in

and -plane, respectively. Gains and 3 dB beam widths ofand -planes are quite close to each other. Some discrepanciesof gains and 3 dB beam widths are partly due to the small mis-alignments of the antenna in millimeter-wave frequencies.


Fig. 6. Simulated and measured return loss of the twin dipole antenna.

Fig. 7. Measured radiation patterns of the twin dipole antenna.

IV. PHASED ARRAY ANTENNA WITH MULTITRANSMISSION

LINE PET CONTROLLED PHASE SHIFTER

For the linear phased array, an array factor is a function of theprogressive phase shift and the element spacing . The arrayfactor is given as

(1)

where

(2)

where is expressed as and is beam scanning angle.is the number of elements.The progressive phase shift causes the radiation emitted from

the array to have a constant phase front that is pointing at theangle . This beam scanning angle ( ) is also a function ofand , given by

(3)

The array factor in (1) can also be expressed as (4) below inan alternate, compact and closed form whose function and their

distributions are more recognizable [11]. It is assumed that thereference point is the physical center of the array.

(4)

The total field of array is equal to the field of a single elementpositioned at the origin multiplied by an array factor, which isexpressed as

(5)

From (4), the maximum value of array factor is [11]. Hencemaximum achievable gain of the array can be found from (4)and (5), which is expressed as

(6)

In (6), the effect of mutual coupling between elements isexcluded for the simplicity. Mutual coupling normally degradesarrayed antenna gain. Equation (6) can be used for the gainapproximation of the array. To achieve more accurate calculationincluding mutual coupling effects, a full-wave electromagneticsimulation can be used for antenna array analysis.


Fig. 8. Structure of printed dipole phased array antenna controlled by PET (a) top view and (b) side view.

The structure of 1 8 printed twin dipole phased arrayantenna is shown in Fig. 8. A conventional microstrip powerdivider with binominal impedance transformers is used forfeeding network to cover the wide bandwidth. The bottommetallization provides good ground plane for the microstrip.

To obtain the required phase shift, the 101 microstrip line,which has the same input impedance as the twin dipole antenna,is perturbed with a dielectric perturber actuated by PET. Thelength of dielectric perturber varies linearly from 5 to 35 mmon top of line 2 to line 8 in Fig. 8. The first line is not perturbed.The PET is configured to have no deflection (no perturbation)when a DC voltage of 0 V is applied, and full deflection (fullperturbation) when a DC voltage of 50 V is applied. A 50 milRT/duroid 6010.2 with a dielectric constant of 10.2 is used asthe dielectric perturber.

The amount of phase shift is linearly proportional to thelength of perturber [7], which is expressed as

(7)

where, is the perturber length along the th trans-mission line. represents the differential propagation con-stant expressed as

(8)

where represents the propagation constant of theth perturbed transmission line, which is microstrip in this case.

Since the first perturbed microstrip line (i.e., the second line orline 2) has the minimum perturbed length, the following rela-tionship is obtained.

(9)

Fig. 9. Differential phase shift for 5 mm dielectric perturber controlled by PET.

With a dielectric perturber of 5 mm, Fig. 9 shows that a dif-ferential phase shift of 88.8 takes place with a 2 dB insertionloss. Narrower microstrip line generates larger phase shift butthe insertion loss is increased. Hence, a proper microstrip line’swidth should be chosen for having a good phase shift as well aslow insertion loss.

Table I summarizes the design and measured parameters forthe twin dipole phased array. The parameter values in Table Iare useful in analytical calculations of the scanning angle ( ),maximally achievable gain, and optimum element spacing ( )of the phased array.

According to (7) and (8), the perturber’s length can bedetermined for a desired phase shift. A length of 5 mmdielectric perturber produces about 88.8 differential phaseshift. Accordingly, the length of each neighboring perturbedline is increased by 5 mm. The length of perturber for thefinal microstrip ( ) is about 35 mm, which givesa differential phase shift of 621.6 .


TABLE IPARAMETER VALUES OF THE TWIN DIPOLE PHASED ARRAY

TABLE IICOMPARISON AMONG ANALYTICAL, SIMULATION, AND MEASURED RESULTS OF THE 1� 8 PHASED ARRAY

Fig. 10. Measured return loss of the printed twin dipole phased array antenna.

IE3D analysis shows that a progressive phase shift of 88.8gives around beam scanning with low side lobe levels.An analytical scan angle can also be obtained using (3), andmaximally achievable gain of the phased array can be obtainedfrom (6). The maximum spacing ( ) between elements to avoidgrating lobes is expressed as

(10)

From analytical equations in (3), (6) and (10) and the parametersin Table I, the calculated , maximally achievable gain, andmaximum spacing are calculated to be 19.47 , 16.73 dBi, and7.5 mm, respectively. The results agree very well with IE3Dsimulation as given in Table II.

Measured return loss of the 1 8 twin dipole array isplotted in Fig. 10. The measured return loss is about 41.9 dB at30.3 GHz for the unperturbed twin dipole phased array antenna.With perturbation by the dielectric perturber, the return loss isabout 31.8 dB at 30.7 GHz, which shows a 0.4 GHz frequencyshift compared with the unperturbed result. For a bandwidthfrom 30 to 31.5 GHz, a measured return loss is better than15 dB.

V. PHASED ARRAY MEASUREMENTS

The phased array is measured in an anechoic chamber. Asshown in Fig. 8, the antenna is arrayed for the -plane beamscanning. To accomplish bidirectional beam scanning, two tri-angular perturbers are used side by side [12]. PET actuation forthe dielectric perturber is configured as 0 V for no perturbation(no PET deflection) and 50 V for full perturbation (full PET de-flection). The measured twin dipole phased array antenna gainwithout perturbation (0 V for PET) is about 14.4 dBi with a 3 dBbeam width of 6 as shown in Fig. 11. The fully perturbed an-tenna with a dielectric perturber controlled by PET shows about42 ( ) beam scanning with the gain of 12.2 dBi.Side lobe levels of the steered beam are more than 11 dB downcompared with main beam. The gains of steered beams are about2.2 dB down due to the insertion loss incurred by dielectric per-turbation. The beam can be dynamically steered depending onthe voltages applied to PET because the amount of phase shiftchanges according to the applied voltages on PET as shown inFig. 9.

The comparison among analytical, simulation, and measuredresults of the phased array are exhibited in Table II. Beam scan-ning angle is following closely among analytical, IE3D simula-


Fig. 11. Measured H-plane radiation pattern for twin dipole phased array antenna at 30 GHz. Measured beam scanning is from �20 to +22 with fullperturbation.

tion, and measured results. Measured unperturbed gain is about2.3 dB lower than analytical or IE3D simulated data. This is dueto the insertion loss of power divider and the mutual coupling ef-fects among elements, which normally degrades antenna gain.The measured gains of steered beams are about 2.2 dB downcompared to that of unperturbed beam due to the insertion lossincurred by dielectric perturbation.

VI. CONCLUSION

A new printed twin dipole phased array antenna is developedat 30 GHz using a multitransmission line tunable phase shiftercontrolled by a PET. The new twin dipole antenna is designedusing a microstrip-fed CPS Tee junction. To construct theTee junction, CCPS is used to have a physical discontinuityat CPS while fields are continuous all over the transmissionline. The Tee junction effectively splits power to each CPSoutput port with low insertion loss. The PET actuated phaseshifter requires only one (one-directional beam scanning) ortwo (bi-directional beam scanning) applied voltages to producethe progressive phase shift. A PET controlled phase shifter istested and optimized for the proper phase shift with minimalinsertion loss. The twin dipole phased array antenna shows a42 ( ) beam scanning with more than 11 dBside lobe suppression across the scan. The phased array shouldfind many applications in wireless communications and radarsystems.

ACKNOWLEDGMENT

The authors would like to thank C. Wang of Texas A&M Uni-versity for technical assistance.

REFERENCES

[1] A. Nesic, S. Jovanovic, and V. Brankovic, “Design of printed dipolesnear the third resonance,” in Proc. IEEE Int. Antennas and PropagationSymp. Dig., vol. 2, Atlanta, GA, 1998, pp. 928–931.

[2] M. Scott, “A printed dipole for wide-scanning array application,” inProc. IEEE 11th Int. Conf. Antennas and Propagation, vol. 1, 2001, pp.37–40.

[3] G. A. Evtioushkine, J. W. Kim, and K. S. Han, “Very wideband printeddipole antenna array,” Electron. Lett., vol. 34, no. 24, pp. 2292–2293,Nov. 1998.

[4] L. Zhu and K. Wu, “Model-based characterization of CPS-fed printeddipole for innovative design of uniplanar integrated antenna,” IEEE Mi-crowave and Guided Wave Lett., vol. 9, pp. 342–344, Sept. 1999.

[5] K. L. Deng, C. C. Meng, S. S. Lu, H. D. Lee, and H. Wang, “A fullymonolithic twin dipole antenna mixer on a GaAs substrate,” in Proc.Asia Pacific Microwave Conf. Dig., Sydney, NSW, Australia, 2000, pp.54–57.

[6] T. Y. Yun and K. Chang, “A phased-array antenna using a multi-linephase shifter controlled by a piezoelectric transducer,” in IEEE Int. Mi-crowave Symp. Dig., vol. 2, Boston, MA, 2000, pp. 831–833.

[7] , “A low-cost 8 to 26.5 GHz phased array antenna using a piezoelec-tric transducer controlled phase shifter,” IEEE Trans. Antennas Prop-agat., vol. 49, pp. 1290–1298, Sept. 2001.

[8] Y. H. Suh and K. Chang, “A microsatrip fed coplanar stripline Tee junc-tion using coupled coplanar stripline,” in Proc. IEEE Int. MicrowaveSymp. Dig., vol. 2, Phoenix, AZ, 2001, pp. 611–614.

[9] IE3D, 8.1 ed., Zeland Software Inc., 2001.[10] Y. H. Suh and K. Chang, “A wideband coplanar stripline to microstrip

transition,” IEEE Microwave and Wireless Components Lett., vol. 11,pp. 28–29, Jan. 2001.

[11] C. A. Balanis, Antenna Theory Analysis and Design, 2nd ed. NewYork: Wiley.

[12] T. Y. Yun, C. Wang, P. Zepeda, C. T. Rodenbeck, M. R. Coutant, M. Y.Li, and K. Chang, “A 10- to 21-GHz, low-cost, multifrequency, and full-duplex phased-array antenna system,” IEEE Trans. Antennas Propagat.,vol. 50, pp. 641–650, May 2002.

Young-Ho Suh (S’01–M’02) received the B.S degreein electrical and control engineering from Hong-IkUniversity, Seoul, Korea, in 1992, and the M.S andPh.D. degrees in electrical engineering from TexasA&M University, College Station, TX, in 1998, and2002, respectively.

From 1992 to 1996, he worked for LG-HoneywellCo. Ltd., Seoul, Korea, as a Research Engineer.From 1996 to 1998, he worked on developing robustwireless communication systems for GSM receiverunder multipath fading channel for his M.S degree.

From 1998 to 2002, he was a Research Assistant in the Electromagnetics andMicrowave Laboratory, Department of Electrical Engineering, Texas A&MUniversity, College Station, TX, where he was involved in rectenna designfor wireless power transmissions, phased array antennas, and coplanar trans-mission line circuit components development. In May 2002, he joined MimixBroadband Inc., Houston, TX, as a Senior Microwave Design Engineer, wherehe is working on state-of-the-art microwave/millimeter-wave active circuitdesigns including low noise/power amplifiers, receivers, transmitters, and trans-ceiver modules for LMDS, point-to-point, point-to-multipoint radio systemsin Ka band using GaAs MMICs. His research area includes state-of-the-artmillimeter-wave transceiver modules, transitions between dissimilar transmis-sion lines, uniplanar transmission line analysis and components development,microwave power transmission, antennas for wireless communications, andphased array antennas.


Kai Chang (S’75–M’76–SM’85–F’91) received theB.S.E.E. degree from the National Taiwan Univer-sity, Taipei, Taiwan, R.O.C., the M.S. degree from theState University of New York at Stony Brook, and thePh.D. degree from the University of Michigan, AnnArbor, in 1970, 1972, and 1976, respectively.

From 1972 to 1976, he worked for the MicrowaveSolid-State Circuits Group, Cooley ElectronicsLaboratory, University of Michigan, as a ResearchAssistant. From 1976 to 1978, he was employedby Shared Applications, Inc., Ann Arbor, where he

worked in computer simulation of microwave circuits and microwave tubes.From 1978 to 1981, he worked for the Electron Dynamics Division, HughesAircraft Company, Torrance, CA, where he was involved in the researchand development of millimeter-wave solid-state devices and circuits, powercombiners, oscillators and transmitters. From 1981 to 1985, he worked forthe TRW Electronics and Defense, Redondo Beach, CA, as a Section Head,developing state-of-the-art millimeter-wave integrated circuits and subsystemsincluding mixers, VCOs, transmitters, amplifiers, modulators, upconverters,switches, multipliers, receivers, and transceivers. He joined the ElectricalEngineering Department of Texas A&M University in August 1985 as anAssociate Professor and was promoted to a Professor in 1988. In January 1990,he was appointed E-Systems Endowed Professor of Electrical Engineering.He authored and coauthored several books “Microwave Solid-State Circuitsand Applications” (New York: Wiley, 1994), “Microwave Ring Circuits andAntennas” (New York: Wiley, 1996), “Integrated Active Antennas and SpatialPower Combining” (New York: Wiley, 1996), and “RF and Microwave WirelessSystems” (New York: Wiley, 2000). He served as the editor of the four-volume“Handbook of Microwave and Optical Components” (New York: Wiley, 1989and 1990). He is the Editor of the Microwave and Optical Technology Lettersand the Wiley Book Series in Microwave and Optical Engineering. He haspublished over 350 technical papers and several book chapters in the areas ofmicrowave and millimeter-wave devices, circuits, and antennas. His currentinterests are in microwave and millimeter-wave devices and circuits, microwaveintegrated circuits, integrated antennas, wideband and active antennas, phasedarrays, microwave power transmission, and microwave optical interactions.

Dr. Chang received the Special Achievement Award from TRW in 1984, theHalliburton Professor Award in 1988, the Distinguished Teaching Award in1989, the Distinguished Research Award in 1992, and the TEES Fellow Awardin 1996 from the Texas A&M University.


Physical Limitations of Antennas in a Lossy MediumAnders Karlsson

Abstract—The dissipated power and the directivity of antennasin a homogeneous, lossy medium are systematically analyzed in thispaper. The antennas are ideal and located inside a lossless sphere.In the lossy space outside the sphere, the electromagnetic fields areexpanded in a complete set of vector wave functions. The radia-tion efficiency, the directivity, and the power gain are defined forantennas in a lossy medium, and the optimal values of these quan-tities are derived. Simple relations between the maximal number ofports, or channels, an antenna can use and the optimal directivityand gain of the antenna are presented.

Index Terms—Antenna theory, lossy systems.

I. INTRODUCTION

I N some applications there is a need for wireless communica-tion with devices in lossy materials. A conductive medium

is a low-pass filter for the electromagnetic waves, and one isthen often forced to use low frequencies, or equivalently, longwavelengths. If the space for the antenna is limited it results inan antenna that is small compared to the wavelength. The draw-back is that small antennas in lossy materials consume muchpower, due to the ohmic losses in the near-zone of the antenna.Hence, the design of the antenna and the choice of frequencyare delicate problems, where two power loss mechanisms withcounteracting frequency dependences are involved. This powerproblem is addressed in this paper.

Antennas in lossy materials are found in various areas. In geo-physical applications underground antennas are used, e.g., inbore holes. In marine technology antennas are used for commu-nication with underwater objects. In medical applications thereis an increased usage of wireless communication with implants.Implants, e.g., pacemakers, have limited power supply and it isimportant to use power efficient antennas.

Some of the results in this paper are based on the resultsobtained by Chu [4] and Harrington [8], who investigatedphysical limitations for antennas in free space. Chu derivedthe optimal value of the directivity and the optimal value of theratio between the directivity and the -value of omni-directionalantennas and Harrington derived the corresponding results forgeneral antennas. There are a number of other articles thataddress the optimization of the -value of an antenna, cf. [5],[7], and [12].

For a lossy material it is the dissipated power, rather than the-value, that is the most important quantity in the design of an

antenna. In this paper, the radiation efficiency, the directivity,

Manuscript received May 9, 2003; revised August 6, 2003. This work wassupported by the Competence Center for Circuit Design at Lund University.

The author is with the Department of Electroscience, Lund Institute of Tech-nology, S-221 00 Lund, Sweden (e-mail: [email protected]).


and the power gain of antennas are defined and studied for thesimplified geometry where the antenna is enclosed in a loss-less sphere. The optimal values of these three quantities are themain results in this paper. The optimal value of the directivity isshown to be related to the maximum number of ports, or chan-nels, of the antenna, a result that holds also in a lossless medium.It is emphasized that in a lossy medium the magnetic dipole isthe most radiation efficient antenna, a well known and impor-tant result, cf. [11].

II. PRELIMINARIES

The antennas are confined in a spherical, lossless region,denoted . They are idealized in the sense that there are no

ohmic losses in . The volume is denoted and isan infinite, homogeneous, conducting medium with a complexpermittivity

(2.1)

where the time-dependence is assumed. The correspondingwave number is denoted and is given by

(2.2)

The permeability is assumed to be real. The waveimpedance in reads

(2.3)

III. GENERAL ANTENNAS IN CONDUCTING MEDIA

In the exterior region the electric field is expanded inspherical vector waves , also referred to as partialwaves. These waves satisfy Maxwell’s equations and are com-plete on a spherical surface. The details of the spherical vectorwaves are given in Appendix A. The expansion reads

(3.1)

The corresponding magnetic field is given by the induction law

(3.2)

0018-926X/04$20.00 © 2004 IEEE


where . Here, is the index for the twodifferent wave types (TE and TM), for waves that areeven with respect to the azimuthal angle and for thewaves that are odd with respect to is the indexfor the polar direction, and is the index for theazimuthal angle. For only the partial waves withare nonzero, cf., (A.2). The expansion in (3.1) covers all possibletypes of time harmonic sources inside .

A. Classification

Antennas that radiate partial waves with are referredto as magnetic antennas, since the reactive part of their radiatedcomplex power is positive, i.e., inductive. Antennas radiatingpartial waves with are referred to as electric antennas,since they are capacitive when they are small compared to thewavelength.

The expansion coefficients in the expansion (3.1) cantheoretically be altered independently of each other. Hence,each partial wave corresponds to an independent port of the an-tenna. The maximum number of ports, or channels, an antennacan use is then equal to the maximum number of partial wavesthe antenna can radiate.

The following classification of antennas is used in this paper:

Partial wave antenna—antenna that radiates only onepartial wave . The antenna has one port.Magnetic multipole antenna of order An antenna thatradiates partial waves with and index . The max-imum number of ports is .Electric multipole antenna of order An antenna that ra-diates partial waves with and index . The maximumnumber of ports is .Magnetic antenna of order An antenna that radiatespartial waves with and with . Themaximum number of ports is .Electric antenna of order An antenna that radiatespartial waves with and with . Themaximum number of ports is .Combined antenna of order An antenna that radi-ates partial waves with and . Themaximum number of ports is .

B. Rotation of an Antenna

If an antenna is rotated, the new set of radiated partial wavesis determined by the rotational matrix for the vector waves, cf.[3]. That matrix is diagonal in the index and in the index , butnot in the other two indices and . Thus, a magnetic multipoleantenna of index is still a magnetic multipole antenna of index, after it is rotated. This type of invariance under rotation is

true for all types of antennas in Section III-A, except for thepartial wave antenna. The invariance is utilized in Section IV todetermine the optimal values of the directivity and power gain.A partial wave antenna that radiates the partial wave , isunder a rotation transformed to an antenna that radiates severalpartial waves , where can be both , , and cantake the values .

C. The Power Flow

The complex power radiated from an antenna is givenby

(3.3)

where is the surface is the radial unit vector,and is the complex conjugate of the magnetic field. Thecomplex power is decomposed as

(3.4)

The active part of the power is the power dissipated in theregion , whereas and are the time averages of thestored magnetic and electric energies in the exterior region.

The impedance and admittance of the antenna are relatedto the complex power by the power relation

(3.5)

where and are the complex current and voltage that feedsthe antenna, respectively. The star denotes complex conjugate.For a nonideal antenna the powers inside should be addedto the left-hand side of (3.5).

The complex power radiated from a combined antenna oforder follows from (A.4) and (A.5), and from (3.1)–(3.3)

(3.6)

The complex powers of the other types of antennas in Sec-tion III-A are special cases of (3.6). The normalized complexpower, , of multipole antennas of order dependsonly on the indices and . If the transmitted complex powerof such an antenna is denoted and the correspondingimpedance is denoted then

(3.7)

where is the wave impedance.

D. Asymptotic Values of and

When the asymptotic behavior of the Hankel func-tions, (A.6), implies that the asymptotic values of the radiatedcomplex power, cf., (3.6) and of the impedance, cf., (3.5), are

(3.8)

KARLSSON: PHYSICAL LIMITATIONS OF ANTENNAS IN A LOSSY MEDIUM 2029

As the limiting values of the Hankel functions yield

(3.9)

The asymptotic values in (3.8) and (3.9) are valid for all of theantennas in Section III-A. The values are illustrated in Fig. 1.

In the rest of the pape,r only active power will be consideredand for this reason the term power is understood to mean activepower.

E. Far-Field and Directivity

The far-field amplitude is defined as

(3.10)

The far-field amplitude of a combined antenna of order isgiven by the asymptotic values of the spherical Hankel func-tions, cf., (A.6)

(3.11)

The far-field amplitude of the antenna corresponding to (3.1) isthus completely defined by the coefficients .

The directivity is defined in the same way as for a losslessmedium, cf., [10]. The directivity of the combined antenna isobtained from the far-field amplitude of the antenna and fromthe orthogonality of the vector spherical harmonics, cf., (3.12),as shown in (A.4), at the bottom of the page, where

, and where max is with respect to and .Hence, also the directivity is completely defined by the expan-sion coefficients. The far-field amplitudes and the directivitiesof the other antennas in Section III-A follow from (3.11) and(3.12).

F. Radiation Efficiency and Power Gain

For antennas in a lossless space the radiation efficiency, ,is defined as the ratio of the power radiated from the antenna tothe power put into the antenna. This definition is not applicablehere since the antenna is ideal and hence, the efficiency wouldbe one. A possible alternative definition for an ideal antenna ina lossy material is the quotient , where is thepower radiated from the antenna and is the power radiatedthrough a spherical surface of radius . That ratio indicates how

Fig. 1. Argument of the impedance, arg(Z) of magnetic multipole antennasof order l = 1 (lower dashed line), 2 (middle dashed), and 3 (upper dashed)and for the electric multipole antennas of order l = 1 (upper solid line), 2(middle solid), and 3 (lower solid). The frequency is 400 MHz, " = 50, and� = 1 S/m, corresponding to an argument of the wave impedance arg(�) =0:37 rad. The asymptotic values in (3.8) and (3.9) are reached when a! 0 and1, respectively.

much of the power fed to the antenna is radiated in the far-zone.In the far-zone

(3.13)

for a combined antenna of order , as seen from (3.6) and(A.6). The radiated powers of the other types of antennas arespecial cases of this expression. In order to have a definitionof radiation efficiency that is independent of the radius , thefollowing dimensionless quantity is used

(3.14)

where

(3.15)

The radiated power at a distance from an antenna is expressedin terms of the radiation efficiency and the input power, , as

. The notation is in accordancewith most antenna literature. It should not be confused with thenotation for the wave impedance.

(3.12)


The asymptotic value of for large radii is

(3.16)

which is in agreement with the graph of the efficiency in Fig. 2.The radiation efficiencies of multipole antennas of order aredenoted and according to (3.6), (3.14), and (3.15) theyread as in (3.17), shown at the bottom of the page.

The product of the directivity and the radiation efficiency,, is proportional to the quotient of the maximum power

flow density in the far-zone and the input power to the antenna.It is referred to as the power gain of the antenna and thenotation

(3.18)

is adopted. This definition is in concordance with the powergain of antennas in lossless media, also referred to as themaximum value of the gain, cf. [2]. The notations andare below used for the power gains of magnetic and electricantennas, respectively.

IV. OPTIMIZATION

Optimization of an antenna is in this context to find theamplitudes of the radiated partial waves such that aspecified quantity is optimized. The techniques used by Chuand Harrington, cf. [4] and [8] can be used to derive the optimal(i.e., maximal) values of the directivity and of the power gain,

, of general spherical antennas in a lossy medium. Harringtonshowed that the optimal value of the directivity for a com-bined antenna of order in vacuum is , i.e., half ofthe number of ports for the antenna. That proof holds also forconductive media. For convenience a derivation of the optimaldirectivity, analogous to the one given by Harrington, is givenin Appendix B. The other derivations are left to the reader.Optimization of the radiation efficiency is to minimize thepower fed to an antenna for a given power flow in the far-zone,regardless of the directivity. Optimization of the directivity isto maximize the power flow density in one direction in thefar-zone, for a given total power flow in the far-zone, regardlessof the power fed to the antenna. Optimization of the power gainis to maximize the power flow density in one direction in thefar-zone, for a given power fed to the antenna.

For a lossy material it can be shown from (3.17) that the op-timal value of the radiation efficiency, , for any antenna isthe one obtained for a magnetic dipole. This is seen in Fig. 2.

Fig. 2. Radiation efficiency, � , of six different multipole antennas. Thedashed curves are for the magnetic antennas and the solid curves are for electricantennas. The curves are for l = 1 (upper), l = 2 (middle), and l = 3 (lower).These curves emphasize that the magnetic dipole is the most radiation efficientantenna. When a = 5 mm the magnetic dipole is approximately 10 dB moreefficient than the electric dipole. The frequency is 400 MHz, " = 50, and� = 1 S/m.

In Appendix B it is shown that the optimal directivity of anelectric or magnetic antenna of order is

The corresponding value for a combined antenna of orderis .

For magnetic and electric multipole antennas of order thecorresponding results for the directivities and , respec-tively, are

(4.1)

The relation

(4.2)

is a result of the fact that sets of partial waves of different indexare independent of each other. It is notable that the optimal valueof the directivity of an electric or a magnetic multipole antennaof order one, i.e., a dipole antenna, is 1.5. This value is the sameas the directivity of each partial wave antenna of order one. Forhigher order antennas the directivity of a partial wave antennaof order is always smaller than the maximum directivity of themultipole antenna of order .

(3.17)


The optimal directivity of a general antenna that consists ofa combination of independent partial wave antennas, wherethe maximum order of any of the antennas is , has a lowerand upper bound

Equality is only achieved for a combined antenna of order .Notice also that two times the optimal value of the directivity isan upper bound for the number of independent ports an antennacan have.

Next the optimal power gain is presented. Using thesame method as in Appendix B the optimal values of for amagnetic antenna and an electric antenna of order can bederived

(4.3)

where and are given by (3.17) and (4.1), andis the optimal power gain of a multipole antenna of order . Theoptimal value of the power gain of a combined antenna of order

equals the sum of the optimal gains of the electric and themagnetic antenna of order , i.e.

(4.4)

According to (3.16), the asymptotic values as are

(4.5)

V. NUMERICAL EXAMPLES

From the formulas in this paper it is straightforward towrite short programs that illustrate the difference between theantennas in Section III-A. The three graphs given here are forantennas at 400 MHz, located in a material that is similar tomuscles in a body. The conceivable application is implanteddevices with wireless communication, even though the infinitelossy region is somewhat unrealistic. The conductivityis S/m and the relative permittivity is . InFig. 1 the phase of the impedance of six different multipoleantennas is plotted as a function of the radius . The argumentof the wave impedance of the material in is 0.37 radians.It is seen that the asymptotic values in (3.8) and (3.9) areapproached for large and small values of , respectively. InFig. 2 the radiation efficiency is given as a functionof for the same six multipole antennas. The figure clearlyshows that for a small radius the magnetic dipole is themost efficient antenna. For a radius mm it is morethan 20 dB more efficient than the electric dipole, and 30 dBbetter than the magnetic quadrupole . In Fig. 3, thepower gain is plotted for electric and magnetic antennaswith , and . One always obtains a larger gain

Fig. 3. Power gain G of three magnetic and three electric antennas. Thedashed curves are for the magnetic antennas and the solid curves for the electricantennas. The curves are for l = 1 (lower), l = 2 (middle), and l =

3 (upper). The frequency is 400 MHz, " = 50, and � = 1 S/m.

by adding higher order multipoles, but for small antennas theimprovement compared to the dipole antennas is negligible.Graphs like that in Fig. 3 indicate what order, , one shoulduse for an electric or magnetic antenna. In that way they alsoindicate the number of useful ports of the antenna.

VI. CONCLUSION

The main results in the paper are the optimal values of the ra-diation efficiency, the directivity, and the power gain of antennasconfined in a lossless sphere. Only ideal antennas are treated inthis paper. Real antennas have ohmic losses in the wires that re-duce the radiation efficiency as well as the power gain. However,that power problem is associated with the actual antenna designand is out of the scope of this paper. A comprehensive study ofthe design of antennas in lossy materials is found in [9].

The purpose with the optimal values of the radiation effi-ciency, the directivity, and the power gain is to give the an-tenna designer relative measures and theoretical limitations ofthe properties of antennas. Optimization of the radiation effi-ciency of an antenna is to minimize the dissipated power fora given power flow in the far-zone. The most radiation efficientantenna is the magnetic dipole. The radius of the sphere shouldbe as large as possible.

Optimization of the directivity of an antenna is to maximizethe power flow density in one direction in the far-zone for agiven total power flow in the far-zone. For an electric antennaor magnetic antenna of order the optimal directivity is

and the amplitudes of the radiated partialwaves are given by (B.5). The maximum number of ports the an-tenna can use is twice the optimal directivity. The optimal valueof the directivity is independent of frequency and of the materialin . In theory one can achieve any directivity, even for smallantennas, by a suitable choice of . However, for a small an-tenna the dissipated losses increase very rapidly with andit costs a lot of power to obtain high directivity.


Optimization of the power gain of an antenna is to maximizethe power flow density in one direction for a given input powerto the antenna. For an electric antenna or magnetic antenna oforder the optimal power gain is given by (4.3). The powergain increases with increasing . A graph like that in Fig. 3indicates the most suitable value of .

APPENDIX AVECTOR WAVES

The definition of spherical vector waves can be found in dif-ferent textbooks, e.g., [6] and [8]. In this paper they are definedusing vector spherical harmonics, cf. [1]

(A.1)

The following definition of the spherical harmonics is used:

(A.2)

where and take the values

(A.3)

In the current application the index will never take the value0, since there are no monopole antennas. The vector sphericalharmonics constitute an orthogonal set of vector function on theunit sphere

(A.4)

where the integration is over the unit sphere and where. The outgoing divergence-free spherical vector

waves are defined by (A.5), show at the bottom of the page,where is the spherical Hankel function ofthe second kind. The asymptotic behavior in the far-zone andthe limiting values in the near-zone of the spherical Hankelfunctions are

(A.6)

APPENDIX BOPTIMAL DIRECTIVITY

The optimization problems of finding the maximum value thedirectivity, is a multivariable optimization problem. First as-sume the following function of variables

(B.1)

where are given real numbers and are given positive realnumbers. This function has a maximum when all of its first orderderivatives with respect to are zero. That leadsto the following relations for the variables

(B.2)

The corresponding maximum value of is

(B.3)

Now consider electric antennas and magnetic antennas of orderand let and denote the corresponding directivities.

Without loss of generality the maximum power flow density isassumed to be in some direction given by the spherical angles

and , and in that direction the polarization of the corre-sponding wave is assumed to be parallel to some unit vector .The optimal value of the directivity is independent of the angles

and , and of the vector , due to the invariance under ro-tation described in Section III-B. If , thenand are identified as the real quantities

(B.4)

According to (B.2), the optimal directivity is obtained when

(B.5)

The optimal value of the directivity is given by (B.3)

(B.6)

Next, is expressed in terms of and as. Since is independent of one may integrate

(B.6) in from 0 to . The result is

(B.7)

(A.5)


where . Furthermore, is also independent ofand , and the relation above can be integrated over the unit

sphere. The orthonormality of the vector spherical harmonics,(A.4), results in

(B.8)

This is in accordance with the result in [8].Notice that that (B.5) and (B.6) can be generalized. Assume

a general antenna consisting of independent partial waveantennas that are to be fed so that the directivity function

, cf. [10], is optimized in a prescribed direction andwith a prescribed polarization of the radiated wave. Then aslight modification of (B.5) and (B.6) gives the amplitudes ofthe antennas and the value of the optimal directivity function.It also follows that the mean value of the optimal directivityfunction, with respect to and , is . Hence, the optimalvalue of the directivity is always greater than or equal to .

REFERENCES

[1] G. Arfken, Mathematical Methods for Physicists, 3rd ed. Orlando, FL:Academic Press, 1985.

[2] C. A. Balanis, Antenna Theory, 2nd ed. New York: Wiley, 1997.[3] A. Boström, G. Kristensson, and S. Ström, “Transformation properties

of plane, spherical and cylindrical scalar and vector wave functions,”in Field Representations and Introduction to Scattering, V. V. Varadan,A. Lakhtakia, and V. K. Varadan, Eds, Amsterdam: Elsevier SciencePublishers, 1991, ch. 4, pp. 165–210.

[4] L. J. Chu, “Physical limitations of omni-directional antennas,” Appl.Phys., vol. 19, pp. 1163–1175, 1948.

[5] R. E. Collin, “Minimum Q of small antennas,” J. Elect. Waves Applicat.,vol. 12, pp. 1369–1393, 1998.

[6] J. E. Hansen, Ed., Spherical Near-Field Antenna Measurements. ser.Number 26 in IEE electromagnetic waves series, Stevenage, U.K.: Pere-grinus, 1988, ISBN: 0-86 341-110-X.

[7] R. C. Hansen, “Fundamental limitations in antennas,” Proc. IEEE, vol.69, no. 2, pp. 170–182, 1981.

[8] R. F. Harrington, Time Harmonic Electromagnetic Fields. New York:McGraw-Hill, 1961.

[9] R. W. P. King and G. S. Smith, Antennas in Matter, Cambridge, London,U.K.: MIT Press, 1981.

[10] J. D. Kraus, Antennas, 2nd ed. New York: McGraw-Hill, 1988.[11] H. A. Wheeler, “Fundamental limitations of a small vlf antenna for sub-

marines,” IRE Trans. Antennas Propagat., vol. 6, pp. 123–125, 1958.[12] , “Small antennas,” IEEE Trans. Antennas Propagat., vol. 23, no.

4, pp. 462–469, 1975.

Anders Karlsson was born in 1955, Gothenburg,Sweden. He received the M.Sc. and Ph.D. degreesfrom Chalmers University of Technology, Gothen-burg, Sweden, in 1979 and 1984, respectively.

Since 2000, he has been a Professor at the De-partment of Electroscience, Lund University, Lund,Sweden. His research acivities comprehend scat-tering and propagation of waves, inverse problems,and time-domain methods. Currently, he is involvedin projects concerning propagation of light in blood,wireless communication with implants, and design

of passive components on silicon.


Minimum Norm Mutual Coupling CompensationWith Applications in Direction of

Arrival EstimationC. K. Edwin Lau, Raviraj S. Adve, Senior Member, IEEE, and Tapan K. Sarkar, Fellow, IEEE

Abstract—This paper introduces a new mutual coupling com-pensation method based on the minimum norm solution to an un-derdetermined system of equations. The crucial advantage overprevious techniques is that the formulation is valid independentof the type of antenna element and provides good results in situa-tions where signal strengths vary considerably. In using the matrixpencil algorithm to estimate the directions of arrival, the examplesshow that the proposed method results in significantly lower biasthan the traditional open circuit method. The analysis of mutualcoupling is also applied in the context of a Code Division MultipleAccess communication system.

Index Terms—Code division multiaccess, direction of arrival es-timation, matrix pencil, MUSIC, mutual coupling compensation.

I. INTRODUCTION

D IRECTION of arrival (DOA) estimation is an importantfeature of smart antenna arrays. It could serve as a fun-

damental building block for applications such as space divi-sion multiple access (SDMA) and Enhanced 911 (E911), theproposed wireless emergency service [1]. Several algorithmshave been proposed for DOA estimation, including the popularMUSIC-type techniques, ESPRIT [1] and matrix pencil (MP)[2]–[4]. These signal processing algorithms have been shownto provide accurate estimates, even in moderate signal to noise(SNR) conditions.

The problem is that these signal processing algorithms gen-erally ignore the electromagnetic behavior of the receiving an-tenna. The receiver is assumed to be an ideal, equispaced, lineararray of isotropic point sensors. In this case, the array sam-ples, but does not reradiate the incident signals. Each signalcan be associated with a linear phase front, the slope of whichis directly related to the DOA. Most signal processing tech-niques rely heavily on this assumption. In practice, this idealsituation cannot be met. The elements of the array must be ofsome nonzero size. The elements sample and reradiate the inci-dent fields, causing mutual coupling. Mutual coupling distortsthe linear phase front of the incoming signal, significantly de-grading performance [5]–[7]. Only in the case of a single in-coming signal is the phase front somewhat retained. However,

Manuscript received July 23, 2002; revised September 7, 2003. This researchwas supported by a grant from the Nortel Institute for Telecommunications, Uni-versity of Toronto.

C. K. E. Lau and R. S. Adve are with the University of Toronto, Toronto, ONM5S 3G4, Canada (e-mail: [email protected]).

T. K. Sarkar is with Syracuse University, Syracuse, NY 13244 USA.Digital Object Identifier 10.1109/TAP.2004.832511

for arrays with strong mutual coupling, the phase front is sig-nificantly corrupted and the DOA estimate is inaccurate. Anypractical implementation of DOA estimation therefore requirescompensation for mutual coupling.

Research into compensating for the mutual coupling has beenbased mainly on the idea of using open circuit voltages, firstproposed by Gupta and Ksienski [5]. The authors argue that dueto the lack of a terminal current, the open circuit voltages are freeof mutual coupling. However, as shown in [7], this only reducesthe effects of mutual coupling. The technique presented there ismore effective in suppressing mutual coupling effects [7], [8].

A big drawback with the approaches of [5] and [7], is thatthey are valid for only linear dipoles. The work of [5] isvalid only for a linear array of half wavelength dipoles spacedapart by half a wavelength. The work of [7] is restricted tolinear arrays of linear dipoles, though of arbitrary length andspacing. In this paper we introduce the use of a minimum normtechnique, based on the technique in [7], for general arrays witharbitrary elements. As an aside we also extend the open circuittechnique of [5] to arbitrary arrays. The method of moments(MoM) is used to accurately model the interactions betweenantenna elements. In the minimum norm approach, the MoMadmittance matrix is used to estimate the incident fields, withminimum energy, that would generate the received voltages.Unlike in [9], this technique does not require the solution tothe entire MoM problem. The compensation matrix dependsonly on the MoM admittance matrix and can be calculateda priori to reduce computation load.

In this paper, we use the MP [2] and the popular MUSIC [1],[10] DOA estimation algorithms to compare various compensa-tion methods. Section II presents the model for mutual couplingusing antenna analysis based on the MoM. This eventually leadsto the formulation of minimum norm mutual coupling compen-sation method. Section III presents examples illustrating the per-formance of the open circuit and the minimum norm methodsin case of a equispaced, linear array of dipoles. Section IV endswith some conclusions and a summary of the contributions pre-sented here.

II. MUTUAL COUPLING AND COMPENSATION

Most DOA estimation algorithms including MP and MUSICassume an ideal, linear array of isotropic sensors. Unfortunately,such an ideal sensor is clearly not realizable. A practical antennaarray comprises elements of some physical size. Such elements

0018-926X/04$20.00 © 2004 IEEE

LAU et al.: MINIMUM NORM MUTUAL COUPLING COMPENSATION WITH APPLICATIONS 2035

sample and reradiate incident fields that interact with other ele-ments, i.e., the elements are mutually coupled. Mutual couplingseverely degrades the accuracy of the DOA estimator [6]. Anyimplementation of DOA estimation must account for the mutualcoupling between elements.

In a practical antenna array, the received signals are the volt-ages measured across the load at the port of each element. Todeal with mutual coupling, researchers originally proposed pro-cessing these measured voltages to obtain the open circuit volt-ages, the voltages if all the ports were open circuited [5], [6].Open circuiting the ports reduces the currents on the elements,consequently the reradiated fields and therefore the mutual cou-pling. However, as shown in [7], this methodology is valid onlywhen all signals have similar strengths. In [7], we use a MoManalysis to compensate for mutual coupling. That technique isvery effective, but is valid only for a linear array of paralleldipoles.

We present here a technique that is theoretically valid for allkinds of arrays. Based on a minimum norm solution to an un-determined system of MoM equations, the technique makes noassumptions regarding the type of antenna, or the spacing be-tween elements. However, for simplicity, this methodology ispresented here for a linear array of dipoles. We begin with abrief review of the analysis technique, as the MoM analysis fordipole arrays is well known [7], [11]. The review included heresets the stage for the minimum norm solution.

A. System Model

We begin with the general formulation of the MoM basedon subdomain basis functions for a receiving antenna array of

-elements. The central assumption is that only a single basisfunction contributes to the current at the port of each element inthe array. The incident electric field is related to the currents onthe antenna through a linear operator [12]

(1)

The current is approximated by a set of subdomain basis func-tions, , with basis functions per element, i.e.

(2)

where is the th current coefficient. Using a set of testingfunctions, , and a convenient definition of inner product, (1)can be reduced to a matrix equation

(3)

(4)

where the th element of and the th element ofthe matrix are

(5)

(6)

The matrix is the matrix with zero en-tries other than the diagonal entries, where the th

Fig. 1. Linear array of wire dipoles terminated in loads Z .

entry corresponds to the port. This th entry is the loadimpedance of the corresponding element. The matrix isthe MoM impedance matrix. Assuming a single basis functioncorresponds to the current at a port of the array, from (4), themeasured voltages, affected by mutual coupling are given by

(7)

The matrix is the submatrix of corre-sponding to the ports. , a compressed version of , isthe diagonal matrix of port impedances. is amatrix of dimensionless entries. Note that the entries ofare directly related to the incident fields and are free of mutualcoupling.

In this paper, this general formulation is applied to a lineararray of dipoles. It must be emphasized that this choice is notfundamental to the theoretical development here and is madeonly for purposes of illustration. Consider a wire dipole antennaarray of -directed elements as shown in Fig. 1. Each elementhas a centrally located port terminated in impedance . To an-alyze this array we use sinusoidal basis functions. Each elementis divided into segments of equal length. To satisfy therequirement that only a single basis function corresponds to thecurrent on the array is chosen to be odd. Based on a Galerkinformulation, the weighting and testing functions are the same.The entries for the MoM voltage and impedance matrices areavailable in [7], [11].

B. Open Circuit Voltages

The principal idea of [5] is to use the open circuit voltagesinstead of the measured voltages for further signal processing.However, the theory is valid only for half wavelength dipoleswith half wavelength spacing. In the more general case, one canuse the MoM analysis in conjunction with the Thevenin andNorton equivalent circuits to obtain the open circuit voltages.Define the MoM admittance matrix to be the inverse ofthe impedance matrix . Note that this is not the same ma-trix, , in (4). Also define a new matrix whoseentries are those rows and columns of the MoM admittance ma-trix that correspond to ports, i.e.

(8)

(9)

(10)


Fig. 2. Example 3.1. MP, using uncompensated voltages.

The open circuit voltages are then related to the short circuitcurrents as

(11)

and the measured voltages to the short circuit currents as

(12)

Eliminating the short circuit currents from (11) and (12) yieldsthe open circuit voltages

(13)

In the following sections the open circuit voltages we refer toare obtained from the measured voltages using (13).

C. Minimum Norm Compensation Formulation

As shown in [7], using the open circuit voltages only some-what reduces the effects of mutual coupling. In [7], we recon-struct a part of the MoM voltage vector under the assumption ofa linear dipole array. The motivation comes from the fact that,from (5), the MoM voltages are directly related to the incidentfields and so are free of mutual coupling.

In the general case, from (7), the equation relating the mea-sured and MoM voltages is underdetermined and thecannot be reconstructed exactly. However, one can find the min-imum norm solution to this equation. This solution providesthe vector with the minimum two-norm (minimum energy) thatwould result in the received voltages. The resulting vector is anestimate of the MoM voltage vector. Using (7), the minimumnorm solution to the MoM voltage vector is

(14)

where is the conjugate transpose (Hermitian) of matrix .Entries in corresponding to the ports may be used forfurther signal processing.

Fig. 3. Example 3.1. MP, using open circuit voltages.

Fig. 4. Example 3.1. MP, using minimum norm compensation.

Physically, the compensation procedure may be interpretedas finding the signal with minimum energy that results in themeasured voltages. Since the MoM analysis and so the matrix

may be obtained a priori, the computation load to use (14)is no greater than finding the open circuit voltages or using thetechnique of [7]. In the following section, we compare the per-formance of the two compensation methods in various settings.

III. NUMERICAL EXAMPLES

In this section, we present numerical examples to illustrate theworkings of the two compensation techniques, the open circuitand minimum norm methods. The application here is DOA es-timation. The first two examples deal with the DOA estimationof multiple signals and demonstrates the impact of mutual cou-pling and compares the two compensation techniques. The thirdexample deals with the impact of mutual coupling on code di-vision multiple access (CDMA) communications in particular,and the effectiveness of mutual coupling compensation in this


TABLE ICOMPARING OPEN CIRCUIT AND MINIMUM NORM TECHNIQUES. EQUAL SIGNAL STRENGTHS

case. Due to mutual coupling, the signal level at each elementmay be different. The SNR is defined here as the average SNRat all ports of the array, i.e., in adding white, complex Gaussiannoise at each element, the power level is chosen to set an av-erage SNR. In all examples using MP, the pencil parameter isset to .

A. Three Signals of Equal Strength

This example uses a seven element array with interelementspacing of . The MoM analysis uses 7 unknowns per ele-ment, i.e., a total of 49 unknowns are used. The array receivesthree signals from 40 , 90 and 140 . Each signal has a SNR of1 dB. The MP algorithm uses only a single snapshot. The plotsshown here use the results of 1000 independent trials.

Fig. 2 shows a histogram of the results of using MP withoutany compensation for mutual coupling. 38 times, the estima-tion procedure fails completely by resulting in imaginary an-gles. This happens because MP estimates the complex phase

before estimating the direction . In 38 instances,the argument to the function becomes greater than 1.As is clearly seen in the figure, the DOA estimation is very poorwith very large errors.

Figs. 3 and 4 plots the performance after compensation formutual coupling. Fig. 3 plots the use of open circuit voltageswhile Fig. 4 plots the results of using the minimum norm tech-nique. In both figures, the hugely improved performance overthe uncompensated case is very clear. Neither technique resultsin any imaginary angles. Note that because of the accurate per-formance, we can estimate a standard deviation, which for allcases is approximately 3.5 .

As Table I shows, the crucial difference between the twocompensation techniques is in the bias. The bias resultingfrom using the minimum norm compensation approach issignificantly smaller than using the open circuit voltages. Thisis because using the open circuit voltages only implies the lackof a terminal current. Physically, there is still a nonzero currenton the dipole arms. These currents reradiate, resulting in someresidual mutual coupling.

Fig. 5 explains the improved performance of the minimumnorm technique over the open circuit approach. The figureplots of the phase front of the three incoming signals in thevarious scenarios of this example. It plots the phase at eachelement in the ideal case, in the case of no mutual couplingcompensation, using the open circuit approach and using theminimum norm solution. Without compensation, the phaseinformation is significantly corrupted, explaining the erroneousresults. Both compensation techniques correct this somewhat.However, clearly the minimum norm solution is better than

Fig. 5. Example 3.1. Phase front of three incoming signals.

Fig. 6. Example 3.2. MP, using uncompensated voltages.

using the open circuit voltages. This explains, from the phasepoint of view, why the two compensation methods work andwhy the proposed approach is better than the traditional opencircuit approach.

B. Three Signals of Unequal Strength

In this example we use the same array as in the first examplewith the the signal bearings at 40 , 70 , and 140 , with SNR’s of7, 15, and 5 dB respectively. We use 10 000 independent trials.


Fig. 7. Example 3.2. MP, using open circuit voltages.

Fig. 8. Example 3.2. MP, using minimum norm compensation.

Fig. 6 shows a histogram of MP estimate without any mutualcoupling compensation. 2269 estimates result in imaginary an-gles. Clearly the remaining estimates are not of any practicaluse.

Figs. 7 and 8 are results of using MP compensated with theopen circuit and minimum norm approaches respectively. Bothcompensation methods improve the estimation dramatically. Allthe imaginary angles are recovered. Similar to the previous ex-ample, the open circuit method exhibits a larger bias than theminimum norm approach. The bias is even stronger in this ex-ample than the last one as the signal at 70 is relatively strongand closer to the 40 signal.

Fig. 9 shows the pseudo-spectrum generated by MUSICwithout compensation, using the open circuit voltages and withminimum norm compensation. In all cases, 15 time samplesare used to estimate the covariance matrix. As can be seen,with either compensation technique, the resolution improvesand the bias is reduced. Again, the bias in the estimation isless with minimum norm method than that with open circuitmethod. This is in agreement with the examples presented

Fig. 9. Example 3.2. MUSIC.

Fig. 10. Example 3.2. MUSIC. The middle signal is at 60 .

for the MP algorithm. If the signal at 70 is moved to 60 ,as shown in Fig. 10, the results are even more dramatic. If nocompensation is used, the signals at 40 and 60 merge. Butafter the compensation, the two spikes are recovered. Again,using the open circuit voltages results in a greater bias than theminimum norm method.

Figs. 11 and 12 show the results if the strength of the signal at70 is increased to 25 dB. The results are shown in . We can seefrom the figures that the bias in not significant when using theminimum norm method in Fig. 12. When using the open circuitmethod, the bias in the weaker signals is 2 and 3.7 . Table IIsummarizes our statistical findings of this example.

C. Mutual Coupling Compensation in CDMA Communications

One motivation for this research is position location inwireless communication systems. Here we focus on a CDMAsystem. In applying the MP technique to a practical array in aCDMA based communication setting, a curious fact emerges.


Fig. 11. Example 3.2. MP, using open circuit voltages. Signal at 70 is 25 dB.

The CDMA processing gain provides some resistance to mu-tual coupling. This is because, after the matched filter, thereis effectively only one signal plus relatively weak residualinterference. With only one signal impinging on the array, thelinear phase front is not fatally corrupted and it is possibleto estimate the DOA. This is true particularly of arrays withmoderate mutual coupling.

To illustrate this effect, we use the same example as in Sec-tion III-A. However, each signal is spread with a spreading gainof 128. We use four signal samples per chip. For a fair compar-ison, the power of each signal is reduced by the spreading gain.Using the filter matched to the first signal, two of three signalsare suppressed. Note that in using MP to estimate the DOA ofthis signal after the matched filter, we set the number of signalsto one, i.e., . This also eliminates a drawback associatedwith MP, the restriction on the number of signals that can beestimated simultaneously [13]. MP is applied without compen-sating for mutual coupling.

Fig. 13 plots the histogram of the resulting estimates. Incomparison to Fig. 2 the accuracy is dramatically improved.No estimate results in imaginary angles. In fact, the accuracyis comparable to using the open circuit voltages as in Fig. 3.

It must be emphasized that this resistance to mutual couplingis only an approximation. Depending on the accuracy required,compensation for mutual coupling can still play an importantrole. Fig. 14 plots the results of using the minimum norm ap-proach. The performance is improved with significantly reducedbias.

IV. CONCLUSION

Practical implementations of DOA estimation must deal withthe problem of mutual coupling between antenna elements. Thework of [7] introduced the concept of reconstructing a part ofthe MoM voltage vector. We extend this concept here and de-velop a very effective technique based on the minimum normsolution to an underdetermined system of equations. The ap-proach is to find the signals, with minimum energy, that would

Fig. 12. Example 3.2. MP, using minimum norm compensation. Signal at 70is 25 dB.

Fig. 13. Example 3.3. CDMA/MP, using uncompensated voltages.

create the mutually coupled measured signals. The overhead as-sociated with the compensation procedure is limited to a matrixmultiplication.

In testing the proposed approach, the technique proves to bemore accurate than the classical open circuit approach. The min-imum norm technique reduces the bias in the estimates becausethe phase response is reconstructed more accurately than whenusing open circuit voltages.

In applying DOA estimation specifically to CDMA com-munications a curious fact emerges. If DOA estimation isapplied after the matched filter, the CDMA spreading gain re-sults in the desired signal plus residual interference. The phasefront of a single signal is not significantly corrupted and sothe resulting DOA estimation, without compensation is fairlyaccurate. However, one cannot conclude that mutual couplingcompensation is not required. Applying compensation furtherimproves performance. Since the additional cost is restricted toa matrix multiplication, the resulting performance gains wouldprobably outweigh the cost of implementing mutual couplingcompensation.


TABLE IICOMPARING OPEN CIRCUIT AND MINIMUM NORM TECHNIQUES. UNEQUAL SIGNAL STRENGTHS.

Fig. 14. Example 3.3. CDMA/MP, using minimum norm compensation.

In summary, we have presented a practical and accurate min-imum norm mutual coupling compensation method. The newapproach proves to more accurate than the traditional open cir-cuit approach. This method can theoretically also be applied toarrays of arbitrary elements.

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[3] J. E. F. del Rio and T. K. Sarkar, “Comparison between the matrix pencilmethod and the Fourier transform for high-resolution spectral estima-tion,” Digital Signal Processing: A Review Journal, vol. 6, pp. 108–125,1996.

[4] R. S. Adve, O. M. Pereira-Filho, T. K. Sarkar, and S. M. Rao, “Extrapo-lation of time domain responses from three dimensional objects utilizingthe matrix pencil technique,” IEEE Trans. Antennas Propagat., vol. 45,pp. 147–156, Jan. 1997.

[5] I. J. Gupta and A. A. Ksienski, “Effect of mutual coupling on the per-formance of adaptive array,” IEEE Trans. Antennas Propagat., vol. 31,pp. 785–91, Sept. 1983.

[6] C.-C. Yeh, M.-L. Leou, and D. R. Ucci, “Bearing estimations with mu-tual coupling present,” IEEE Trans. Antennas Propagat., vol. 37, pp.1332–5, Oct. 1989.

[7] R. S. Adve and T. K. Sarkar, “Compensation for the effects of mu-tual coupling on direct data domain algorithms,” IEEE Trans. AntennasPropagat., vol. 48, pp. 86–94, Jan. 2000.

[8] M. Ali and P. Wahid, “Analysis of mutual coupling effect in adaptivearray antennas,” in Proc. IEEE Antennas and Propagation Soc. Int.Symp., June 2002.

[9] K. M. Pasala and E. M. Friel, “Mutual coupling effects and their reduc-tion in wideband direction of arrival estimation,” IEEE Trans. Aerospaceand Electron. Syst., vol. 30, pp. 1116–1122, Apr. 1994.

[10] R. O. Schmidt, “Multiple emitter location and signal parameter estima-tion,” IEEE Trans. Antennas Propagat., vol. 34, pp. 276–290, Mar. 1986.

[11] B. J. Strait, T. K. Sarkar, and D. C. Kuo, “Special programs foranalysis of radiation by wire antennas,” Syracuse Univ., Tech. Rep.AFCRL-TR-73-0399, 1973.

[12] R. F. Harrington, Field Computation by Moment Methods. Melbourne,FL: Kreiger, 1982.

[13] C. K. E. Lau, R. S. Adve, and T. K. Sarkar, “Combined CDMA andmatrix pencil direction of arrival estimation,” in Proc. IEEE VehicularTechnology Conf., 2002, pp. 496–499.

Edwin C. K. Lau received the B.A.Sc. and M.A.Sc.degrees, both in electrical engineering, from the Uni-versity of Toronto, Toronto, ON, Canada, in 2000 and2003, respectively.

He is one of the participants of the Communica-tions and Information Technology Ontario (CITO)Research Partnerships Program. His area of researchincludes retrodirective antennas, microwave circuitand antenna design, and direction of arrival estima-tion algorithm.

Raviraj S. Adve (S’88–M’97–SM’03) received theB.Tech. from the Indian Institute of Technology,Bombay, in 1990 and the Ph.D. degree from SyracuseUniversity, Syracuse, NY, in 1996, all in electricalengineering. His dissertation, on the impact ofmutual coupling on the performance of adaptiveantenna arrays, received the Syracuse University“Outstanding Dissertation Award” in 1997.

From 1997 to August 2000, he was a SeniorResearch Engineer with Research Associates for De-fense Conversion (RADC) Inc., Marcy, NY, working

on contract with the Air Force Research Laboratory, Sensors Directorate,Signal Processing Branch, Rome, NY. He is currently an Assistant Professorin the Department of Electrical and Computer Engineering, University ofToronto. He has also been a consultant to Stiefvater Consultants. His researchinterests are in practical adaptive signal processing algorithms for wirelesscommunication and airborne radar systems.


Tapan K. Sarkar (S’69–M’76–SM’81–F’92) re-ceived the B.Tech. degree from the Indian Instituteof Technology, Kharagpur, in 1969, the M.Sc.E.degree from the University of New Brunswick,Fredericton, NB, Canada, in 1971, and the M.S. andPh.D. degrees from Syracuse University, Syracuse,NY, in 1975.

From 1975 to 1976, he was with the TACO Divi-sion, General Instruments Corporation. He was withthe Rochester Institute of Technology, Rochester, NY,from 1976 to 1985. He was a Research Fellow at the

Gordon McKay Laboratory, Harvard University, Cambridge, MA, from 1977to 1978. He is now a Professor in the Department of Electrical and ComputerEngineering, Syracuse University. He has authored or coauthored more than210 journal articles and numerous conference papers and has written 28 chap-ters in books and ten books, including his most recent, Iterative and Self Adap-tive Finite-Elements in Electromagnetic Modeling (Boston, MA: Artech House,1998). His current research interests deal with numerical solutions of operatorequations arising in electromagnetics and signal processing with application tosystem design.

Dr. Sarkar is a Registered Professional Engineer in the State of New York. Heis a member of Sigma Xi and the International Union of Radio Science Com-missions A and B. He received one of the ”best solution” awards in May 1977at the Rome Air Development Center (RADC) Spectral Estimation Workshop.He received the Best Paper Award of the IEEE Transactions on Electromag-netic Compatibility in 1979 and in the 1997 National Radar Conference. Hereceived the College of Engineering Research Award in 1996 and the Chan-cellor’s Citation for Excellence in Research in 1998 at Syracuse University. Hereceived the title Docteur Honoris Causa from Universite Blaise Pascal, Cler-mont Ferrand, France in 1998 and the medal of the city of Clermont Ferrand,France, in 2000. He was an Associate Editor for feature articles of the IEEE An-tennas and Propagation Society Newsletter, and he was the Technical ProgramChairman for the 1988 IEEE Antennas and Propagation Society InternationalSymposium and URSI Radio Science Meeting. He is on the editorial board ofJournal of Electromagnetic Waves and Applications and Microwave and OpticalTechnology Letters. He has been appointed a U.S. Research Council Represen-tative to many URSI General Assemblies. He was the Chairman of the Inter-commission Working Group of International URSI on Time Domain Metrologyfrom 1990 to 1996.


A Phase-Space Beam Summation Formulation forUltrawide-Band Radiation

Amir Shlivinski, Member, IEEE, Ehud Heyman, Fellow, IEEE, Amir Boag, Senior Member, IEEE, andChristine Letrou, Member, IEEE

Abstract—A new discrete phase space Gaussian beam (GB)summation representation for ultrawide-band (UWB) radiationfrom an aperture source distribution is presented. The formula-tion is based on the theory of the windowed Fourier transform(WFT) frames, wherein we introduce a novel relation between thefrequency and the frame overcompleteness. With this procedure,the discrete lattice of beams that are emitted by the aperturesatisfies the main requirement of being frequency independent,so that only a single set of beams needs to be traced through themedium for all the frequencies in the band. It is also shown thata properly tuned class of iso-diffracting (ID) Gaussian-windowsprovides the “snuggest” frame representation for all frequen-cies, thus generating stable and localized expansion coefficients.Furthermore, due to the ID property, the resulting GBs prop-agators are fully described by frequency independent matriceswhose calculation in the ambient environment need to be doneonly once for all frequencies. Consequently, the theory may alsobe expressed directly in the time-domain as will be presentedelsewhere. The localization implied by the new formulation isdemonstrated numerically for an UWB focused aperture. It isshown that the algorithm extracts the local radiation propertiesof the aperture source and enhances only those beams that con-form with these properties, i.e., those residing near the phasespace Lagrange manifold. Further localization is due to the factthe algorithm accounts only for beams that pass within a fewbeamwidths vicinity of the observation point. It is thus shownthat the total number of beams is much smaller than the LandauPollak bound on the aperture’s degrees of freedom.

Index Terms—Beam summation representations, frame theory,Gaussian beams (GBs), phase space, ultrawide-band (UWB), win-dowed Fourier transform (WFT).

NOMENCLATURE

GB Gaussian beam.ID isodiffracting.UWB ultrawide-band.WFT windowed Fourier transform.

Manuscript received November 21, 2001; revised June 12, 2003. The work ofE. Heyman was supported in part by the Israel Science Foundation under Grant216/02 and in part by the Air Force Office of Scientific Research (AFOSR) underGrant F49620-01-C-0018. The work of A. Boag was supported in part by theIsrael Science Foundation under Grant 577/00.

A. Shlivinski was with the School of Electrical Engineering, Tel Aviv Uni-versity, Tel Aviv 69978, Israel. He is now with the Department of ElectricalEngineering, University of Kassel, 34109 Kassel, Germany.

E. Heyman and A. Boag are with the School of Electrical Engineering TelAviv University, Tel Aviv 69978, Israel.

C. Letrou is with GET/INT, CNRS SAMOVAR, UMR 5157, France.Digital Object Identifier 10.1109/TAP.2004.832513

I. INTRODUCTION

B EAM based phase-space formulations are an importanttool in the wave theory since they provide a systematic

framework for ray-based construction of spectrally uniformlocal solutions in complex configurations [1]–[3]. In these for-mulations, the field is expanded into a phase-space spectrum ofbeams that emanate at a given set of points and directions in thesource domain, and thereafter are tracked locally in the medium(cf. Fig. 1). The advantages of the beam formulations over themore traditional representations are: 1) unlike the plane waves,the beam propagators can be tracked locally in inhomogeneousmedia or through interactions with interfaces, and unlike rays,they are insensitive to the geometrical optics (GOs) transitionzones; 2) the formulations are a priori localized in the vicinityof the phase-space skeleton of GOs (the so-called Lagrangemanifold; see Section IV-C) since only those beam propagatorsthat pass near the observation point actually contribute there.Thus, beam representations combine the algorithmical ease ofGOs with the uniform features of spectral representations, andtherefore have been used recently in various applications [4].

An important property of these formulations is that the spec-trum of beam propagators is overcomplete and thus may be apriori discretized as, for example, in the Gabor series repre-sentation. This attractive feature has led to the utilization ofGabor-based beam algorithms in various applications involvingradiation, scattering and inverse scattering in complex environ-ments [1], [5]–[12].

The Gabor representation is critically complete, i.e., thephase-space grid is constrained by the Gabor condition

, where and are the step sizes of the spatial andspectral discretization [see (9)]. Consequently, it suffers fromtwo inherent difficulties:

a) Nonlocality and instability of the expansion coefficients:The “analysis function” (the dual or biorthogonal func-tion), which is used to calculate the expansion coeffi-cients tends to be highly nonlocal and irregular, and incertain cases it is even not integrable (see Fig. 3(f)and [13]–[17]), hence, the Gabor coefficients represent anonlocal and unstable sampling of the field [1], [3].

b) Frequency-dependent beam lattice: The beam latticeobtained in the Gabor representation changes with fre-quency [18], hence a different set of beams needs to betracked for each frequency (as opposed to the situation inFig. 1). This difficulty stems from the Gabor conditionon the phase space grid: choosing to be constant for allfrequencies (i.e., constant beam initiation points) resultsin frequency dependent beam directions, and vice versa[see discussion accompanying (20), (21)].

0018-926X/04$20.00 © 2004 IEEE

SHLIVINSKI et al.: PHASE-SPACE BEAM SUMMATION FORMULATION FOR ULTRAWIDE-BAND RADIATION 2043

Fig. 1. Discrete phase space beam summation representation for widebandradiation from an extended aperture source.

Difficulty (b) makes the conventional Gabor-based beam formu-lation inapplicable for UWB applications. Difficulty (a) makesit inconvenient even for monochromatic fields.

For monochromatic applications, difficulty (a) has been cir-cumvented recently by using a frame-based beam summationrepresentation [19], [20]. The overcomplete nature of this rep-resentation smoothes out and localizes the dual function, endingup with stable and local coefficients at the expense of having tocalculate more coefficients and trace more beam propagators.This poses a tradeoff in the choice of the oversampling ratioversus the stability and localization of the representation. A rea-sonable solution has been found at an oversampling of order 4/3or larger for one dimensional (1-D) apertures.

In this paper, we introduce a novel scheme wherein theframe formulation accommodates the difficulty under (b) forUWB fields [21], [22]. The scheme is based on the fact that theovercomplete frame removes the Gabor constraint and henceby a proper scaling of the overcompletness with the frequency(see Section III-A) one may construct a frequency independentbeam lattice, so that the same set of beams is used for theentire relevant frequency spectrum as schematized in Fig. 1. InSection IV, it is shown further that the iso-diffracting Gaussianbeams (ID-GBs) provide the “snuggest” frame basis for allfrequencies (thus providing local and stable coefficients). Thesewindows, which in fact have been introduced in a differentcontext [23]–[26], also simplify the beam calculations since,the resulting ID-GB propagators are fully described by fre-quency independent matrices whose calculation in the ambientenvironment need to be done only once for all frequencies.Consequently, these beams can be transformed in closed forminto the time domain, where they describe the so-called IDpulsed beams [23]–[28]. Based on this property, we have alsointroduced in [29] a new discrete phase space beam summationrepresentation for short-pulse fields directly in the time domain(full papers will be published elsewhere).

The paper is organized as follows: Section II reviews therelevant elements of frame theory, starting with general frames(Section II-A) and then concentrating on the windowed Fourier

Fig. 2. Coverage of an overcomplete phase space grid with � = 0:5 using theGaussian window (14) with � = 0:5 ; 1; and 2 (dotted, solid, and dash-dottedlines, respectively). Heavy dots: Grid points.

transform (WFT) frames that form the basis of the beam sum-mation representation (Sections II-B and II-C). Using analysisand numerical examples, we identify the relevant frame pa-rameters for a snug and stable representation. The UWB beamsummation representation is presented in Section III, startingwith the formulation of the frequency independent beam lattice(Section III-A) followed by the parameterization of the ID-GBwindows for an UWB snuggest representation and a thoroughanalysis of the algorithm (Section IV). The localization andfrequency independence issues implied by the new formula-tion are demonstrated numerically in Section V for an UWBfocused aperture. It is shown that the algorithm extracts thelocal radiation properties of the aperture source and enhancesonly those beams that conform with these properties, i.e., thoseresiding near the phase space Lagrange manifold. The pre-sentation concludes in Section VI with an extensive summaryof the algorithm, the various considerations in choosing theexpansion parameters.

II. ELEMENTS OF FRAME FORMULATIONS

This section presents a brief review of the relevant theory offrames, starting in Section II-A with general frames and thenproceeding in Section II-B with a detailed analysis of the WFTframes that form the basis of our beam summation representa-tion. Extensive treatments of frame theory can be found in [16],[17], yet the results are presented here with a new slant that ismore relevant to our UWB wave analysis.

A. Frames, Dual Frames and Frame Representation of Signals

The theory of frames has been introduced originally in [30],but it has gained renewed interest recently since it provides aframework for an advanced phase-space signal processing [16],[17].

Definition [17, Sec. 3.2]: A family of functions , ,in a Hilbert space is called a frame if there exist “framebounds” such that for all

(1)

In the context of this paper, we shall be interested in the functionspace with the inner product for

, where the asterisk denotes complex conjugate. For the


Fig. 3. Exact and approximate dual functions '(x) and ' (x) (solid and dotted lines) corresponding to the Gaussian window (14) for 6 different values of � .In all cases, the phase space grid has been chosen to match to the window according to (16), and the unit cell boundaries at��x=2 are depicted as two vertical lines.

2-D aperture distributions considered later on in this paper, theframe will be obtained by a Cartesian multiplication of the 1-Dframes in each of the Cartesian coordinates.

Generally, the set is overcomplete, thus, it is not orthog-onal and not even a basis. It does not have to be normalized andthe bounds and may depend on the relative magnitude ofthe elements in .

The frame operator is defined as

(2)

Clearly is self adjoint. Rephrasing (1) in operator conventions,yields the bounds on ,

[17, Sec. 3.2], where is the identity operator. Sincehas a lower bound , it has an inverse. Applying to

, yields the set

(3)

which is also a frame with bounds , and frame operator[17, Proposition 3.2.3]. is termed the dual frame andis the dual frame operator.

A frame representation for is given by [17, eq. 3.2.8]

(4)

with a dual expression using . It should be noted thatthe frame representation (4) is not unique, i.e., there are othersets of coefficients that can be used in (4) instead ofand still express , yet the latter minimizes the norm of thecoefficient series [17, Proposition 3.2.4].

The frame representation requires a choice of an appropriateframe for a given application, and a calculation of itsdual via (3). The computations involve the inversion of

, which can be performed via several methods, e.g., an iter-ative Neumann series procedure [17, Sec. 3.2] or a projectionof on a finite Hilbert space (e.g., a set of sufficiently densesampling points) discussed in the Appendix . The frame calcu-lations to be presented below have been performed via the latterapproach, yet, we shall briefly comment on the former since itexplains some properties of frames, which will be used in ourUWB beam representation of Sections III and IV.

Recasting as where andis a constant, leads to a Neumann series expansion of

(5)

This series converges if . Since from (1), it follows that the series in (5) con-

verges if [31]. The optimal value that mini-mizes is readily found to be

(6)

The series converges fast if the ratio tends to 1 so thattends to 0. Such frame is called snug, whereas if

the frame is tight [17]. For a snug frame, one may use theterm in (5), i.e.

(7)


In general however, and are not known analytically, hencethe calculation of for each frequency may be time con-suming. In our UWB formulation, we shall therefore choose aparameter range where one can use an analytic approximationfor and [see (11)]. This not only simplifies the frequencydomain calculations but also leads to closed form expressionsin the time domain theory [29].

B. WFT Frames

In two types of frames are mostly used: the waveletframes and the WFT frames (also termed Weyl-Heisenbergframes or Gabor frames). In this paper, we utilize the latter.Choosing a proper window function , the ele-ments of the frame and of its dual are given by [16], [17]

(8a)

(8b)

where we use conveniently the vector index .The parameters define the spatial and spectral displace-ment units, i.e, is centered at the lattice point inthe phase-space. For the set to constitute a frame,it is necessary that the unit-cell area be smaller than , i.e.

(9)

The parameter describes the overcompleteness or the redun-dancy of the frame ( is the oversampling factor). The frameis overcomplete for and it is critically complete in theGabor limit , where it becomes a basis. A necessary con-dition on [17, Proposition 3.4.1] basically states that the phasespace should be covered without “holes”, i.e., and all itstranslations should provide a full coverage of the real axis withno gaps, with a similar condition on its Fourier transform .

The dual window needs to be calculated for a givenand a phase space lattice . One numerical approach isoutlined in the Appendix . An exact reference solution can also becomputed via the Zak transform for rational oversampling [33].Though not used in computations below, the Neumann seriesapproach (5)–(7) is helpful here since it explains properties thatwill be employed in constructing the UWB beam representationof Sections III and IV. In general, finding of (6) involveselaborate calculations for the frame bounds and (seeAppendix). For WFT frames, we may use the bounds [17,eq.. 3.4.2]

(10)

to find an approximation to (6):, so that the term in the Neumann series

(5) for becomes (cf. (7))

(11)

is not only the limit of for , but it also approxi-mates over a wide range of provided that the window ismatched to the lattice over that range as discussed in (12). Weprefer (11) with over the “optimal” value in (7) since unlike

, (11) is known analytically and does not require numericalcalculations; this becomes essential when the frame expansionis used for UWB fields where needs to be calculated foreach frequency in the band. Equation (11) also yields simple an-alytic expressions when the formulation is transformed to obtaina beam representation in the time domain [29].

The properties of the frame and of its dual depend on twomain parameters: the “overcompleteness” parameter of (9)and the “matching” parameter that describes how well the spa-tial and spectral distributions of fit into the phase space lattice.It is defined as

(12)

where andare the spatial and spectral widths

of . When , the spatial and spectral window coverageof the unit cell are balanced, but if or ,the window is spatially narrow or wide, respectively (i.e.,spectrally wide or narrow), with respect to the unit cell (seeFig. 2). As we shall show in Figs. 3 and 5, for a given , thesnuggest frame and, thus, the most localized dual functionare obtained when .

Finally, in view of (4), the WFT frame representation foris given by

(13a)

(13b)

From (13b), are samples of the WFT of with respect tothe “analysis function” at the phase space points ,while (13a) synthesizes in terms of all the phase space trans-lations of the “synthesis function” .

In general, it is desired that and its dual be localizedboth spatially and spectrally, so that (13a) expresses in termof localized phase space constituents with local and stablecoefficients . Balian-Low theorem [17, Theorem 4.1.1] and[32] implies in this case is required to be strictly less than 1.

C. Special Case: Gaussian Windows

We consider the Gaussian window

(14)

which is normalized to . Noting that the spatial andspectral widths are and , respec-tively, the matching parameter (12) is found to be

(15)

where the second expression follows via (9). The cell dimen-sions that “match” this window are obtained by setting ,giving

(16)


Fig. 4. Relative error k' � ' k=k'k between the exact ' and itsapproximation ' of (11), as a function of � .

The dual function corresponding to (14) is depicted in Fig. 3for a wide range of the overcompleteness parameter . For each

, the cell dimensions have been chosen via (16) to “match” thewindow, and the resulting cell’s boundaries at are shownas vertical lines. The exact functions , calculated by the methodin the Appendix , are shown by the full lines, while the dottedlines are the approximate functions of (11), which basicallyreplicate up to a proportionality constant. One observes thatin the range , provides a good approximation of ,while for , becomes increasingly less localized andin the limit it tends to the Gabor biorthogonal function[1], [13], [14], which is seen to be nonlocal and irregular. Thequality of the approximation is further examined in Fig. 4 whichdepicts the relative error as a function of .

The role of the matching parameter in (12) is explored inFig. 5, which depicts the numerically calculated forand for three values of : , 1, and 2 (i.e., havebeen chosen such that is spatially narrow, matched, and widewith respect to the lattice, respectively; cf. Fig. 2). As expected,the smoothest and most localized is obtained for . Thisobservation is further supported by Fig. 6, which shows that fora given the frame is snuggest with minimal [see (6)]when .

III. WIDEBAND BEAM SUMMATION FORMULATION

The WFT frame discussed above is used now to construct theUWB discrete beam summation representation. Such represen-tation should satisfy the following requirements: (a) It shouldutilize a frequency independent beam lattice so that the beamaxes do not have to be retraced for each frequency. (b) Thewindow function should properly scale with the frequency toprovide the snuggest frame for the entire frequency band. (c)The resulting GB propagators should be trackable in the am-bient environment.

The method is presented in the context of radiation into theuniform half space due to a given UWB scalar fielddistribution in the plane, with frequency band

. The coordinate conventions in the 3-D spaceare with denoting the coordinates trans-verse to , and the radiated field is denoted as (Fig. 1).

Fig. 5. Exact dual function ' corresponding to the Gaussian window (14) for� = 0:5with three values of �: � = 0:5 ; 1; and 2 (dotted, solid and dash-dottedlines, respectively).

Fig. 6. Frame-bound ratio B=A as a function of �, calculated numerically(via the method in the Appendix ) for the Gaussian window (14) with � =

0:5 (dash-dotted line) and � = 0:75 (solid line). For each � , B=A is minimal(snuggest frame) at � = 1.

We use the caret to denote frequency domain field constituentswith a suppressed time-dependence ; time-domain con-stituents which will be considered in subsequent publicationshave no caret.

The method is presented here in a general format, while ex-plicit expressions for the ID Gaussian window functions, whichprovide an UWB snug frame representation will be given inSection IV.

A. Frequency Independent Phase-Space Beam Lattice

We start by defining the plane wave spectrum of the initialdistribution, denoted by a tilde

(17)

where is the wave number and is the wave speed.The frequency normalized spectral wavenumber

is used instead of the conventional spectral wavenumbersince is a pure geometrical constituent that

defines the spectral propagation direction in a frequency inde-pendent sense.


We therefore define the spatial-spectral phase space as

(18)

The UWB phase space grid is, thus, defined in the domain by

(19)where are the unit cell dimensions, and we useconveniently a vector index notation

. In general, the unit cell dimensionsalong the and axes need not be the same as long as theysatisfy the overcompleteness condition (21) in each coordinate.Here, however, we shall not utilize these options. This griddefines the origins and directions of the beam lattice henceit is required to have the same grid for all frequencies in therelevant band, while providing an overcomplete coverage ofthe domain.

To construct this grid we first choose a reference frequencysuch that (the choice of will be discussed in

Section IV.B), and then choose to be critically complete at, i.e., it satisfies the Gabor condition at

(20)

The same grid is then used for all . The unit cell area isgiven by

(21)

Thus, scaling the overcompleteness parameter with the fre-quency such that yields a frequency independentbeam lattice for all .

B. Wideband Beam Expansion of the Field

Next, for a proper window function and its dual, we introduce the WFT frames in for all

[cf. (8a) and (8b)]

(22a)

(22b)

The window and its dual are obtained from aCartesian multiplication of the 1-D functions, i.e.,

. Note that in general, and do not haveto be the same as long as they are valid windows in each coordi-nate for all the relevant frequencies. Nevertheless, here assumethat .

Referring to the synthesis (13), the WFT frame representationof the field in the plane for all is given by

(23a)

(23b)

Assuming that is localized around , the coefficientsare the local spectrum of sampled at the grid points .

The radiated field for is obtained now by replacingin (23a) by the “beam propagators” , giving

(24)

where denote the fields radiated into by the synthesiswindows . They may be described, for example, by theKirchhoff integral

(25)

where is the free space Green’s function,with .

Alternatively, can be expressed by a plane waverepresentation

(26)

where is the spectrum (17) of

, with being the spectrum of the “mother” , andis the spectral wavenumber in the direction.

If is wide on a wavelength scale then behave likecollimated beams whose axes emerge from the points in the

plane in the frequency independent directionsdefined via . Propagating beamsoccur only for where is the spectral width of

( for collimated beams). For , propagatetangentially along the plane and decay exponentiallywith . In practice, we ignore all the evanescent beams and usein (24) only the beams with , thereby expressing theradiated field as a discrete superposition of beams that emergefrom all lattice points in the aperture plane and in all reallattice directions .

IV. ID GAUSSIAN WINDOWS

The ID Gaussian windows are the preferable basis functionsfor the UWB beam synthesis, either in the frequency or timedomains, because they have the following properties:

1) Their width is scaled with the frequency in a specificfashion, termed “isodiffracting” [see (31)] that provides thesnuggest frame for all frequencies in the relevant band [see(33)].2) They give rise to GB propagators that can be tracked lo-cally in the ambient environment. In view of the ID scaling,the beam propagation through inhomogeneous nondispersivemedia or through interfaces need to be calculated only at areference frequency, and can then extrapolated to all otherfrequencies [23], [25], [26].3) The corresponding time-domain phase-space propagatorshave closed form expressions, known as the ID pulsed beams[23], [25], [26].


The ID windows have the general form [25], [34]where is a frequency independent, com-

plex symmetric matrix such that is positive definite andis the transpose of . The positive definiteness of

guarantees that has a Gaussian decay as the distancefrom the origin increases.

For the sake of simplicity, we consider here the special caseof a real symmetric window

(27)

where is a frequency-independent parameter whoseoptimal value for the present application will be determinedfollowing (33). This parameter is the collimation (or Rayleigh)distance of the GB that emerges from this window [see (30)],and the term ID implies that this distance is the same for allfrequencies.

A. The Phase-Space Propagators: ID-GBs

The phase-space propagators are calculated by sub-

stituting into (26). For large (collimated beams), theintegral can be evaluated asymptotically as detailed in [34],[35]. For a given , the result can be expressed in the mostphysically appealing format by utilizing the beam coordinates

defined for a given phase-space point : isa coordinate along the beam axis that emerges from in the

plane in the direction as discussed after (26).The coordinates transverse to the beam axisare chosen such that the projection of on the planecoincides with the direction of , whilewhere the over-circle denotes unit vectors along the corre-sponding axes. With this choice, the linear phase impliedby the window function in (22a) is operative in the direc-tion but not in the direction. These coordinates are relatedto the system coordinates via

(28)

Utilizing these coordinates, we find by saddle point integrationof (26) that is a GB of the form

(29)

This expression is an astigmatic GB with principle axes ,, 2, with waist at and collimation distances

. The astigmatism is caused by the beam tilt which reducesthe effective initial beamwidth in the direction by a factor

. The collimation distances are given, accordingly,by

(30)

These determines the waists , the beamwidths ,the wavefront radii of curvature and the diffraction an-gles in the cuts, via

(31)

Thus, since we are interested in collimated beams with narrowspectral spread that can be tracked analytically via paraxialmodels, we shall choose for all in the relevantfrequency band and for the largest relevant tilt angle in thesource data (see further discussion in Section IV-B).

The expression in (29) can be extended to UWB propagationin inhomogeneous media or transmission through curved inter-faces. The result has the generic ID-GB form [23], [25], [26]

(32)

This expression describes the beam along its (generally curved)propagation trajectory, which is the GOs ray that emerges from

in a direction specified by , with and being the axialand transversal coordinates. Here is the wavespeed alongthis axis while the 2 2 complex symmetric matrices and

are found by solving a Ricatti type equation along the axis.It can readily be shown that is positive definite for all, hence the quadratic form in the exponent exhibits a

Gaussian decay as increases away from the beam axis, andthat [26].

It thus follows that the ID-GB is determined by the frequencyindependent functions and which need to be solvedonce for all frequencies.

B. Choosing the Frame Parameters

The optimal value of is determined by considering thematching parameter in (12). Using

and , where, yields the frequency independent result

. Setting for a “balanced” window which providesthe snuggest frame (see Section II-B and Fig. 5), we obtain

(33)

where in the second and third expressions we utilized (20).Thus, the ID Gaussian window (27) with related to and


Fig. 7. Relative error of the GB approximation (29) to the exact beampropagators, shown as a function of the tilt angle �nnn for several values of thecollimation parameter kb, at k = 0:1. Without loss of generality, the beamsare tilted only in the x direction (i.e., ��nnn = (sin �nnn; 0)). The error iscalculated at a distance z = 400 from the source by integrating the beamprofile normal to the axis along the beam coordinate x (the field structurealong x is not affected by the beam tilt; see (30).

via (33), provides the snuggest WFT frame for all frequenciesin the band.

This leads to the following considerations for choosing theframe parameters , and :

First, is chosen to be sufficiently large so that will becollimated even at the lowest end of the frequency spectrum tojustify the GB solution. Recalling (31), should satisfy

(34)

It follows that the choice of depends on and on the sourcespectral (directional) spread. If the source has a wide spectralrange that give rise to large propagation angles, it is requiredto choose the beam collimation to be sufficiently largeso that GB be collimated even at the largest in the sourcespectrum.

Another factor that should be considered in this context isthat the quality of the GB approximation (29) to the exact beampropagators (given by either the Kirchhoff or the plane wave in-tegrals in (25)–(26) with Gaussian initial condition) deterioratesfor large tilt angles . This is demonstrated in Fig. 7 which alsoshows that for a given , the quality of the GB approximationimproves by increasing the collimation . Thus, it is also re-quired to choose to be sufficiently large so that from Fig. 7the GB approximation of the propagators at the largest is ac-curate to a desired level.

Next, we set where is a constant tobe chosen, as a tradeoff between the desired accuracy and thenumerical efficacy, see the discussion in Section V. From (21),

, so that should be as small as possibleto minimize the oversampling. For analytic simplicity, on theother hand, should be sufficiently large making smallenough for all so that the dual ID window canbe approximated by the 2-D extension of (11)

(35)

Fig. 8. Ray trajectories for the focused aperture in (40). The figure alsoshows four observation points where the field will be tested: (a) (near zone)rrr = (xxx; z) = (0;50), (b) (cusp point) (0, 400), (c) (far zone) (0,600), and(d) (shadow zone) (200 360). The phase space observation manifolds O (rrr)corresponding to these points are depicted in Fig. 11.

where we also used . Choosing, for example,yields hence from Fig. 4 the error in

calculating the expansion coefficients at via the approx-imate of (35) is bounded by 4%, and it becomes smallerfor as gets smaller. The field reconstructionerror is also bounded by this level: see Figs. 10 and 12–14.Choosing a larger improves the accuracy (e.g., yields

and a 2% error bound). Note that actually Fig. 4depicts the error for the 1-D frames considered in Section II-C;for the 2-D frames considered here, the error is larger by a factorof roughly 1.5.

Clearly for the cases noted above, is localized (ap-proximately a Gaussian) and the expansion coefficients are localand stable as required.

Finally, it is important to note though that the UWB beamexpansion is valid in the entire band even if is taken to besmaller, yet the approximation (35) can be used only for thelower frequencies where is small enough (say,

).Once and are determined subject to these considera-

tions, the lattice is determined via (33). Note thatis proportional to , hence in choosing one should alsoconsider the desired degree of spatial localization which setsa limit on .

C. Phase-Space Localization

An important feature of the phase-space representation is thea priori localization of and around well defined regions inthe domain. We assume that the initial distribution is givenby

(36)

where the “amplitude” and “phase” functions and areslowly varying functions of on a wavelength scale. The localdirection of radiation at a given and is given by

(37)


Fig. 9. Coefficients a at k = 0:25 and 0.5, shown in the m ;n plane for m ;n = (0; 0). The expansion parameter b has been taken as: (a,b) b = jRj=2 =200, (c,d) b = jRj = 400, and (e,f) b = 2jRj = 800. The corresponding grids (�x; ��) are determined by b via (33). The gray scale is in decibels.

In the continuous phase space of (18), this relation defines theso called Lagrange submanifold (the phase-space skeletonof GOs).

As mentioned after (23b), the coefficients are samplesof the local spectrum of with respect to the window atthe phase space grid points . If in (23b) is wide on thewavelength scale, then it senses the local radiation properties of

. Consequently the coefficients are nonnegligible only for

points near the discrete Lagrange manifold [1], [3], [25],[34], [36]

(38)

The number of nonnegligible elements near depends on thewidth of . This limits the number of beams that are excited bythe aperture.


Fig. 10. Real part of the reconstructed field in the aperture, shown only for x � 0 (for x < 0, the field is symmetrical). (a) k = 0:4 and (b) k = 0:5. Full lines:the exact field. Dashed and dotted lines: the reconstructed field using the exact and the approximated dual function (the analysis window) ' , respectively. Thedashed lines are almost indistinguishable from the full lines).

The effective range of summation in (24) is constrained fur-ther since only those beams that pass near actually contributeto the field. For a given in a homogeneous region, these beamscorrespond to in the vicinity of the discrete observationmanifold [1], [25], [34]

(39)

which simply relates the initiation points and directionsof these beams. The width of the contributing zone neardepends on the spatial width of at the observation point. Aswill be demonstrated in Section V.C, it is sufficient to accountonly for those beams that pass within the three beam-widthsvicinity of .

The simultaneous constraints above implement the a priorilocalization.


As an example, we consider the radiation from an UWB fo-cusing aperture distribution

(40)

and zero otherwise, with frequency band and. The parameter is the radius of curvature of the

wavefront in the aperture and it is taken here to be ,so that the field focuses to a cusp at (see Fig. 8). Thewidth of the aperture is . Such apertures are usuallycharacterized by the Fresnel number [39], [40]

(41)

where is the Fresnel length, while the -numberdefines the spectral width. Thus, the given distribution is

characterized by large Fresnel numbers ranging from 16 atto 32 at .

A. Wideband Phase Space Coefficients

We choose the reference frequency to be , i.e., .This value is smaller than recommended after (35),

Fig. 11. Observation manifoldsO corresponding to some typical observationpoints. They are plotted on the phase space map of the coefficients a for thecase b = 400 and k = 0:5 in Fig. 9(d). The indexes (a,b,c,d) correspond to theobservation points in Fig. 8.

which leads to . It was chosen in order to explorethe error in calculating the expansion coefficients usingthe approximate of (35) instead of the exact . With thischoice, using the approximate yields noticeable error in thefield for (where ) while being sufficientlyaccurate for , whereas using the exact (which iscalculated numerically for each ) yields accurate results ev-erywhere (see numerical experiments in Figs. 10 and 12–14).

We used three choices of the window parameterand . For , the beam propagators start

to diverge around , before the focal zone, while forthey are still collimated there. Note that in all cases the

beams are collimated and satisfy [see discussionafter (35)]. For each , the cell dimensions are calculatedvia (33).

The expansion coefficients have been calculated using(23b) and the results are shown in Fig. 9 for the three valuesof and for two frequencies. The coefficients are shown in the

plane for a slice of the phase space where. In all the figures, the dominant coefficients are concen-

trated near the Lagrange manifold of (38), which yields, for


Fig. 12. Radiated field at three frequencies k = 0.25, 0.4, and 0.5, shown in the z = 200 plane for x � 0 (the field for x < 0 is symmetrical). The algorithmutilizes threshold level " = 0.03 and summation over all beams passing within a three beamwidths vicinity of the observation point. Dashed and dotted lines: Thefield synthesized by using the exact and the approximated dual function (the analysis window) ' , respectively. Solid line: Exact reference solution.

Fig. 13. As in Fig. 12, but for z = 360.

the distribution in (40), . In the discrete phase space,this condition becomes using (33)

(42)

Note that this condition is frequency independent, but the widthof the strip of coefficients and their magnitudes depend on thefrequency. Also note the difference in the unit cell dimensionsfor the different values of .

One also observes large contributions along the phase spacelines corresponding to the diffraction at the endpoint of the aperture. They were termed diffraction manifolds[1], [25], [36].

The phase space characteristics can be explored analyticallyby evaluating the coefficients approximately, usingfrom (35). Substituting into (23b) yields the following integralfor the coefficients

(43)

For grid points that are far away from the aperture boundaries,the end points effect in the integral can be neglected leading tothe closed form result for :

(44)

One may readily verify that the dominant coefficients in (44) areindeed aligned along the discretized Lagrange manifold of (42).

B. Reconstruction of the Aperture Field

The quality of the representation is explored first by consid-ering the field reconstruction in the aperture. We apply thresh-olding to the set , i.e., we consider only the significant coef-ficients that satisfy where is a small errorparameter, thus reducing the number of elements in the summa-tion. Specifically we use , i.e., we neglect the terms inFig. 9 below .

We compare reconstruction using the exact coefficients(dashed lines), and using the approximate ones that are obtainedby processing the data with the approximate of (35) [dottedlines]. In the former case, the error is mainly due to the thresh-olding at a level [for the reconstructed field isalmost indistinguishable from the exact field (solid line)]. In thelatter, there is a noticeable error for , which is due to theerror in (35) for , whereas for where ,the reconstruction error is smaller than 6% as predicted in Fig. 4and the discussion after (35). One also observes a weak Gibbseffects at the aperture truncation points . We there-fore consider here only and 0.5 since the quality of theapproximate reconstruction is excellent for .

In view of these results it is recommended to use oreven in order to obtain accurate results while using theapproximate dual function of (35).


Fig. 14. As Fig. 12, but for z = 600.

C. Radiated Field

The radiated field has been calculated via the beam summa-tion formula (24) using the ID-GB propagators of (29), andthe results are shown in Figs. 12–14. In view of our UWB phasespace construction, the beam trajectories and the beam param-eters are frequency independent. The summation includes onlythe propagating beams with [see discussion after (26)].As noted in Section V-B, the number of beams excited is reducesby applying thresholding to at an error level (i.e.,

).Furthermore, only those beams that pass “near” a given

observation point are summed in (24): As follows from thenumerical results below, it is sufficient to include only thosepassing within the three beam-widths vicinity of , wherethe beamwidth is given in (31). Viewed from a phase-spaceperspective, the beams that pass near are defined by theobservation manifold of (39), which is illustrated inFig. 11 for the four observation points (a)–(d) that are indicated inFig. 8, and the algorithm selects the beams that are located withina three beam-widths vicinity of . Dominant contributionsat a given are therefore obtained only from those phase-spacepoints in the vicinity of the intersection of of with theLagrange and diffraction manifolds of the beam amplitudes(shown by the gray scale in Fig. 11).

Considering as an example the near zone point (a), the beamcontributions obtained from the intersection of in Fig. 11with the Lagrange manifold describe the GOs field near the aper-ture center, while those obtained from the intersection with thediffraction manifolds describe edge diffraction contributions at. At the focal point (b), on the other hand, in Fig. 11 is

essentially parallel to the Lagrange manifold and the field is de-scribed by significant beam contributions from the entire aper-ture. Finally, at point (d) in the shadow zone, the beams along thecorresponding in Fig. 11 are weakly excited hence thereare no sgnificant GOs contributions but there are contributionsfrom the intersection of with the diffraction manifolds. Thetotal numbers of beams used in the formulation are summarizedin Table I. For further discussion, see Section V-D.

Figs. 12–14 depict cross sectional cuts of the UWB field cal-culated at several distances in the near, intermediate and farzones, and at three different frequencies and

. The dashed and dotted lines compare, respectively,the fields calculated by using the exact and the approximated

TABLE INUMBER OF BEAMS USED IN OUR ALGORITHM. 2ND COLUMN: NUMBER OF

EXCITED BEAMS AFTER THRESHOLDING AT LEVEL " = 0:03. 3RD-6TH

COLUMNS: NUMBER OF BEAMS USED TO SYNTHESIZE THE FIELD AT THE

OBSERVATION POINTS (A)–(D) IN FIG. 8

of (35), respectively. As in Fig. 10, the former provides ac-curate beam amplitudes and excellent agreement with the exactsolution (full lines) for the entire frequency band: the minor no-ticeable discrepancies are due to the thresholding of at a levelof , and are essentially eliminated by using .Using the approximate leads to accurate results forbut a noticeable error for . Note that the beam formula-tion provides a good representation for the field near the shadowboundary of edge diffraction (see Fig. 12 around ), forthe field in the focal zone in Fig. 13, and for the transition to thefar zone in Fig. 14.

D. The Phase Space Degrees of Freedom

The manifestation of localization in discrete phase-spacerepresentations like the discrete Fourier transform or the beamsummation, can be quantified in view of the Landau-Pollak(LP) bound that specifies the number of discrete degrees offreedom required for a given field to be approximated to aprescribed accuracy [3]. This issue is explored here in thecontext of our UWB beam summation representation. Clearly,the oversampling increases the number of terms, but on theother hand, the snug localization extracts the local features ofthe field and renders the phase space coefficients highly local-ized near the Lagrange manifold. Consequently the number ofdegrees of freedom may be significantly reduced relative to theLP bound. Further savings are achieved for UWB fields sincethe beams need to be traced only once at a reference frequency.

We start with the LP dimension thatdefines the degrees of freedom in a discrete representation ofa 2-D (aperture) domain, with and being the spatial andspectral cross sections of the field distribution, respectively [3].Setting yields

(45)


This expression describes a complete phase space coverage andit is obtained, for example, by assigning visible spectrumpoints to each of the spatial points in the aperture where

.A smaller number is obtained if one includes only the GOs

radiation from the aperture and excludes edge diffraction. Re-ferring to the aperture source distribution in (40) we use

and obtain

(46)

where is the Fresnel number of (41).In the beam summation formulation, the actual number of

elements is further reduced since it is sensitive to the localstructure of the phase space. We consider only the nonnegligiblecoefficients that satisfy where is a smallerror parameter. Referring to Fig. 9, the number of phasespace coefficients is then a sum of elementsalong the discrete Lagrange (or GOs) manifold of (38) and

elements near the diffraction manifold. The latter willbe neglected in the estimates in view of the bound

, which applies asymptotically for large apertures (is proportional to the linear dimension while isproportional to the area). From (44), the relevant coefficientsnear are enclosed within a 4-D tube

(47)

For a given , (47) represents the interior of a circleof radius in theplane, which is centered at on . Thenumber of points in that circle is equal to its area

. Hence accounting for all thepoints in the aperture yields an estimate on

(48)

with , where the second expressionfollows from (33). It follows that the minimal number of ele-ments is obtained with .

Comparing (48) with (46) one obtains the asymptotic relation, i.e., the phase space beam representa-

tion consists, asymptotically, of a smaller number of elements.This property follows from the fact that the beam formulationextracts the local radiation property of the source and thus in-cludes only the elements along the Lagrange manifold, whereasLP is a global bound ( bounds the spectral cross section of theentire aperture field).

One should also note the term appearing in (48) whichrepresents the frame oversampling.

Finally, the number of elements needed to describe the fieldat a given may be further reduced if out of all the phase spacepoints in (48) one keeps only those corresponding to beams thatpass near , as discussed in Section V-C, and summarized inTable I.

Applying (48) to the present configuration, using thresholdlevel we obtain and for

and , respectively. These numbers are smaller than

those in the second column of Table I for the beams that are actu-ally excited by the aperture at the same threshold level. The dif-ference is essentially due to the edge diffraction beams that areincluded in the calculations of Table I but not in (48). Recallingthe discussion before (47), it is expected that for larger aperturesthe relative difference between the estimate for in (48)and the results of Table I will diminish.

VI. SUMMARY AND CONCLUDING REMARKS

A novel discrete phase space beam summation representationfor UWB radiation from extended source distributions was in-troduced. The representation is based on a WFT frame analysisof the aperture source distribution. The formulation comprisesthe following key features:

1) It utilizes a frequency independent beam lattice (unlikethe conventional Gabor scheme), emerging from a discreteset of points and orientations in the aperture. This importantfeature is achieved by introducing relation (21) between theovercompleteness and the frequency.2) The ID-GB are shown to provide snuggest frames forall frequencies, provided that the beam parameter and thephase-space grid are related via (33).3) The ID-GB propagators are also fully described by fre-quency independent parameters [see (28) and (32)]. Con-sequently, the calculation of the ID-GB propagation in theambient environment need to be done only once for allfrequencies.4) The expansion coefficients are samples of the WFT ofthe source distribution with respect to the dual (or “analysis”)window. An important parameter is the reference frequencywhich is larger than , the highest frequency in the sourcespectrum. If is chosen to be sufficiently large (with ), the overcompleteness is greater than

for all and the dual window can be approximatedby the ID-GB window as in (35). This greatly simplifies thecalculations since the dual function does not have to be calcu-lated numerically for each frequency in the band (the numer-ical procedure for calculating the dual function numericallyis described in the Appendix). Furthermore, in this case thedual function can be transformed in closed form to the timedomain, thus providing a starting point for the new time do-main theory that has been briefly reported in [37], [38].5) Following item 4, the overcompleteness poses a tradeoffbetween analytical simplicity and numerical efficacy. Refer-ring to [19], [20] is sufficient for expansion coef-ficient localization. If, like in our case, an approximate dualfunction is desired, we have found that is preferablesince in this case the error due to the approximate dual func-tion is bounded by 6% at , and it is lower forwhere (Fig. 4 and numerical examples in Sec-tion V). Choosing reduces the error bound at to3%. Large increases the overcompleteness and the numberof elements, but simplifies the calculations of the widebandexpansion coefficients (item 4) as well as the field calcula-tions (items 1 and 3). Furthermore, the snug localization ofthe formulation renders the phase space representation highlylocalized near the Lagrange manifold (the GOs skeleton),


hence, depending on the source properties, the number of el-ements in the expansion can be significantly reduced relativeto the LP bound (see Section V-D).

In view of the frequency independence of the various charac-teristics above, the formulation can be transformed in closedform into the time domain, where the propagators are the IDpulsed-beams and the expansion amplitudes are found via a newdiscrete local slant-stack (or Radon) transform. Initial results ofthese new formulations have been reported in [29], [37], [38],but full details will be published separately.

Referring to item 1, we note that the oversampling increasesat lower frequencies so that the numerical efficiency there de-creases (see item 5). Thus, although the bandwidth of the for-mulation above can be arbitrarily large, we have introduced in[37], [38] a modified formulation for excitations with bandwidthlarger than one octave, . For these cases, wedivide the excitation band into a hierarchy of one-octave sub-bands, and apply the UWB formulation above in each bandwhile choosing the parameters in each band such that the spatialand spectral discretizations are obtained by a multilevel binarydecimation of those at the highest band. The result is a self-con-sistent representation wherein the beam sets at the lower bandsare decimated subsets of those at the highest band, so that onlythe set of beam-propagators at the highest frequency band needsto be traced, while for the lower bands one may use properlydecimated subsets. The resulting multiband algorithms in theUWB frequency domain and in the time domain have been re-ported in [37], [38], but full papers are in preparation.

We are also engaged in extending the expansion algorithmto circular cylinders, mainly in connection with applications toindoor propagation. This subject will be reported elsewhere.

APPENDIX

DUAL FRAME CALCULATION VIA AN EXPANSION IN A FINITE

HILBERT SPACE

We present the numerical algorithm used for calculating thedual window of the WFT frames shown in Figs. 3 and 5 andof the frame bounds in Fig. 6. Two other approaches, namelythe Neumann series and the Zak transform, have been discussedafter (8b).

The dual window is defined in (8b) by , whichin view of (2) yields explicitly with

. To calculate , we expand it as ,where , is a valid expansion set in a finiteHilbert space that covers the support of (i.e., canbe a basis, a set of sampling delta functions, or a frame set), and

are unknown coefficients. We therefore obtain

(49)

Recalling that for , this equation is satisfiedif

(50)

where , is a column vector andis a column vector of nulls except for the term correspondingto that equals 1. has columns but an infinite

number of rows, however, it may be truncated since from [41]we have for . This property holds for anyframe, yet it may readily be verified for the present WFT frameby recalling the essentially finite support of . Since isa frame set, (50) is overdetermined and its solution may be ob-tained by the singular value decomposition (SVD) [42], yielding

, being the pseudo inverse obtained after trun-cating the low singular values (see [43]).

A convenient choice for the expansion set is the Galerkin setin which case is Hermitian. In

this case, the largest and smallest singular values of are theupper and lower frame bounds and , respectively [41].

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Amir Shlivinski (S’98–M’04) was born in Tel-Aviv,Israel, in February 1969. He received the B.Sc. (cumlaude), M.Sc. (summa cum laude), and Ph.D. (withdistinction) degrees in electrical engineering, all fromTel-Aviv University, Israel, in 1991, 1997, and 2003,respectively.

From 1991 to 1999, he worked as a research anddevelopment Electromagnetic Engineer, and between1999 and 2003, he was a full-time Ph.D. student and aTeaching Assistant at Tel-Aviv University. Currently,he is a Postdoctorate Fellow at the Department of

Electrical Engineering, University of Kassel, Kassel, Germany. His main fieldsof interest are electromagnetics, wave theory and antenna theory, with emphasison analytic methods and time-domain phenomena.

Ehud Heyman (S’80–M’82–SM’88–F’01) wasborn in Tel Aviv, Israel, in 1952. He received theB.Sc. degree in electrical engineering from Tel AvivUniversity, Israel (summa cum laude) as Valedicto-rian, the M.Sc. degree in electrical engineering (withdistinction) from The Technion—Israel Institute ofTechnology, Haifa, and the Ph.D. degree in electro-physics from the Polytechnic Institute of New York(now Polytechnic University), Brooklyn, in 1977,1979, and 1982, respectively.

In 1983, he joined the Department of PhysicalElectronics of the Faculty of Engineering, Tel Aviv University where he isnow a Professor of electromagnetic theory and Heads the School of ElectricalEngineering. From 1991 to 1992, he was on sabbatical at Northeastern Uni-versity, Boston, MA, the Massachusetts Institute of Technology, Cambridge,and the A. J. Devaney Association, Boston. He spent several summers as aVisiting Professor at various universities. He has published over 80 articles andhas been an Invited Speaker at many international conferences. His researchinterests involve analytic methods in wave theory, including high-frequency andtime-domain techniques for propagation and scattering, short-pulse antennasand pulsed beams, inverse scattering and target identification, imaging andsynthetic aperture radar propagation in random medium.

Prof. Heyman is a Member of Sigma Xi and the Chairman of the Israeli Na-tional Committee for Radio Sciences (URSI). He is an Associate Editor of theIEEE Press Series on Electromagnetic Waves and was an Associate Editor ofthe IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. While at the Poly-technic Institute he was a Research Fellow and later a Postdoctoral Fellow, aswell as a Rothschild, a Fullbright, and a Hebrew Technical Institute Fellow.

Amir Boag (S’89–M’91–SM’96) received the B.Sc.degree in electrical engineering and the B.A. degreein physics (both summa cum laude) in 1983, theM.Sc. degree in electrical engineering in 1985, andthe Ph.D. degree in electrical engineering in 1991from The Technion—Israel Institute of Technology,Haifa.

From 1991 to 1992, he was on the Faculty of theDepartment of Electrical Engineering, The Tech-nion—Israel Institute of Technology. From 1992to 1994, he was a Visiting Assistant Professor withthe Electromagnetic Communication Laboratory,

Department of Electrical and Computer Engineering, University of Illinois atUrbana-Champaign. In 1994, he joined Israel Aircraft Industries as a ResearchEngineer where he became a Manager of the Electromagnetics Department in1997. Since 1999, he has been with the Department of Physical Electronics, theSchool of Electrical Engineering, Tel Aviv University, Israel. He has publishedmore than 40 journal articles and presented more than 70 conference papers onelectromagnetics and acoustics. His research interests are in electromagnetictheory, wave scattering, and design of antennas and optical devices.

Christine Letrou (M’96) received the Diplômed’Ingénieur of Institut National des Télécommu-nications (INT), Evry, France, in 1982 and theDocteur-Ingénieur and Ph.D. degrees from Uni-versité Paris XI, Orsay, France, in 1985 and 1988,respectively.

Since then, she has been an Assistant Professorat INT in charge of microwaves, electromagneticsand antennas, teaching, and research activities. Cur-rently, she is also a Member of the CNRS LaboratorySAMOVAR (UMR 5157), Paris, France. Her main

research interests are in phase-space methods development, antennas andquasioptical devices and systems design, and propagation modeling for highbit rate communication systems.


Theoretical Considerations in the Optimization ofSurface Waves on a Planar Structure

Samir F. Mahmoud, Senior Member, IEEE, Yahia M. M. Antar, Fellow, IEEE, Hany F. Hammad, andAl P. Freundorfer, Senior Member, IEEE

Abstract—The problem of optimum excitation of surface waveson a grounded dielectric slab by means of slots in the ground planeis considered. By adopting a two-dimensional (2-D) model, anal-ysis lead to closed forms for the power launched as surface wavesand power leaked as radiation. Input admittance of a single slotsource and mutual admittance between two slots are derived andutilized to design a three element Yagi array of slots to achieve aprescribed ratio of forward to backward surface wave power. As adevelopment of the 2-D model, we allow finite extent of slot exci-tation by assuming a Gaussian E-field distribution across the slot.The effect of the Gaussian width on the excited surface wave poweris studied. The analysis is relevant to the study of surface waveson printed circuits. Specifically, it applies to the implementation ofpower combiners based on quasioptical slab beam that have beenrecently introduced in the literature for use in the millimeter waveband.

Index Terms—Millimeter wave power combiners, planar struc-tures, quasioptical power combiners, surface waves.

I. INTRODUCTION

EXCITATION of surface waves on planar integrated mi-crowave circuits is often considered as an adversary effect

causing power loss and undesired coupling. However thereexist situations when the main objective is to efficiently excitea surface wave with least possible leakage, or radiated power.One recent example of these situations is the implementationof quasioptical slab beam power combiners in the millimeterband, in which surface waves are the means of power transport[1]–[4]. These combiners depend on the efficient excitationof the dominant surface wave mode inside a dielectric slab.Recent investigations by the authors [3], [4] have suggestedthe use of coplanar waveguide (CPW) driven slots as the mostsuitable surface wave launchers for monolithic fabrication inthe millimeter regime. In this paper an attempt is made toestablish the theoretical foundation for operation of the slabbeam power combiner. Accordingly we seek to maximize thesurface wave excited by a slot dipole on the ground plane ofa grounded dielectric slab. Starting with a two-dimensional(2-D) model of the grounded slab and the slot, a rigoroustheory is presented in Section II that leads to closed formsfor the excited surface wave and leakage powers. The input

Manuscript received January 5, 2003; revised July 9, 2003.S. F. Mahmoud is with the Electrical Engineering Department, Kuwait Uni-

versity, Kuwait (e-mail: [email protected]).Y. M. M. Antar is with the Department of Electrical and Computer Engi-

neering, Royal Military College of Canada, Station Forces Kingston, ON K7K7B4, Canada (e-mail: [email protected]).

H. F. Hammad and A. P. Freundorfer are with the Electrical and ComputerEngineering Department, Queen’s University, Kingston, ON K7L 3N6, Canada.


(a)

(b)

Fig. 1(a). 2-D model of a grounded dielectric slab of relative permittivity "with a slot source in the ground plane. (b). Grounded dielectric slab fed by aYagi array of three slots. The middle slot of width “s” is the driven one and theother two are director and reflector slots having widths s and s , respectively.The spacing between the driven slot and the director/reflector slots are “c ”=c ,respectively.

admittance of a single slot and mutual admittance betweentwo slots are derived in Section III. Numerical examples aregiven in Section IV including design examples of Yagi slotarrays that achieve high surface wave front to back ratio. InSection V, we alleviate the assumption of uniformly excitedslot by allowing a Gaussian distribution of the E-field insidethe slot. This breaks the two dimensionality of the problem,but still allows the derivation of closed forms for the surfacewave and radiated powers. Comparison is made between thepresent theory and experimental implementations as given bythe authors in earlier work [4].

II. THEORY OF GROUNDED SLAB EXCITATION BY A SLOT

SOURCE (2-D MODEL)

We start by considering a 2-D model of the problem wherea grounded dielectric slab of uniform thickness is assumedto extend infinitely parallel to the - plane. A -directed slotof width “ ” (along ) in the ground plane is used to excitethe slab as shown in Fig. 1(a). The slot itself is excited bya uniform -oriented electric field and therefore, acts asan infinite magnetic line source of magnetic current given by

(volts). To limit the problem to the 2-Dmodel, we assume that the slot is infinite in the direction and

0018-926X/04$20.00 © 2004 IEEE


is uniformly excited. In Section V, we consider a finite slotexcitation. The excited fields can be expressed by a discrete setof (say ) surface wave modes traveling along as well as acontinuous spectrum over the transverse wavenumber (along )that account for radiated fields. This is known as the transversespectral representation for the fields [5]–[8]. This representationis adopted here since it facilitates the determination of themodal amplitudes as will be seen. Since the slot acts as a

-oriented uniform magnetic line source, fields are obviously ofTM type with magnetic field and electric field components

and . Assuming a time harmonic excitation of the form, and that the source lies in planes, general

expressions for the total fields in the directions are

(1)

(2)

where the superscript and signs apply for and, respectively, and , and, are surface wave mode

and pseudomode amplitudes. The integration is taken over thetransverse wavenumber . Here , are themagnetic -field component and the vector electric field of the

th surface wave mode; namely for the surface wave mode

(3)

and .Here, , ,

the attenuation rate of the th mode in air, andis the free space wavenumber.

On the other hand, , are the fields of a pseu-domode [6]–[8] with a transverse real wavenumber . A pseu-domode is the result of an incident plane wave on the slab anda reflected one; namely

(4)

where: , and . Ob-viously, (3) and (4) ensure the continuity of the -magnetic fieldacross the interface . The associated electric field is ob-tainable from (3) and (4) through Maxwell’s equations. Thecontinuity of requires that be continuous

across leading to the modal equation for a surface waveand determines for a pseudomode as

(5)

Obviously, the range of the spectrum representsfields with active radiation power while the rangecorresponds to the evanescent part of the field. While a pseudo-mode does not satisfy the radiation condition on its own, the sumof pseudomodes making up the radiated and evanescent fieldsdoes satisfy the radiation condition, as it should [6]. The modalelectric field vector is .

Orthogonality relationships can be established among the sur-face wave modes and pseudomodes and can be expressed by [8]

(6)

where the unless whence it is equal to unity,while is the usual Dirac Delta function. In addition there isorthogonality between a surface wave mode and a pseudomode.The surface wave factor is easily obtained as

(7)

where the modal fields have been normalized such that.

The second integration in (6) is a bit more difficult to evaluate.Following [8], it is useful to change that integration to a contourintegral as follows:

(8)

Evaluating the right-hand side (RHS) and using the identity, we get

in (6) as

(9)

It is worth noting that, in the absence of any loss in the struc-ture, is real and stands for the surface wave power flowingalong . Meanwhile is real and represents spectral powerdensity over the range , while in the range ,

is pure imaginary and so is , which then represents re-active power density.

MAHMOUD et al.: THEORETICAL CONSIDERATIONS IN THE OPTIMIZATION OF SURFACE WAVES 2059

Now consider a -directed infinite slot of infinitesimallynarrow width ; Fig. 1, over which a uniform exists.This acts as a magnetic line source of magnetic current

. The fields generated by this sourcetake the general form in (1) and (2) and the coefficients and

are to be determined from the electric field discontinuityand the magnetic field continuity across the source; namely

(10)

Inserting and from (1) and (2) and using the orthogonalityrelations in (6), we get the modal amplitudes as

(11)

Now, we are able to obtain both the guided surface wave power(in both ) and radiation power in simple summation and in-tegral form, respectively. Namely

(12)

and

(13)

where and are given by (7) and (9), respectively.It is constructive at this point to compare between the

transverse wavenumber spectral representation with the moreconventional longitudinal wavenumber spectral representation.The latter has been used for several decades by Wait [9], Fullerand Wait [10] and many others, and recently pursued by Bhat-tacharyya [11]. While the transverse spectral representationexplicitly displays the surface wave and the radiation spectraseparately, the longitudinal spectral representation does not.However, the latter representation can be converted to theformer by changing the path of integration on the real axis ofthe longitudinal wavenumber (say ) to the complex planewhere the poles contribution gives the surface wave modes andthe branch cut gives the radiation part [9]–[11].

III. SLOT ADMITTANCE

The slot admittance (Siemens per unit length along ) is givenby

(14)

where is given by (1). It turns out that the slot con-ductance is given, as expected, by the sum of surface waveand radiation powers divided by . As for the slot susceptance

, per unit length, care should be taken in evaluating (14) by

accounting for the finite width of the slot and the variation ofover the slot. Therefore

(15)where which is real over the range ofintegration. Performing the integrals over and , and substi-tuting for from (9), we arrive at

(16)Note, that the square bracketed term accounts for the finite widthof the slot. As tends to , this term behaves as , renderingthe whole integrand to behave as , which ensures a conver-gent integral.

The results obtained so far also allow us to determine themutual admittance between two parallel slots of givenwidths and given spacing “ .” For “ ” sufficiently larger thanslot widths, the mutual admittance is given by

(17)

where is the magnetic field at slot 2 due to a voltageapplied to slot 1. Note that in the range

, and in the range .

IV. NUMERICAL RESULTS

The percentage power launched in surface waves relative tothe total power delivered by the source is computed from (12)and (13) versus normalized frequency for different values of therelative permittivity . The frequency is limited to allow forthe propagation of only a single surface wave mode. Defininga normalized frequency as , a single modeoperation occurs when . Results in Fig. 2 show a mono-tonic increase of the percentage surface wave power with upto about where there is a broad maximum whose valueincreases with the substrate . For example a peak value of 88%is attained for , and 68% for .

The slot conductance and susceptance per one freespace wavelength along are plotted versus in Figs. 3 and4. The conductance displays a peak around and thepeak value depends on the substrate relative dielectric constant,while has a maximum slope near the peak of . While theslot conductance is independent of the slot width for a narrowenough slot, the slot susceptance changes considerably with theslot width as seen in Fig. 4. The break up of the slot in surfacewave and radiation components; and , are plotted inFig. 5 for two values of . The surface wave conductance at-tains its maximum around . For , both and


Fig. 2. Percentage surface wave power excited by a slot source versusnormalized frequency F � k d

p" � 1. The relative permitivity " is a

varying parameter.

are reduced, but is reduced at a slightly higher rate.This explains the shift of the maximum percentage surface wavepower to higher values of than 1.6 (see Fig. 2). These obser-vations agree with experimental work conducted by the authorsin [4].

A. Design of a Yagi Slot Array

Having obtained the self and mutual admittance of slots, onecan design an array of such slots to achieve maximum front( ) to backward ( direction) ratio of excited surface wave.A three-element Yagi slot array is shown in Fig. 1(b). It is com-posed of one fed slot, of width “ ” and parasitic director andreflector slots of widths and . The separation between thefed slot and each of the director and reflector slots is denoted by

and , respectively. The results of two design examples areshown in Fig. 6 where the relative forward and backward surfacewave powers are plotted against frequency. Each array is numer-ically optimized with respect to the slot widths and the spacingbetween the elements for a maximum front to back ratio of sur-face wave power at the center frequency. As seen in Fig. 6, It ispossible to achieve a front to back ratio better than 20 dB over abandwidth of 2.7% (Design 1) or 4.4% (Design 2) around thenormalized frequency .

The theoretical results displayed in Fig. 6 can be comparedto simulated and measured data reported by the authors in [4,Fig. 12], where a three element Yagi array launcher is imple-mented on a Duroid substrate with 2.54 mm thick-ness. The resulting at a frequency of 11.8 GHz.A comparison between the present theory, represented by de-sign 1 and 2 of Fig. 6 and the simulated results in [4] is given inTable I in terms of the bandwidth of the 20 dB and 15 dB front toback ratio. Although the theory is based on a 2-D model, thereis a good agreement with the simulated results in [4].

V. FIELDS OF A FINITE LENGTH SLOT

So far we have carried out a 2-D analysis. Now we wish toconsider a more realistic situation where the slot source, whichis equivalent to a magnetic line source, is effectively of finitelength along . We shall continue to assume that the slab ishaving infinite width in the direction. The magnetic current

Fig. 3. Slot conductance in milli-mho per a free space wavelength versusnormalized frequency F . The relative permitivity " is a varying parameter.

Fig. 4. Slot conductance and susceptance versus F for " = 9:8. The slotsusceptance varies with the relative slot width parameter s=d.

Fig. 5. Surface wave and radiation conductance of a slot versus F . " takesthe values 3 and 9.8.

on the ground surface is considered to have a Gaussianform; i.e.

(18)


Fig. 6. Forward and backward surface wave power versus frequency f=f fortwo designs. The center frequency corresponds to F = 1:9 and " = 9:8.With reference to Fig. 1(b), the parameters of the two designs are: Design 1:s = 0:1d, s = 0:1d, and s = 0:15d, and c = 2:066d, and c = 0:817d.Design 2: s = 0:1d, s = 0:125d, and s = 0:2d, and c = 2:066d andc = 0:817d.

TABLE IFBR BANDWIDTH OF THREE ELEMENT YAGI; COMPARISON BETWEEN

THEORY AND SIMULATION

which is a source of effectively a finite length. The fields gener-ated by this source are no longer independent of . Working inthe spectral domain we use the Fourier transform to get

(19)

The fields generated by this source vary along asinstead of being independent of as before. The surface wavemodes and pseudomodes are now a mixture of both TM andTE to parts. As it is well known the TE surface wave existsonly when . Considering first the TM to fields allcomponents can be derived from only. Similar to the modeexpansion in (1) we can write as

(20)

whose -inverse Fourier transform is

(21)

In (20), and are the -componentof the electric field of the th surface wave mode and apseudomode with transverse wavenumber . The longi-tudinal wavenumbers are: and

.

The other field components; , , , and are derivedfrom (20) through well-known relations [8]. The normalizingfactors and are still defined as in (6), but now theyare functions of . After some manipulations, we get

(22)

where and are those given by (7) and (9). Theamplitudes of the surface wave modes and pseudomodes areobtained by using (18) and (19), and applying the boundaryconditions (10)

(23)

and

(24)

Combining these equations with (20) and (21), we obtain aclosed form expression for . Next considering the TE waves,similar expressions for the TE part of the spectrum can bederived, with replacing . Namely [(23), (24)] apply toTE modes after multiplying the RHS by . We arenow in a position to obtain the fields and powers launched assurface waves. Assuming the propagation of a single TM mode,we can derive the component of inside the dielectriclayer as

(25)

A similar expression exists for the TE field component withan extra term inside the integral term. This manifests thefact that this component vanishes when .

Expression (25) can be evaluated for and to get [12]

(26)

where is the Gaussian beam width at a distancefrom the slot, and . A similar expression is

obtained for the TE field component that has an extra mul-tiplying . It is seen that the Gaussian beamwidthincreases linearly with and the field magnitude decays with

as as expected for the diffraction of a Gaussian beam[12].


Now turning attention to the surface wave power, we canwrite

(27)

where should now be taken as the sum of of the TMand TE field. Note however that the TE mode does not existwhen .

Substituting from (25) in (27), we get after some manipula-tions

(28)

where

It is instructive to note that in the limit , tendsto one and tends to zero. In this case, (28) reduces to (12)which applies to an infinite uniformly excited slot. Thus, theterms and account for the Gaussian distribution of thesource. This suggests that we define a Gaussian SW efficiencyequal to . Namely

(29)

We can define an effective length of the Gaussian source as thelength of a uniformly excited slot that would produce the sameTM surface wave power as the Gaussian excited slot. Denotingthat length by , we have from (28)

(30)

A plot of the surface wave conductance versusnormalized frequency is shown for different and differentnormalized Gaussian width in Fig. 7. Here is the ef-fective wavenumber on the dielectric slab and is taken equal to

. It is worth noting here that the TE mode contri-bution to the power is zero for . For the example givenin Fig. 7, the contribution of the TE mode is less than 5% up to

for and less than 8.5% for . Asit is the case with uniform excitation (Fig. 5), the surface waveconductance peaks around . It increases with exceptfor low values of . The Gaussian efficiency parameter definedin [(29), (30)] is plotted in Fig. 8. It is seen that , which isalso equal to , increases with up to a satura-tion level. The variation with depends on the value of asseen in the figure.

Fig. 7.Surface wave conductance versus F for a Gaussian excited slot withk w = �=2 and �. Two values of " ; 3.0 and 9.8, are considered.

Fig. 8. Gaussian efficiency versus F for different slot Gaussian width k w =

�=2 and � and " = 9:8 and 3.0.

VI. CONCLUSION

Rigorous analysis of surface wave excitation and radiationfrom a grounded dielectric slab driven by a slot source has beenpresented. The analysis is relevant to the design of quasiopticalslab beam power combiners that use surface waves to transportpower on a dielectric slab. Adopting a 2-D model of the slab andthe slot, closed form expressions for the surface wave and radia-tion powers have been derived. In addition, slot self-admittanceand mutual admittance of two parallel slots have been derived.This facilitates the design of Yagi slot arrays aiming at achievingmaximum front to back ratio of excited surface waves.Numerical results show that ratio better than 20 dB canbe achieved over a bandwidth of 4%. The analytical results aresupported by previously published simulation and experimentalwork by the authors [4]. In order to improve the 2-D model, thecase of a Gaussian -excited slot is treated. Analysis shows thatthe surface wave power decays linearly with distance traveledalong the slab for much greater than the Gaussian beamwidth.An effective length of the Gaussian slot is derived. Although thepresent theory has been applied to a single homogeneous dielec-tric slab, extension to an inhomogeneous slab, or a multiplayerslab, is straightforward. Such extension should lead to the studyof the interplay between surface wave and radiated powers onprinted circuits. This study is underway.


REFERENCES

[1] J. Harvey, E. R. Brown, D. B. Rutledge, and R. A. York, “Spatial powercombining for high-power transmitters,” IEEE Microwave Mag., pp.48–59, Dec. 2000.

[2] A. R. Perkons, Y. Qian, and T. Itoh, “TM surface wave power combiningby planar active-lens amplifier,” IEEE Trans. Microwave Theory Tech.,vol. 46, pp. 775–783, June 1998.

[3] H. F. Hammad, A. P. Freundorfer, and Y. M. M. Antar, “CPW slot an-tenna for TM slab mode excitation,” presented at the Proc. IEEE Antennaand Propagation and URSI Int. Symp., Boston, MA, July 2000.

[4] H. F. Hammad, Y. M. M. Antar, A. P. Freundorfer, and S. F. Mahmoud,“Uni-planar CPW-fed slot launchers for efficient TM0 surface wave ex-citation ,” IEEE Trans. Microwave Theory Tech, vol. 51, pp. 1234–1240,Apr. 2003.

[5] E. Bahar, “Scattering of VLF radio waves in curved earth-ionospherewaveguide,” Radio Sci., vol. 3, no. 2, pp. 145–154, 1968.

[6] V. V. Shevchenco, Continuous Transitions in Open Waveguides: TheGolem Press, 1971, ch. 1, sec. 1–7.

[7] S. F. Mahmoud and J. C. Beal, “Scattering of surface waves at a dielectricdiscontinuity on a planar waveguide,” IEEE Trans. Microwave TheoryTech., vol. MTT-23, pp. 193–198, 1975.

[8] S. F. Mahmoud, “Electromagnetic waveguides; theory and applications,”in IEE Electromagnetic Waves Series 32. Stevenage, U.K.: Peregrinus,1991, sec. 4.3.

[9] J. R. Wait, Electromagnetic Waves in Stratified Media. New York:Pergamon, 1970, ch. 6, pp. 33–35.

[10] J. A. Fuller and J. R. Wait, “A pulsed dipole in the earth,” J. Appl. Phys.,vol. 10, pp. 238–270, 1976.

[11] A. K. Bhattacharyya, “Characteristics of space and surface waves in amultilayered structure,” IEEE Trans. Antennas Propagat., vol. 38, pp.1231–1238, Aug. 1990.

[12] L. C. Shen and J. A. Kong, Applied Electromagnetism, 2nd ed. Boston,MA: PWS-Kent, 1987, sec. 8.3.

ACKNOWLEDGMENT

S. F. Mahmoud acknowledges the support of Kuwait Uni-versity for providing him with Sabbatical leave to perform thisresearch.

Samir F. Mahmoud (S’69–M’73–SM’83) graduatedfrom the Electronic Engineering Department, CairoUniversity, Cairo, Egypt, in 1964 and received theM.Sc. and Ph.D. degrees from the Electrical Engi-neering Department, Queen’s University, Kingston,ON, Canada, in 1970 and 1973, respectively.

During academic year 1973 to 1974, he was aVisiting Research Fellow at the Cooperative Institutefor Research in Environmental Sciences (CIRES),Boulder, CO, doing research on communication intunnels. He spent two sabbatical years, 1980 to 1982,

between Queen Mary College, London and the British Aerospace, Stevenage,U.K., where he was involved in design of antennas for satellite communication.Currently he is a Full Professor at the Electrical Engineering Department,Kuwait University. Recently, he has visited several places including Interuni-versity Micro-Electronics Centre (IMEC), Leuven, Belgium, and spent asabbatical leave at Queen’s University and the Royal Military College ofCanada, Kingston, ON Canada, from 2001 to 2002. His research activities havebeen in the areas of antennas, geophysics, tunnel communication, e.m waveinteraction with composite materials and microwave integrated circuits.

Dr. Mahmoud is a Fellow of the Institution of Electrical Engineers (IEE),London, U.K. He was a recipient of the Best IEEE/ Microwave Theory Tech-nology Paper for 2003.

Yahia M. M. Antar (S’73–M’76–SM’85–F’00) wasborn on November 18, 1946, in Meit Temmama,Egypt. He received the B.Sc. (Hons.) degree in 1966from Alexandria University, Egypt, and the M.Sc.and Ph.D. degrees from the University of Manitoba,Winnipeg, Canada, in 1971 and 1975, respectively,all in electrical engineering.

In 1966, he joined the Faculty of Engineeringat Alexandria University,where he was involved inteaching and research. At the University of Manitobahe held a University Fellowship, an NRC Postgrad-

uate and Postdoctoral Fellowships. From 1976 to 1977, he was with the Facultyof Engineering, University of Regina. In June 1977, he was awarded a VisitingFellowship from the Government of Canada to work at the CommunicationsResearch Centre, Department of Communications, Shirley’s Bay, Ottawa,where he was involved in research and development of satellite technology withthe Space Electronics group. In May 1979, he joined the Division of ElectricalEngineering, National Research Council of Canada, Ottawa, where he workedon polarization radar applications in remote sensing of precipitation, radio wavepropagation, electromagnetic scattering and radar cross section investigations.In November 1987, he joined the staff of the Department of Electrical andComputer Engineering, Royal Military College of Canada, Kingston, ON,Canada, where he is now a Professor of electrical and computer engineering.He is presently the Chairman of the Canadian National Commission (CNC),International Scientific Radio Union (URSI), holds adjunct appointment at theUniversity of Manitoba, and has a cross appointment at Queen’s Universityin Kingston. He has authored or coauthored over 100 journal papers on thesetopics, and supervised or cosupervised over 45 Ph.D. and M.Sc. theses at theRoyal Military College and Queen’s University, of which three have receivedthe Governor General Gold Medal. His current research interests includepolarization studies, integrated antennas, microwave, and millimeter wavecircuits.

Dr. Antar is a Fellow of the Engineering Institute of Canada (FEIC). Hereceived the 2003 RMC Excellence in Research Prize. In May 2002, he be-came the holder of a Canada Research Chair (CRC) in Electromagnetic Engi-neering. He is an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS

AND PROPAGATION and Associate Editor (Features) of the IEEE ANTENNAS AND


Hany F. Hammad received the B.Sc. degree withhonors from Ain Shames University, Cairo, Egypt, in1994 and the M.Sc. and Ph.D. degrees from Queen’sUniversity, Kingston, ON, Canada, in 1997 and 2002,respectively. His Ph.D. thesis was ranked as the “Out-standing Thesis of Engineering and Applied ScienceDivision” at Queen’s University.

His research areas of interests are the analysis anddesign of antennas and microwave integrated circuits.

Al P. Freundorfer (M’90) received the B.A.Sc.,M.A.Sc., and Ph.D. degrees from the Universityof Toronto, ON, Canada, in 1981, 1983, and 1989,respectively.

In 1990, he joined the Department of ElectricalEngineering, Queen’s University, Kingston, ON,Canada. Since then he has done work in nonlinearoptics of organic crystals, coherent optical networkanalysis as well as microwave integrated circuits.Currently he is focusing his attention on monolithicmicrowave circuits used in lightwave systems with

bit rates in excess of 20 Gb/s and on monolithic millimeter wave integratedcircuits used in wireless communications.


Generalized System Function Analysis of Exterior andInterior Resonances of Antenna and

Scattering ProblemsLong Li and Chang-Hong Liang, Senior Member, IEEE

Abstract—The generalized system function ( ), directly as-sociated with radiated and scattered fields, is presented to effec-tively analyze the exterior and interior resonances of antenna andscattering systems in this paper. ( ) is constructed by using themodel-based parameter estimation technique combined with thecomplex frequency theory. The behaviors of the exterior and inte-rior resonances can be distinguished by analyzing the character-istics of pole-zero of ( ) in a finite operational frequency band.The intensity of the exterior resonance can be effectively estimatedin terms of values and residues at the complex resonant frequen-cies. The truly scattered fields from a closed conducting region canbe obtained by eliminating the poles corresponding to the interiorresonances from ( ). Some examples of the practical antennaarrays and scattering systems are given to illustrate the applica-tion and validity of the proposed approach in this paper.

Index Terms—Generalized system function, exterior andinterior resonances, complex resonant frequency, factor,model-based parameter estimation (MBPE).

I. INTRODUCTION

WITH the increasingly complicated electromagnetic envi-ronment, the interaction and mutual coupling between

antennas and scatterers become more and more severe thatsometimes give rise to the strong electromagnetic oscillationphenomena. Therefore, the study of resonance behaviors in theelectromagnetic compatibility (EMC) has been an interestingand challenging problem for years [1]–[6]. The -field integralequation (EFIE) and the -field integral equation (MFIE)have been used extensively to analyze antennas radiation andscattering from perfectly conducting bodies. It is well knownthat bodies with closed conducting regions can support inte-rior resonance at certain discrete frequencies where both the

-field and -field integral equations fail to calculate thescattered (external) field [7]. Theoretically, the undeterminablecomponent of the surface current associated with the cavitymode does not radiate. However, due to truncation error andnumerical error effects, at these frequencies the cavity modeis both very weakly excited and radiated very weakly, so thematrix problem was found to have a different structure fromthat of the functional equation problem [8]. Some techniques

Manuscript received July 11, 2003; revised October 25, 2003. This work wassupported by the National Natural Science Foundation of China under Contract69931030.

The authors are with the School of Electronic Engineering, Xidian Uni-versity, Xi’an 710071, Shaanxi, China; (e-mail: [email protected];[email protected]).


have been proposed for dealing with these numerical problems[8]–[12]. Most of these methods generate a system of equationsthat has a unique solution for the current and external fields atall frequencies.

The complex resonant frequency presented in circuittheory [13], [14] is firstly introduced to antenna and scat-tering systems in this paper, which relates the real resonantfrequency with radiated or scattered losses. The generalizedsystem function, , directly associated with radiated andscattered fields, is presented to effectively analyze the exteriorand interior resonances of the antenna and scattering problemsin this paper, which is constructed by using model-based pa-rameter estimation (MBPE) technique. The MBPE [15]–[19]is a form of “smart” curve fitting, with broad applications toa fast analysis of radiation patterns or RCS of antennas orscatterers in a widely operating bandwidth. By analyzing thecharacteristics of poles and zeros of, we can determine theexterior and interior resonant frequencies of antenna and scat-tering systems efficiently. The complex frequency method forcalculating antenna or scattering external is also presentedin this paper. Furthermore, The exterior resonance strength canbe effectively estimated by the values of and residues at thecomplex resonant frequencies. The truly scattered fields froma closed conducting region can be got by eliminating the polescorresponding to the interior resonances from the generalizedsystem function. Some examples and discussion, paralleldipoles antenna, two conducting objects scattering system andan infinitely long elliptical cylinder scattering problem aregiven in this paper.

II. COMPLEX RESONANT FREQUENCY

For an arbitrary lossy resonant system, the complex resonantfrequency [13], [14] can be introduced and written as

(1)

where is a real resonant frequency of the system, re-pressents the losses of the resonant system. In general sense,the electric field can be written as

(2)

With the presence of the losses, the energy stored in the resonantsystem will decay at a rate proportional to the average energypresented at any time, so that

(3)

0018-926X/04$20.00 © 2004 IEEE

LI AND LIANG: GENERALIZED SYSTEM FUNCTION ANALYSIS OF EXTERIOR AND INTERIOR RESONANCES 2065

where is the average energy present at . But the rate ofdecrease of must equal the power loss, so that

(4)

In addition, an important parameter specifying selectivity, andperformance in general, of a resonant system is the qualityfactor, . A general definition of applicable to all resonantsystem is

(5)

Substituting (5) into (4), we can easily get

(6)

Therefore, the general expression of is

(7)

It can be seen that the introduction of unifies the resonantfrequency and of a resonant system, and each complexresonant frequency corresponds to one resonant mode. It iswell known that antenna or scattering system is essentiallyequivalent to a lossy network. Assume the system media arelossless, the loss represents the radiated or scattered powerfrom the antennas or scattering bodies. The average stored energy

denotes the sum of stored electric field and magnetic fieldenergies around the antennas or scatterers, which is independentof the radiated energies from the antennas or scatterers [20].Therefore, the complex resonant frequency is applicable tonot only the resonant cavity in the closed system, but also theantenna and scattering resonant problems in the open system. Ifthe system media are lossless, the complex resonant frequency

corresponding to a nonradiated mode (cavity mode) willreduce to the real resonant frequency .

III. GENERALIZED SYSTEM FUNCTION CONSTRUCTED

BY MBPE

The MBPE is a smart curve fitting technique [15]–[19], whichhas been widely applied to the fast analysis of radiation pat-terns or RCS of antennas or scatterers over a wide frequencyband. MBPE makes use of low-order analytical formulas as fit-ting models, while the unknown coefficients for the fitting modelare obtained by matching it to multipoint sampled values [18]or fitting it to frequency derivatives of the function at one ortwo frequencies [15]. In this paper, MBPE is mainly used toconstruct the generalized system function associated with elec-tromagnetic fields in the complex frequency domain. Accordingto the observed objects, one form of a fitting model that is com-monly employed in MBPE is represented by Padé rational func-tion as follows:

(8)

where and represent thecoefficients of numerator and denominator polynomials, respec-tively. Note that or can be normalized to 1 in denominatorcoefficients. Thus, (8) has unknown complexcoefficients. represents the complex frequency . It is ob-vious that MBPE utilizes the rational function approximationand extends it into complex frequency domain, which providesan appropriate tool for analyzing the resonance characteristicsof antenna and scattering systems from the point of view of com-plex frequency. According to the uniform approximation theory[21], the error of MBPE interpolation is minimum whenor , and the properties of existence and uniquenessof rational function approximation can be demonstrated [22].

Based on the theory of signals and systems, we know a partic-ularly important and useful class of linear time-invariant (LTI)systems is those for which the input and output satisfy a linearconstant-coefficient differential equation of the form [23]

(9)

where and represent the input and output time func-tions, respectively. Taking the Laplace transform of (9), weobtain

(10)is commonly referred to as the system function or, alterna-

tively, the transfer function. Many properties of LTI systems areclosely associated with the characteristics of the system functionin the plane. It is very interesting that (10) is consistent with (8)formed by MBPE in mathematical representation. In physicalsense, (10) represents the system function, which is the Laplacetransform of impulse response of LTI systems. In the analysis ofantenna or scattering electromagnetic systems, the ideal sourcemodels [24] of voltage, current, or unit plane wave are com-monly utilized as the excitation functions, and the frequency re-sponses of antenna properties, such as the current distribution

, input impedance , radiation patterns ,RCS, or near fields , etc., can be thought of asthe output functions. In this case, the output functions just cor-respond to the impulse responses of the antenna or scatteringsystem in time domain. If we make use of MBPE techniqueto approximate the output function frequency responses, (8) ischaracterized by the system function. Therefore, the general-ized system function directly associated with the radiatedor scattered fields can be constructed by MBPE technique ina limited operational bandwidth with a model containing a fi-nite number of suitably chosen complex poles, which describesthe intrinsic characteristics of the antenna or scattering systems.Equation (8) can be further factored into the form (let )

(11)


where is scale factor and are thecomplex poles and zeros of the generalized system function, re-spectively. We know the denominator polynomial of rep-resents the characteristic polynomial of antenna or scatteringsystems. The zeros of the denominator polynomial, namely thepoles of , define the locations of the natural resonances ofantenna or scattering systems, involving the exterior and inte-rior resonances. It is worth pointing out that only the exteriorresonance can be interpreted as intrinsic to the scatterer in prin-ciple. According to the stability of the electromagnetic systems,we know the true poles of the systems should reside in the lefthalf of the complex frequency plane. The validity of the com-plex poles obtained by (11) will be discussed in the followingsection. It is assumed that the poles are all simple. This has beennumerically substantiated. A partial fraction expansion yields

(12)

where represents the complex pole and is the corre-sponding residue. Therefore, many properties of antenna andscattering electromagnetic systems can be characterized by afew pole locations with the corresponding residues.

When the antenna and scattering systems are regarded as themultiport networks, assumed input ports and output ports,the generalized system function matrix can be similarly con-structed by MBPE based on the linear superposition principle,which can be expressed as

......

......

...

(13)

Namely

(14)

where represents the generalized system function ma-trix, and is referred to as the subsystem function. Alltrue poles of the generalized system function matrix define thenatural resonances of the antenna or scattering systems.is a square matrix when , and thus the poles are the solu-tions to the following:

(15)

where symbol indicates taking the determinant of matrix.While combining MoM with MBPE technique deals with theantenna or scattering problems, the generalized impedance ma-trix can be obtained in -plane in terms of the theory ofgeneralized networks [24], which describes the intrinsic charac-teristics of the system structures and is independent of the com-plicate excitations and loads. Therefore, the complex poles ofthe antenna or scattering systems are determined by

(16)

Fig. 1. Two parallel dipoles system.

Fig. 2. Frequency response of system function (far E-field) magnitude.

IV. EXTERIOR AND INTERIOR RESONANT FREQUENCIES AND

FACTOR

We know that a finite number of suitably chosen complexpoles of define the natural resonances of the antenna orscattering electromagnetic open systems. Assume the pole

, the corresponding partial fraction of the gen-eralized system function can be expressed as

(17)

The time response corresponding to the complex pole is

(18)

Comparing (18) with (2), we can see that the residue rep-resents the complex magnitude of the electric field at , ifthe complex frequency response of the electric field at onepoint in space is chosen as the generalized system function. Therelationship of the complex pole with the complex resonant fre-quency presented in the previous section is

(19)


TABLE IZEROS, POLES, AND RESIDUES OF THE GENERALIZED SYSTEM FUNCTION

Fig. 3. Frequency response of the near E-field magnitude.

Therefore, the resonant frequency and of the electromagneticopen system can be easily obtained

(20)

Obviously, by calculating the complex poles of the generalizedsystem function based on the physical models, we can di-rectly get the resonant frequency and corresponding . It iswell-known that the quantitative analysis of electric and mag-netic field energies stored in the near-field zone of the antennasor scatterers is very difficult to give, and thus the calculationof antenna or scattering external has also been an interestingand challenging problem for years [20], [25]–[27]. In this paper,the complex frequency method combined with the generalizedsystem function is used to calculate the antenna or scattering

efficiently, which has been illustrated by the later numericaltests.

For a scattering from a closed perfect conducting region, thetotal current flowing on the surface is not determined by EFIE orMFIE and the incident external field at the interior resonant fre-quencies. Theoretically, the undeterminable component of thesurface current that associated with the cavity mode does notradiate. Therefore, the poles corresponding to the interior res-onances should locate on the positive imaginary axis of theplane in principle. However, due to truncation error and numer-ical error effects, the cavity mode is both very weakly excitedand radiated very weakly [8]. So these poles do not strictly occuron the imaginary axis of the plane but reside in the left half ofthe plane off imaginary axis very small. To get the truly scat-tered field, these poles corresponding to the interior resonancesmust be eliminated.

Fig. 4. Comparison of the electric field magnitude distribution around dipolesat resonance with nonresonance (a) resonance and (b) nonresonance.

By analyzing the characteristics of poles and zeros of the gen-eralized system function and combining with adaptabilityof MBPE, we can accurately predict the occurrence of resonancephenomena and determine the exterior and interior resonant fre-quencies of the antenna and scattering systems. The intensity ofresonance can be effectively estimated by the values of andresidues at the complex resonant frequencies. Only when boththe external resonant and the residues are larger, are the res-onance phenomena characterized by the strong peak field in thenear region and large frequency sensitivity in the far field regionof the antennas and scattering bodies.

It should be pointed out that the complex poles referred abovemust be the true and stable poles of the antenna or scatteringelectromagnetic systems. A discussion on the validity of thepoles of the generalized system function constructed by MBPEis given here. On the one hand, according to the stability of thepractical antenna and scattering systems, these complex poles


must locate in the left half of the plane. Therefore, the poles oc-curred in the right half of the plane must be invalid poles of thesystem. On the other hand, in MBPE, if the Padé rational func-tions with different numerator and denominator ordersare used to construct the generalized system function, we mightget some different complex poles. One knows that the true polesare corresponding to the complex natural exterior resonant fre-quency of systems, which should be independent of the form offitting function and the orders of the rational function. Thus, thelocations of the true poles are stable or invariant. However, theother poles besides those residing in the right half of the planewill vary with the orders of the rational functions, which must beinvalid poles and referred to as “parasitical” poles. In addition,as previous discussion, the poles corresponding to the interiorresonances are not strictly occur on the imaginary axis of theplane but reside in the left half of the plane off imaginary axisvery small. The truly external scattered field can be obtained byeliminating these poles from (12).

V. APPLICATIONS AND DISCUSSION

In the following examples, some special resonance behaviorswill be analyzed by the generalized system functions directlyassociated with the radiated or scattered fields of some local re-gions, involving far fields and near fields. The results of numer-ical tests show that the exterior resonance phenomena are veryremarkable by virtue of the strong interaction and mutual cou-pling between antennas or scatterers, and the interior resonancebehaviors in EFIE give rise to a false scattered field.

A. Test 1 Parallel Dipole Antennas

Consider the two parallel dipoles system shown in Fig. 1.Dipole 1 will be excited by the ideal voltage source, and theterminal of dipole 2 shorted. The length both of them is

m, with radius m. The distance between themis m. The frequency response of the radiated electricfield of observation point at in the far zoneis chosen as the output function, i.e., the generalized systemfunction . The MBPE technique is applied to the antennassystem over a frequency range of 15–25 MHz, using the radiatedelectric field data obtained from a numerically rigorous methodof moments (MoM) computer codes based on EFIE. The Padérational function is chosen to set the numerator orderand the denominator order .

Fig. 2 shows the frequency response of the generalizedsystem function constructed by MBPE, with comparisonsbeing made of the MoM result. As can be seen from Fig. 2, thetwo curves are nearly graphically indistinguishable. In this case,only six sampling frequencies are required for the MBPE tech-nique. The actual sampling points that were used are indicatedby dots on the plots contained in Fig. 2. It is interesting thatall of the fitting frequencies are sampled before the resonantfrequency, but the resonant behavior can be found efficientlydue to the adaptability of MBPE technique. It is worth pointingout that MoM direct calculation using 102 unknowns took 28 sto calculate the frequency response at 100 frequency valuesfrom 15 to 25 MHz. MBPE took a total of 2 s to generate thesolutions with 0.1 MHz increment, poles and zeros.

Fig. 5. Two conducting bodies scattering system.

Fig. 6. Frequency responses of the generalized system function matrix(scattered near field. (a) E component and (b) E component.

The characteristics of zeros, poles, and correspondingresidues of the generalized system function are shown inTable I. Note that the data in the table have been transformedfrom to (MHz).

The facts show that there are two true and stable complexpoles in the antenna system within the finite operation frequencyband, which are marked by asterisks in the Table I. On thebasis of the complex frequency theory presented in the previoussection, from (20), the exterior resonant frequencies and ofthe two parallel dipoles system are obtained, respectively, asfollows:


TABLE IIZEROS, POLES, AND RESIDUES OF THE SUB-SYSTEM FUNCTION H (s)

TABLE IIIZEROS, POLES, AND RESIDUES OF THE SUB-SYSTEM FUNCTION H (s)

Fig. 7. Scattered electric field magnitude distributions inXZ and Y Z planeat resonance (18.169 MHz). (a) XZ plane and (b) Y Z plane.

It can be found that the exterior resonance behavior to occurat the frequency 19.6921 MHz with high , as shown inFig. 2. It is worth pointing out that the calculation of antenna

is definite and efficient by using the complex frequencymethod. To demonstrate the validity of , a classical formula

Fig. 8. Plane wave incident upon an infinite long perfect conducting ellipticalcylinder.

for finding presented in [20], [24], based on the Fosterreactance theorem

has been used to calculate the antenna of the two paralleldipoles system. Utilizing a first-order accurate difference ap-proximation to the partial frequency derivative of the reactancematrix , we obtain at the resonant frequency19.6921 MHz, which is very closed to the result of the complexfrequency method. It can be seen that the other resonant mode,18.4607 MHz, makes a little contribution to the resonance be-havior in this case for the low .

To further understand the behavior of the resonance, wecalculated the frequency response of the electric field magnitudeat the observation point in the vicinity of the dipole 1 indicatedby dot in Fig. 1. It can be seen that the behavior of the resonanceis also remarkably embodied by the phenomenon of strongpeak field in the near zone of the antenna system, as shown


TABLE IVZEROS, POLES, AND RESIDUES OF THE GENERALIZED SYSTEM FUNCTION (E )

in Fig. 3. The comparison of the electric field magnitudedistribution in the xz-plane in the near zone of the two paralleldipoles system at resonance with nonresonance is given inthe Fig. 4(a) and (b), respectively. It should be noted that theelectric fields are much stronger at resonance than those atnonresonance in the same excitation, but only accumulatingin the vicinity of the dipoles. The symmetric distribution ofthe electric fields magnitude at resonance implies the balanceof the electric field energy and magnetic field energy storedin the antenna open system physically.

B. Test 2 Two Conducting Bodies Scattering System

Consider the two perfectly conducting bodies scatteringsystem shown in Fig. 5, which is excited by the normalized

-polar plane wave , and the direction of prop-agation is . The sizes of two perfect conducting bodies are

m, m, m, andthe two conducting bodies are m apart. The scatteredelectric fields of the observation point, , inthe near zone of the scattering bodies are chosen as the outputfunctions. Because the scattered electric field in the near regionhas two main components and , the generalizedsystem function matrix can be constructed by MBPE techniquein two directions in order to analyze the exterior resonancecharacteristics of near fields. The Padé rational function ischosen to have the same numerator order and denomi-nator order . From Fig. 6, it can be seen that the solid linecalculated by MoM is mostly hidden by the MBPE curve.

The zeros, poles and residues of the subsystem functions ofand directions are calculated and shown in Tables II and III,

respectively.From Tables II and III, we found that there exist two true and

stable complex poles in the scattering system, which representthe external natural resonances and marked by asterisks respec-tively. Based on the theory of the complex frequency, we caneasily get the exterior resonant frequencies and scattering asfollows:

According to the values and the corresponding residues, wecan estimate that the resonance phenomenon of strong peak nearfield would appear at the frequency 18.169 MHz, as shown inFig. 6. The magnitude distributions of the scattered electric fieldin and plane at resonance (18.169 MHz) are shown inFig. 7(a) and (b), respectively. It is interesting that the scattered

Fig. 9. Frequency response of backscattered RCS.

electric fields accumulate mainly in the region between the twoconducting bodies, being strong at both sides and weak at center,with stand-wave-like distribution.

C. Test 3 Infinitely Long Elliptical Cylinder ScatteringProblem

Consider an infinite long perfect conducting ellipticalcylinder scattered by transverse magnetic (TM) plane waveincident in the direction, as shown in Fig. 8. The cross sectionis an ellipse with semimajor axis 1.0 meter and semiminor axis0.25 meter. A method of moments formulation of the EFIEis used to obtain the scattered electric field. We have utilizedpluses as expansion functions and delta functions as weightingfunctions. Using 180 unknowns, we numerically found twointerior resonances to occur at 327.33 and 435.42 MHz withinthe bandwidth from 200 to 500 MHz. The radar cross section(RCS) for backscatter as a function of the frequency for theelliptical cylinder is shown in Fig. 9. It can be seen that thebackscattered fields are uncorrected in the frequencies near theinterior resonances. The locally magnified figures show clearlythe RCS has a sharp dip at the two interior resonant frequencies.

We choose the frequency response of the backscattered elec-tric field as the generalized system function, . The MBPEtechnique is applied to the scattering problem over a frequencyrange 200–500 MHz, using the backscattered electric field data


Fig. 10. Comparison of the modified system function response with the resultscalculated by CFIE and EFIE: (a) Full frequency band, (b) near interior resonantfrequency 1, and (c) near interior resonant frequency 2.

obtained from the MoM formulation of EFIE, which are trans-formed into RCS and indicated by circles on the plots con-tained in Fig. 9. The Padé rational function is chosen to havea numerator order and a denominator order .In this case, the MBPE-calculated curve is excellent agreementwith the MoM result, including those in the vicinity of the in-terior resonances. It is worth pointing out that MoM requires avery larger number of evaluations in order to resolve these in-terior resonance behaviors. However, the MBPE with

and fitting to the sampling values accurately modelsboth resonances. MoM direct solution took 4175 s to calculatethe frequency response with 0.01 MHz increment from 200 to500 MHz. The MBPE technique took a total of 32 s to generatethe solutions, poles, and zeros. Table IV shows the zeros, poles,and corresponding residues of the generalized system function.

By analyzing the characteristics of the poles and combiningwith the complex frequency theory discussed previously, wefound two “true” complex poles occur in the left half of theplane but off the imaginary axis very small, which just definethe interior resonant frequencies and are marked by asterisksin the Table IV. It can be seen the residues corresponding to thetwo poles are very small, which imply the scattered contributionfrom the interior resonances should become especially small. Aspointed out in [8], the scattered contribution results from the nu-merical error in EFIE. To get the truly scattered field, we modifythe generalized system function by eliminating those poles cor-responding to the interior resonances from (12). The comparisonof the modified system function response with the results calcu-lated by the combined field integral equation (CFIE) is shownin Fig. 10. The two curves are nearly graphically indistinguish-able, and the phenomena of the interior resonance are removedsuccessfully.

VI. CONCLUSION

This paper has presented the generalized system functionwhich directly associate with radiated and scattered fields

to give an efficient analysis of the exterior and interior reso-nances of antenna and scattering problems. is constructedby using the MBPE technique combined with the complexfrequency theory. The behaviors of the exterior and interior res-onances can be distinguished by analyzing the characteristicsof pole-zero of in a finite operational frequency band.The exterior complex resonant frequencies must reside in theleft half of the plane off the imaginary axis. The imaginarypart of is related to the radiated or scattering losses, i.e.,

. The intensity of the exterior resonance can be estimatedeffectively in terms of values and residues at the complexresonant frequencies. The interior resonant frequencies occuron the positive imaginary axis of the plane theoretically, butdue to truncation error and numerical error effects, the internal(cavity) modes are both very weakly excited and radiatedvery weakly. Therefore, those poles corresponding to interiorresonances also locate in the left half of the plane but offimaginary axis very small. It is shown that only exterior polescan be interpreted as intrinsic to the scatterer. The truly scat-tered fields from a closed conducting region can be obtainedsimply by eliminating the poles corresponding to the interiorresonances from the generalized system function.

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[22] P. J. Davis, Interpolation and Approximation. London, U.K.: Blaisdell,1963.

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Long Li was born in Anshun, Guizhou, China, inJanuary 1977. He received the B.Eng. degree inelectromagnetic field and microwave technologyfrom Xidian University, Xi’an, China, in 1998.Since 1999, he has taken a combined Master-Doctorprogram and is working toward the Ph.D. degreein the National Key Laboratory of Antenna andMicrowave Technology, Xidian University.

His research interests include computational elec-tromagnetics, slot antenna array, hybrid algorithmsand electromagnetic compatibility.

Chang-Hong Liang (M’80–SM’83) was born inShanghai, China, in December 1943. He gradu-ated from Xidian University (Formerly NorthwestTelecommunications Institute), Xi’an, China, in1965, and continued his graduate studies until 1967.

From 1980 to 1982, he worked at Syracuse Uni-versity, New York, as a Visiting Scholar. Since 1986,he has been a Professor and Ph.D. student advisorin the School of Electronic Engineering, Xidian Uni-versity, where he is also a Director of the AcademicCommittee of National Key Lab of Antenna and Mi-

crowave Technology. He has published numerous papers and proceeding arti-cles, is the author of five books. He is an Editor-in-Chief of the Journal of XidianUniversity. He has wide research interests, which include computational mi-crowave and computational electromagnetics, microwave network theory, mi-crowave measurement method and data processing, lossy variational electro-magnetics, electromagnetic inverse scattering, electromagnetic compatibility.

Prof. Liang is a Fellow of the Chinese Institute of Electronics (CIE), and hasreceived the titles of “National Distinguished Contribution,” “National Excel-lent Teacher,” etc.


MIMO Wireless Communication ChannelPhenomenology

Daniel W. Bliss, Member, IEEE, Amanda M. Chan, and Nicholas B. Chang

Abstract—Wireless communication using multiple-input mul-tiple-output (MIMO) systems enables increased spectral efficiencyand link reliability for a given total transmit power. Increasedcapacity is achieved by introducing additional spatial channelswhich are exploited using space-time coding. The spatial diversityimproves the link reliability by reducing the adverse effects oflink fading and shadowing. The choice of coding and the resultingperformance improvement are dependent upon the channelphenomenology. In this paper, experimental channel-probingestimates are reported for outdoor environments near the per-sonal communication services frequency allocation (1790 MHz).A simple channel parameterization is introduced. Channel dis-tance metrics are introduced. Because the bandwidth of thechannel-probing signal (1.3 MHz) is sufficient to resolve somedelays in outdoor environments, frequency-selective fading is alsoinvestigated. Channel complexity and channel stationarity are in-vestigated. Complexity is associated with channel-matrix singularvalue distributions. Stationarity is associated with the stability ofchannel singular value and singular vector structure over time.

Index Terms—Channel coding, information theory, multipathchannels, multiple-input multiple-output (MIMO) systems.

I. INTRODUCTION

MULTIPLE-INPUT multiple-output (MIMO) systems area natural extension of developments in antenna array

communication. While the advantages of multiple receive an-tennas, such as gain and spatial diversity, have been known andexploited for some time [1]–[3], the use of transmit diversityhas been investigated more recently [4], [5]. Finally, the advan-tages of MIMO communication, exploiting the physical channelbetween many transmit and receive antennas, are currently re-ceiving significant attention [6]–[8]. Because MIMO commu-nication capacity is dependent upon channel phenomenology,studying and parameterizing this phenomenology is of signifi-cant interest [9]–[19].

This paper makes a number of contributions to this areaof study. First, while most experimental results have focusedon indoor phenomenology, the phenomenology investigatedhere focuses on outdoor environments. Second, results forboth stationary and vehicle-mounted moving transmitters are

Manuscript received March 19, 2003; revised September 27, 2003. This workwas supported by the U.S. Air Force under Air Force Contract F19628-00-C-0002.

D. W. Bliss and A. M. Chan are with Advanced Sensor TechniquesGroup, MIT Lincoln Laboratory, Lexington, MA 02420-9185 USA (e-mail:[email protected], [email protected].

N. Chang is with the Department of Electrical Engineering and ComputerScience, University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail:[email protected]).


reported. Third, experimental phenomenological results are re-ported for both 4 4 and relatively large 8 8 MIMO systems,including channel stationarity, both in time and frequency.Fourth, two metrics of channel variation are introduced. Onemetric provides a measure of capacity loss assuming thatreceiver beamformers are constructed using incorrect channelestimates, which is useful to determine performance lossesdue to channel nonstationarity (either in time or frequency).The other metric is sensitive to the shape of the channel eigen-value distribution, which is appropriate for space-time codingoptimization, assuming a uniformed transmitter (UT) (thatis transmitters without channel state information). Finally, asimple channel parameterization is provided which empiricallymatches channel eigenvalue distributions well and provides asimple approach to generate representative simulated channelsfor space-time coding optimization.

MIMO systems provide a number of advantages over single-antenna communication. Sensitivity to fading is reduced by thespatial diversity provided by multiple spatial paths. Under certainenvironmental conditions, the power requirements associatedwith high spectral-efficiency communication can be significantlyreduced by avoiding the compressive region of the informationtheoretic capacity bound. This is done by distributing energyamongst multipath modes in the environment. Spectral efficiencyis defined as the total number of bits per second per Hztransmitted from one array to the other. Because MIMO systemsuse antenna arrays, interference can be mitigated naturally.

In this paper, outdoor MIMO channel phenomenology nearthe PCS frequency allocation, 1.79 GHz, is discussed. Thechannel-probing signal has a bandwidth of 1.3 MHz. Thisbandwidth is sufficient to resolve some delays, inducing fre-quency-selective fading in outdoor environments. In Sections IIand III, information theoretic capacity of MIMO communi-cation systems and channel estimation are reviewed. Channeldifference metrics are introduced in Section IV. Performanceof MIMO communication systems and optimal selectionof space-time coding are dependent upon the complexityof the channel [20], [21]. This phenomenology for outdoorenvironments is investigated using MIMO channel-probingexperiments. The results are interpreted using a simple parame-terization introduced in Section V. The channel phenomenologyexperiments are described in Section VI, and the experimentalresults, reporting estimates of channel complexity and station-arity, are discussed in Section VII.

II. CHANNEL CAPACITY

The information theoretic capacity of MIMO systems hasbeen discussed widely [6]–[8]. It is assumed for the sake of the

0018-926X/04$20.00 © 2004 IEEE


following discussion that the receiver can accurately estimate apseudostationary channel. Given this assumption, there are twotypes of spectral-efficiency bounds: informed transmitter (IT)and UT, depending on whether or not channel estimates are fedback to the transmitter.

For narrowband MIMO systems, the coupling between thetransmitter and receiver can be modeled using

(1)

where is the (number of receive by transmitantenna) channel matrix, containing the complex attenuationbetween each transmit and receive antenna, is anmatrix containing the samples of the transmit array vector,

is an matrix containing the samples of thecomplex receive-array output, and is an matrixcontaining zero-mean complex Gaussian noise. It is often usefulto investigate the structure of the channel matrix and themean-square attenuation independently. This can be achievedby studying the root-mean-square normalized channel matrix

(2)

(3)

where is the mean-square transmitter-to-receiver attenuation,is the normalized channel matrix, and indicates the

Frobenius norm.

A. IT

There are a variety of possible transmitter constraints. Hereit is assumed that the fundamental limitation is the total powertransmitted. The optimization of the noise-nor-malized transmit covariance matrix is constrained by thetotal noise-normalized transmit power . Allowing differenttransmit powers at each antenna, this constraint can be enforcedusing the form . The results of the channel-spec-tral-efficiency bounds discussions presented in [8] are repeatedhere. The capacity can be achieved if the channel is known byboth the transmitter and receiver, giving

(4)

where the notation indicates determinant, indicates Her-mitian conjugate, and indicates an identity matrix of size

. Solving for the optimal , the resulting capacity is givenby

(5)

where is an diagonal matrix with entries, whose values are the top eigenvalues

of . The values must satisfy

(6)

If (6) is not satisfied for some , it will not be satisfied for anysmaller .

B. UT

If the channel is not known at the transmitter, then the optimaltransmission strategy is to transmit equal power with each an-tenna, , [7]. Assuming that the receiver canaccurately estimate the channel, but the transmitter does not at-tempt to optimize its output to compensate for the channel, themaximum spectral efficiency is given by

(7)

This is a common transmit constraint as it may be difficult toprovide the transmitter channel estimates.

Similarly to the IT case, the UT spectral-efficiency boundis purely a function of the channel-matrix singular values. Ex-pressing the channel matrix with a singular vector decomposi-tion, , the capacity is a function of eigenvalues, butnot of the eigenvectors, of

(8)

where the singular-value entries of the diagonal matrix aregiven by .

C. Frequency-Selective Channels

In environments where there is frequency-selective fading,the channel matrix is a function of frequency . As has beendiscussed in [22], the resulting capacity is a function of thisfading structure. Exploiting the fact that frequency channels areorthogonal, the capacity in frequency-selective fading can becalculated using an extension of (5) and (7). For the UT, thisleads to the frequency-selective spectral-efficiency bound

(9)

where the distance between frequency samples is given by ,and -bin frequency-partitioned channel matrix is given by

. . .(10)

The approximation is exact if the supported delay range of thechannel is sampled sufficiently.

BLISS et al.: MIMO WIRELESS COMMUNICATION CHANNEL PHENOMENOLOGY 2075

For the IT channel capacity, power is optimally distributedamongst both spatial modes and frequency channels. The ca-pacity can be expressed

(11)

which is maximized by (5) with the appropriate substitutionsfor the frequency-selective channel, and diagonal entries inin (6) are selected from the eigenvalues of . Because ofthe block diagonal structure of , thespace-frequency noise-normalized transmit covariance matrix

is a block diagonal matrix, normalized so that .

III. ESTIMATION

The Gaussian probability density function for a multivariate,signal-in-the-mean, statistical model of the received signal isgiven by

(12)

where is the noise covariance matrix. The maximum-likeli-hood estimate of is given by

(13)

assuming that the reference signals in are known andis nonsingular. As one might intuit from the structure of (13),if a signal’s sole use is channel estimation, then the choice of

, such that is proportional to the identity matrix (that isequal-power orthogonal signals) is optimal for channel probingin finite signal-to-noise ratio (SNR) environments. However, ifjoint channel and signal detection is used, then orthogonal sig-nals are not necessarily optimal for link performance.

The previous channel-estimation discussion explicitly as-sumed flat fading. However, the frequency-selective channelscan be estimated by first estimating a finite impulse-responseMIMO channel which can be transformed to the frequencydomain.

A finite impulse-response extension of (1) is given by intro-ducing delayed copies of at delays

...(14)

so that the transmit matrix has dimension . Theresulting wideband channel matrix has the dimension

(15)

Using this form, an effective channel filter is associated witheach transmit-to-receive antenna link. Assuming regular delay

sampling, the explicit frequency-selective form can be con-structed using a discrete Fourier transform

(16)

or equivalently

(17)

where the -point discrete Fourier transform is represented byand the Kronecker product is represented by .

IV. CHANNEL DIFFERENCE METRICS

A variety of metrics are possible. Here, two metrics are dis-cussed. Both metrics are ad hoc, but are motivated by limitingforms of the information theoretic capacity.

The first metric, discussed in Section IV-A, is sensitive to thedifferences in channel eigenvalue distributions. While there arean unlimited number of channel eigenvalue distributions thatcan provide a particular capacity, for a given mean channel at-tenuation and power, performance of space-time codes is sensi-tive to the shape of the distribution. Because the optimization ofUT space-time codes depends upon the eigenvalue distributionbut not the eigenvector structure, the metric introduced in Sec-tion IV-A is an appropriate metric for investigating this issue.Specifically, space-time codes must select a rate versus redun-dancy operating point [20], [21]. The optimal operating pointis a function of the shape of the channel eigenvalue distribution.This metric is used to estimate the channel parameter introducedin Section V-C.

The second metric, discussed in Section IV-B, is sensitiveto differences in both the singular-value distribution and thechannel eigenvector structure. In general, MIMO receiversemploy some sort of beamformer to coherently combine thesignals impinging upon each receive antenna. In dynamicenvironments (either in time or frequency) channel estimatescan quickly become inaccurate. A measure of the adverseeffects of using these “stale” estimates is provided by thismetric. Effectively, this metric provides a measure of thefractional capacity loss in the low SNR (or equivalently lowspectral-efficiency) limit. Because performance in the low SNRlimit is not affected by interference introduced by the othertransmit antennas, MIMO systems operating at higher SNR willexperience greater interference and thus worse performance.Consequently, this metric is an optimistic estimate of theexpected performance due to dynamic channels.

A. Eigenvalue-Based Metric

As was mentioned in Section II, MIMO capacity is only afunction of the channel singular values. Equivalently, capacityis invariant under channel-matrix transformations of the form

(18)


where and are arbitrary unitary matrices. Conse-quently, for some applications it is useful to employ a metricwhich is also invariant under this transformation. Because ca-pacity is a function of the structure of the channel singular-valuedistribution, the metric should be sensitive to this structure.

The channel capacity is a function of . A natural metricwould employ the distance between the capacity for two channelmatrices at the same average total received power, that is, thesame

(19)

However, there are two problems with this definition. First, thedifference is a function of . Second, there is degeneracy in

singular values that gives a particular capacity. To addressthe first issue the difference can be investigated in a high SNRlimit, resulting in

(20)

where indicates the th largest eigenvalue of . To in-crease the sensitivity to the shape of the eigenvalue distribution,the metric is defined to be the Euclidean difference, assumingthat each eigenvalue is associated with an orthogonal dimension,resulting in

(21)

B. Fractional Receiver-Loss Metric

In this section a power-weighted mean metric is intro-duced. The metric takes into account both the eigenvalue andeigenvector structure of the channels. It is motivated by the ef-fect of receive beamformer mismatch on capacity. Starting with(7), the low SNR UT capacity approximation is given by

(22)

where is the column of the channel matrix associated withtransmitter . In the low SNR limit, the optimal receive beam-former is given by the matched response given in . If someother beamformer is employed, , then signal energy is lost,adversely affecting the capacity

(23)

One possible reason that a beamformer might use the wrongmatched spatial filter is channel nonstationarity. Assuming theSNR is sufficiently low, the fractional capacity loss is given by

(24)

which is the power-weighted mean estimate, whereis defined to be the inner product between the “good”

and “bad” unit-norm array responses for the th transmitter. Itis generally desirable for metrics to be symmetric with respectto and , thus avoiding moral attributions with regard tochannel matrices. Using the previous discussion as motivation,a symmetric form of fractional capacity loss is given by

(25)

where the “power-weighted” average is evaluated overtransmitters.

The metric presented in (25) provides an estimate of the lossin capacity if the incorrect channel is assumed in a low SNR en-vironment. In general, the loss of capacity is much more signifi-cant if operating in a high spectral efficiency, and therefore highSNR regime. If only spatial mitigation is employed (as opposedto a combination of spatial processing and multiuser detection[23], [24]), a slight channel mismatch will introduce significantinterference, and thus strongly adversely affect demodulationperformance.

V. CHANNEL PHENOMENOLOGY

A. Singular Values

The singular-value distribution of , or the related eigen-value distribution of , is a useful tool for understandingthe expected performance of MIMO communication systems.From the discussion in Section II, it can be seen that thechannel capacity is a function of channel singular values, butnot the singular-vector structure of the channel. Thus, channelphenomenology can be investigated by studying the statisticsof channel singular-value distributions.


B. Channel Parameterization

A commonly employed model assumes the channel is pro-portional to a matrix, , where the entries are independentlydrawn from a unit-norm complex circular Gaussian distribu-tion. While the distribution is convenient, it does suffer froma singular-value distribution that is overly optimistic for manyenvironments. One solution is to introduce spatial correlationsusing the transformation [10]–[12]. Whilethis approach is limited [8], it produces simply more realisticchannels than the uncorrelated Gaussian model.

The spatial correlation matrices can be factored so thatand , where and are unitary

matrices, and and are positive semidefinite diagonalmatrices. When the arrays are located in environments that aresignificantly different, then correlations seen by one array willtypically be much stronger than the other, and the effect of eitherthe left or right will dominate the shaping of the channel ma-trix singular-value distribution. Conversely, if the environmentsare similar then one would expect that . In prac-tice, similar channel matrix singular-value distributions can beachieved given either assumption. However, the required valuesof and , are, of course, different. For the experiments dis-cussed in this paper, both arrays are in similar environments anda symmetric form seems a reasonable model. Assuming that thenumber of transmit and receive antennas are equal and have sim-ilar spatial correlation characteristics, the diagonal matrices canbe set equal, , producing the new randomchannel matrix

(26)

(27)

where is used to set overall scale, is given by the size of, and and indicate random unitary matrices. The form

of given here is somewhat arbitrary, but has the satisfyingcharacteristics that as a rank-one channel matrix is pro-duced, and as a spatially uncorrelated Gaussian matrixis produced; thus, the parameterization can easily approximate,in a statistical sense, nearly all environments. This stochasticchannel parameterization has the advantage that it is not depen-dent upon the particular causes of the correlation, or details ofthe arrays or environment. The normalization for is chosenso that the expected value of is .

The model can be related to the ergodic or mean capacity [8](averaged over an ensemble channel). Exploiting the fact thatMIMO capacity is convex cap, a bound on the mean capacity isgiven by

(28)

This bound is not necessarily tight, but is useful for illustratingthe effects of channel parameter value on capacity. In Fig. 1,

Fig. 1. Ratio of bounds on mean UT capacity of � = 0:2, 0.4, 0.6 to � = 1.

the ratio of capacity bounds for 0.2, 0.4, and 0.6 for a 4 4MIMO system is displayed. In practice, the ratio of bounds tendsto produce slightly optimistic capacity results at values of . Theessential features are accurate. Assuming that the space-timecoding takes the channel statistics into account for values of0.6 or greater, performance loss is not overwhelming. A secondinteresting feature is that at very high SNR the ratio of capacityslowly approaches 1. This is because at very high SNR evenstrongly attenuated channel modes become useful. Modelingapproaches that introduce reduced “effective” numbers of an-tennas do not reproduce this phenomenon well.

C. Channel Parameter Estimation

An estimate for associated with particular transmit and re-ceive locations is given by minimizing the mean-square metricgiven in (21)

(29)

where indicates the estimated value of . Here the expecta-tion, denoted by , indicates averaging is over an ensembleof for a given and an ensemble of for given transmit andreceiver sites.

It is worth noting that this approach does not necessarily pro-vide an unbiased estimate of . Estimates of , using the metricintroduced in here, are dependent upon the received SNR. Toreduce the bias, one can add complex Gaussian noise to

to produce , mimicking the integrated SNR of the estimateof .

Data presented here has sufficiently high SNR such that canbe estimated within .

VI. EXPERIMENT

The experimental system employed is a slightly modifiedversion of the Massachusetts Institute of Technology (MIT)Lincoln Laboratory system used previously [3], [23], [25]–[27].The transmit array consists of up to eight arbitrary waveformtransmitters. The transmitters can support up to a 2 MHzbandwidth. These transmitters can be used independently, astwo groups of four coherent transmitters or as a single co-herent group of eight transmitters. The transmit systems canbe deployed in the laboratory or in vehicles. When operatingcoherently as a multiantenna transmit system, the individual


TABLE ILIST OF TRANSMIT SITES

transmitters can send independent sequences using a commonlocal oscillator. Synchronization between transmitters and re-ceiver and transmitter geolocation is provided by GPS receiversin the transmitters and receivers.

The MIT Lincoln Laboratory array receiver system is a high-performance 16-channel receiver system that can operate overa range of 20 MHz to 2 GHz, supporting a bandwidth of up to8 MHz. The receiver can be deployed in the laboratory or in astationary “bread truck.”

A. MIT Campus Experiment

The experiments were performed during July and August2002 on and near the MIT campus. These outdoor exper-iments were performed in a frequency allocation near thePCS band (1.79 GHz). The transmitters periodically emitted1.7 s bursts containing a combination of channel-probing andspace-time-coding waveforms. A variety of coding and inter-ference regimes were explored for both moving and stationarytransmitters. The space-time-coding results are beyond thescope of this paper and are discussed elsewhere [23], [24].Channel-probing sequences using both four and eight transmit-ters were used.

The receive antenna array was placed on top of a tall one-storybuilding (Brookline St. and Henry St.) surrounded by two- andthree-story buildings with a parking lot along one side. Differentfour or eight antenna subsets of the 16-channel receiver wereused to improve statistical significance. The nearly linear re-ceive array had a total aperture of less than 8 m, arranged as threesubapertures of less than 1.5 m each. The transmit arrays werelocated on the top of vehicles within 2 km of the receive array.On each vehicle four antennas were approximately located atthe vertices of a square, with separation of greater than two tothree wavelengths. When operating as an eight-element trans-mitter, two adjacent parked vehicles were used, connected by acable that distributed a local oscillator signal.

The channel-probing sequence supports a bandwidth of1.3 MHz with a length of 1.7 ms repeated ten times. Allfour or eight transmitters emitted nearly orthogonal signalssimultaneously.

Fig. 2. Scatter plot of mean-squared SISO link attenuation, a , versus linkrange for the outdoor environment near the PCS frequency allocation. The errorbars indicate a range of �1 standard deviation of the estimates at a given site.

Fig. 3. CDF of channel a estimates, normalized by the mean a for each site,for SISO, 4� 4 and 8� 8 MIMO systems.

VII. EXPERIMENTAL RESULTS

Channel-complexity and channel-stationarity performanceresults are presented in this section. A list of transmit sitesused for these results is presented in Table I. The table includesdistance between transmitter and receiver, velocity of trans-mitter, the number of transmit antennas, and the estimatedfor the transmit site. Uncertainty in is determined using thebootstrap technique [28]. Cumulative distribution functions(CDF) reported here are evaluated over appropriate entriesfrom Table I. The systematic uncertainty in the estimation ofcaused by estimation bias, given the model, is less than 0.02.

A. Attenuation

The peak-normalized mean-squared single-input single-output (SISO) attenuation (path gain) averaged over transmitand receive antenna pairs for a given transmit site is displayedin Fig. 2 for the outdoor environment. The uncertainty in theestimate is evaluated using a bootstrap technique.

B. Channel Complexity

Channel complexity is presented using three different ap-proaches. Variation in estimates, eigenvalue CDFs, andestimate CDFs are presented.

In Fig. 3, CDFs of estimates nor-malized by mean for each transmit site are displayed. CDFsare displayed for narrowband SISO, 4 4, and 8 8 MIMOsystems. As one would expect, because of the spatial diversity,the variation in mean antenna-pair received power decreasesdramatically as the number of antenna pairs increases. This


Fig. 4. CDF of narrowband channel eigenvalue distributions for 4� 4 MIMOsystems: (a) simulated Gaussian channel and (b) experimental results.

demonstrates one of the most important statistical effects thatMIMO links exploit to improve communication link robustness.For example, if one wanted to operate with a probability of 0.9to close the link, one would have to operate the SISO link withan excess SISO SNR ( ) margin of over 15 dB. The MIMOsystems received the added benefit of array gain, which is notaccounted for in the figure.

In Figs. 4 and 5, CDFs of eigenvalues are presented for 4 4and 8 8 mean-squared-channel-matrix-element-normalizednarrowband channel matrices, . Both simulatedGaussian channels and experimental results are displayed. Su-perficially, the distributions of the simulated and experimentaldistributions are similar. However, closer inspection reveals thatthe experimental distributions cover a greater range of eigen-values. This is the result of the steeper channel-eigenvaluesdistribution that is observed in the experimental data comparedto the simulated Gaussian channel. The experimental CDFs areevaluated over all site lists. Some care must be taken in inter-preting these figures because eigenvalues are not independent.Nonetheless, the steepness of the CDFs is remarkable. Onemight interpret this to indicate that optimized space-time codesshould operate with a relatively high probability of success.

The CDFs for estimates are presented in Fig. 6. The meanvalues of for each environment are

While one might expect smaller variation in the 8 8 systemsbecause of the much larger number of paths, this effect may havebeen exaggerated in Fig. 6 because of the limited number of8 8 sites available in the experiment.

Fig. 5. CDF of narrowband channel eigenvalue distributions for 8� 8 MIMOsystems: (a) simulated Gaussian channel and (b) experimental results.

Fig. 6. CDF of � estimates for 4� 4 and 8� 8 MIMO systems.

The values of the channel-complexity parameter, , are, ofcourse, dependent upon the details of the environment andthe geometry of the transmit and receive arrays. As can beseen in Table I, the values of vary from one transmit siteto another transmit site. Furthermore, one would anticipatesignificantly different values of in unlike environments,such as the open plains of the Midwest or in highly elevatedtowers. The dependency upon array geometries is somewhatless clear. Because the arrays employed in this experimentare spatially undersampled, the received signal experiencessignificant spatial aliasing. Increasing the array aperture mayhelp resolve closely spaced scatterers; this occurs at the expenseof folding other widely spaced scatterers back on similararray responses. Consequently, while perturbations in arraygeometries certainly affect particular received signals, theseperturbations are not expected to affect strongly the statisticalproperties of the channel; thus values of are not expectedto be a strong function of array geometry.


Fig. 7. Eigenvalues, �, of HH as a function of time for (a) stationary and(b) moving transmitters. The same overall attenuation, estimated at t = 0, isused for all time samples.

This point can be demonstrated by constraining the choiceof receive antennas used for calculating . By excluding or re-quiring antennas to be consecutive and calculating under theseconstraints, sensitivity to antenna separation can be investigated.In the following table, three sets of constraints are implemented.In the first column, all receive antennas are used. In the secondcolumn, employed antennas are separated by at least two un-used antennas. In the third column, only consecutive antennasare used.

There is a slight bias for greater antenna separation to producelarger values of , which is consistent with the expectation thatgreater antenna separation produces more random channel ma-trices. However, this trend is very subtle, and in all cases, theresults are statistically consistent with being independent of an-tenna separation at these relatively large antenna separations.

C. Channel Stationarity

The temporal variation of eigenvalues of for stationaryand moving transmitters is displayed in Fig. 7. In this figure thenormalization is fixed, allowing for overall shifts in attenuation.As one would expect, the eigenvalues of the moving transmittervary significantly more than those of the stationary environment.However, the eigenvalues of the stationary transmitter do varysomewhat. While the transmitters and receivers are physicallystationary, the environment does move. This effect is particu-larly noticeable near busy roads. Furthermore, while the mul-tiple antennas are driven using the same local oscillator, given

Fig. 8. Example time variation of power-weighted mean cos �, fH(t );H(t)g, for stationary and moving 4� 4 MIMO systems.

Fig. 9. CDF of time variation of power-weighted mean cos �, fH(t );H(t)g, for stationary 4� 4 MIMO system. Contours of CDFprobabilities of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 are displayed.Because there is little variation, all curves are compressed near of 1.

the commercial grade transmitters, there are always some smallrelative-frequency offsets. The example variation is given fortransmit sites #7 and #14.

While the moving-transmitter eigenvalues fluctuate morethan those of the stationary transmitter, the values are re-markably stable in time. Conversely, an example of the timevariation of the power-weighted mean metric [from(25)], displayed in Fig. 8, varies significantly for the movingtransmitter within 10 ms. This indicates that the eigenvectorstructure varies significantly, while the distribution of eigen-values tends to be more stable. In the example, the stationarytransmitter is located at site #7, and the moving transmitter islocated at #14. Over the same period, the stationary transmitteris relatively stable. CDFs for stationary and moving transmittersare displayed in Figs. 9 and 10. In the figures, 4 4 MIMOexperiment sites with a speed less than or equal to 0.2 m/s wereconsidered to be “stationary” (sites: 7, 9, 16, and 18), and thosewith speeds greater than 5 m/s were considered to be “moving”(sites: 10, 12–15, 17).

As was discussed in Section IV-B, the performance implica-tions of a particular value of depend upon the operating SNRand the receiver design. At low SNR, the fractional UT capacityloss due to receiver mismatch is given directly by the value of

. At high SNR, if interference mitigation is primarily achievedthrough spatial antenna processing, then the performance losscan be significantly worse. This is because contamination frominterfering transmit antennas is allowed to overwhelm the in-tended signals at the outputs of inaccurate beamformers. Fur-thermore, the significant variation of the moving transmitter isan indication that implementing an IT MIMO system would be


Fig. 10. CDF of time variation of power-weighted mean cos �, fH(t );H(t)g, for moving 4� 4 MIMO system. Contours of CDFprobabilities of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 are displayed.

Fig. 11. Example of frequency-selective variation of the power-weightedmean cos �, fH(f );H(f)g.

Fig. 12. CDF of frequency-selective variation of the power-weighted meancos �, fH(f );H(f)g. Contours of CDF probabilities of 0.1, 0.2, 0.3, 0.4,0.5, 0.6, 0.7, 0.8, and 0.9 are displayed.

very challenging for the moving transmitter, but might be viablefor some stationary MIMO systems.

D. Frequency-Selective Fading

An example of the frequency variation of the power-weightedmean is displayed in Fig. 11. The variation is indicatedusing the metric presented in (25). In the example, the stationarytransmitter is located at site #7. Relatively small frequencyoffsets induce significant changes in . TheCDF of the frequency-selective channel variation is displayedin Fig. 12 (using sites: 7, 9, 16, and 18). This sensitivity indi-cates that there is significant resolved delay spread and that tosafely operate using the narrowband assumption, bandwidthsless than 100 kHz should be employed. It is worth noting that

delay spread, and the resulting frequency-selective fading, isboth a function of environment and link length. Consequently,some care must be taken in interpreting this result.

VIII. SUMMARY

In this paper, outdoor MIMO channel phenomenology wasdiscussed. Data from an experiment performed on and nearthe MIT campus was used to study the phenomenology. Thephenomenology was investigated from the perspective of thesingular-value distributions of the channel matrices. A channelparameterization approach was introduced. Two channel-dif-ference metrics were introduced. The first was used to estimatethe channel parameter. The second metric was employed todemonstrate significant channel variation both as a functionof time and frequency.

ACKNOWLEDGMENT

Opinions, interpretations, conclusions, and recommendationsare those of the authors and are not necessarily endorsed by theUnited States Government.

The authors would like to thank the excellent MIT LincolnLaboratory staff involved in the MIMO experiment, in par-ticular S. Tobin, J. Nowak, L. Duter, J. Mann, B. Downing,P. Priestner, B. Devine, T. Tavilla, A. McKellips and G. Hatke.The authors would also like to thank the MIT New TechnologyInitiative Committee for their support. The authors would alsolike to thank K. Forsythe, A. Yegulalp, and D. Ryan of MITLincoln Laboratory and V. Tarokh of Harvard University fortheir thoughtful comments. The authors would like to thankN. Sunkavally of MIT for his contributions to the experimentand the analysis.

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[14] M. Stoytchev and H. Safar, “Statistics of the MIMO radio channel inindoor environments,” in Proc. IEEE Vehicular Technology Conf., vol.3, 2001, pp. 1804–1808.

[15] D. P. McNamara, M. A. Beach, and P. N. Fletcher, “Experimental in-vestigation of the temporal variation of MIMO channels,” in Proc. IEEEVehicular Technology Conf., vol. 2, 2001, pp. 1063–1067.

[16] C. C. Martin, N. R. Sollenberger, and J. H. Winters, “MIMO radiochannel measurements: Performance comparison of antenna configu-rations,” in Proc. IEEE Vehicular Technology Conf., vol. 2, 2001, pp.1225–1229.

[17] J. F. Kepler, T. P. Krauss, and S. Mukthavaram, “Delay spread measure-ments on a wideband MIMO channel at 3.7 GHz,” in Proc. IEEE Vehic-ular Technology Conf., vol. 4, 2002, pp. 2498–2502.

[18] T. Svantesson, “A double-bounce channel model for multi-polarizedMIMO systems,” in Proc. IEEE Vehicular Technology Conf., vol. 2,2002, pp. 691–695.

[19] J. P. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen, andF. Frederiksen, “A stochastic MIMO radio channel model with ex-perimental validation,” IEEE J. Select. Areas Commun., vol. 20, pp.1211–1226, August 2002.

[20] L. Zheng and D. Tse, “Optimal diversity-multiplexing tradeoff in mul-tiple antenna fading channels,” in Proc. 35th Asilomar Conf. Signals,Systems & Computers, Pacific Grove, CA, Nov. 2001.

[21] H. E. Gamal, “On the robustness of space-time coding,” IEEE Trans.Signal Processing, vol. 50, pp. 2417–2428, Oct. 2002.

[22] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding for wirelesscommunications,” IEEE Trans. Commun., vol. 46, pp. 357–366, Mar.1998.

[23] D. W. Bliss, P. H. Wu, and A. M. Chan, “Multichannel multiuser detec-tion of space-time turbo codes: Experimental performance results,” inProc. 36th Asilomar Conf. Signals, Systems & Computers, Pacific Grove,CA, Nov. 2002.

[24] D. W. Bliss, “Robust MIMO wireless communication in the presence ofinterference using ad hoc antenna arrays,” presented at the Proc. MilitaryCommunications Conf., MILCOM 2003, Boston, MA, Oct. 2003.

[25] , “Angle of arrival estimation in the presence of multiple accessinterference for CDMA cellular phone systems,” in Proc. 2000 IEEESensor Array and Multichannel Signal Processing Workshop, Cam-bridge, MA, Ma. 2000.

[26] C. M. Keller and D. W. Bliss, “Cellular and PCS propagation measure-ments and statistical models for urban multipath on an antenna array,”in Proc. 2000 IEEE Sensor Array and Multichannel Signal ProcessingWorkshop, Cambridge, MA, Mar. 2000, pp. 32–36.

[27] D. W. Bliss, “Robust MIMO wireless communication in the presence ofinterference using ad hoc antenna arrays,” presented at the MILCOM,Boston, MA, Oct. 2003.

[28] B. Efron, The Jackknife, the Bootstrap and Other ResamplingPlans. Bristol, U.K.: Society for Industrial and Applied Mathe-matics, 1982.

Daniel W. Bliss (M’97) received the B.S.E.E. degreein electrical engineering from Arizona State Univer-sity, Tuscon, in 1989 and the M.S. and Ph.D. degreesin physics from the University of California at SanDiego, in 1995 and 1997, respectively.

Employed by General Dynamics from 1989 to1991, he designed avionics for the Atlas-Centaurlaunch vehicle and performed research and devel-opment of fault-tolerant avionics. As a memberof the Superconducting Magnet Group at GeneralDynamics from 1991 to 1993, he performed mag-

netic field calculations and optimization for high-energy particle acceleratorsuperconducting magnets. His doctoral work rom 1993 to 1997, was in thearea of high-energy particle physics, searching for bound states of gluons,studying the two-photon production of hadronic final states, and investigatinginnovative techniques for lattice gauge theory calculations. Since 1997, hehas been employed by MIT Lincoln Laboratory, where he is currently a StaffMember at in the Advanced Sensor Techniques Group, where he focuses onmultiantenna adaptive signal processing, primarily for communication systems,and on parameter estimation bounds, primarily for geolocation. His currentresearch topics include algorithm development for multichannel multiuserdetectors (MCMUD) and information theoretic bounds and space-time codingfor MIMO communication systems.

Amanda M. Chan received the B.S.E.E. andM.S.E.E. degrees in electrical engineering from theUniversity of Michigan, Ann Arbor, in 2000 and2002, respectively.

Currently, she is an Associate Staff Memberin the Advanced Sensor Techniques Group, MITLincoln Laboratory, Lexington, MA. Her interestsare in channel phenomenology. She has previouslyworked with implementation of synthetic aperturegeolocation of cellular phones. Most recently, shehas worked on the implementation of MIMO channel

parameterization.

Nicholas B. Chang received the B.S.E. degree inelectrical engineering (magna cum laude) fromPrinceton University, Princeton, NJ and the M.S.E.degree in electrical engineering from the Univer-sity of Michigan, Ann Arbor, in 2002 and 2004,respectively.

He worked for MIT Lincoln Laboratory, Lex-ington, MA, in 2001 and 2002, focusing on syntheticaperture geolocation of wireless systems and channelphenomenology of MIMO communications systems.He is currently a Graduate Student in the Department

of Electrical Engineering and Computer Science, University of Michigan, AnnArbor.

Mr. Chang is a Member of Tau Beta Pi and Sigma Xi, the Scientific ResearchSociety.


Service Oriented Statistics of Interruption Time Dueto Rainfall in Earth-Space Communication Systems

Emilio Matricciani

Abstract—The paper reports and discusses simulated statisticsobtained by the synthetic storm technique, in the Po Valley,Northern Italy, on the interruptions (outages), due to rainfall,observed in contiguous (clock) periods of the day of duration ,with 1 min 24 h. The results refer to a 32 slant path at11.6 GHz, although the main conclusions are independent of car-rier frequency and of site, and are based on a large experimentalrain rate database (10.6 years of observation). The results can beused to assess the quality and unavailability of services of duration

during a day in earth-space communication systems affectedby rain attenuation. A distinction is made and discussed betweenchannel unavailability and service unavailability. The numericalresults and the best fit and extrapolation formulas derived mightprovide a rough approximation to the same statistics at differentelevation angles, clock intervals and carrier frequencies, for siteswith the same climate of the Po Valley.

Index Terms—Channel unavailability, diurnal cycles, fadeduration, microwave propagation, rain attenuation, serviceunavailability.

I. INTRODUCTION

THE design of satellite systems working in frequency bandsaffected by rain attenuation, (dB), are based, today, only

on the long term probability distribution (i.e., fraction of time)that is exceeded in an average year (or in the worst

month). In an age of a large variety of services offered, or to beoffered, to users from satellites or from troposphere platforms,it would be useful to match system design to the time of theday and to the expected duration of the services offered, e.g.,internet sessions, digital video and audio broadcasting. The ap-plication of forward error correction (FEC) codes, as currentlydone, may make rainfall attenuation a less severe problem for asatellite system in the 12-GHz band, if the system is designedby taking into account not and its low values (i.e., therange – ) and thus very short fade durations usuallyconsidered in telephony services, a concept we can call channelunavailability, but the number of interruptions of a maximumduration that a user can tolerate for a certain service, a TV orradio show, etc., a concept we can call service unavailability.

At higher frequencies (e.g., 20–30 and 40–50 GHz frequencybands), coding gain may be, however, largely ineffective, bothfor rain attenuation and for the “quasistatic” extra attenuationdue to other sources of fading, such as oxygen, water vaporand clouds. Rain attenuation, however, is likely to cause longrandom interruptions as the results below show.

Manuscript received March 3, 2003; revised October 27, 2003.The author is with the Dipartimento di Elettronica e Informazione, Politec-

nico di Milano, 20133 Milan, Italy (e-mail: [email protected]).Digital Object Identifier 10.1109/TAP.2004.832374

The necessity to distinguish between channel unavailabilityand service unavailability calls for statistics on the number (orprobability) of outages of a given duration in specific times ofthe day, i.e., statistics on the interruptions of a service becauseof excessive rain attenuation.

To author’s knowledge, however, this kind of statistics,or service grades, and a clear distinction between channelunavailability and service unavailability, have not yet beenestablished. Once this is done, the kind of statistics and thescaling methods proposed in this paper could be used to designa satellite system based on service unavailability rather than onchannel unavailability.

The assessment of these statistics for a geographical area anda satellite system requires long term experiments and thus a verylarge economic and human effort. Few experimental data col-lected many years ago in relatively short observations and forlong intervals of the day (e.g., four hours), are available in theopen literature [1]–[5]. Reliable simulations, based on physicalprediction models and a long observation, are then welcome.One such method is the “synthetic storm” technique [6] and thelong observation is our database of rain-rate time series.

For each rainstorm, physically described by a rain rate timeseries collected at a site with a rain gauge, the synthetic stormtechnique can generate a rain attenuation time series at any car-rier frequency and polarization, for any slant path with elevationangle larger than about 10 . The synthetic storm technique wassuccessfully tested to predict conventional long term ’s[6], long term statistics of fade duration [7], long term ’srelative to contiguous periods of the day of four hours [8]. Ifsimulated rain attenuation time series are compared to simulta-neous real measurements (e.g., as done for Spino d’Adda in a37.8 slant path to Italsat, at 18.77 GHz [9]), the agreement isvery good, especially when the rain storm motion is parallel toprojection to ground of the slant path. When the motion is notparallel to projection to ground, the simulated time series are sorealistic that they might as well be measured in a long obser-vation. In fact, the synthetic storm technique yields results thatare averaged over all the rain storms velocity field. These indi-rect tests are of considerable importance for the present workbecause, in our opinion, they suggest that our simulations canprovide results that remind experimental ones.

An earth-space microwave radio link is also affected byfading due to clouds, water vapor and oxygen. The fading dueto these phenomena can be large (e.g., see [10]), but morestatic than rain attenuation. They must be taken, of course,into account in a full design of the communication system. Atmicrowaves, however, rainfall is still the major random physicalcause of fading that can affect a channel for intervals of time

0018-926X/04$20.00 © 2004 IEEE


comparable to the duration of many services, so that the serviceoriented statistics of interruptions reported below refer to rainattenuation only.

We have applied the synthetic storm technique to simulate11.6-GHz rain attenuation time series in a 32 slant path inNorthern Italy by means of a large and reliable database of rainrate time series. As discussed above, we think that the resultscan be considered a good estimate of true measurements of rainattenuation in an earth-space channel in the 12 GHz band, or inhigher microwave frequency bands, once the results are scaled.

In [8], [11] we simulated, tested and discussed long termprobability distributions concerning contiguous (clock) periodsof equal duration (min) in a day, for several values of . Inthe present paper, we present detailed long term statistics on thenumber and probabilities of service interruption. The results arethen useful to assess the quality and unavailability (i.e., inter-ruptions or outages) of services of duration in a day.

Next section sets the stage for our simulations, Section IIIreports the simulated results, Sections IV and V provide em-pirical formulas useful to generate predictions at different sites,frequencies, duration of contiguous periods and elevation an-gles, and Section VI draws some conclusions.

II. RADIO LINK SIMULATIONS AND EXPERIMENTAL

RAINFALL DATABASE

Once the radio link geometric and radio electrical parametershave been specified and the synthetic storm technique has beenapplied to a rain rate time series, we get a rain attenuation timeseries [6]. The simulations refer to the following parameters.

a) Carrier frequency at 11.6 GHz, circular polarization,although the main conclusions are independent of fre-quency and polarization.

b) 32 slant path at Spino d’Adda (45.5 , 9.5 , 84 mabove sea level), a site in a flat countryside near Milan,with rain conditions typical of the Po Valley. The ele-vation angle and carrier frequency are those of a formerradio link to geostationary satellite Sirio (15 ), used inthe 1970s and 1980s for radio propagation experiments inthe 12-GHz band at Spino d’Adda.

c) The duration of the contiguous (clock) periods assumesthe following values: 1, 5, 10, 15, 30, 60 (1 h) min, and2, 4, 6, 12, 24 h, synchronized with 00:00 hours of theUniversal Time (UT).

Fig. 1 shows an example of a simulated time series, whichshows a 2-h interruption between 16:00 and 18:00 h UT(17:00–19:00 local time). The type of interruption shownreminds of fade duration for a fixed interval [12], but it is notexactly the same, for two reasons: (a) we have observed fixedcontiguous periods of the day because the services (e.g., TV,audio broadcasting) start independently of weather conditionsand may be synchronized to the hour, to 15 min past the hour,etc., and (b) the fade level exceeded in these intervals can belarger than dB (threshold) at both the start and end times,while fade durations statistics always concern the same valueof rain attenuation at both times.

For system design the results provide data on interruptions(outages) of duration when the built-in power margin,

Fig. 1. Example of a rain attenuation time series generated by the syntheticstorm technique at 11.6 GHz, circular polarization, in a 32 slant path at Spinod’Adda. For 1 dB threshold, for instance, a 2-h partial outage occurs between14:00 and 16:00 UT (15:00 and 17:00 local time), a 2-h full outage occursbetween 16:00 and 18:00 UT (17:00 and 19:00) and another (synchronized) 2-hpartial outage at 18:00 UT (19:00).

(dB), is continuously less than in any clock period of dura-tion . For example, for , the results yield the long termprobability distribution that is exceeded in any 1-h period ofthe day. Now, since in a day there are 24 such contiguous pe-riods, the statistics have been averaged over all the total numberof 1-h contiguous intervals in the observation period.

The results have been derived from a statistically reliable setof rain rate time series (for a definition of a rain storm andits duration, as measured with a rain gauge, see [6]) collectedat Spino d’Adda, with a continuous observation from October1979 to December 1982 (326 rain storms) and from May 1992to December 1997 (804 rain storms), and with a random obser-vation from 1983 to 1987 (103 rain events), 1233 rain stormsin total, a very large database. Of the 1983–87 period we haveestimated an equivalent continuous observation period of about885 600 min (1.7 years). The total observation period amountsto 5 531 040 min, about 10.6 equivalent years, i.e., the order ofmagnitude of the life of a commercial satellite, and the total time

is 225 425 min, i.e., 4.1% of an average year. As aconsequence the total number of contiguous periods observed isgiven by 5 531 040 (min) divided by (min), e.g., 92 184 h for

.

III. SIMULATED RESULTS

In this section we show the simulated results and discuss theimportant connection between the long term and the prob-ability distribution of 1-min long fades.

A. Statistics for Different Values of

Table I reports the overall statistics on the number of outages(interruptions) of duration in a day, for a given rain attenu-ation threshold (dB), for thresholds up to 10 dB. We like toshow these numerical values because they can be useful to sim-ulate systems directly, and also because future predictions de-rived from theoretical models could be compared to the resultsof this rather large database.

The trends shown in Table I are physically justified: the longeris , the less likely is a full outage. For instance, for threshold

MATRICCIANI: SERVICE ORIENTED STATISTICS OF INTERRUPTION TIME DUE TO RAINFALL 2085

TABLE INUMBER OF OUTAGES (INTERRUPTIONS) FOR CONTIGUOUS (CLOCK) PERIODS OF DURATION T (min) OF THE DAY, AS A FUNCTION OF RAIN ATTENUATION

THRESHOLD A (dB) AT 11.6 GHz, IN A 32 SLANT PATH, AT SPINO d’ADDA, IN 10.6 YEARS OF OBSERVATION

the 24-h interval was never in outage conditions(not explicitly shown in Table I because we found no outages).On the contrary the two 12-h intervals experienced 14 outages(interruptions), the 1440 1-min intervals experienced 225 425outages for and 890 for , in 10.6 yearsof observation. By dividing the data of Table I by 10.6 we getthe average number of service interruptions in a year, given that

.

For the system designer, however, probabilities are moremeaningful. If, for each , the data of Table I are divided by

, we obtain the long term probabilities, drawnin Fig. 2 for , that a service of duration isnot provided continuously to a user in any of the contiguousperiods of duration , given that . As it clearlyappears, the longer the service duration , the less likely it maybe interrupted for the entire period. Notice, however, that thesestatistics, once they are complemented to unity, cannot be readas availability statistics (except for values of of the orderof 1 min, see Section III-B), because the “availability” wouldsurely include shorter intervals of partial outages (see Fig. 1and Section IV-B).

The probabilities discussed above are long term results aver-aged in a day, so that they do not provide hints on diurnal peri-odic cycles, as those found in the long term ’s [11]. Figs. 3and 4 show some examples of these cycles for and

, respectively, and for some values of threshold .Also shown, as horizontal lines, the equivalent uniform distri-bution of outages, i.e., the total number of outages divided bythe number of contiguous periods in a day. We notice that thesimulated distribution is not uniform. Only some intervals showa partial uniform distribution. The curves become smoother, of

Fig. 2. Probability that rain attenuation A (dB), predicted by the syntheticstorm technique, is exceeded in an average year at 11.6 GHz, in a 32 slantpath at Spino d’Adda. Observation time is 10.6 years.

course, when larger values of are considered (Fig. 4,). These results agree with those reported in [8] and [11]

and show, once more, that the more intense fades, in the PoValley, tend to occur in the afternoon and in the evening. Theconsequence on system design is obvious, as service quality sig-nificantly depends not only on service duration, but also on thetime the service is started.


Fig. 3. Number of occurrences of full outages in clock intervals of 5 min along a day for thresholds 0 dB (upper figure), 1 dB (middle) and 3 dB (lower) at11.6 GHz, 32 slant path. The local time is one hour ahead of Universal Time (UT). Notice that, to show more clearly the cycles, 24+ 6 repeated hours are shown.Numerical values are reported at the beginning of the interval. Observation time is 10.6 years. Horizontal dotted lines yield the equivalent uniform distributions.

Fig. 4. Number of occurrences of full outages in clock intervals of 15 min along a day for thresholds 0 dB (upper figure), 1 dB (middle) and 3 dB (lower) at11.6 GHz, 32 slant path. The local time is one hour ahead of Universal Time (UT). Notice that, to show more clearly the cycles, 24+ 6 repeated hours are shown.Numerical values are reported at the beginning of the interval. Observation time is 10.6 years. Horizontal dotted lines yield the equivalent uniform distributions.

B. Relation Between and the Probability Distribution of1-min Long Fades

The results for shown in Table I,, once referred to the period of observation (10.6 years

or 5 531 040 min), yield the long-term probability distribution(or fraction of time) that a given rain attenuation is exceeded.This is the long term mentioned in the introduction andcurrently used to design satellite systems, and obtained by mea-surements or predictions. In fact, it is very likely, from a phys-ical point of view, that during an interval as short as 1 min, thechannel is fully affected by rain attenuation, i.e., continuously,so that, for a given threshold , the number of outages in anyshorter interval would be obtained by scaling the results accord-ingly and, as a consequence of being the 1-min interval fullyattenuated, the relative frequencies would not change. For in-

stance, for (the usual sampling time of rain attenuationmeasurements) we would simply obtain:

(1)

In our opinion this is one of the physical reasons why thepredictions of the synthetic storm technique derived by meansof 1-min rain rate time series yield a good estimate of the mea-surements [6], even if rain rate time series with smaller timeresolutions can improve the estimates of the highest peaks ofrain attenuation or its fastest rates of change [9].

Some experimental evidence supports this assumption. To-gether with full outages, we have also counted the number ofpartial outages, i.e., any interruption that lasts less than min.For example, Fig. 1 shows 2 partial outages for .


A rough estimate of the interval at which partial outages be-come full outages, for a given threshold exceeded, can be the(interpolated) value of for which the number of occurrencesof partial outages equals the number of the full outages, i.e., par-tial and full outages are equally likely. Fig. 5 shows the curveobtained for our database. We can notice that for ,

so that intervals of about one hour are equallylikely to be fully or partially attenuated. For larger than about1–2 dB a clear “knee” appears so that is approximately 3or 4 min, regardless of the threshold . Fig. 5 predicts

for and for , thelatter being a threshold and a power margin very large for anypractical application today. These figures are significantly largerthan 1 min, the physical resolution of the simulated rain atten-uation time series, so that 1-min intervals should be very likelyfully attenuated.

Another physical justification can be found in the frequencyextension of the power spectrum of rain attenuation which fallsoff with a 20 dB/decade slope [13] up to a frequency of theorder of few hundredths of hertz, i.e., periods of the order of 1 or2 min. Beyond this frequency range, the spectrum is dominatedby another physical phenomenon, namely turbulence scintilla-tion [14], more responsible of fade durations of the order of fewseconds, and here not of concern.

In conclusions, with some justified confidence, we cansuppose that in a radio link, with a built-in power margin(dB) and affected by rain attenuation (dB), clock intervalsof 1 min, or less, are very likely to be either fully unavailablewhen (threshold is very likely to be continuouslyexceeded during the interval ), or fully available when(threshold is very likely to be never exceeded during theinterval ).

IV. EMPIRICAL MODELS

In this section we first model the number of full interruptions,and afterwards we extend the concept of service unavailabilityto include some partial outages.

A. Number of Full Interruptions

For numerical calculations and extrapolations, the columndata of Table I for can be modeled by simple math-ematical functions of the power-law type, with a 0.1 dB -axistranslation to include the threshold . The constantsbelow were determined by minimizing the root mean square(rms) error.

The model of the number of outages, as a function of thethreshold (dB) and interval (min), for , isgiven by:

(2)

with an rms error less than 10% for any , and for thresholds upto 10 dB. The number of interruptions in an average year is givenby and the corresponding long term probabilitydistributions are obtained by dividing (2) by , Sec-tion II, i.e.

(3)

Fig. 5. Interpolated value of T (min) for which the number of occurrences ofpartial outages equals the number of the full outages (partial and full outagesare equally likely).

For (recall that means both the value of theattenuation and the threshold exceeded in the expression of any

, and thus the value to insert in the case is 0)and , for instance, (3) yields , i.e., 4.6%,to be compared with the exact value 4.1% (corresponding to225 425 min) of Section I. For and ,we get : since there are 52 560 contiguousintervals of 10 min in 1 year, (3) yields

interruptions in an average year (Table I, accordingly, gives).

B. Number of Synchronized Partial Interruptions

Equation (2), or the data shown in Table I, can be used toextend the concept of service unavailability by taking into ac-count synchronized partial outages, i.e., by considering the in-terruptions that last an interval , with the maximumcontinuous interruption tolerated by users (e.g., Table I). Themeaning of a synchronized partial outage of duration is thefollowing: it is a continuous interval during which the service isunavailable for a time less than the maximum tolerated by users,but always synchronized with 00:00 UT. Fig. 1, for instance,shows, for , a synchronized partial outage starting at18:00 UT.

Let us see this point by an example taken from Table I. Con-sider . We have 70 outages of 15 min , and151 outages of 10 min (by reading the corresponding row ofTable I). Now, since a 15-min outage includes some 10-min in-terruptions, namely only two out of three, there must have been

synchronized partial outages of10 min . In fact during a period of , a 10-mininterval (and thus a possible 10-min interruption) can start at

, , of the next 15-min period. The 10-mininterruption starting at , however, cannot give riseto a synchronized 15-min interruption because it is split into


Fig. 6. Model derived percentage of synchronized partial outages (durationT ), as a function of rain attenuation threshold, for several values of clock periodT , for a fixed ratio T =T = 0:8. Notice that when T = 1:25 min, T =

1 min. All curves intersect at A = 0:9 dB.

twointerruptions belonging to different 15-min contiguous pe-riods. This splitting is taken into account by the presence of themultiplying factor , also justified by a uniformdistribution within a short interval (Figs. 3 and 4).

Of course, the results on synchronized partial outages do notinclude 10-min partial outages, which are not synchronized withthe clock periods. They are neither present, of course, in the10-min statistics of Table I. In conclusion, from our results wecan easily estimate only the number of partial outages that startat the beginning of the time slot of duration , and at multiplesof , as long as the ending time of the partial interruption iswithin the period .

Since (2) is a function of both and , we can find thenumber of the synchronized partial outages, , as

(4)

Fig. 6, as an example, shows the percentage of partial outages, for , for several values of . We see,

for instance, that for and , the numberof synchronized partial outages, of duration

, is about 38% of the total number of 15-min interrup-tions (about 74 according to (2), or 70 according to Table I),i.e., . By (4) we can calculate all intermediatecases between 1 min and 15 min, as necessary.

V. SCALING TO OTHER SITES, ELEVATION ANGLES

AND FREQUENCIES

The results of Section IV can be used to scale long term’s, measured or calculated from one of the several pre-

diction models available in the literature, to a certainwith different values of , at least to sites with thesame climate of the Po Valley. As discussed in Section III, wecan assume that the reference time of the measured or predicted

is, to a rough approximation, equal to about 1 min, andthen apply (3) twice: the first time with and the

second time with as parameter. The ratio between the latterand the former yields the scaling ratio to apply to current mea-surements, or predictions, for obtaining new outage statistics forclock periods of , i.e.

(5)

In (5) we have used the notation to refer to our reference32 slant path at 11.6 GHz. Now the probability distribution ofthe full outages, for a given threshold and clock interval ,is given by

(6)

Equations (5) and (6) may provide a good estimate of themeasurements at sites other than Spino d’Adda, but with thesame type of climate, because of the following argument: al-though (2) and (3) apply to this latter site, for a given A theratio (5) can be more independent of site and particular rainyconditions. At another locality, the occurrences of 1-min and

interruptions, for a given , can be largely differentof those found at Spino d’Adda, but it is likely that both occur-rences change in the same way so as to give a ratio that remindsof that given by (5). A ratio gets rid of possible common mul-tiplying factors. Partial outages can be scaled in the same way.This provisional conclusion, however, should be tested againstreal data.

Notice that (5) applies only to 11.6 GHz rain attenuation andto a 32 slant path. To apply it to slant paths with differentradio electrical and geometric parameters we have to scale rainattenuation.

To scale rain attenuation from a reference elevation angle(32 in our case) to a different elevation angle , we can sup-pose that (dB) is, to a first approximation, proportional to the(average) rainy path length. Hence, for a fixed rain height (asassumed in [6]), we get

(7)

As for frequency scaling, at microwaves we know that, for agiven probability, rain attenuation (dB) exceeded at a carrierfrequency can be empirically related to rain attenuation(dB) exceeded at a carrier frequency , in the same slant path,by the power law [15]

(8)

In conclusion, rain attenuation , in decibels, at carrierfrequency (GHz) in a slant path with elevation angle , canbe scaled according to (5)–(8), if, in (5), we insert the corre-sponding value of rain attenuation at 11.6 GHz for a 32 slantpath, i.e., the value

(9)


In (9), is a constant, once and are fixed. Equation (5) canthus be used to find the ratio to scale to . Forinstance, if we want to scale rain attenuation measured [or pre-dicted, we must know ] at in a slantpath (zenith) to a different , from (9) we get , so thatrain attenuation to be inserted into (5) is reduced to 22%. Now,we can calculate the scaling ratio for . For instance,10 dB in a 90 slant path at 40 GHz yields 2.2 dB at 11.6 GHzin the reference 32 slant path. Now, if , from (5)we get , so that, from (6),

. From this probability (or relative fre-quency) it is then possible to find the number of 10-min interrup-tions, or synchronized partial interruptions, in an average year,according to Section IV.

VI. DISCUSSION AND CONCLUSION

The simulated results obtained by the synthetic storm tech-nique can be considered, in our opinion, as experimental ones.They provide statistics on the interruptions (outages), due torainfall, in contiguous (clock) periods of the day of duration .The results are then useful to assess the quality and unavail-ability of services of duration during a day, e.g., in a satellitebroadcasting system, or in a system using troposphere “geo-stationary” platforms 20 km aloft, in brief, in any earth-spaceradio link. The numerical results at 11.6 GHz in a 32 slant path(Table I) and the best fit and extrapolation formulas (2)–(9) canprovide a rough approximation to the same statistics for otherelevation angles, clock periods and carrier frequencies, at siteswith the same climate of the Po Valley.

As for broadcasting system design, the data in Table I show,for instance, that a TV program that lasts 1 h is likely to beinterrupted (“blocked”) in Northern Italy fortimes in an average year, if no built-in power margin or FECgain are available (case ). For a power margin of 3 dB(case ), in an average year (data of Table I, dividedby 10.6) there is one interruption of 30 min, 49 interruptions of5 min, 365 interruptions of 1 min.

Now, let us consider a probability of bit error or less(e.g., see [16], p. 450), and code the bit stream with one of thelatest turbo codes with code rate 1/2 and 18 iterations. Assumingthat in clear-sky conditions there are enough power and band-width for the doubled bit rate, the turbo code yields a gain ofabout 10 dB in the ratio between the energy per bit and theone-sided white noise power density , , e.g., see [16,p. 282, fig. 7.4]. Then, in an average year, in the Milan area,by providing a system power margin (relative to clear-sky con-ditions) only by means of the most effective turbo code, in anaverage year there would be 84 interruptions of 1 min (Table I,

, 890/10.6), 8.4 interruptions of 5 min (some ofwhich can include, of course, 5 contiguous 1-min interruptions),2.3 interruptions of 10 min, etc. These numbers provide resultsaveraged over a day (the uniform distribution of Figs. 3 and 4),but in specific parts of the day, as Figs. 3 and 4 show, these valuescan be exceeded several times, especially in the afternoon and inthe evening at Spino d’Adda, and a finer system design shouldconsider this fact.

The application of FEC coding schemes, as currently done,may make rainfall attenuation a less severe problem for a satel-lite broadcasting system, if the system is designed by taking intoaccount not the conventional probability distribution andits low values (i.e., the range – ) and thus very shortfade durations usually considered in telephony services (channelunavailability), but the number of interruptions of a maximumduration that a user can tolerate for a certain service, a TV orradio show, etc., (service unavailability). Once a clear distinc-tion between channel unavailability and service unavailabilityis established, the statistics and the scaling methods proposed inthis paper could be used to design an earth-space system basedon service unavailability rather than on channel unavailability.

At higher frequencies (e.g., 30-40-50 GHz), coding gain maybe, however, largely ineffective, both for the “quasistatic” extraattenuation due to oxygen, water vapor and clouds, and for rainattenuation. For the latter, long random interruptions will be ex-perienced, as Table I, would show, once were scaled.

REFERENCES

[1] J. Goldhirsh, “Cumulative slant path rain attenuation statistics associatedwith the comstar beacon at 28.56 GHz for Wallops Island, VA,” IEEETrans. Antennas Propagat., vol. 27, pp. 752–758, 1979.

[2] H. W. Arnold, D. C. Cox, and A. J. Rustako, “Rain attenuation at 10–30GHz along earth-space paths: elevation angle, frequency, seasonal, anddiurnal effects,” IEEE Trans. Commun., vol. 29, pp. 716–721, 1981.

[3] H. Fukuchi, T. Kozu, K. Nakamura, J. Awaka, H. Inomata, and Y. Otsu,“Centimeter wave propagation experiments using the beacon signals ofCS and BSE satellites,” IEEE Trans. Antennas Propagat., vol. 31, pp.603–613, 1983.

[4] M. Mauri, “Hourly variation of attenuation statistics at 11–12 GHz asseen with SIRIO,” presented at the 32 Congr. Scientifico Int. per l’Elet-tronica, Rome, 1985, pp. 81–85.

[5] P. J. I. de Maagt, S. I. E. Touw, J. Dijk, G. Brussaard, L. J. M. Wijde-mans, and J. E. Allnut, “Diurnal variations of 11.2 GHz attenuation ona satellite path in Indonesia,” Electron. Lett., vol. 29, pp. 2149–2150,1993.

[6] E. Matricciani, “Physical-mathematical model of the dynamics of rainattenuation based on rain rate time series and a two-layer vertical struc-ture of precipitation,” Radio Science, vol. 31, pp. 281–295, 1996.

[7] , “Prediction of fade durations due to rain in satellite communica-tion systems,” Radio Science, vol. 32, pp. 935–941, 1997.

[8] , “Diurnal distribution of rain attenuation in communication andbroadcasting satellite systems at 11.6 GHz in Italy,” IEEE Trans. Broad-casting, vol. 44, pp. 250–258, 1998.

[9] E. Matricciani, M. Mauri, and C. Riva, “A rain rate data base useful tosimulate reliable rain attenuation time series for applications to satelliteand tropospheric communication systems,” in Proc. Eur. Conf. WirelessTechnology (ECWT 2002), Milan, Sept. 26–27, 2002, pp. 265–268.

[10] S. Ventouras, I. Otung, and C. Wrench, “Simulation of satellite systemsoperating at Ka-band and above using experimental time series of tro-pospheric attenuation,” in Proc. Inst. Elect. Eng. Colloquium on Simu-lation and Modeling of Satellite Systems, London, U.K., Apr. 2002, pp.11/1–11/5.

[11] E. Matricciani, “An assessment of rain attenuation impact on satellitecommunication: matching service quality and system design to the timeof the day,” Space Commun., vol. 16, pp. 195–205, 2000.

[12] A. Safaai-Jazi, H. Ajaz, and W. L. Stutzman, “Empirical models for rainfade time on Ku- and Ka-band satellite links,” IEEE Trans. AntennasPropagat., vol. 43, pp. 1411–1415, 1995.

[13] E. Matricciani, “Physical-mathematical model of dynamics of rain at-tenuation with application to power spectrum,” Electron. Lett., vol. 30,pp. 522–524, 1994.

[14] E. Matricciani, M. Mauri, and C. Riva, “Scintillation and simultaneousrain attenuation at 12.5 GHz to satellite Olympus,” Radio Science, vol.22, pp. 1861–1866, 1997.

[15] G. Drufuca, “Rain attenuation statistics for frequencies above 10 GHzfrom raingauges observations,” J. Rech. Atmosperique, pp. 399–411,1974.

[16] T. Pratt, C. W. Bostian, and J. E. Allnutt, Satellite Communications, 2nded. New York: Wiley, 2003.


Emilio Matricciani was born in Italy in 1952. Afterserving in the army, he received the Laurea degree inelectronics engineering from Politecnico di Milano,Milan, Italy, in 1978.

He joined Politecnico di Milano in 1978 as arecipient of a research scholarship in satellite com-munications, and in 1981, he became an AssistantProfessor of electrical communications. In 1987,he joined the Università di Padova, Padua, Italy,as an Associate professor of microwaves. Since1991, he has been with Politecnico di Milano as

an Associate Professor of electrical communications. In the year 2001, hequalified as a Full Professor of telecommunications. He has been involved inthe experiments conducted with the Italian satellite Sirio in the 1970s in the12–14 and 18 GHz bands, and afterwards, in the planning and conductingthe experiments with Italsat in the 20–30 and 40–50 GHz bands, in the 1980sand 1990s. His actual research interests include satellite communicationsfor fixed and mobile systems, radio wave propagation, history of scienceand technology. In addition to the institutional activities, he teaches thefundamental aspects of communicating scientific and technical informationto undergraduate, graduate, master, and doctorate students.


Full-Wave Analysis of Dielectric Frequency-SelectiveSurfaces Using a Vectorial Modal Method

Angela Coves, Benito Gimeno, Jordi Gil, Miguel V. Andrés, Member, IEEE, Angel A. San Blas, andVicente E. Boria, Senior Member, IEEE

Abstract—A novel vectorial modal method is presented forstudying guidance and scattering of frequency-selective structuresbased on lossy all-dielectric multilayered waveguide gratingsfor both TE and TM polarizations. The wave equation for thetransverse magnetic field is written in terms of a linear differentialoperator satisfying an eigenvalue equation. The definition of anauxiliary problem whose eigenvectors satisfy an orthogonalityrelationship allows for a matrix representation of the eigenvalueequation. Our proposed technique has been applied to the studyof lossy all-dielectric periodic guiding media with periodicity inone dimension. This method yields the propagation constants andfield distributions in such media. The reflection and transmissioncoefficients of a single layer under a plane-wave excitation canbe obtained by imposing the boundary conditions. Study of thescattering parameters of the whole multilayered structure isaccomplished by the cascade connection of components as charac-terized by their scattering parameters. Results obtained with thismethod for the propagation characteristics of a one-dimensionalperiodic dielectric medium are compared with those presented byother authors, and results for the scattering of several dielectricfrequency-selective surfaces (DFSS) are compared with boththeoretical and experimental results presented in the literature,finding a very good agreement. A symmetrical band-stop filterwith a single waveguide grating is also demonstrated theoretically.

Index Terms—Band-stop filter, dielectric grating, frequency-selective surfaces, Galerkin method, orthogonality relationship,vectorial modal method.

I. INTRODUCTION

THE frequency-selective characteristics of multilayeredperiodic structures, both dielectric [1]–[5] and metallic

[6]–[8] are of considerable importance in electromagnetics.Periodic screens have been used in the last decades in many ap-plications as filters, radomes and polarizers of electromagneticwaves. A dielectric layer with periodically varying dielectricconstant can be used as an alternative way to obtain a frequencyselective surface. Dielectric waveguide gratings can be usedin the microwave-frequency band as dichroic subreflectors in

Manuscript received September 25, 2002; revised February 4, 2003. Thiswork was supported by the Ministerio de Ciencia y Tecnología, Spanish Gov-ernment, under Research Projects TIC2000-0591-C03-01 and TIC2000-0591-C03-03.

A. Coves and A. A. San Blas are with the Departamento de Física y Arquitec-tura de Computadores, Universidad Miguel Hernández de Elche, 03202 Elche(Alicante), Spain.

B. Gimeno and M. V. Andrés are with the Departamento de Física Aplicada yElectromagnetismo- I.C.M.U.V., Universidad de Valencia, 46100 Burjasot (Va-lencia), Spain.

J. Gil and V. E. Boria are with Departamento de Comunicaciones, UniversidadPolitécnica de Valencia, 46022 Valencia, Spain.


large reflecting antennas [9]. Thin-film dielectric structurescontaining a periodic variation along the film have recently beenof considerable interest in integrated optics [10], because of theimportant role they play in applications such as beam-to-sur-face-wave couplers, filters, distributed feedback amplifiers andlasers, nonlinear generation of second harmonics, and beamreflection on steering devices of the Bragg type. Eielectricfrequency-selective surfaces (DFFS) have been analyzed witha variety of different methods, both analytical and numerical.Rigorous analytical methods [1], [11] for studying dielectricperiodic structures are limited however to gratings with specialsimple groove shapes. Numerical methods [12], on the otherhand, have made possible the analysis of periodic structures ofsignificant geometry/material complexity in the periodic cell.

The numerical method presented in this paper is based on anovel vectorial modal method [13], [14] for studying guidanceand scattering by lossy all-dielectric guiding periodic structures.For the structures considered here, each layer is either a peri-odic dielectric grating, formed by any number of lossy dielec-tric slabs, or a uniform dielectric slab, and all grating layershave the same periodicity. This formulation has been applied tothe accurate analysis of the modal spectrum (propagation con-stants and fields distribution) of dielectric periodic structuresfor both TE and TM polarizations, and results are successfullycompared with those presented in the technical literature. Thereflection and transmission coefficients of the structures undera plane-wave excitation are obtained with the generalized scat-tering matrix (GSM) theory [15], [16] and successfully com-pared with both theoretical and experimental results obtainedby other authors.

It must be emphasized that the complexity of the new methodproposed in this paper does not increase with the number of di-electric slabs present in the unit cells, such as it usually happenswith other classical analysis techniques (e.g., the TRM tech-nique [17]). Furthermore, the presence of losses in the dielectricslabs can be easily considered by simply introducing a complexpermittivity in the formulation derived.

The formalism used to obtain the modal spectrum of lossydielectric periodic structures and the orthogonality relationsof the modes is presented in Section II. In Section III-A,numerical results obtained using this formalism for the propa-gation constants and the fields distribution in the grating regionare compared with those presented in the bibliography. Thespectral response of several DFSS are obtained and comparedwith both numerical and experimental results obtained by otherauthors in Section III-B, and a symmetrical band-stop filterwith a single waveguide grating is also proved theoretically.

0018-926X/04$20.00 © 2004 IEEE


Fig. 1. Characteristic unit cell of a periodic dielectric medium.

II. THEORY

The first objective of this section is to obtain the modalspectrum of a general lossy all-dielectric periodic guidingmedium uniform in the axis. Next, making use of thismultimodal characterization, the boundary conditions betweenadjacent layers will be imposed by means of the GSM approachin order to obtain the scattering parameters of a one-dimensional(1-D) dielectric FSS.

Next we present the theoretical bases of the method forstudying a dielectric 1-D periodic medium in the directionwith translational invariance along the axis (see Fig. 1). Weassume that the electric and magnetic fields in this medium canbe expressed as a linear superposition of fields with explicitharmonic dependence on the coordinate (we assume that thetime dependence is always implicit and has an harmonic form

for all vector fields)

(1)

(2)

where is the propagation constant, and , represent the trans-verse components to the direction of the electric and mag-netic fields, respectively, and , are the components in the

direction. The transverse components of the fields, when nosources are present, satisfy the vector wave equations [13], [14]

(3)

(4)

being the relative complex permittivity of the pe-riodic medium, whereas is the free-space wavenumber

. In these equations, we can identify in square bracketsthe differential operators governing the transverse componentsevolution along the axis.

For our purposes, it is more interesting to rewrite (3) and (4)as a pure 2-D eigenvalue problem for the differential evolutionoperators and

(5)

where and are the eigenvectors of the and operators,respectively; these operators are nonself-adjoint operators. The

eigenvectors of nonself-adjoint operators do not satisfy a stan-dard orthogonality condition; this prevents the possibility ofexpanding arbitrary functions in terms of the operator eigen-vectors, which is in fact one of our objectives: obtain a modalbasis to represent the electromagnetic field. But we can stillrepresent (3) and (4) in a matrix form if such equations areexpressed in an auxiliary system which can be used to derivethe matrix form of the eigenvalue problem, whose eigenvectorssatisfy an orthogonality relationship of the form

(6)

For the auxiliary system we have used the modes correspondingto a homogeneous medium of relative dielectric permittivityas the auxiliary basis. The geometry of the problem is shownin Fig. 1; the periodicity D of the structure has been chosenin the direction, being the problem uniform in the axis.The medium is assumed as nonmagnetic and periodicin the direction, so it is fully characterized by its complexrelative permittivity in the unit cell defined as

(7)

where is the Heaviside function, is the number oflossy dielectric slabs inside the periodic cell (which can be anarbitrary number), and the i-th dielectric slab is centered at point

and its width is , as it is shown in Fig. 1. The more adequateauxiliary basis functions in this case are the well-known Floquetharmonics, i.e., the eigenfunctions of the evolution operatorcorresponding to a periodic structure immersed in a homoge-neous medium of relative permittivity [6]. For a medium in-variant with the coordinate , the auxiliary basis of the Floquetmodes is formed by the set of TE and TM family modes [6]

(8)

(9)

COVES et al.: FULL-WAVE ANALYSIS OF DFSS USING VECTORIAL MODAL METHOD 2093

where

(10)

and is the angle of incidence of the fundamental harmonicassociated with the structure excitation. These modes satisfy theorthogonality relationship (6)

(11)

Thus the modes of the real problem can be expanded in termsof the auxiliary system as

(12)

where and are the complex coefficients of the modal ex-pansion for the transverse magnetic and electric fields of the thmode, respectively. We are certainly concerned with the matrixrepresentation of the linear operator of the real problem . Thematrix elements of the operator in the basis will thenbe easily obtained by applying the standard Galerkin momentmethod [18]. By inserting the first equation of (12) into the firstequation of (5), and applying the linear properties of , we find

(13)

The next step in the application of the Galerkin procedure isto choose a set of weighting functions , and to take theinner product for each yielding the following linear matrixeigenvalue problem

(14)were the matrix elements of the operator are obtained asfollows:

(15)

For practical purposes, it is convenient to introduce the differ-ence operator , resulting

(16)

Thus, the matrix elements of the operator are ob-tained by means of (15)

(17)

where the first term is diagonal because the operator is ex-pressed in its own orthogonal basis. However, it can be easilyshown that there is no coupling among TE and TM modes whenevaluating the integrals of the matrix elements of the oper-ator. Thus, the problem can be easily separated into two TE andTM polarizations. The integrals involved in the evaluation of thematrix elements of contain the relative dielectric permittivity

function, which is a discontinuous function, and also its trans-verse gradient. Some confusion may exist in the evaluation ofthese integrals due to the discontinuous nature of the dielectricfunction and its transverse gradient. In [19] it is described theway to solve these integrals correctly for the case of a rectan-gular waveguide with a relative permittivity function defined asa sum of lossy dielectric slabs. For the particular case of a 1-Dperiodic dielectric medium, these integrals have been analyti-cally calculated for each polarization obtaining

(18)

(19)

where the summation includes all the lossy dielectric slabs in-side the periodic cell. As a consequence, only a diagonalizationprocess for each polarization has to be performed numericallyfor obtaining the propagation constants and the magnetic fieldsof the periodic medium at each frequency point.

At this point, it is important to remark that we have trans-formed the differential eigenvalue (5) for the operator , whichis responsible of the evolution of the transverse magnetic field,into a linear matrix eigenvalue problem. An analog equation forthe operator , responsible of the evolution of the transverseelectric field, can be derived

(20)

Thus, the information contained in the above matrix (14) and(20) is the same as in the differential equations shown in (5).Diagonalization of the matrix yields the squared of the

th mode propagation constant (the th eigenvalue of ),and also its transverse magnetic amplitude through theknowledge of the th eigenvector . It is important to notethat the diagonalization of not only provides us with thepropagation constants (eigenvalues) and transverse magneticamplitudes (eigenvectors) of the modes, but also with thetransverse electric fields of the modes, which are related to

through constraints derived from Maxwell’s equations [20].This fact is very important from a computational point-of view,because the diagonalization process for the matrix is onlyrequested in the numerical implementation of this method.Therefore, it is not necessary to implement the numericaldiagonalization of the matrix corresponding to the electricoperator .

However, the matrix has an infinite number of elements.In order to develop a realistic method, we have to work witha finite set of well-known auxiliary fields. Unfortunately, thereare no general conditions that guarantee the convergence of theexpansions. This convergence will depend on both the nature of


the operator and the auxiliary problem chosen to define the or-thogonal basis. In general, we observe that the modes are betterdescribed by increasing the number of auxiliary modes. In thesame way, auxiliary bases that can represent the most relevantfeatures of the real problem produce faster convergence. In anycase, numerical convergence tests must be done by sweeping thenumber of auxiliary modes over meaningful ranges and studyingthe stability of solutions.

Finally, the transverse electric field of the th mode is relatedto through constraints given by Maxwell’s equations, whichin this case can be expressed in terms of the modal characteristicimpedances [6]

(21)

(22)

where these modal characteristic impedances are given by

(23)

(24)

Note that the TM characteristic impedance varies with thecoordinate, that is, the characteristic impedance is not a con-stant because the relative permittivity is a function of thecoordinate.

Once obtained the fields and propagation constants in allregions of the structure, both homogeneous and periodic, theproblem is reduced to obtain the scattering parameters of thestructure. To this end, the boundary conditions between adjacentlayers will be imposed, obtaining the GSM at each interfacebetween adjacent layers of the structure, i.e., the amplitudes ofreflected and transmitted modes. For a propagation distancethrough a layer with propagation constants , the multimodescattering matrix is defined as the scattering matrix of a uniformtransmission line of length [16]. Then, we construct the GSMof the global structure by means of the cascaded connectionof the individual GSMs of the interfaces and the scatteringmatrices corresponding to the propagation through the layers,following the technique described in [16]. The global GSMyields the amplitudes of the scattered modes, which are re-flected and transmitted by the structure, considering an incidentplane wave with a unit amplitude. Finally, the reflectanceand transmittance of the structure are easily obtained.

In the described theory, when all modes are included, thematrix is infinitely-dimensional. In order to develop a realisticmethod, we have to work with a finite set of well-knownauxiliary fields to expand the modes of each periodic dielectriclayer in terms of the modes of the auxiliary basis functions.On the other hand, also the multimode scattering matrix isinfinitely-dimensional. The most straightforward way to reducethe scattering problem to a computationally tractable form isto truncate the individual layer multimode scattering matricesat a finite size which is large enough to allow for accuratecalculation of the scattered modes, reflected and transmitted,which are significant in the overall solution but small enough tobe tractable for numerical calculation. Then, for each particular

case, a study of convergence must be performed in order to reachan accurate solution for both the propagation characteristicsin each periodic layer, and the scattering parameters of theoverall structure.

III. NUMERICAL AND EXPERIMENTAL RESULTS

A. Accuracy and Convergence Properties of the ModalExpansion in Periodic Dielectric Media

In order to compare our results for the propagation charac-teristics of an infinite periodic dielectric structure with thoseobtained previously by other authors, we first examine an in-finite periodic structure (see Fig. 1) with five dielectric slabswithin the unit cell of period . The parame-ters are , , , ,

, , , ,, , and . In the

calculations the auxiliary system used was an infinite homoge-neous dielectric medium with with the same periodof the real problem. Results of the convergence study of the so-lutions of this infinite periodic medium are shown in Fig. (2a)and (b). In these figures it is represented, for TE [Fig. 2(a)] andTM [Fig. 2(b)] polarization, the normalized propagation con-stant of the first and third mode (left-hand axis), andthe sixth mode (right-hand axis) as a function of the number

of auxiliary modes, which in this case are the Floquet modesof an infinite homogeneous medium with relative permittivity

. Results are obtained for a frequency of 10 GHz andfor a normalized Floquet wavenumber . For TEpolarization it can be seen that for all modes represented the con-vergence is reached with only 15 modes of the auxiliary basis,taking 0.01 s per frequency point for obtaining the first fifteenmodes (in a Pentium II@350 MHz processor). Nevertheless, ahigher number of auxiliary modes is needed for the case of TMpolarization in order to reach the convergence of the solutions,which in this case is auxiliary Floquet modes, whichtakes again only 0.01 s per point.

For the same problem considered before, Fig. 3 shows thecurves of the normalized propagation constant as a func-tion of the normalized Floquet wavenumber of thefirst and second mode for both TE and TM polarizations. In thisfigure the results calculated in [21] with the transverse resonantmethod (TRM) are also shown, and an excellent agreement isobserved.

The present method also allows the calculation of the fieldpatterns. The electric field distribution of the first and secondmode along the characteristic cell for the geometry of Fig. 2is shown in Fig. 4. Fig. 4(a) shows the behavior of the elec-tric field normalized with respect to its maximum value for thefirst and second TE modes as a function of the normalized co-ordinate for values of (straight line) and 1(dashed line). In this figure we have also represented the resultscalculated in [21] for the first TE mode at and

, showing an excellent agreement with our re-sults. The normalized magnetic field for the first and second TMmode is shown in Fig. 4(b) as a function of for values of

(straight line) and 1 (dashed line). We have alsorepresented the results calculated in [22] for the first TM mode at


Fig. 2. Study of convergence of the normalized propagation constant j�=k j as a function of the number N of auxiliary modes for the first and third mode(left-hand axis), and sixth mode (right-hand axis) for both (a) TE and (b) TM polarizations of an infinite periodic dielectric medium with five dielectric slabs withinthe unit cell. Parameters: D = 17:99 mm, " = 1:0, " = 1:28, " = 2:56, " = 1:28, " = 1:0, ~" = 1:0, l = 5:105 mm, l = 2:09 mm,l = 3:60 mm, l = 2:09 mm and l = 5:105 mm. Results obtained for a normalized Floquet wavenumber (k D=�) = 0 and a working frequency of10 GHz.

Fig. 3. Normalized propagation constant j�=k j as a function of thenormalized Floquet wavenumber (k D=�) of the first and second mode forboth TE and TM polarizations of the geometry detailed in Fig. 2. Comparisonbetween our results (lines) and results obtained with the TRM method [21](dots).

, and for the second TM mode at ,showing also a good agreement with our results.

B. Plane-Wave Scattering by a DFSS

Following the analysis of Section II, we have carried out nu-merical studies of the reflection and transmission coefficients ofseveral dielectric FSSs in comparison with theoretical and ex-perimental results obtained by other authors. The first case an-alyzed is a periodic dielectric grating formed by two dielectricslabs within the unit cell of period and thick-ness immersed in air under a TE plane waveincidence at ; we have set ,

, and . Results of the convergencestudy of the reflection coefficient of this structure are shown inFig. 5. In this figure it is represented the reflection coefficientat three different frequencies as a function of the number of

modes included in the construction of the GSMs. The re-flectance for lower frequency is represented inthe right-hand axis. The second and third frequency, representedin the left-hand axis, are resonant frequencies of the grating. Forall the cases considered, the reflected and transmitted fields werefound to conserve power to within one part in , that is, thetotal active reflected and transmitted power coefficients relatedto the propagation modes differ from unity by less than .For the numerical results shown in the rest of this section, it wassufficient to take modes in the construction of the mul-timode scattering matrices.

The frequency dependence of the reflectance of the grating isshown in Fig. 6; our results are compared with those calculatedin [17] with the TRM. For this case we have chosenand , taking 0.01 s per frequency point, including theCPU time for the computation of the modes and the scatteringanalysis. The agreement between both methods is excellent.

The second case analyzed is the transmittance of a periodicdielectric grating immersed in air for a TM-polarized wave atnormal incidence. The structure is characterized by a periodiccell of period , with two dielectric slabs withparameters , (plexiglas), and

(air); the width of the grating is (seeFig. 7) and . In the calculations a loss tangent of

(lossy dielectric) was used, and we have con-sidered an angle of incidence on the structure of to takeinto account the possible asymmetry of the experimental setupas explained in [23]. We have chosen and ,taking 0.33 s per frequency point. Fig. 7 compares the calcu-lated response using this method with the experimentally mea-sured response, finding a good agreement.

The third case analyzed is a multilayered periodic structurewith two grating layers and six homogeneous layers [seeFig. 8(a)] between air and a substrate. The parameters of thisstructure are , , , ,

, , and . Thegrating thicknesses are and


Fig. 4. Field distributions along the unit cell of the first and second mode for the geometry of Fig. 2. (a) Distribution of the normalized electric field for the firstand second TE mode at (k D=�) = 0 (straight line) and 1 (dashed line). Results calculated in [21] are shown with dots. (b) Distribution of the normalizedmagnetic field for the first and second TM mode at (k D=�) = 0 (straight line) and 1 (dashed line). Results calculated in [22] are also shown with dots.

Fig. 5. Convergence study of the reflectance of a periodic dielectric gratingin air as a function of the number of modes M included in the construction ofthe GSMs. The results are presented for three different frequencies under a TEplane wave incidence at � = 45 . The grating has a period D = 1:0 mm andthickness h = 1:713 mm with parameters: l = l = 0:5 mm, " = 2:56," = 1:44, ~" = 1:44.

Fig. 6. Frequency dependence of the reflectance of the periodic dielectricgrating detailed in Fig. 5 under a TE plane wave incidence at � = 45 .Comparison between our results (line) and those calculated in [17] with theTRM (dots).

. In Fig. 8(b) we compareour results for the transmittance of the structure for a normal

Fig. 7. Calculated and measured transmittance of a periodic dielectric gratingimmersed in air for a TM-polarized incident wave. The grating period is D =30:0mm and the thickness is h = 8:7mm. Parameters: l = l = 15:0mm," = 2:59, " = 1:0, ~" = 1:0. Comparison between our results (lines)and the experimental ones (dots) presented in [23].

TE-polarized incident wave with those obtained in [24] with therigorous coupled-wave theory, showing a good concordance.For this case we chose and , taking 0.11 s perfrequency point.

Finally, a novel design of a reflection (band-stop) filterfor TE-polarization has been performed using the presentedtheory. The design of a reflection filter involves the selectionof the filter parameters such as the thickness of each layereither homogeneous or periodic and the dielectric permittivitydistribution within the unit cell of the periodic layers, in orderto achieve symmetrical line shapes and reduced reflectancearound the central wavelength, for a particular polarizationand angle of the incident wave. A simple reflection filterwith a single-layer waveguide grating has been designed for anormal TE-polarized incident wave, containing a rectangulargrating composed of two dielectric materials whose reflectanceis shown in Fig. 9, with the structure illustrated in the inset.The spectral response shown in Fig. 9 can be predicted usingclassical unmodulated slab waveguide theory. For the greaterpart of the spectrum, the grating has the reflectance of a thinfilm with a dielectric constant equal to the average dielectricconstant of the grating. The grating thickness has been chosen


Fig. 8. Transmittance of a multilayered periodic structure with two grating layers (layers 1 and 7) and six homogeneous layers between air and a substrate.Comparison between our results (line) and those calculated in [24] with the rigorous coupled-wave theory (dots).

Fig. 9. Reflection filter spectral response with the structure illustrated in theinset. The peak frequency is centered at 15 GHz. The filter parameters are: D =

11:28 mm, h = 4:37 mm, " = 6:13 (E-glass), " = 3:7 (silica), l =

l = D=2.

to be half-wavelength for the central wavelength of the filter, sothe spectral response shows a reduced reflectance around thiswavelength. But at a specific frequency (resonance frequency)the diffractive character of the grating enables the incident waveto excite a leaky mode supportable by the grating, resulting in atransmission null, as shown in Fig. 9 [25]. The approximate valuefor the resonance frequency location can be predicted imposingthe phase-match condition for the equivalent unmodulated slabwaveguide , where is the propagationconstant of the unmodulated waveguide in the y-direction,and is the wavevector provided by the grating. At thisfrequency the waveguide mode is excited, and this mode willreradiate plane waves into the air regions above and belowthe layer through the same space harmonic, thereby acting asa leaky wave. The reradiated waves interfere with the directlyreflected and transmitted fields, and when the two componentsabove the layer add in phase, strong reflection takes place.For a given thickness and dielectric materials of the grating,the resonance frequency can be adjusted choosing an adequatevalue of the grating period such that it coincides with the centralfrequency of the filter, resulting in a symmetrical reflection

filter with small sideband reflection. To maximize the efficiencyof this device, a subwavelength grating period is chosen inorder to permit that only the zero order Floquet mode topropagates in free space. The filter has been designed in thespectral range of 13–17 GHz, for the peak frequency centeredat 15 GHz, having the following parameters: ,

, (E-glass), (silica),and . The spectral response of the

filter in Fig. 9 shows that in the range of 13–17 GHz(except around the band-stop frequency), and in therange of 14.88–15.12 GHz, thus being the bandwidth (fullwidth at half maximum of the reflected power)(1.6%). In the calculations we have chosen and ,taking 0.03 s per frequency point.

IV. CONCLUSION

P

A vectorial modal method has been applied to analyze 1-Dperiodic dielectric media with any number of lossy dielectricslabs within the unit cell for both TE and TM polarizations.This formulation allows the study of electromagnetic scatteringfrom multilayered periodic structures by means of the GSMtheory. The study of a wide variety of configurations withsmall computational cost has been performed. Furthermore,we have tested the theory by comparison with theoretical andexperimental results found in the technical literature, showinga good agreement. A reflection filter using practical materialshas been designed for a central frequency of 15 GHz, showinglow symmetrical sidebands in the range of 13–17 GHz. Newtechniques for filter design are opened applying the filtersynthesis theory with transmission lines for band-stop, band-pass and low-pass filters with periodic structures. In future, weare also interested in the analysis of the 3-D oblique incidenceupon the dielectric grating.

ACKNOWLEDGMENT

The authors would like to thank the Reviewers for their usefulcomments and suggestions.


REFERENCES

[1] T. K. Gaylord and M. G. Moharam, “Analysis and applications ofoptical diffraction by gratings,” Proc. IEEE, vol. 73, pp. 894–938,May 1985.

[2] M. K. Moaveni, “Plane wave diffraction by dielectric gratings, finite-difference formulation,” IEEE Trans. Antennas Propagat., vol. 37, pp.1026–1031, Aug. 1989.

[3] J. C. W. A. Costa and A. J. Giarola, “Analysis of the selective behaviorof multilayer structures with a dielectric grating,” IEEE Trans. AntennasPropagat., vol. 43, pp. 529–533, May 1995.

[4] C. Zuffada and T. Cwik, “Synthesis of novel all-dielectric grating filtersusing genetic algorithms,” IEEE Trans. Antennas Propagat., vol. 46, pp.657–663, May 1998.

[5] A. Coves, B. Gimeno, D. Camilleri, M. V. Andres, A. San Blas, andV. E. Boria, “Scattering by dielectric frequency-selective surfacesusing a vectorial modal method,” in IEEE AP-S Int. Symp. andURSI National Radio Sci. Meeting, San Antonio, TX, June 16–21,2002, pp. 580–583.

[6] R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequencyselective surfaces—a review,” Proc. IEEE, vol. 76, pp. 1593–1615, Dec.1988.

[7] K. Kobayashi, “Diffraction of a plane wave by a thick strip grating,”IEEE Trans. Antennas Propagat., vol. 37, pp. 459–470, Apr. 1989.

[8] T. Cwik and R. Mittra, “The cascade connection of planar periodic sur-faces and lossy dielectric layers to form an arbitrary periodic screen,”Proc. IEEE, vol. 76, pp. 1593–1615, Dec. 1988.

[9] V. D. Agrawal and W. A. Imbriale, “Design of a dichroic Cassegrainsubreflector,” IEEE Trans. Antennas Propagat., vol. 27, pp. 466–473,1979.

[10] S. S. Wang and R. Magnusson, “Design of waveguide-grating filters withsymmetrical line shapes and low sidebands,” Opt. Lett., vol. 19, no. 12,pp. 919–921, Jun. 1994.

[11] S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectricwaveguides,” IEEE Trans. Microwave Theory Techniques, vol. 23, pp.123–133, Jan. 1975.

[12] W. P. Pinello, R. Lee, and A. C. Cangellaris, “Finite element modeling ofelectro-magnetic wave interactions with periodic dielectric structures,”IEEE Trans. Microwave Theory Tech., vol. 42, pp. 2294–2301, Dec.1994.

[13] E. Silvestre, M. V. Andres, and P. Andres, “Biorthonormal-basis methodfor the vector description of optical-fiber modes,” J. Lightwave Technol.,vol. 16, no. 5, pp. 923–928, May 1998.

[14] E. Silvestre, M. A. Abian, B. Gimeno, A. Ferrando, M. V. Andres, andV. E. Boria, “Analysis of inhomogeneously filled waveguides using abi-orthonormal-basis method,” IEEE Trans. Microwave Theory Tech.,vol. 48, pp. 589–596, Apr. 2000.

[15] R. C. Hall, R. Mittra, and K. M. Mitzner, “Analysis of multilayered peri-odic structures using generalized scattering matrix theory,” IEEE Trans.Antennas Propagat., vol. 36, pp. 511–517, Apr. 1988.

[16] T. S. Chu and T. Itoh, “Generalized scattering matrix method for anal-ysis of cascaded and offset microstrip step discontinuities,” IEEE Trans.Microwave Theory Tech., vol. 34, pp. 280–284, Feb. 1986.

[17] H. L. Bertoni, L. H. S. Cheo, and T. Tamir, “Frequency-selective reflec-tion and transmission by a periodic dielectric layer,” IEEE Trans. An-tennas Propagat., vol. 37, pp. 78–83, Jan. 1989.

[18] D. G. Dudley, Mathematical Foundations for Electromagnetic Theory,1st ed. Piscataway, NJ: IEEE Press, 1994.

[19] A. Coves, B. Gimeno, M. V. Andres, J. A. Montsoriu, and E. Silvestre,“Evaluation of Discontinuities in Modal Vectorial Methods,” , submittedfor publication.

[20] R. E. Collin, Field Theory of Guided Waves, 2nd ed. New York: IEEEPress, 1991.

[21] J. C. W. A. Costa and A. J. Giarola, “Electromagnetic wave propaga-tion in multilayer dielectric periodic structures,” IEEE Trans. AntennasPropagat., vol. 41, pp. 1432–1438, Oct. 1993.

[22] , “Wave propagation in multilayer dielectric periodic structures,”in IEEE AP-S Int. Symp. Digest, vol. 4, Ann Arbor, MI, 1993, pp.1964–1967.

[23] S. Tibuleac, R. Magnusson, T. A. Maldonado, P. P. Young, and T. R.Holzheimer, “Dielectric frequency-selective structures incorporatingwaveguide gratings,” IEEE Trans. Microwave Theory Tech., vol. 48, pp.553–561, Apr. 2000.

[24] R. Magnusson and S. S. Wang, “Transmission bandpass guided-moderesonance filters,” Appl. Opt., vol. 34, no. 35, pp. 8106–8109, Dec. 1995.

[25] T. Tamir and F. Y. Kou, “Varieties of leaky waves and their excitation,”IEEE J. Quant. Electron., vol. 22, pp. 544–551, 1986.

Angela Coves was born in Elche, Spain, on May 20,1976. She received the Licenciada degree in physicsfrom the Universidad de Valencia, Valencia, Spain,in 1999, where she is currently working toward thePh.D.

She is currently with the Departamento de FísicaAplicada y Electromagnetismo, Universidad deValencia. Since 2001, she has been an AssistantProfessor with the Departamento de Física y Ar-quitectura de Computadores, Universidad MiguelHernández de Elche, Elche (Alicante), Spain. Her

current research interests include numerical methods in computer-aided tech-niques for the analysis of microwave passive components such as waveguidestructures and multilayered periodic structures including dielectric media.

Benito Gimeno was born in Valencia, Spain, on Jan-uary 29, 1964. He received the Licenciado degree inPhysics in 1987 and the Ph.D. degree in 1992, bothfrom the Universidad de Valencia, Spain.

He was a Fellow at the Universidad de Valenciafrom 1987 to 1990. Since 1990, he served as Assis-tant Professor in the Departmento de Física Aplicaday Electromagnetismo at the Universidad de Valencia,where he became Associate Professor in 1997.He was working at European Space Research andTechnology Centre of the European Space Agency

(ESTEC) as a Research Fellow during 1994 and 1995. In 2003, he obtaineda fellowship from the Spanish Government for a short stay (three months) atthe Universita degli Studi di Pavia, Pavia, Italy, as a Visiting Scientific. Heis currently with the Departamento de Física Aplicada y Electromagnetismo-I.C.M.U.V., Universidad de Valencia, Burjasot (Valencia), Spain. His currentresearch interests include the areas of computer-aided techniques for analysisof microwave passive components, waveguide and cavities structures includingdielectric resonators and photonic band-gap crystals.

Jordi Gil was born in Valencia, Spain, on April 27,1977. He received the Licenciado degree in physicsfrom the Universidad de Valencia, in 2000, where heis currently working toward the Ph.D.

He is currently working at the Ingegneria dei Sis-temi IDS-S.p.A., Pisa, Italy. Since 2001, he has beena young researcher in the frame of the EC projectMMCODEF “Millimeter-wave and MicrowaveComponents Design Framework for Ground andSpace Multimedia Network” in collaboration withthe European Space Agency (ESA). His current

research interests include numerical methods in computer-aided techniques forthe analysis of microwave passive components such as waveguide structuresand multilayered periodic structures including dielectric media.

Miguel V. Andrés (M’91) was born in Valencia,Spain, in 1957. He received the Licenciado en Físicadegree in 1979 and the Doctor en Física (Ph.D.) de-gree in 1985, both from the Universidad de Valencia.

From 1983, he served successively as AssistantProfessor and Lecturer in the Departamento deFísica Aplicada, Universidad de Valencia. From1984 to 1987, he was visiting for several periods theDepartment of Physics, University of Surrey, U.K.,as a Research Fellow. Until 1984, he was engagedin research on microwave surface waveguides. His

current research interests are optical fiber devices and systems for signalprocessing and sensor applications, and electromagnetic wave propagation inmicrowave and optical waveguides and devices.

Dr. M. V. Andrés is a Member of the Institute of Physics (IOP).


Angel A. San Blas was born in Fortaleny (Valencia),Spain, on September 20, 1976. He received the Inge-niero de Telecomunicación degree from the Univer-sidad Politécnica de Valencia, in November 2000.

In 2001, he was awarded a researcher position inthe Departamento de Comunicaciones, UniversidadPolitécnica de Valencia, where he worked for twoyears. In 2003, he joined the Departamento deFísica y Arquitectura de Computadores, UniversidadMiguel Hernández de Elche, Spain, where he iscurrently an Assistant Professor. His current research

interests include the analysis of discontinuities in waveguide structures, designof microwave filters, and coaxial excitation of microwave devices.

Vicente E. Boria (SM’02) was born in Valencia,Spain, on May 18, 1970. He received the Ingenierode Telecomunicación degree with first-class honorsand the “Doctor Ingeniero de Telecomunicación”degree from the Universidad Politécnica de Valencia,Valencia, Spain, in 1993 and 1997, respectively.

In 1993, he joined the Departamento de Comu-nicaciones, Universidad Politécnica de Valencia,where he was an Assistant Lecturer from 1993 to1995, a Lecturer from 1996 to 1997, an AssociateProfessor from 1998 to 2002, and became Full

Professor in 2003. In 1995 and 1996, he was awarded a Spanish Traineeposition at ESTEC-ESA, Noordwijk, the Netherlands, where he worked inthe area of electromagnetic analysis and design of waveguide devices. Hehas authored or coauthored over 20 papers in refereed international technicaljournals and over 70 papers in international conference proceedings in his areasof research interest. His current research interests include numerical methodsfor the analysis of waveguide and scattering structures, automated design ofwaveguide components, radiating systems, measurement techniques and powereffects in passive systems.

In 1993, Dr. Boria received from the Spanish “Ministerio de Educación yCiencia” the First National Prize of Telecommunication Engineering Studies forhis outstanding student record. In 2001, he received from the Social Council ofUniversidad Politécnica de Valencia the First Research Prize for his outstandingactivity during the period 1995 to 2000. Since 1992, he has been a member ofthe IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and ofthe IEEE Antennas and Propagation Society (IEEE AP-S). Since 2003, he hasbeen a member of the Technical Committee of the IEEE-MTT InternationalMicrowave Symposium and of the European Microwave Conference.


On the Interaction Between Electric and MagneticCurrents in Stratified Media

Daniel Llorens del Río and Juan R. Mosig, Fellow, IEEE

Abstract—The presence of both electric and magnetic currentelements embedded in stratified media is necessary to model manyproblems of interest in current integrated circuit and printedantenna technology. The integral equation formulation as appliedto these problems is reviewed. Special attention is given to theGreen’s dyadic for the electric field generated by a magneticcurrent element. The fact that spectral-domain transmission lineGreen’s functions for a multilayered structure are closed formintegrable along the axis normal to the stratification is exploitedto greatly improve the efficiency and the accuracy of the method.Theory and implementation are demonstrated in two practicalproblems: 1) arbitrarily shaped apertures in thick conductingscreens, and 2) a metallic airbridge over a slot line.

Index Terms—Integral equations, slot antennas, apertures inthick screens, airbridges.

I. INTRODUCTION

RECENT progress in printed circuit technology hasmade pervasive the use of multiple ground-plane, mul-

tiple-via connected circuits [1]. Coplanar waveguide (CPW)has finally become the transmission line of choice for mostmillimeter-wave printed circuit applications [2], for examplewhen integrated with slot antennas. In most cases, the CPWis used together with airbridges or combined with via-holesinterconnects.

In all these structures, the most efficient approach is tomodel slots and apertures with equivalent magnetic currentsand metallic interconnections (vias, airbridges) with equivalentelectric currents. The scattered fields would then be obtained,invoking linearity, as convolutions of the equivalent currentswith the adequate Green’s functions

(1a)

(1b)

A particularly interesting situation is encountered when mod-eling slots in thick metallic planes. An approximate method todeal with this problem, termed the “Delta” approach, has re-cently been presented in [3]. In its final formulation, this methodonly provides a first order approximation to the effect of finitethickness and neglects the effect of the walls, particularly thepossible coupling between the walls at both sides of the slot.Other methods [4] only consider the walls approximately. The

Manuscript received June 2, 2003; revisedAugust 22, 2003. This work wassupported by the ESA/ESTEC under Contract 14062/00/NL/GD.

The authors are with the Laboratory of Electromagnetics and Acoustics(LEMA), Swiss Federal Institute of Technology, EPFL-Ecublens Lausanne,CH-1015, Switzerland (e-mail: [email protected]).


Fig. 1. Aperture in thick screen: original problem. Above and below the screenlie arbitrary, laterally open multilayered media.

thickness effect is particularly important in millimetric CPWcircuits that use compact filters [5] where very narrow slots arecommon. As it will be shown in this paper, an efficient and gen-eral modeling approach is to use vertical electric currents in thelateral metallic walls and horizontal magnetic currents in theapertures themselves. Therefore, we have again a mixed situa-tion where the four Green’s functions in (1) are needed.

In the case of multilayered media, and are usu-ally written in a mixed potential form [6]. For and ,it is more advantageous, if electric and magnetic cells do notoverlap, to keep the field formulation. The strong singularitiesof these functions will be reduced to absolutely integrable sin-gularities with a preprocessing of the vertical coordinate in thespectral domain, which was pioneered in [7] and extended inscope in [8] and [9]. This preprocessing greatly increases theefficiency and accuracy of the method, but requires that all un-knowns be exactly perpendicular or parallel to the axis of strati-fication (a “2.5D” geometry). Most printed technology productsmatch this requirement.

II. MODELING APERTURES IN THICK SCREENS

To illustrate the general strategy proposed in this paper to dealwith interactions, we start by considering the “thick slot”problem of Fig. 1, where a stratified medium includes a thickground plane in which a slot has been created. Thisproblem is usually solved by using equivalence principles to re-duce it to a more tractable configuration. Fig. 2 shows the stan-dard strategy, which divides the problem in three parts linkedby magnetic currents. In the layered media above and below thethick slot, standard MoM integral equation approaches may beused. The medium representing the slot is now a cavity (or awaveguide closed at both ends); here, apropiate modal expan-sions or cavity/waveguide Green’s functions should be used.

0018-926X/04$20.00 © 2004 IEEE

LLORENS DEL RÍO AND MOSIG: ON THE INTERACTION BETWEEN ELECTRIC AND MAGNETIC CURRENTS IN STRATIFIED MEDIA 2101

Fig. 2. Equivalent problem with cavity or closed waveguide.

Fig. 3. Equivalent problem with laterally open parallel plate medium andwall electric current unknowns. Numbers 1–5 refer to regions where equivalentcurrents J or M do exist.

This method has two main inconvenients. Firstly, the modes ofthe waveguide must be computed, which is difficult to do fora waveguide of arbitrary section. Secondly, the modal expan-sion must be matched at the apertures to the multilayered mediaabove and below, which requires the computation of extra cou-pling integrals.

In this paper, we propose a new strategy for creating an equiv-alent problem. The geometry is again divided into three regions,and the media above and below are the same. However, in themedium representing the thick slot (the “internal” medium), theequivalent is now a parallel plate waveguide (PPWG). To ac-count for the metallic walls, we add unknown vertical electriccurrents embedded in the PPWG. With this strategy, we need tocompute interactions between horizontal magnetic currents andvertical electric currents. As this situation arises in many otherconfigurations (for instance, airbridges over slot lines) this is themain concern of this paper.

A. Formulation

It is apparent that in either of these strategies, the com-putational burden will at least double when compared toa zero-thickness analysis, owing to the slot aperture beingmeshed twice. For a general solution to the problem, this isimpossible to avoid. Still, a blind attack on the problem wouldnot be excessively time-consuming for this reason, but forthe complex, nonanalytical dependence of the space domainPPWG’s Green’s function on the three coordinates , and(Fig. 3).

For the sake of simplicity, we shall assume that the mediaabove and below the slot are devoid of scatterers which couldsupport additional induced electric and magnetic currents. Also

we assume that incident excitation waves exist outside the slot,on both sides. Therefore, the only unknowns will be magneticcurrents above and below the slot, and electric current

on the vertical walls of the slot.Accordingly, the equivalent thick slot problem of Fig. 3 yields

three coupled integral equations. The first two impose conti-nuity of electric and magnetic tangential field at lower and upperapertures:

(2a)

(2b)

The third integral equation forces the electric tangential fieldto vanish on electric walls

(2c)

In these equations, the currents in the upper and lower sides ofthe slot and the superindices of the Green’s functionsrefer to the regions of the problem as indicated (Fig. 3). Thisformulation can be easily generalized to a more complicatedgeometry. Any external media can be theoretically handled byproviding appropiate Green’s functions and/or byincluding additional unknown currents as required.

The magnetic and electric equivalent currents are then ex-panded as linear combinations of rooftop basis functions and

and tested with a Galerkin approach. This results in a momentblock matrix of the form

(3)

The symmetry of , and submatrices stems, respectively,from the following reciprocity relationships between reactionterms [10]

(4a)

(4b)

(4c)

The matrix of (3) is not, however, symmetric. If, instead of astrict Galerkin approach, a set of test functions is used,a symmetric matrix results, as noticed by Harrington [6]. Thisis equivalent to multiplying the first two (block) rows of system(3) by .

The submatrices contain terms of the type. They are computed with a dual mixed potential expres-

sion involving potential Green’s functions , which canpresent an integrable singularity of type . It is remarkablethat each individual Green’s function in (2b) will divergeas , so a double aperture formulation such as the oneused here will not work with vanishingly small screen thick-ness. Even with moderately small thickness, it will be advanta-geous to extract not only the singular part of each reaction term


, but also the contribution of the closest im-ages. This is clearly a limitation of the double-aperture formula-tion, as has already been pointed [11]. The “Delta” formulation[3] is specifically designed to handle the limiting case.

The submatrix involves terms of the form .These are computed by putting in full mixed potentialform, according to the formulation in [12]. Rectangular rooftopsare used to expand in an orthogonal mesh.1 The classicalSommerfeld form for is used, as in [8], but the computa-tion of [13] is never necessary, because it can be substi-tuted by proper application of reciprocity relation (4c). We notethat the line integrals inherent in our choice of the Sommerfeldform of do not appear in the particular case of a homoge-neously filled PPWG. Otherwise, the spectral integration tech-nique, which is detailed in the next section for the elements of

, is also applied to the computation of the elements of . Theintegrated functions to be tabulated present at most a logarithmicsingularity.

B. Submatrices

These contain elements of the form , whichare handled with a pure field formulation. For a generalizedPPWG (which may be filled with a number of stratified dielec-tric layers), the spectral Green’s dyadic has the form

(5)

The various components are obtained from the well knownspectral TL model, where the stratified medium is representedby two (TE and TM) equivalent transmission line networks, andall field quantities can be expressed in terms of voltages andcurrents in this network, under either voltage or cur-rent excitation [14]. We have cast them in the followingform, ready for numerical implementation:

(6a)

(6b)

(6c)

(6d)

(6e)

(6f)

(6g)

whose symmetry properties

(7a)

(7b)

(7c)

1We shall use the conditions that• e = ee(z; �) = ee (z)e (�); (separability)• e ? z or e k z (orthogonality).

A mesh of the kind needed is pictured in Fig. 8.

correspond to spatial domain rotational symmetry

(8a)

(8b)

(8c)

Making use of the identities (14) in Appendix I, at most fourSommerfeld transform operations are needed to obtain the trans-verse components , and only one to obtain

(vertical magnetic currents are not considered).These operations are

(9a)

(9b)

(9c)

(9d)

(9e)

Therefore, the azimuthal dependence of is evaluated (in-expensively) during matrix fill.

All off-diagonal components are singular at the originas , with bounded, when . This can begathered from the behavior of the free space dyadic (with

)

(10)

In principle, as discussed in previous sections, every Green’sfunction will depend on three space coordinates, , and .However, the vertical regularity of the structure makes it pos-sible to take this dependence into account in the spectral do-main. For it is well known that the existence of a TL model forthe stratified medium ensures that the dependence with respectto the vertical coordinates will be of the form

(11)

where parameters do not depend on the vertical co-ordinates once source and observer layers are fixed. Formulaeof this type, that explicitly reveal the exponential dependenceon both vertical coordinates and for any transmission lineparameter ( or ) have been presented in [8] and, more re-cently, again in [9].

The transverse Fourier transform does not affect the verticaldependence; thus, any spatial integration along can be car-ried over in the spectral domain, analytically. This effectivelyeliminates the spatial Green’s functions dependence on thesecoordinates, avoiding cumbersome three-dimensional spatialinterpolation during matrix-of-moments fill.

In the particular case of elements of the typeis fixed because it corresponds to a horizontal


TABLE IGREEN’S FUNCTIONS FOR THE COMPUTATION OF ELEMENTS OF �

�H

AND THEIR SINGULARITIES AT THE ORIGIN

magnetic current cell. Therefore, only the -dependence has tobe treated in the spectral domain. With the same requisites oforthogonality of the mesh for the vertical walls, we can write

(12)

The last right-hand term in the series of equalities of (12) isthe key for a successful numerical integration. The new “inte-grated” dyadic has reduced-order singularities. Theexact type of these singularities can be obtained either from the“integrated” spatial static Green dyadic, or from the behavior ofthe spectral dyadic as . The first approach is more prac-tical because it allows to obtain closed analytical expressionsfor the singular part of the reaction term, in a manner similar towhat is done in [15] for potential integrals. This developmentis detailed in Appendix II. The resulting singularity extractionprocedure is necessary for an efficient and accurate computa-tion of when electric and magnetic cells sharean edge. Thanks to the spectral integration in (12), the strongestsingularity of ’s elements is absolutely integrable, whichmakes the technique much easier to apply.

With the spectral integration technique, six Sommerfeldtransforms are required per vertical cell level-magnetic layerpair: four for the horizontally oriented half-rooftopsand one for either increasing or decreasing

vertically oriented half-rooftops. This issummarized in Table I. For example, in the mesh of Fig. 8,which has two vertical cell levels and two magnetic layers, atotal of functions of this type must be tabulated.

C. Discussion

In addition to avoidance of three-dimensional interpolation,the spectral preprocessing technique has four advantages. Theseare not restricted to the interactions between current elements ofdifferent type, but in that case they are particularly important,because a field formulation is employed.

• Integrated Green’s functions have better spectral behaviorand are thus easier to evaluate in the spatial domain.

• Spatial integration becomes simpler, not only because sur-face integrals are reduced to line integrals, but also be-cause they exhibit weaker singularities at the origin

than their nonintegrated counterparts.• Tables may be reused for differently shaped slots, as long

as the vertical structure (on which spectral integration de-pends) remains unchanged.

• Since vertical integrations are analytical, accuracy isincreased.

It is important to make a note about the method used toperform the Sommerfeld inversion. An elliptical integrationpath circumventing the spectral Green’s function singularities[16], combined with the weighted averages algorithm [17] forthe tail of the integral along the real axis, is used throughout.The different Green’s functions to be tabulated are classifiedin vectors according to their singularity type, and transformedin block; therefore, for each of these vectors, the spectral TLmodel is solved only once. This classification is necessarybecause the weighted averages algorithm is an extrapolationmethod that needs information about the asymptotic behaviorof the integrand as to perform optimally.

D. Examples

Two different examples, a rectangular slot and a “dogbone”slot antenna, have been analyzed, built and measured. A series ofbreadboards has been built for each, where slot thickness variesfrom 35 m (printed slot, essentially a zero thickness case) toseveral millimeter.

The rectangular slot antenna is fed by coupling to a 50 mi-crostrip line. Its dimensions are 25 mm 5 mm and it is foundto resonate at 6.32 GHz on mm, substrate,when metal thickness equals 35 m.

The series of measured breadboards included screens withthicknesses of , and mm. Comparison between dif-ferent numerical models and measurements is given in Figs. 4–6for conciseness, comparison for the 3 and 6 mm cases has beenomitted, but it is reported in [18]. Three different theoretical pre-dictions are shown as follows:

• the “Delta” function approach [3];• a rigorous mode-matching cavity model [19];• the technique described in this paper.

Our technique always follows closely the cavity results (whileavoiding the cavity formulation) up to -thick slots, andthat both agree very well with measurements. As expected,the “Delta” approach is good only for thin slots (up to ).It can be seen that even for slot thickness as small as ,the effect on resonant frequency is clearly nonnegligible (a

% displacement) which stresses the interest and need ofthis analysis.

The second model is a “dogbone”-shaped slot antenna, alsofed by coupling to a 50 microstrip line (Fig. 7).


Fig. 4. Rectangular slot antenna: thickness 35 �m � �=1400 at 6 GHz.

Fig. 5. Rectangular slot antenna: thickness 1 mm � �=50 at 6 GHz.

The mesh used in the MoM analysis is shown in Fig. 8.The freely available mesh generator, TRIANGLE,2 was used.The choice of a triangular mesh for the magnetic unknownsis justified by the fact that, opposite to the vertical walls, slotgeometry can be rather complex. It is worth pointing out thata shape like the “dogbone” does not add any complexity toour approach, while the cavity approach would become verycumbersome and, indeed, it has not been considered here.

The breadboard series included screens mm, mm, andmm thick. They were made of brass and gold-plated for

best contact with the printed antenna substrate; their size was14 cm 14 cm, large enough to avoid finite ground plane sizeinfluencing input impedance.

Simulations and measurements are compared in Figs. 9 to11. It can be seen that the numerical analysis follows closely

2http://www-2.cs.cmu.edu/~quake/triangle.html

Fig. 6. Rectangular slot antenna: thickness 10 mm � �=5 at 6 GHz.

Fig. 7. ”Dogbone” slot in thick screen, fed by microstripline. Dimensions inmillimeters: r = 2:5; e = 2:5; l = 34;w = 5; l = 47; w = 2:164; p =

9:5. Substrate: h = 0:76 mm, � = 2:485.

Fig. 8. Mesh for the “dogbone” model with mask of thickness =3 mm, withtwo vertical electric cell levels between the parallel plates. Electric cells in white(microstripline+ sides of dogbone slot), magnetic cells in gray (top and bottomof dogbone).

the behavior indicated by the measurements, as thickness in-creases. The opening of the resonance loop in the Smith chart ischaracteristic.


Fig. 9. ”Dogbone” antenna: thickness 35 �m � �=1700 at 5 GHz.

Fig. 10. ”Dogbone” antenna: thickness 1 mm � �=60 at 5 GHz.

III. AIRBRIDGE MODELING

CPW circuits usually require airbridges for proper operation,to eliminate the unwanted slotline mode [2]. In many circuits,such as the dual-mode filter introduced in [20], or in MEMSdelay lines, the response of the circuit is highly dependent onairbridge dimensions.

Fig. 11. ”Dogbone” antenna: thickness 5 mm � �=12 at 5 GHz.

Despite its very different nature, the IE formulation of theairbridge/slot problem is basically equivalent to that of thethick slot and again the interaction plays an essentialrole. Due to the presence of horizontal electric cells, a newsubmatrix type appears, that contains interactions ofthe form . These are computed again with amixed potential formulation, where the corresponding Green’sfunctions have been integrated in the spectral domain alongthe source coordinate. In addition, line integral terms appearbecause of the airbridge corners [21].

A. Examples

As a test case, a slot loop antenna that resonates at 3 GHzhas been selected. This antenna (Fig. 12) was a model forstudying the radiating element of open structure, integratedreceiver front-ends for submillimeter-wave operation [22]. Inthat application, the diode can be connected in series to the feedline; then, the airbridge (in the symmetrical position) is used toprovide a dc return path. The airbridge should not affect the RFcharacteristics of the slot loop, so it should be comparativelylong.

In the asymmetrical position, the airbridge can be used toconnect to ground a diode that has been placed at the center ofthe loop. The role of the airbridge is wholly different becausenow it must conduct RF signal. The slot loop, which is aboutone wavelength at RF, presents a magnetic current null at thatposition, which allows for a very short airbridge.

For the first case, a series of airbridges of different formfactors, listed in Fig. 13, was built, and antenna impedancewas measured at the end of the CPW feeding line (Fig. 14).An excellent agreement is observed for all cases, as comparedto simulations (Fig. 15). It can be appreciated that for the


Fig. 12. Slot antenna propotype. Dimensions in millimeters: r = 15:92; w = 2; l = 19:89;w = 1:140; s = 0:930;w = 0:126; s = 2:748. Substrate:h = 1:57 mm, � = 2:33.

Fig. 13. Dimensions (in millimeters) for each case of Figs. 14, and 15.

longest airbridge (case #2) the main resonance of the bareloop is least affected.

The asymmetrical placement of the airbridge produces inturn a very complex behavior, with multiple resonances, due tothe introduction of asymmetric modes in the loop. The mainresonance of the bare loop remains unaffected however, be-cause at that frequency the condition imposed by the airbridgematches the natural symmetric configuration of magneticcurrents. Again, comparison of measurement and simulation(Fig. 16) shows good agreement.

IV. CONCLUSION

A technique to compute interactions between electric andmagnetic currents embedded in multilayered media has beendescribed, as applied to the analysis of 1) slots in metalliza-tion screens of finite thickness, and 2) airbridges in slot/CPWcircuits.

Fig. 14. Measured jS j for the slot loop antenna of Fig. 12 with the airbridgesof Fig. 13, with airbridges on the symmetrical position.

For the “thick slot” problem, this is done by solving for mag-netic currents at both sides of the slot and for electric currentsat the walls on its contour. The problem posed by these ver-tical walls inside a parallel-plate medium is ideally suited to amixed spectral-spatial formulation. A pure mixed potential for-mulation has been kept for the electric elements. For the parallelplate medium, this choice leads to less integral types and con-sequently to reduced memory requirements and faster compu-tations. For electric-magnetic interactions, a field formulation isused. The singularity of the mixed-type Green’s functionsis integrated over the source cell (always a magnetic current cell)with a closed formula.

The method presented here has the flexibility required to gobeyond regular slot shapes, where the use of the cavity approachwould be much more involved. This ability has been demon-strated with the analysis of a “dogbone”-shaped aperture.


Fig. 15. Computed jS j for the slot loop antenna of Fig. 12 with the airbridgesof Fig. 13, with airbridges on the symmetrical position.

Fig. 16. Computed and measured jS j for the slot loop antenna of Fig. 12with an airbridge (case #4 of Fig. 13) on the asymmetrical position.

Quite remarkably, the same technique has been applied toa very different problem, namely, the analysis of the effect ofan airbridge short-circuiting a slot loop antenna at different po-sitions. Again, comparison with measurements has shown thevalidity of the method.

APPENDIX ITRANSFORM RELATIONS

(13a)

(13b)

(14a)

(14b)

(14c)

(14d)

(14e)

APPENDIX IISINGULARITY EXTRACTION

When in (12) is a triangular RWG rooftop, the inner inte-gral in (12) is proportional to (substituting the static part of (10)for )

(15a)

(15b)

where (15a) is for vertically oriented and (15b) for horizon-tally oriented , i.e., . These integrals are both analyt-ical. The second one results in an integral over which has thesame mild-type singularity of a regular potential Green’s func-tion [15]. For (15a), if is a constant and

(16)

is substituted in (15a), we obtain

(17)

This is absolutely integrable, as can be shown by considering

(18)

(because ). Notingthat

(19)


we may write, separating from an infinitesimal regionaround , say, a circular sector of radius

(20)

The second term becomes zero as , and the integrand ofthe third term is bounded, so it also vanishes as . Thereremains

(21)

which is immediate.If is a linear function of , the resulting integral can be

seen to be a combination of (17) and a term of the same type as(15b).

ACKNOWLEDGMENT

The authors would like to thank Dr. Lin of ESTEC (EarthObservation Programs) for his support of this work.

REFERENCES

[1] H. Chen, Q. Li, L. Tsang, C.-C. Huang, and V. Jandhyala, “Analysis ofa large number of vias and differential signaling in multilayered struc-tures,” IEEE Trans. Microwave Theory Tech., vol. 51, pp. 818–829, Mar.2003.

[2] R. Simons, Coplanar Waveguide Circuits, Components and Systems, 1sted. New York: Wiley, 2001.

[3] J. R. Mosig, Scattering by arbitrarily-shaped slots in thick conductingscreens: An approximate solution, in IEEE Trans. on Antennas Prop..Accepted for publication to.

[4] D. T. Auckland and R. F. Harrington, “Electromagnetic transmissionthrough a filled slit in a conducting screen of finite thickness, TE case,”IEEE Trans. Microwave Theory Tech., vol. 26, pp. 499–505, July 1978.

[5] J. Sor, Y. Qian, and T. Itoh, “Miniature low-loss CPW periodic structuresfor filter applications,” IEEE Trans. Microwave Theory Tech., vol. 49,pp. 2336–2341, Dec. 2001.

[6] R. F. Harrington, Field Computation by Moment Methods, 1st ed. NewYork: Macmillan, 1968.

[7] M.-J. Tsai, C. Chen, N. G. Alexopoulos, and T.-S. Horng, “Multiple ar-bitrary shape via-hole and air-bridge transitions in multilayered struc-tures,” IEEE Trans. Microwave Theory Tech., vol. 44, pp. 2504–2511,Dec. 1996.

[8] N. Kınayman and M. I. Aksun, “Efficient use of closed-form Green’sfunctions for the analysis of planar geometries with vertical connec-tions,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 593–603, May1997.

[9] M. Vrancken and G. A. E. Vandenbosch, “Hybrid dyadic-mixed poten-tial and combined spectral-space domain integral-equation analysis ofquasi 3-D structures in stratified media,” IEEE Trans. Microwave TheoryTech., vol. 51, pp. 216–225, Jan. 2003.

[10] V. H. Rumsey, “Reaction concept in electromagnetic theory,” PhysicalRev., vol. 94, no. 6, pp. 1483–1491, June 1954.

[11] R. F. Harrington and D. T. Auckland, “Electromagnetic transmissionthrough narrow slots in thick conducting screens,” IEEE Trans. AntennasPropagat., vol. 28, pp. 616–622, Sept. 1980.

[12] J. Chen, A. A. Kishk, and A. W. Glisson, “Application of a new MPIEformulation to the analysis of a dielectric resonator embedded in amultilayered medium coupled to a microstrip circuit,” IEEE Trans.Microwave Theory Tech., vol. 49, pp. 263–279, Feb. 2001.

[13] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radationby surfaces of arbitrary shape in layered media, part II: Implementationand results for contiguous half-spaces,” IEEE Trans. Antennas Prop-agat., vol. 38, pp. 345–352, Mar. 1990.

[14] K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functionsin integral equation formulations,” IEEE Trans. Antennas Propagat., vol.45, pp. 508–519, Mar. 1997.

[15] D. R. Wilton, S. Rao, A. W. Glisson, D. H. Schaubert, O. Al-Bundak,and C. M. Butler, “Potential integrals for uniform and linear source dis-tributions on polygonal and polyhedral domains,” IEEE Trans. AntennasPropagat., vol. 32, pp. 276–281, Mar. 1984.

[16] P. Gay-Balmaz and J. R. Mosig, “Three dimensional planar radiatingstructures in stratified media,” Int. J. Microwave and Millimeter-WaveCAE, vol. 3, no. 5, pp. 330–343, 1997.

[17] J. R. Mosig, “Integral equation technique,” in Numerical Techniquesfor Microwave and Millimeter-Wave Passive Structures, 1st ed, T. Itoh,Ed. New York: Wiley, 1989, ch. 3.

[18] D. Llorens del Río, I. Stevanovic, and J. R. Mosig, “Analysis of printedstructures including thick slots,” presented at the Proc. COST-284Meeting, Budapest, Apr. 2003.

[19] A. Álvarez Melcón, “Applications of the integral equation technique tothe analysis and synthesis of multilayered printed shielded microwavefilters and cavity backed antennas,” Ph.D. dissertation, Ecole Polytech-nique Fédérale de Lausanne, LEMA-DE, 1998.

[20] L. Cohen, H. Baudrand, D. Bajon, and J. Puech, “Full wave analysis ofcoplanar four-poles resonators using odd and even modes,” in Proc. Int.Workshop on Microwave Filters, Toulouse, France, June 2002.

[21] T. M. Grzegorczyk and J. R. Mosig, “Line charge distributions arising inthe integral equation treatment of bent scatterers in stratified media,” inProc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 148, Dec. 2001,pp. 365–368.

[22] P. Otero, G. V. Eleftheriades, and J. R. Mosig, “Integrated modified rect-angular loop slot antenna on substrate lenses for millimeter- and sub-millimeter-wave frequencies mixer applications,” IEEE Trans. AntennasPropagat., vol. 46, pp. 1489–1497, Oct. 1998.

Daniel Llorens del Río received the Electrical Engi-neer degree from the University of Málaga, Málaga,Spain, in 2000. He is currently working toward thePh.D. degree at the Laboratory of Electromagneticsand Acoustics, Ecole Polytechnique Fédérale de Lau-sanne (EPFL), Switzerland.

His research interests are numerical methods andantenna modeling.

Juan R. Mosig (S’76–M’87–SM’94–F’99) wasborn in Cadiz, Spain. He received the ElectricalEngineer degree in 1973 from the UniversidadPolitecnica de Madrid, Spain. In 1976, he joinedthe Laboratory of Electromagnetics and Acoustics(LEMA), Ecole Polytechnique Fédérale de Lausanne(EPFL), Switzerland, from which he obtained thePh.D. degree in 1983.

Since 1991, he has been a Professor at EPFL andsince 2000, the Head of the EPFL Laboratory of Elec-tromagnetics and Acoustics. In 1984, he was a Vis-

iting Research Associate at Rochester Institute of Technology, Rochester, NY.He has also held scientific appointments at universities of Rennes (France), Nice(France), Technical University of Danemark and the University of Colorado atBoulder. He is the author of four chapters in books on microstrip antennas andcircuits and more than 100 reviewed papers. He is co-organizer and lecturer ofyearly short courses in numerical electromagnetics (Europe and USA). He isthe Chairman of a European COST project on antennas and is responsible forseveral research projects for the European Space Agency. His research interestsinclude electromagnetic theory, numerical methods and planar antennas.

Dr. Mosig is a Member of the Swiss Federal Commission for SpaceApplications.


Scattering by Arbitrarily-Shaped Slots in ThickConducting Screens: An Approximate Solution

Juan R. Mosig, Fellow, IEEE

Abstract—In this paper, the integral equation formulation ofthe thick aperture problem is reviewed and then modified tomake it continuously valid for any aperture thickness. Hence,the new proposed thick aperture formulation is free from thedifficulties usually encountered when applying it to a vanishingthickness slot. Afterwards, a simplification of the formulation isproposed, which reduces dramatically the computational burdenwhile providing valid results for apertures whose thicknessesremain small compared with their linear transverse dimensionsbut having otherwise arbitrary shapes and sizes. Preliminarynumerical results confirm the validity of the proposed techniqueand show clearly its advantages.

Index Terms—Thick slots, apertures, conducting screens, inte-gral equations, Green’s functions.

I. INTRODUCTION

ACLASSICAL problem in EM-theory is the scatteringof an electromagnetic wave by an aperture in a con-

ducting thick screen. This problem has countless applicationsin modern technology, ranging from waveguide filters usinginterconnecting wall holes and irises to cavity-backed slot-fedantennas and passing through many problems of field pen-etration through slits and holes, of paramount relevance inelectromagnetic compatibility. In a general case (Fig. 1), thescreen may be curved and have a nonzero thickness, the aper-ture will have arbitrary shape and dimensions and even thelateral metallic walls associated to the aperture rim may havean irregular profile, thus leading to a truly three-dimensional(3-D) problem. In this work, we will concentrate in the casewhere the thick conducting screen is bound by two parallelsurfaces and is locally flat. Even with this simplification, theproblem remains 3-D and for analysis purposes, a reduction totwo dimensions has been traditionally obtained in two ways.With reference to Fig. 1, either the screen thickness is neglectedand then we formulate the problem in two coordinateslocally tangential to the screen, or a translational symmetryalong one tangential coordinate is assumed, and then wework in a 2-D cut of the problem defined by its profile in the

coordinates.Historically, the first model analyzed was the zero-thickness

screen (frequently but improperly called the zero-thickness slotgeometry). This problem can be traced back to Lord Rayleigh

Manuscript received June 23, 2003; revised September 1, 2003.This work was supported by the Swiss “Office Fédérale de l’Education et de

la Science” under Grant European COST-284 Action.The author is with the Laboratory of Electromagnetics and Acoustics

(LEMA), Swiss Federal Institute of Technology, EPFL-Ecublens Lausanne,CH-1015, Switzerland (e-mail: [email protected]).


Fig. 1. General geometry for an arbitrarily shaped aperture in a conductingcurved screen of variable thickness.

in 1897 [1] and was extensively analyzed in a series of classicalpapers (see [2]–[5], to mention but a few) mainly dealing withelectrically small slots. Consequently, quasistatic or low fre-quency approximations were used. On the other hand, specifictechniques were also developed for large apertures, using geo-metrical [6] and spectral [7] theories of diffraction. The rigorousformulation of a zero-thickness aperture with arbitrary size andshape is made through the use of the equivalence theorem andequivalent magnetic currents. This leads to an integral equationproblem solved with the use of dyadic Green’s functions [8].This nowadays classical formulation was extensively discussedin an excellent review paper [9] and is summarized in advancedtextbooks in electromagnetics [10].

The second 2-D model, assuming translational invariance andvalid for long, thin apertures (slits) was solved by using asymp-totic Wiener-Hopf techniques [11] or coupled integral equations[12]. These works deal essentially with thick slots having rect-angular profiles in the plane. The integral equation approach wasextended to arbitrary profiles [13] and was also combined withfinite elements to cope with more general configurations pos-sibly including inhomogeneous media [14], [15].

Back to the general 3-D aperture problem of Fig. 1, it can beformally solved by using equivalence principles leading to a setof coupled equations. Typically, the two outer problems (outsidethe thick slot) will be formulated as integral equations and theinner problem (inside the thick slot) as a cavity problem wherethe Helmholtz equation is to be satisfied. In practice, the numer-ical implementation will be a difficult task, asking in the externalregions for complicated Green’s functions and 2-D-boundaryelements, which must be coupled to 3-D-finite elements insidethe slot. A clever simplified implementation, based on the reci-procity principle [16] has been used to analyze microstrip an-tennas fed through reasonably thick rectangular slots [17].

Finally, it must be mentioned that the circular aperture caseis of particular relevance in optics, and that the thick case has

0018-926X/04$20.00 © 2004 IEEE


Fig. 2. Two arbitrary regions connected through a slot on a conducting screenof finite (a) and zero (b) thickness.

been solved by Roberts in an optical context [18], emphasizingthe determination of plane wave reflection and transmissioncoefficients.

In this paper, the integral equation formulation of the thickaperture problem is reviewed and then modified to make itcontinuously valid for any aperture thickness. Hence, the newproposed thick aperture formulation is free from the difficultiesusually encountered when applying it to a vanishing thicknessslot. Afterwards, a simplification of the formulation is pro-posed, which reduces dramatically the computational burdenwhile providing valid results for apertures whose thicknessesremain small compared with their linear transverse dimensions(or with the square root of their surface) but having otherwisearbitrary shapes and sizes.

II. THEORETICAL BACKGROUND

The procedure leading to the coupled integral equationswhich solve the problem of a thick slot is well known [10],[12], [14], [15]. We will briefly recall it here for the sake ofcompleteness and for introducing the notation used throughoutthis paper. Consider the generic problem of Fig. 2(a), in whichtwo arbitrary inhomogeneous regions and are originallyseparated by a thick conducting wall. The region alsoincludes a set of impressed currents (sources). A portionof the screen is suppressed, leaving a 3-D-hole, which defines anew region , connecting and [Fig. 2(a)]. As stated inthe introduction, in most problems of practical interest the con-ducting screen is limited by two parallel surfaces and is locallyflat. Also, the region is usually a cylindrical volume witharbitrary but constant cross-section in the -plane and with itsaxis parallel to the screen’s normal coordinate . Nevertheless,

the theory which follows is also formally valid for the moregeneral geometry of Fig. 1.

Following the standard procedure, we replace the two open-ings of the thick aperture by zero-thickness conducting surfaces.The two sides of the surface separating regions andwill be denoted and , while the two sides of the surfaceseparating regions and will be denoted and .Now, according to the equivalence theorem, we define un-known equivalent magnetic surface currents in the followingway [Fig. 2(a)]

(1)

Since surface magnetic currents are cross products of unitnormal vectors and electric fields, the continuity of the tan-gential electric field is automatically fulfilled in the interfacesbetween our three regions. The introduction of the conductingsurfaces allows the consideration of three formally independentproblems, one for each region, that are indirectly coupledthrough the equivalent magnetic currents. In particular theregion becomes a cavity fully bounded by conducting walls.We use now the well known concept of “short-circuited exci-tation fields” [19], defined as the fields created bythe impressed sources in the region where they exist (here )but with the aperture opening covered by the conducting sur-face. With the introduction of the scattered fieldsexisting in each region, the boundary conditions imposing thecontinuity of the tangential components of the total magneticfield across the two interfaces are written as

(2)

where, to keep the notation simple, we have avoided to show thecross product with the normal unit vector , but it is understoodfrom now on that we only consider the tangential compo-nents of the fields.

The transposition of these boundary conditions into integralequations should be straightforward. Invoking linearity andsuperposition, we can write the scattered fields due to anyinduced or equivalent source as a convolution of the sourcewith the pertinent dyadic Green’s functions over the source’sdomain of existence . For instance, the magnetic field of amagnetic current is

(3)

where we have introduced the convolution notation . To de-velop the first boundary condition in (2), we remark that the scat-tered magnetic field in the region is that created by ,while in region the fields are due to and to

. When we consider the fields at the interface, thethree above mentioned currents acts through convolution with,respectively, the three Green’s functions

MOSIG: SCATTERING BY ARBITRARILY-SHAPED SLOTS IN THICK CONDUCTING SCREENS 2111

that we abridge as, respectively, , and . TheseGreen’s functions are also “short-circuited,” i.e., they are theGreen’s functions associated to the respective regions when theyare isolated (decoupled) from each other by conducting zerothickness walls placed in the thick aperture surfaces.

Applying an identical reasoning to the second boundarycondition, we can now translate directly the set (2) into thefollowing system of two coupled integral equations:

(4)

The system of equations (4) for the unknowns andfully defines the thick slot problem. Although in theory theycan be used for the general problem of Fig. 1, the calculationof Green’s functions would be too much involved. Hence, wewill restrict from now on our analysis to the simpler geometriesof the kind illustrated in Fig. 2(a), leaving the general problemto numerically intensive techniques like finite elements or finitedifferences.

III. CAVITY GREEN’S FUNCTIONS AND THE

ZERO THICKNESS SLOT

If we start from the very beginning considering a zero thick-ness slot, the cavity region shrinks to a null volume and weonly need to consider two regions and separated by aninterface in whose sides and we define equivalent surfacemagnetic currents and [Fig. 2(b)]. The single integralequation is now

(5)

Therefore, if we solve the system of (4) associated to the thickslot problem in the limiting case of a vanishing slot thickness

, we should end up with the result ,which is the solution of the integral equation (5). Unfortunately,this is not the case in practice, as the cavity Green’s functionsshow a divergent behavior when the cavity thickness vanishes.This fact deserves further consideration and will be investigatednow.

The four cavity Green’s functions correspond to thefour interactions shown in Fig. 3(a). Electromagnetic reciprocityensures that we must have

(6)

where the index reminds us that this Green’s functionscorrespond to a “mutual” interaction between the two parallelsurfaces bounding the slot and hence the cavity. With thissimplified notation, the set of coupled integral equations (4)can be cast into a convenient matricial form

(7)

where electromagnetic reciprocity ensures the symmetry of theGreen’s functions matrix. A further simplification can be used

Fig. 3. (a) Four cavity Green’s functions, (b) a generic situation, and (c) itssolution by images.

if the interior of the cavity is homogeneous or it is symmetri-cally filled with dielectric media, because then we would haveby symmetry

(8)

where the index reminds us that this Green’s functions corre-spond to a “self” interaction of one of the surfaces bounding theslot and hence the cavity with itself.

In all cases, the four cavity Green’s functions corre-spond all to particular cases of the situation depicted in Fig. 3(b).Formally, we can solve this problem by transforming the cavityinto an infinite waveguide. This is achieved by taking imagesof the source respect to both the lower and upper cavity wallsas in Fig. 3(c). But in this situation, it is well-known that allthe images will keep the sign of the original magnetic source.Therefore, all the Green’s functions (which are of theHM-type) will diverge in the limiting case, as all the images co-alesce into a single source of infinite intensity.

This heuristic conclusion will be confirmed later on by a rig-orous analytic development in a more specific geometry. At thistime, let us simply point out the evident consequence: in itscurrent formulation (7), the thick aperture problem cannot besolved numerically in the limiting case of a zero thickness slot,since all the elements in the Green’s function matrix would di-verge. Indeed, numerical difficulties should be expected when


trying to solve (7) for small values of the thickness , and alter-nate forms must be investigated to provide a smooth transitionto the zero-thickness case.

IV. AN ALGEBRAIC INTERLUDE

To throw out some extra light in the problems revealed in theprevious section, let us consider the algebraic counterpart of theintegral equation system (7), namely the linear system

(9)

where the coefficients play the role of the potentially di-verging cavity Green’s functions and . The formalsolution of this system is

(10)

Now, we can easily see that if under a certain conditionall the coefficients diverge but in such a way that the fol-lowing conditions are satisfied:

(11)

then we obtain the limit solution

(12)

which is indeed the solution of the algebraic equation equiva-lent to the zero-thickness slot integral equation (5). The conclu-sion is that the thick slot equation (7) contains as a particularcase the zero-thickness slot solution, if the cavity Green’s func-tions fulfill conditions equivalent to (11). These conditions willbe checked in a coming section. But even with these conditionssatisfied, the presence of the convolution operator prevents theuse of the equation (7) in situations approaching the zero thick-ness case and an improved formulation of the thick slot problemmust be sought after. To get some hints about what must be done,let us progress a further step in the simplification of our problemand move from algebra to arithmetic by introducing a set of nu-merical values for the coefficients, namely

(13)

which reproduce quite faithfully the numerical conditionsarising in a typical thin-slot situation. The corresponding linearsystem is

(14)

A close look to this system with engineer eyes reveals two verysimilar equations (the information about the field values in both

sides of the slot). Therefore, the logical thought is to replace theoriginal equations by their sum and difference

(15)

We also have two close unknowns (the values of the mag-netic currents in both sides of the slot). So, the meaningfulquantities are their average and their deviation from average.Therefore, we replace also the unknowns by their half-sum andhalf-difference

(16)

with the result

(17)

We have here finally uncovered the clue for a successful at-tack to problem. The combination in the first equationof (17) includes both a small coefficient and a small unknownand hence can be safely neglected. Therefore by starting with

, the first equation provides directly the initial guessfor the average value . This is already an excellent esti-mation of the true solutions of the original system (14), namely

. If we need a better estimation pro-viding different values for the unknowns, we just replace

in the second equation and obtain directly , andtherefore and . If still better accu-racy is needed, the cyclic iteration can be pursued indefinitely.Now, coming back to formal algebra, let us symbolize our linearsystem (14) by the matrix equation

(18)

It is easy to show that replacing the original individual equationsby their sum and difference, is equivalent to premultiplication bya matrix and the linear system (15) corresponds to the matrixequation

(19)

By the same token, replacing the original unknowns by theirhalf-sum and half-difference can be also related to this matrixsince

(20)

and therefore the final transformed problem (17), easilyamenable to an iterative solution, is formally given by

(21)

But is just a scaled version of the unitary 45 rotation matrix

(22)


Hence, we conclude that the potentially useful transformation ofour linear system is just achieved by pre- and post-multiplyingby a 45 rotation matrix.

V. THICK SLOT INTEGRAL EQUATIONS AND

ROTATION MATRICES

Let us apply to our thick slot matrix integral equation (7) thepre- and post-multiplications by the rotation matrix as indi-cated in (21). The final result is given in (23), at the bottom ofthe page, where we have introduced the “average” and “devia-tion” values of the magnetic currents in the slot

(24)

The matrix equation (23) looks much more complicated than theoriginal one (7) and it could be feared that we have worsenedour chances. But, as in the numerical example of the previoussection, the first line in the system (23) is the clue, since noneof its elements will diverge when the slot thickness vanishes,if conditions (11) are fulfilled. We can therefore start with theassumption and solve the first equation in the system(23) to obtain a first estimation of . It is remarkable indeedthat if media and in both sides of the slot are identical (for

instance, free space), then we have and then thefirst equation in (23) becomes uncoupled, directly providing theexact value of .

To clarify these ideas, let us fully develop the proposed proce-dure in the case of a slot filled by an homogeneous or symmet-rically disposed dielectric medium, and therefore satisfying thesymmetry condition (8). In this case, the notation can be greatlysimplified by introducing the combinations

(25)

that we can call the “sigma” and “delta” cavity Green’s func-tions ( and ).For a vanishing thickness slot, the sigma Green’s function willdiverge but the delta one will vanish. With this notation, it is astraightforward matter to show that the matrix equation (23) isequivalent to

(26)

This is a great improvement with respect to the original matrixequation (7)! When the slot thickness vanishes, the only diver-

gent term is . Therefore, the second line in (26) automati-cally gives the result , and the first line reduces to thezero-thickness slot equation.

Hence, we can set up the following procedure for thin slots:

Step 1) assumeStep 2) solve a modified zero-thickness slot equation to ob-tain a first estimate of

(27)

Step 3) estimate by solving the equation

Step 4) improve, if necessary, the estimation of bysolving

Step 5) go to Step 3).

It is worth mentioning that all the above steps are single un-coupled integral equations. In most cases, stopping after theStep 2) will be enough to predict the first order deviation fromthe zero-thickness case introduced by a reasonable slot thick-ness. In fact, Step 2) is identical to the zero-thickness slot in-tegral equation (5), but with the Green’s function kernel cor-rected by an additive term . Therefore, if the “delta” cavityGreen’s functions could be approximated by an easily com-putable expression, the Step 2) would provide first correctionsfor thick slots with no increase in the computational complexity.The next section proposes some reasonable expressions for the“delta” Green’s function.

VI. APPROXIMATIONS FOR DELTA AND

SIGMA GREEN’S FUNCTIONS

First of all we move from fields to potentials and introducethe convenient formalism of the “Mixed Potential Integral Equa-tion” [20], [21]. Until now, all the Green’s functions referredin previous section are of the HM-type (magnetic field due toa magnetic current). Therefore any generic convolution in theprevious sections can be expanded in terms of potentials

(28)

where and are the vector and scalar potentials associatedwith transverse magnetic currents and is the equivalentmagnetic charge.

(23)


As it is well known, in free space we have for the mixed po-tential Green’s functions the values

(29)

with the free space scalar Green’s function given by

(30)

The question is how to compute these quantities in the cavitygeometry. The problem is not trivial and will depend obviouslyon the cavity’s shape and on the medium filling it. In general,for arbitrary shaped slots, the answer can be obtained only byintensive numerical procedures. But we may try to introduce apowerful approximation, which should lead to reasonable re-sults if the slot’s transverse dimensions aren’t smaller than theslot thickness: we just neglect the lateral conducting walls ofthe cavity. Although the validity of this assumption can onlybe judged a posteriori, its appeal is enormous. First, the cavitydelta and sigma Green’s functions will have “universal” expres-sions independent of the slot/cavity shape. And secondly, theseexpressions will be reasonably simple.

Fig. 4 shows the parallel plate waveguide configuration whichremains when we neglect the lateral walls. In this case, relations(29) are still valid, but the scalar Green’s function is no longerthe free-space one (30). Its calculation is easily performed inthe spectral domain [20]. For a source located on the lower wall

we get the result

(31)

A partial check of the above result is provided by the fact thatif we let go to infinity (the parallel plate waveguide reducesto its lower plate), we obtain , which isthe expected result, twice the free space value. Keeping nowfinite and particularizing to the values and , we getthe potential versions of our cavity “self” and “mutual” Green’sfunctions (6), (8) and making sums and differences with themwe get the potential versions of our cavity “sigma” and “delta”Green’s functions (25)

(32)

We have here a clear confirmation of our theoretical predictions.While the “self,” “mutual,” and “sigma” cavity Green’s func-tions diverge for a vanishing slot thickness , the “delta”function goes to zero. Moreover, it is straightforward to showthat these Green’s functions fulfill the conditions equivalent to(11). Moving from the spectral domain to the space domain, wecan write the “delta” potential Green’s function as a Sommer-feld integral

(33)

Fig. 4. Approximating arbitrarily shaped cylindrical cavities by a parallel platewaveguide.

Fig. 5. Modulus of the normalized delta Green’s function for several slotthicknesses: t = �=1000 (dashed line), t = �=100 (dotted line), andt = �=10 (dash-dotted line). The straight solid line is the free space Green’sfunction.

where is the radial source-observer distance. A series expan-sion of the hyperbolic tangent in the above equation will result ina series expression for the delta Green’s function. The amazingresult is that the delta Green’s function can be expressed as analternating-sign infinite series identical to the scalar potentialof an electric point charge when both source and observer are inthe mid-plane of the parallel plate waveguide. To obtain specificinformation about the near field (quasistatic) behavior, we lookat the asymptotical spectral behavior for . Since in thiscase the hyperbolic tangent becomes unity, the delta Green’sfunction corresponds in the near field to twice the free spaceGreen’s function . This behavior is confirmed by the numer-ical evaluation of the Sommerfeld integral (33) using well testedalgorithms [22], [23].

Fig. 5 shows the normalized potential delta Green’s functionfor three slot thicknesses of 0.001, 0.01, and 0.1 free

space wavelengths. It is evident at a glance how in the near fieldbehaves as , since the diagonal line in Fig. 5 is

. As a rule of thumb, we could infer from Fig. 5 thatthe delta Green’s functions remain close to twice the free spaceGreen’s function while the radial distance is smaller than theslot thickness (say ). But for greater radial distances,the values of the delta Green’s function decay very fast and itshould be possible to neglect it.

To put these results in perspective, let us consider a slotin a thick conducting screen separating two semi-infinite free


Fig. 6. Thick� �� slot of thickness� =10 illuminated by normally incidentplane wave having its electric field along the y-coordinate.

spaces. The equation to be solved is (27) or rather its mixedpotential MPIE form. Hence, applying (28) to (27) we willget for instance a combination forthe scalar potential. We can easily demonstrate using imagetheory that the potential Green’s functions and ,associated with the seminfinite media, are both given by twicethe free space Green’s function . Therefore,is just . But the correction term also behaves in thenear field as and therefore the total kernel is expected tohave a quasistatic behavior of type . It could be objectedthat an additive “correction” identical to the corrected termshouldn’t be called a correction, being much more than this.But this is only the limiting near-field situation, when thesource-observer distance is smaller than the slot thickness. Forlarger radial distances, the delta Green’s function decays veryfast (Fig. 5) and so does its “correcting” effect.

VII. PRELIMINARY RESULTS

To check the validity of our assumptions and of our proposedequations, several very simple numerical experiments have beenperformed on a rather thick square slot (transverse dimensions

and thickness ) (Fig. 6). The slot has been madein a screen separating two semiinfinite free space regions and itis excited with a normally incident plane wave having its elec-tric field along the -coordinate. The main and more interestingcomponent of the magnetic current is then along . We haveconsidered this component along the two medians of the squareslot, a “longitudinal” one and the transverse one

(Fig. 6). The problem has been first solved with arigorous treatment, where the set of equations (4) is used, to-gether with exact expressions for the Green’s functions in thecavity. This “full wave cavity” model gives then the most accu-rate expressions for the currents and in both sides of theslot, represented by circles and squares in Figs. 7 and 8. Theyshow the expected behavior from a slot. But it mustbe pointed out that the full wave cavity approach is a very timeconsuming method, mainly due to the bad convergence of cavityGreen’s functions and their lack of translational symmetry. Andthe situation will be much worse, not to say untractable, for anarbitrarily shaped slot. Even disregarding the cavity problem,

Fig. 7. Normalized x-component of magnetic current along the line y = � =2over a square � � � aperture of thickness � =10. Normal incidence planewave illumination. �—M ; —M ; �—M ; �—zero-thickness slot. Solidline—real part, dashed line—imaginary part.

Fig. 8. Normalized x-component of magnetic current along the line x = � =2over a square � � � aperture of thickness � =10. Normal incidence planewave illumination. �—M ; —M ; �—M ; �—zero-thickness slot. Solidline—real part, dashed line—imaginary part.

we should expect an important slowdown with respect to thezero thickness case, since we have twice more unknowns.

The snag with the zero-thickness formulation (5) is that itgives unsatisfactory results, since we get a unique current(stars in Figs. 7–8) that only matches the true values in one sideof the aperture (in this case, the excitation side). Using our cor-rected equation (27), we obtain a first estimation for , whichhappens to be an almost perfect average value (diamonds inFigs. 7 and 8). This clearly indicates that the iterative processof Section V will converge very quickly.

Results of these iterations will be reported in a coming paper.Here, we will rather explore how good are the results obtainedwith (27), that doesn’t introduce any numerical overload withrespect to the zero thickness case. To this end, we have con-sidered the same square slot, thick, excited this


Fig. 9. Radiation pattern in E-plane. Full-wave cavity approach (solid line), “Delta approach” (dashed line), Zero-thickness approximation (dash-dotted line).Square � � � aperture of � =10 thickness. Plane wave impinging from bottom with 45 incidence and H-field polarized in x-direction.

time with a plane wave incident at an angle of 45 . Fig. 9 showsthe scattered field radiation pattern obtained with the rigorous“full-wave cavity model,” assuming a zero-thickness slot andwith our “delta approach.” Even without iterations, the deltaapproach already provides a much better result than the zero-thickness approach and with no increase in computational com-plexity. Moreover, accuracy can be increased if needed. Thiswould be essential when looking for precise predictions of nearfield quantities.

VIII. CONCLUSION

In this paper, we present a rigorous integral equation for-mulation of the thick aperture problem providing a smoothtransition to the zero-thickness case, inspired by an analogywith an algebraic problem. The full usefulness of the newformulation is only evident if the cavity Green’s functions canbe easily calculated or at least efficiently approximated. In thispaper, we propose to use as starting point the zero-thicknesscase. Consequently, a logical approximation is to neglect theinternal lateral walls of the slot and to assume that the equiv-alent cavity is a parallel plate waveguide. The final result is anew integral equation whose unknown is the average value ofthe magnetic currents in both sides of the thick slot. And thisnew equation has exactly the same degree of complexity asthe zero thickness slot equation, since the only modificationis the addition of a correcting “delta” term for the Green’sfunctions, that can be analytically approximated and that dis-appears naturally in the zero-thickness limiting case.

It is hard to obtain an a priori estimate of the accuracy of thisapproximate technique. It should logically work better for largeshallow slots and deteriorate as the slot thickness increases. Butthe analysis of rather electrically thick slots should still leadto reasonable results if the slot’s transverse dimensions remainlarge compared with its thickness. In any case, thicknesses ofthe order of a tenth of wavelength should not pose any realchallenge as the results in this paper demonstrate for the scat-tered field. If the knowledge of the average value of the cur-rents is not sufficient, the technique can always be improved bylooping across an iterative process, which provides improved

values of the half-difference and half-sum of the magnetic cur-rents in both sides of the slots. Also, other approximations ofthe cavity Green’s functions could be explored, like the use ofimages respect to the lateral walls or a modal waveguide expan-sion, that should be excellent for very deep and narrow slots.

The formulation presented is this paper is very flexibleand combines naturally well with the integral equation basedmodels currently used for cavity backed antennas, thick irisesin waveguide filters, slot-fed patches and thick coplanar lines.These configurations and many related ones are of paramountrelevance in innovative and emerging applications, where con-ducting wall thickness cannot be any more neglected, becauseof the technology (self-supporting metallic plates rather thanprinted sheets), the frequency (mm- and sub mm-waves) orboth. An intensive numerical exploration of these geometries,including predictions of very sensitive near-field quantities likemultiport scattering parameters, should provide a more detailedappraisal of the scope of this theory and of its accuracy.

ACKNOWLEDGMENT

Thanks are given to Dr. E. Suter, McKinsey Consultants,Geneva, Switzerland, and to Mr. I. Stevanovic, LEMA-EPFL,Switzerland, for helpful discussions and numerical checking ofthe ideas developed in this paper.

REFERENCES

[1] L. Rayleigh, “On the incidence of aerial and electric waves upon smallobstacles in the form of ellipsoids or elliptic cylinders, and on thepassage of electric waves through a circular aperture in a conductingscreen,” Phil. Mag., vol. 44, pp. 28–52, July 1897.

[2] H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev., vol. 66,pp. 163–182, Oct. 1944.

[3] H. Levine and J. Schwinger, “On the theory of difraction by an aperturein an infinite plane screen I,” Phys. Rev., vol. 74, pp. 958–974, 1948.

[4] , “On the theory of difraction by an aperture in an infinite planescreen II,” Phys. Rev., vol. 75, pp. 1423–1432, 1949.

[5] Y. Rahmat-Samii and R. Mittra, “Electromagnetic coupling throughsmall apertures in a conducting screen,” IEEE Trans. Antennas Prop-agat., vol. AP-25, pp. 180–187, Mar. 1977.

[6] J. B. Keller, “Geometrical theory of diffraction,” J. Appl. Phys., vol. 28,pp. 426–444, 1962.

[7] R. Mittra, Y. Rahmat-Samii, and W. Ko, “Spectral theory of diffraction,”Appl. Phys., vol. 10, pp. 1–13, 1976.


[8] C. T. Tai, Generalized Vector and Dyadic Analysis. New York: IEEEPress, 1992.

[9] C. M. Butler, Y. Rahmat-Samii, and R. Mittra, “Electromagnetic pene-tration through apertures in conducting surfaces,” IEEE Trans. AntennasPropagat., vol. AP-26, pp. 82–93, Jan. 1978.

[10] C. Balanis, Advanced Engineering Electromagnetics. New York:Wiley, 1989.

[11] S. C. Kashyap and M. A. K. Hamid, “Diffraction characteristics of aslit in a thick conducting screen,” IEEE Trans. Antennas Propagat., vol.AP-19, pp. 499–507, July 1971.

[12] F. L. Neerhoff and G. Mur, “Diffraction of a plane electromagnetic waveby a slit in a thick screen placed between two different media,” Appl. Sci.Res., vol. 28, pp. 73–88, July 1973.

[13] D. T. Auckland and R. F. Harrington, “A nonmodal formulation for elec-tromagnetic transmission through a filled slot of arbitrary cross-sectionin a thick conducting screen,” IEEE Trans. Microwave Theory Tech., vol.MTT-28, pp. 548–555, June 1980.

[14] J. Jin and J. Volakis, “TM scattering by an inhomogeneously filled aper-ture in a thick conducting plane,” in Proc. Inst. Elect. Eng., vol. 137,June 1990, pp. 153–159.

[15] S. Gedney and R. Mittra, “Electromagnetic transmission through inho-mogeneously filled slots in a thick conducting plane—Arbitrary inci-dence,” IEEE Trans. Electromagn. Compat., vol. 34, pp. 404–415, Nov.1992.

[16] D. M. Pozar, “A microstrip antenna aperture coupled to a microstripline,” Electron. Lett., vol. 21, pp. 49–50, 1985.

[17] P. Haddad and D. Pozar, “Characterization of aperture coupled mi-crostrip patch antenna with thick ground plane,” Electron. Lett., vol. 30,pp. 1106–1107, July 1994.

[18] A. Roberts, “Electromagnetic theory of diffraction by a circular aperturein a thick, perfectly conducting screen,” J. Opt. Soc. Amer. A, Opt. ImageSci., vol. 4, pp. 1970–1983, Oct. 1987.

[19] J. Van Bladel, Electromagnetic Fields. New York: McGraw-Hill, 1964.[20] J. R. Mosig, “Integral equation technique,” in Numerical Techniques for

Microwave and Milimeter Wave Passive Structures. New York: Wiley,1989, ch. 3.

[21] K. Michalski and J. R. Mosig, “Multilayered media Green’s functions inintegral equation formulations,” IEEE Trans. Antennas Propagat., vol.45, pp. 508–519, Mar. 1997.

[22] J. R. Mosig, R. C. Hall, and F. E. Gardiol, “Numerical analysis of mi-crostrip patch antennas,” in Handbook of Microstrip Antennas, London,U.K.: Peregrinus, 1989, ch. 8.

[23] K. Michalski, “Extrapolation methods for sommerfeld integral tails,”IEEE Trans. Antennas Propagat., vol. 46, pp. 1405–1418, Oct. 1998.

Juan R. Mosig (S’76–M’87–SM’94–F’99) wasborn in Cadiz, Spain. He received the ElectricalEngineer degree in 1973 from the UniversidadPolitecnica de Madrid, Spain. In 1976, he joinedthe Laboratory of Electromagnetics and Acoustics(LEMA), Ecole Polytechnique Fédérale de Lausanne(EPFL), Switzerland, from which he obtained thePh.D. degree in 1983.

Since 1991, he has been a Professor at EPFL andsince 2000, the Head of the EPFL Laboratory of Elec-tromagnetics and Acoustics. In 1984, he was a Vis-

iting Research Associate at Rochester Institute of Technology, Rochester, NY.He has also held scientific appointments at universities of Rennes (France), Nice(France), Technical University of Danemark and the University of Colorado atBoulder. He is the author of four chapters in books on microstrip antennas andcircuits and more than 100 reviewed papers. He is co-organizer and lecturer ofyearly short courses in numerical electromagnetics (Europe and USA). He isthe Chairman of a European COST project on antennas and is responsible forseveral research projects for the European Space Agency. His research interestsinclude electromagnetic theory, numerical methods and planar antennas.

Dr. Mosig is a Member of the Swiss Federal Commission for SpaceApplications.


Double Higher Order Method of Moments for SurfaceIntegral Equation Modeling of Metallic and Dielectric

Antennas and ScatterersMiroslav Djordjevic, Member, IEEE, and Branislav M. Notaros, Senior Member, IEEE

Abstract—A novel double higher order Galerkin-type method ofmoments based on higher order geometrical modeling and higherorder current modeling is proposed for surface integral equationanalysis of combined metallic and dielectric antennas and scat-terers of arbitrary shapes. The technique employs generalizedcurvilinear quadrilaterals of arbitrary geometrical orders forthe approximation of geometry (metallic and dielectric surfaces)and hierarchical divergence-conforming polynomial vector basisfunctions of arbitrary orders for the approximation of electricand magnetic surface currents within the elements. The geomet-rical orders and current-approximation orders of the elementsare entirely independent from each other, and can be combinedindependently for the best overall performance of the method indifferent applications. The results obtained by the higher ordertechnique are validated against the analytical solutions and thenumerical results obtained by low-order moment-method tech-niques from literature. The examples show excellent accuracy,flexibility, and efficiency of the new technique at modeling of bothcurrent variation and curvature, and demonstrate advantages oflarge-domain models using curved quadrilaterals of high geomet-rical orders with basis functions of high current-approximationorders over commonly used small-domain models and low-ordertechniques. The reduction in the number of unknowns is by anorder of magnitude when compared to low-order solutions.

Index Terms—Electromagnetic analysis, electromagnetic scat-tering, higher order modeling, integral equations, method ofmoments (MoM).

I. INTRODUCTION

ANTENNAS involved in modern wireless systems areoften composed of metallic and dielectric parts of ar-

bitrary shapes and with arbitrary curvature. There is a clearneed for advanced analysis and design tools for predicting theperformance and optimizing the parameters of such antennasprior to costly prototype development. These tools have tobe based on general computational electromagnetic methodsfor modeling of arbitrary three-dimensional (3-D) combinedmetallic and dielectric structures. In addition, antenna designersdemand that the simulation tools be accurate, fast, reliable, andrun on relatively small computing platforms, such as standarddesktop PCs.

One of the most general approaches to the analysis of metallicand dielectric structures is the surface integral equation (SIE)

Manuscript received February 14, 2003; revised August 4, 2003. This workwas supported by the National Science Foundation under Grant ECS-0115756.

The authors are with the Department of Electrical and Computer Engi-neering, University of Massachusetts Dartmouth, Dartmouth, MA 02747-2300USA (e-mail: [email protected]; [email protected]).


approach, where both electric and magnetic surface currentsare introduced over boundary surfaces between homogeneousparts of the structure, and surface integral equations basedon boundary conditions for both electric and magnetic fieldintensity vectors are solved with current densities as unknowns.The SIEs are discretized by the method of moments (MoM)[1], which gives rise to MoM-SIE modeling techniques [2]–[4].Overall, the MoM-SIE method is an extremely powerful andversatile numerical methodology for electromagnetic-fieldsimulation in antenna and scattering applications that involveperfectly conducting and penetrable (dielectric and linearmagnetic) materials.

However, practically all the existing 3-D MoM-SIE simu-lation tools for metallic/dielectric structures are low-order orsmall-domain (subdomain) techniques—the structure is mod-eled by surface geometrical elements (boundary elements) thatare electrically very small and the electric and magnetic currentsover the elements are approximated by low-order (zeroth-orderand first-order) basis functions. More precisely, the boundaryelements (patches) are on the order of in each dimension,

being the wavelength in the medium. This results in a verylarge number of unknowns (unknown current-distribution coef-ficients) needed to obtain results of satisfactory accuracy, withall the associated problems and enormous requirements in com-putational resources. In addition, commonly used boundary ele-ments are in the form of flat triangular and quadrilateral patches,and thus they do not provide enough flexibility and efficiency inmodeling of structures with pronounced curvature.

An alternative which can greatly reduce the number of un-knowns for a given problem and enhance further the accuracyand efficiency of the MoM-SIE analysis in antenna/scatteringapplications is the higher order or large-domain computationalapproach. According to this approach, a structure is approxi-mated by a number of as large as possible geometrical elements,and the approximation of current components within individualelements is in the form of a single (two-fold) functional seriesof sufficiently high order. Only relatively recently the computa-tional electromagnetics (CEM) community has started to exten-sively investigate and employ higher order surface and volumeelements and higher order basis functions in the frame of MoM,including both the SIE formulation [5]–[9] and volume integralequation (VIE) formulation [10]–[15], and the finite elementmethod (FEM) [6], [16]–[20].

For MoM-SIE modeling of general structures that may pos-sess arbitrary curvature, it is essential to have both higher ordergeometrical flexibility for curvature modeling and higher order

0018-926X/04$20.00 © 2004 IEEE

DJORDJEVIC AND NOTAROS: DOUBLE HIGHER ORDER MoM FOR SIE MODELING 2119

current-approximation flexibility for current modeling in thesame method. In other words, if higher order (large-domain)basis functions for currents are used on flat patches, many smallpatches may be required for the geometrical precision of themodel, and then higher order basis functions actually reduceto low-order functions (on small patches). On the other hand,geometrical flexibility of curved patches can be fully exploitedonly if they can be made electrically large, which implies theuse of higher order current expansions within the elements aswell. Finally, in order to make the modeling of realistic struc-tures optimal, it is convenient to have elements of different or-ders and sizes combined together in the same model. If all ofthese requirements are to be satisfied, implementation of hier-archical-type higher order polynomial basis functions for theapproximation of electric and magnetic surface currents overcurved boundary elements seems to be the right choice.

This paper proposes a novel higher order (large-domain)PC-oriented Galerkin-type MoM-SIE technique for 3-D elec-tromagnetics based on higher order geometrical modeling andhigher order current modeling, which we refer to as a doublehigher order method. The surface elements proposed for the ap-proximation of geometry (metallic and dielectric surfaces) aregeneralized curvilinear quadrilaterals of arbitrary geometricalorders. The basis functions proposed for the approximation ofcurrents within the elements are hierarchical divergence-con-forming polynomial vector basis functions of arbitrary orders.The proposed technique represents a generalization of theMoM-SIE technique [9], where bilinear quadrilateral surfaceelements (boundary elements of the first geometrical order)are used with higher order polynomial current expansions.The new method enables excellent curvature modeling (e.g.,a sphere is practically perfectly modeled by only six curvedquadrilateral boundary elements of the fourth geometricalorder) and excellent current-distribution modeling (e.g., usingthe eighth-order polynomial current-approximation in the twoparametric coordinates on a quadrilateral boundary element).This enables using large curved MoM quadrilaterals that are onthe order of (e.g., ) in each dimension as buildingblocks for modeling of the electromagnetic structure (i.e.,the boundary elements can be by two orders of magnitudelarger in area than traditional low-order boundary elements).Element orders in the model, however, can also be low, so thatthe lower order modeling approach is actually included in thehigher order modeling. The geometrical orders and current-ap-proximation orders of the elements are entirely independentfrom each other, and the two sets of parameters of the doublehigher order model can be combined independently for the bestoverall performance of the method. Because the proposed basisfunctions are hierarchical, a whole spectrum of element sizeswith the corresponding current-approximation orders can beused at the same time in a single simulation model of a com-plex structure. Additionally, each individual element can havedrastically different edge lengths, enabling a whole range of“regular” and “irregular” element shapes (e.g., square-shaped,rectangular, strip-like, trapezoidal, triangle-like, etc.) to beused in a simulation model as well. Some preliminary results

of double-higher order MoM modeling of purely metallicstructures are presented in [21].

This paper is organized as follows. Section II presents themathematical development of the proposed boundary elementsand describes numerical components of the new double higherorder MoM-SIE technique. This includes the derivation of sur-face integral equations for electric and magnetic surface currentdensity vectors as unknown quantities, development of general-ized Galerkin impedances (the system matrix elements) for arbi-trary boundary elements (i.e., for any choice of surface elementsfor geometrical modeling and any choice of divergence-con-forming basis functions for current modeling), generation ofgeneralized curvilinear quadrilateral elements for higher ordermodeling of geometry, implementation of hierarchical polyno-mial vector basis functions for higher order modeling of currentsover the quadrilaterals, and evaluation of generalized Galerkinimpedances for the new proposed double higher order quadri-lateral elements. In Section III, the accuracy, convergence, andefficiency of the new MoM-SIE technique are evaluated anddiscussed in several characteristic examples. The results ob-tained by the higher order MoM are compared with the an-alytical solutions and the numerical results obtained by low-order MoM techniques from literature. Numerical examples in-clude a dihedral corner reflector, a metallic spherical scatterer(analyzed using six different higher order models), a dielectricspherical scatterer (analyzed using five different higher ordermodels), and a circular cylinder of finite length with attachedwire monopoles. The examples show excellent flexibility andefficiency of the new technique at modeling of both currentvariation and curvature, and demonstrate its advantages overlow-order MoM techniques.

II. NOVEL DOUBLE HIGHER ORDER MOM FOR

ELECTROMAGNETIC MODELING

A. Surface Integral Equation Formulation

Consider an electromagnetic system consisting of arbitrarilyshaped metallic and dielectric bodies. Let the system be excitedby a time-harmonic electromagnetic field of complex field-in-tensities and , and angular frequency . This field maybe a combination of incident plane waves or the impressed fieldof one or more concentrated generators. According to the sur-face equivalence principle (generalized Huygens’ principle), wecan break the entire system into subsystems, each representingone of the dielectric regions (domains), together with the be-longing metallic surfaces, with the remaining space being filledwith the same medium. One of the domains is the external spacesurrounding the structure. The scattered electric and magneticfields, and , in each domain can be expressed in terms ofthe equivalent (artificial) surface electric current, of density ,and equivalent (artificial) surface magnetic currents, of density

, which are placed on the boundary surface of the domain,with the objective to produce a zero total field in the surroundingspace. On the metallic surfaces, only the surface electric currents

exist (these are actual currents) and .


The boundary conditions for the tangential components of thetotal (incident plus scattered) electric and magnetic field vectorson the boundary surface between any two adjacent dielectricdomains (domains 1 and 2) yield

(1)

(2)

where we assume that the incident (impressed) field is presentonly in domain 1. On the conducting bodies, the boundary con-ditions (1) and (2) reduce to only, so for metallicsurfaces in domain 1 we have

(3)

The scattered electric field in the region of complex permit-tivity and complex permeability is expressed in terms of theelectric and magnetic current densities as follows:

(4)

(5)

(6)

while the scattered magnetic field is obtained as

(7)

(8)

(9)

In the above expressions, and are the magnetic and elec-tric vector potentials, and and are the electric and magneticscalar potentials, respectively. The potentials are given by

(10)

(11)

(12)

(13)

where is the boundary surface of the considered domain, andthe Green’s function for the unbounded homogeneous mediumof parameters and

(14)

being the propagation coefficient in the medium and thedistance of the field point from the source point.

Having in mind the integral expressions for fields and in(4)–13, (1)–(3) represent a set of coupled electric/magnetic fieldintegral equations (EFIE/MFIE) for and as unknowns,which can be discretized and solved using the MoM.

B. Generalized Galerkin Impedances for Arbitrary SurfaceElements

Assume first that all the surfaces (metallic and dielectric)in the system are approximated by a number of arbitrary sur-face elements. Let us approximate the surface electric and mag-netic current density vectors, and , over every elementin the model by a convenient set of basis functions with un-known complex current-distribution coefficients. In order to de-termine these coefficients, the EFIE/MFIE system in (1)–(3) istested by means of the Galerkin method, i.e., using the samefunctions used for current expansion. The four types of general-ized Galerkin impedances (the system matrix elements) corre-sponding to the four combinations of electric- and magnetic-cur-rent testing functions and defined on the th surfaceelement and the electric- and magnetic-current basis func-tions and defined on the th element in the model aregiven by

(15)

(16)

(17)

(18)

The generalized voltages (the excitation column-matrix ele-ments) are evaluated as

(19)

(20)

Substituting (5) into (15), expanding , and ap-plying the surface divergence theorem leads to the following ex-pressions for electric/electric Galerkin impedances:

(21)

where is the outward normal to the boundary contourof the surface . When the divergence-conforming current ex-pansion on boundary elements is used, the last term in (21) isidentically equal to zero, because the normal components oftesting functions are either zero at the element edges or thetwo contributions of the elements sharing an edge exactly cancelout in the final expressions for generalized impedances. Finally,


expressing the potentials in (21) in terms of the electric-currentbasis function over the th surface element , we obtain

(22)

Similarly, starting with (6) and (11), expanding ,and performing a cyclic permutation of the scalar triple product,the expression for electric/magnetic generalized impedances in(16) can be transformed to read

(23)

By duality, the magnetic/electric and magnetic/magnetic gen-eralized Galerkin impedances in (17) and (18) have the same re-spective forms as those in (23) and (22), and are given by

(24)

(25)

Equations (22)–(25) provide general expressions for MoMgeneralized impedances for solving the EFIE/MFIE in (1)–(3)using any type of surface discretization and any adopted setof divergence-conforming basis functions in the context of theGalerkin method. In what follows, we shall restrict our attentionto the specific higher order MoM technique proposed for anal-ysis of electromagnetic radiation and scattering in this paper.

C. Higher Order Geometrical Modeling

As basic building blocks for geometrical modeling of3-D electromagnetic structures of arbitrary shape and mate-rial composition, we propose generalized curved parametricquadrilaterals of higher (theoretically arbitrary) geometricalorders (Fig. 1). A generalized quadrilateral is determined by

points (interpolation nodes) arbitrarilypositioned in space, where and ( , ) aregeometrical orders of the element along - and - parametriccoordinates, respectively (note that the orders do not need to be

Fig. 1. Generalized parametric quadrilateral of geometrical orders K andK (K ;K � 1). M = (K + 1)(K + 1) is the total number ofinterpolation points.

the same within an element). The quadrilateral can be describedanalytically as

(26)where are the position vectors of the interpo-lation nodes, are Lagrange-type interpolation polyno-mials satisfying the Kronecker delta relation ,with and representing the parametric coordinates of theth node, and are constant vector coefficients related to

. For more details on geometrical propertiesof parametric elements (in the context of FEM) the reader isreferred to [22], [23].

In this paper, we use the equidistant distribution of interpola-tion nodes along each coordinate in the parametric space, whilethe use of specific nonequidistant node distributions, whichwould provide additional modeling flexibility and accuracy insome applications, is possible as well. In addition, any otherchoice of higher order surface expansions for geometricalmodeling that can be represented as a double sum of 2-D powerfunctions (e.g., parametric quadrilaterals using splinefunctions for describing the geometry) can also readily beimplemented in our method for electromagnetic analysis.

Note that, in general, the surface tangent is discontinuouson the boundary of two attached curved generalized parametricquadrilateral elements defined by (26), regardless of the geo-metrical orders and of the quadrilaterals. However, thisgeometrical discontinuity across the boundaries of adjacent ele-ments becomes less pronounced as the elements of higher ge-ometrical orders are used. For instance, when approximatinga circular cylinder using 32 interpolation points along its cir-cumference and three different geometrical models constructedfrom: (A) 32 first-order elements; (B) 16second-order elements; and (C) eight fourth-order elements per cylinder circumference,the angles between the surface tangents of the neighboring ele-ments at the junctions in models (A), (B), and (C) are 168.750 ,179.787 , and 180.011 , respectively, compared to the exact180 . If a more accurate model is needed, one can increase thetotal number and/or geometrical orders of patches. Note alsothat this geometrical problem is not present if the geometry is


described in terms of spline functions, which can provide con-tinuous surface tangents across the edges shared by curved el-ements (e.g., third-order splines used to solve scattering frombodies of revolution in [24]).

All the geometries considered as examples in this paper aremodeled using specialized geometrical preprocessor codes, andno general meshers are employed. Development and discussionsof general geometrical preprocessors for mesh generation foran arbitrary geometry using higher order surface elements isbeyond the scope of this paper.

D. Higher Order Basis Functions for Electric and MagneticCurrents

Electric and magnetic surface current density vectors overevery generalized quadrilateral in the model are represented as

(27)

(28)

where are divergence-conforming hierarchical-type vectorbasis functions defined as

, even, odd

(29)

Parameters and are the adopted degrees of the poly-nomial current approximation, which are entirely independentfrom the element geometrical orders ( and ), and ,

, , and are unknown current-distribution coeffi-cients. The unitary vectors and in (29) are obtained as

(30)

with given in (26), and is the Jacobian of the covariant trans-formation

(31)

Note, that the sum limits in (27) and (28) that correspond tothe variations of a current density vector component in the di-rection across that component are by one smaller than the orderscorresponding to the variations in the other parametric coordi-nate. This mixed-order arrangement, which ensures equal ap-proximation orders for surface charge densities correspondingto the - and -directed current basis functions, has been found

to be a preferable choice for modeling of surface currents in allapplications. It enables considerable reductions in the overallnumber of unknowns, at no expense in terms of the accuracy ofcurrent and charge modeling over surfaces. An excellent theo-retical elaboration of this approach (in the context of FEM) canbe found in [25].

E. Generalized Galerkin Impedances for Higher OrderQuadrilateral Elements

The unknown coefficients and in (27) and (28) aredetermined by solving the EFIE/MFIE system with the gener-alized Galerkin impedances given in (22)–(25), which we nowspecialize for the implementation of generalized curved quadri-lateral elements of arbitrary geometrical orders, (26), and hier-archical divergence-conforming polynomial vector basis func-tions of arbitrary current-approximation orders, (29). Withoutthe loss of generality, we consider only the -components ofbasis and testing functions. Furthermore, we consider the func-tions in the following simplified form:

(32)

where are the simple 2-D power functions

(33)

The generalized Galerkin impedances corresponding to thecomplete basis functions in (29) can be obtained as a linearcombination of those corresponding to the simplified functionsin (32) and (33). In addition, the impedances for any higherorder set of basis functions of divergence-conforming polyno-mial type can also be constructed as a linear combination of theimpedances for the simple power functions in (32) and (33). Anotable example may be higher order hierarchical basis func-tions with improved orthogonality properties constructed fromultraspherical and Chebyshev polynomials [26], [27] (note thatthe technique presented in [26], [27] is restricted to bilinearquadrilaterals (elements with ) only, as well asthat these basis functions, being more complicated than the reg-ular polynomials, require larger MoM matrix filling times, andare therefore impractical when iterative solvers are not used).

Upon substituting (32) into (22), the electric/electric imped-ances corresponding to the testing function defined by indexes

and on the th quadrilateral and the basis function de-fined by indexes and on the th quadrilateral become

(34)


where and are the current-approximation ordersof the th quadrilateral along the - and -coordinate, respec-tively, and are the corresponding orders for the thquadrilateral, and the integration limits in both quadrilaterals are

and . The source-to-field distanceis computed as

(35)

Taking into account the parametric representation of thequadrilateral surface element, (26), then leads to the finalexpression:

(36)

where and are the geometrical orders along the -and -coordinate, respectively, and are the geometricalvector coefficients in the polynomial expansion of the thquadrilateral, , , and are the correspondingparameters for the th quadrilateral in the model, and is thebasic Galerkin potential integral given by

(37)

Similarly, using (32) and expanding the gradient of Green’sfunction, the electric/magnetic impedances in (23) are trans-formed to

(38)

Using (26) then yields

(39)

where is the basic Galerkin field integral evaluated as

(40)

Note that only two types of scalar basic Galerkin integrals,and in (37) and (40), are needed for the entire Galerkin

impedance matrix. Moreover, only -integrals are sufficientfor purely metallic structures. These integrals are evaluatedonly once for any pair, and , of quadrilateral elements inthe model. Rapid and accurate combined numerical/analyticalmethods are developed for the integration over curved higherorder generalized quadrilateral surfaces, for the - and -inte-grals. When the distance in (35) is relatively small (or zero),the procedure of extracting the (quasi)singularity is performed[28]. As can be expected, the problems with the (quasi)singularintegration are more pronounced with the field integrals. Effi-cient algorithms for recursive construction of the generalizedGalerkin impedances and the EFIE/MFIE system matrix areused in order to avoid redundant operations related to the sum-mation indexes in the Gauss–Legendre integration formulas, aswell as the indexes and for current expansions and andfor geometrical representations within the impedances.

Starting with the generalized voltages given in (19) and (20),several models of lumped and distributed excitations and loads[29] are included in the proposed MoM technique (loads are in-troduced using the concept of a compensating electric field). Theresulting system of linear algebraic equations with complex un-knowns and is solved classically, by the Gaussian elim-ination. By postprocessing of these coefficients, the currentsand over any generalized quadrilateral in the model andfields and in any dielectric region (including the far field)are obtained.

III. NUMERICAL RESULTS AND DISCUSSION

A. Dihedral Corner Reflector

As an example of structures with flat surfaces, consider thescattering from a metallic 90 dihedral corner reflector. The twoplates, each being large, are modeled by atotal of bilinear quadrilateral elements

, which in this case reduce to squares, with the polynomialdegrees in all of the elements. Without the useof symmetry, this results in unknowns. Fig. 2shows the radar cross-section (RCS) of the reflector in the fullazimuthal (horizontal) plane for the vertical polarization of theincident plane wave. The results obtained by the higher order


Fig. 2. Radar cross-section of a 90 dihedral corner reflector, in the fullhorizontal plane, for the vertical polarization of the incident plane wave,obtained by the higher order MoM and by the low-order MoM from [30].

Fig. 3. Radar cross-section of a dihedral corner reflector for four differentorders (2, 4, 6, and 8) of the polynomial approximation of currents in the higherorder MoM.

MoM are compared with the low-order MoM results from [30](the number of unknowns is not specified in [30]), and an excel-lent agreement is observed. Note that the quadrilaterals in thehigher order model are on a side.

The convergence analysis of the higher order current approxi-mation is performed for this example. Four different levels of thepolynomial approximation of currents are adopted: (1)

; (2) ;(3) ; and (4)

. The corresponding RCS results are shownin Fig. 3. We observe excellent convergence properties of thepolynomial basis functions, the RCS prediction average abso-lute differences between levels (1) and (2), (2) and (3), and (3)and (4), being 7.6, 3.4, and 0.3 dB, respectively. In specific, notethat even the second-order current approximation yields accu-rate result for the lobes at the directions perpendicular to thedihedral sides. Additionally, with the fourth-order basis func-tions, the dominant double-reflected fields in the forward re-gion of the reflector are also predicted reasonably accurately. Fi-nally, the sixth-order (or higher) current-approximation modeladds the accuracy in the computation of fields in the back re-gion of the reflector as well. Note also that the estimated number

Fig. 4. Four geometrical models of a spherical scatterer constructed from (a)96, (b) 216, (c) 384, and (d) 600 bilinear quadrilaterals (K = K = 1).

of unknowns, based on a topological analysis, for a commonlow-order MoM solution with the reflector subdivided into trian-gular patches with Rao–Wilton–Glisson (RWG) basis functions[31] is around 12000, which is about 10 times the number of un-knowns required by the higher order MoM and .

B. Metallic Spherical Scatterer

As an example of curved metallic structures, consider a spher-ical metallic scatterer of radius illuminated by an inci-dent plane electromagnetic wave in the frequency range 10–600MHz. In the first set of experiments, the first-order geomet-rical modeling is employed ( in all elements).Four different geometrical models constructed from (1)

bilinear quadrilaterals [Fig. 4(a)], (2)bilinear quadrilaterals [Fig. 4(b)], (3)

bilinear quadrilaterals [Fig. 4(c)], and (4)bilinear quadrilaterals [Fig. 4(d)]

are implemented, with the second-order current approximationin every element in all of the four models.

The total numbers of unknowns without the use of symmetryin models (1), (2), (3), and (4) amount to 768, 1728,3072, and 4800, respectively.

Shown in Fig. 5 is the RCS of the sphere, normalized to thesphere cross-section area, as a function of . The results ob-tained by the higher order MoM are compared with the analyt-ical solution in the form of Mie’s series. An excellent agree-ment between the numerical results obtained with the model (4)and analytical results is observed with the average absolute RCSprediction error less than 3%, while models (1), (2), and (3) pro-vide acceptable results only up to the frequency at which0.53, 1, and 1.6, respectively [the results obtained by the model(1) are not shown in Fig. 5]. Note that an increase in the cur-rent-approximation orders and in models (1)–(3) doesnot yield better results at higher frequencies, meaning that the


Fig. 5. Normalized radar cross-section [RCS=(a �)] of a metallic sphere, forthree higher order MoM models employing the first-order geometrical modelingin Fig. 4(b)–(d), respectively, along with the exact solution (Mie’s series).

Fig. 6. Induced electric surface current over the surface of the modelin Fig. 4(b) at two frequencies, corresponding to (a) a=� = 0:6 and (b)a=� = 1:2.

errors in the RCS prediction using these models are a conse-quence of the inaccuracy in geometrical modeling of the spheresurface. Note also that, even though this is an almost small-do-main application of the proposed large-domain method, wherea large number (600) of elements (with relatively low currentapproximation orders) is needed for the sphere surface to be ge-ometrically accurately represented by parametric surfaces of thefirst geometrical order, the largest quadrilateral elements in themodel (4) are on a side at the highest frequency consid-ered, which is still considerably above the usual small-domainlimit of .

For an additional insight into the correlation of errors inmodeling of geometry and errors in modeling of currents,Fig. 6 shows the induced electric surface current over thesurface of the model (2) at two frequencies, corresponding to(a) and (b) . We observe that, while themutual orientation of quadrilateral elements in the model at thefrequency (a) does not influence the surface current distributionover the sphere surface, the interconnections and surface-tan-gent discontinuities between quadrilaterals at the frequency(b) act like wedges, and a nonphysical current distribution is

Fig. 7. Two geometrical models of a spherical scatterer constructed from (a)six and (b) 24 generalized quadrilaterals of the fourth geometrical order (K =K = 4).

Fig. 8. Normalized radar cross-section [RCS=(a �)] of a metallic sphere,for two higher order MoM models employing the fourth-order geometricalmodeling in Fig. 7(a) and (b), respectively, along with the exact solution (Mie’sseries).

obtained that follows the geometry of the quadrilateral mesh,where the variations of the current density magnitude clearlyindicate the boundaries of the quadrilaterals constituting themodel. These variations, of course, do not exist on the surfaceof the actual spherical scatterer. In other words, the error inmodeling of curvature expressed in terms of the wavelengthis negligible at the frequency (a), while at the frequency (b),it can not be ignored. The same conclusion is then translatedfrom the current distribution consideration to the far field andRCS computation at frequencies (a) and (b), as can be observedfrom Fig. 5.

In the second set of experiments, the fourth-order geometricalmodeling is employed ( 4 in all elements). Thesphere surface is first approximated by (A) 6 fourth-orderquadrilaterals [Fig. 7(a)] in conjunction with the eighth-ordercurrent approximation ( 8) in each element andthen by (B) 24 fourth-order quadrilaterals[Fig. 7(b)] with the sixth-order current approximation (

6) in each element. This results in a total of 768and 1728 unknowns in models (A) and (B), respectively, withno symmetry used.

Fig. 8 shows the simulated RCS of the sphere obtained bythe two geometrically higher order MoM models, as comparedwith the exact solution calculated in terms of Mie’s series. Weobserve an excellent agreement between the numerical results


Fig. 9. Normalized radar cross-section [RCS=(a �)] of a dielectric(" = 2:25) sphere, for three higher order MoM models employing thefirst-order geometrical modeling in Fig. 4(a)–(c), respectively, along with theexact solution (Mie’s series).

obtained with the model (A) and analytical results up to the fre-quency at which and the curved quadrilateral ele-ments in the model are approximately across. In particular,the maximum absolute RCS prediction error is less than 1% for

(quadrilaterals are maximally across), andthen increases slightly for . With the model(B), an excellent agreement with the exact solution is obtainedin the entire frequency range considered, with the maximum ab-solute RCS prediction error less than 0.5% for andless than 3% for .

Note that all the results for scattering from metallic spherespresented in this subsection are obtained by solving the EFIE(3) and no treatment of internal resonances is applied. The newdouble higher order method appears to yield equally accurateresults at the internal resonances of the sphere, even though thecondition number of the MoM matrix is very large at these fre-quencies. The RCS solution is sensitive to internal resonancesonly when the current approximation orders are not sufficient,which is also in agreement with the previous results [32].

C. Dielectric Spherical Scatterer

As an example of curved dielectric structures, consider aspherical dielectric scatterer 1 m in radius in the frequencyrange 10–600 MHz. The relative permittivity of the dielectricis (polyethylene). Shown in Fig. 9 is the RCS of thesphere calculated using the first-order geometrical modeling

, with the sphere surface being approximatedby means of (1) bilinear quadrilaterals [Fig. 4(a)], (2)

bilinear quadrilaterals [Fig. 4(b)], and (3)bilinear quadrilaterals [Fig. 4(c)], along with the analytical so-lution in the form of Mie’s series. The adopted electric andmagnetic current approximation orders in models (1), (2), and(3) are 4, 2, and 2 and the resulting total numbersof unknowns 6144, 3456, and 6144, respectively. Weobserve that the RCS predictions are slightly shifted towardhigher frequencies with all the three models, the frequencyshift being the most pronounced with the model (1) at higherfrequencies. The fact that the geometrical models are inscribedinto the sphere certainly contributes to this shift of the results.

Fig. 10. Normalized radar cross-section [RCS=(a �)] of a dielectric (" =2:25) sphere, for two higher order MoM models employing the fourth-ordergeometrical modeling in Fig. 7(a) and (b), respectively, along with the exactsolution (Mie’s series).

Note, however, that a very good agreement can be observedbetween the numerical results obtained by the model (3) andthe analytical results in the entire frequency range considered.Note also that the numerical results in Fig. 9 obtained by any ofthe three models in Fig. 4(a)–(c) are significantly more accuratethan the corresponding numerical results obtained with thesame models for the metallic sphere (Fig. 5), which can beattributed to the fact that inaccuracies in modeling of surfacesof penetrable (dielectric) bodies do not degrade the overallanalysis results as significantly as in the case of nonpenetrable(metallic) bodies.

Fig. 10 shows the RCS of the dielectric sphere evaluated usingthe two fourth-order geometrical modelsshown in Fig. 7. In the model (A), the adopted electric andmagnetic current approximation orders are

, while in the model (B), these orders are setto be . We observe that, ascompared to the exact solution (Mie’s series), the model (A)performs well up to the frequency at whichand the curved quadrilateral elements in the model are about

or across .Furthermore, the maximum absolute RCS prediction error isless than 2% for , with the maximum lengthof curved quadrilateral elements not exceeding

. The model (B) provides an accurate RCS predic-tion in the entire frequency range considered (quadrilaterals are

across at the highest frequency), with the maximumabsolute error less than 1% for (maximumside dimension of quadrilaterals is about ) and a slightlyincreased error in the rest of the frequency range considered dueto a minimal frequency shift of the results.

D. Wire Monopoles Attached to a Metallic Cylinder

As an example of antennas with curved surfaces, consider asystem of wire monopoles attached to a metallic cylinder. Theradius of the cylinder is 10 cm and its height 22 cm. The systemis analyzed in two configurations: (1) with a single 12-cmmonopole antenna attached to the cylinder and (2) with an 8-cmdriven monopole and 44-cm parasitic monopole attached to the


Fig. 11. Circular cylinder of finite length with attached wire monopoles,modeled by 32 biquadratic (K = K = 2) quadrilaterals and two wires.

cylinder, as indicated in Fig. 11. The radii of the driven andpassive monopoles are 1 and 2 mm, respectively. The antennasystem is analyzed at the frequency of 833 MHz.

Shown in Fig. 11 is the simulated geometrical model of thestructure. The cylinder is modeled using 28 and 32 second-order

quadrilateral surface elements in configura-tions (1) and (2), respectively. Each monopole is modeled bya single straight wire segment. The driven monopole is fed bya point delta generator at its base. Note that the triangle-likecurved quadrilaterals are used around the wire-to-surface con-nections in order to easily enable current continuity across junc-tions. Note also that the flexibility of the generalized quadrilat-erals at approximating both the curvature of the surface and thecurvature of the edges of the cylinder, along with their flexibilityto accommodate for degenerate quadrilateral shapes, enable theeffective modeling of the cylinder with two junctions by meansof only 32 surface elements. Note finally that neither the factthat the two adjacent outer edges of the quadrilaterals approx-imating the bases of the cylinder form an angle of 180 at thequadrilateral vertex they share nor the fact that the quadrilat-eral edges in the wire-to-surface junctions are extremely short(on the order of the wire radius) as compared to the other threeedges of the quadrilateral do not deteriorate the accuracy of thecurrent modeling and the overall accuracy of the analysis.

The results for the radiated far field obtained by the higherorder MoM are compared with the results obtained by thelow-order MoM from [33]. The patterns are shown in Fig. 12for the configuration (1) and Fig. 13 for the configuration (2).The two-fold symmetry is used in both MoM approaches and avery good agreement of the two sets of results is observed. Thediscrepancy between the results is less than 3.5% in the entirepattern range in Fig. 13 and is practically nonexistent in Fig. 12.The simulation results for the monopole antenna impedance forthe two configurations are given in Figs. 12 and 13 as well. Weobserve that the impedances computed by the two methods alsoagree very well. Note that the numbers of unknowns requiredby the higher order MoM, 49 for the configuration (1) and62 for the configuration (2), are considerably smaller than thecorresponding numbers of unknowns required by the low-orderMoM [33], 936 and 986.

Fig. 12. Normalized far field pattern and the antenna input impedanceof the antenna system in Fig. 11 with only one monopole antenna present[configuration (1)], obtained by the higher order MoM and by the low-orderMoM from [33].

Fig. 13. Normalized far field pattern and the antenna input impedance ofthe antenna system in Fig. 11 with both a driven monopole and a parasiticmonopole present [configuration (2)], obtained by the higher order MoM andby the low-order MoM from [33].

IV. CONCLUSION

This paper has proposed a highly efficient and accuratedouble higher order PC-oriented Galerkin-type MoM for mod-eling of arbitrary metallic and dielectric antennas and scatterers.The method is based on higher order geometrical modelingand higher order current modeling in the context of the SIEformulation for combined metallic (perfectly conducting)and dielectric (penetrable) structures. It employs generalizedcurvilinear quadrilaterals of arbitrary geometrical orders for theapproximation of geometry (metallic and dielectric surfaces)and hierarchical divergence-conforming polynomial vectorbasis functions of arbitrary orders for the approximation ofelectric and magnetic surface currents within the elements. Thegeometrical orders and current-approximation orders of theelements are entirely independent from each other, and can becombined independently for the best overall performance ofthe method in different applications. The paper has presentedthe mathematical and computational development of the newMoM-SIE technique, including the evaluation of generalized


Galerkin impedances (MoM matrix elements) for double higherorder quadrilateral boundary elements.

The accuracy, convergence, and efficiency of the newMoM-SIE technique have been demonstrated in several char-acteristic examples. The results obtained by the higher orderMoM have been validated against the analytical solutionsand the numerical results obtained by low-order MoM tech-niques from literature. The flexibility of the new techniquehas allowed for a very effective modeling of a dihedral cornerreflector, a metallic spherical scatterer, a dielectric spher-ical scatterer, and a circular cylinder of finite length withattached wire monopoles by means of only a few large flatand curved quadrilateral boundary elements and a minimalnumber of unknowns. All the examples have shown excellentflexibility and efficiency of the new technique at modelingof both current variation and curvature. The examples havedemonstrated advantages of large-domain models using curvedquadrilaterals of high geometrical orders with basis functionsof high current-approximation orders over commonly usedsmall-domain models and existing low-order techniques fromliterature (the reduction in the number of unknowns is by anorder of magnitude when compared to low-order solutions),but also over almost small-domain models that represent lowerorder versions of the proposed large-domain, high-order (moreprecisely, low-to-high order) technique. Finally, it has beendemonstrated that both components of the double higher ordermodeling, i.e., higher order geometrical modeling and higherorder current modeling, are essential for accurate and efficientMoM-SIE analysis of general antenna (scattering) structures.

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Miroslav Djordjevic (S’00–M’04) was bornin Cuprija, Serbia and Montenegro (former Yu-goslavia), in 1973. He received the Dipl.Ing. (B.S.)degree from the University of Belgrade, Belgrade,Serbia and Montenegro, in 1998, the M.S. degreefrom the University of California, Los Angeles(UCLA), in 2000, and the Ph.D. degree from theUniversity of Massachusetts (UMass) Dartmouth, in2004.

From 1998 to 2000, he was a Graduate Student Re-searcher at the Antenna Research and Measurement

(ARAM) Laboratory, UCLA. Since 2000 to 2003, he was a Research Assistantat UMass where he is currently a Postdoctoral Associate. His research interestsare in higher order modeling, hybrid methods, and analysis of vehicle mountedantennas.

Branislav M. Notaros (M’00–SM’03) was bornin Zrenjanin, Yugoslavia, in 1965. He received theDipl.Ing. (B.Sc.), M.Sc., and Ph.D. degrees in elec-trical engineering from the University of Belgrade,Belgrade, Yugoslavia, in 1988, 1992, and 1995,respectively.

He is currently an Assistant Professor of electricaland computer engineering with the University ofMassachusetts Dartmouth. From 1996 to 1998, hewas an Assistant Professor with the Departmentof Electrical Engineering, University of Belgrade.

He spent the 1998 to 1999 academic year as a Visiting Research Associatewith the University of Colorado at Boulder. His teaching activities are in thearea of theoretical and applied electromagnetics. He is the Co-Director of theTelecommunications Laboratory, Advanced Technology and ManufacturingCenter, University of Massachusetts Dartmouth. He has authored or coauthored15 journal papers, 40 conference papers, a book chapter, five universitytextbooks and workbooks, and a conceptual assessment tool for electromag-netics education. His research interests are predominantly in computationalelectromagnetics and antenna design.

Dr. Notaros was the recipient of the 1999 Institution of Electrical Engineers(IEE) Marconi Premium.


Loop-Tree Implementation of the Adaptive IntegralMethod (AIM) for Numerically-Stable, Broadband,

Fast Electromagnetic ModelingVladimir I. Okhmatovski, Member, IEEE, Jason D. Morsey, Member, IEEE, and Andreas C. Cangellaris, Fellow, IEEE

Abstract—The adaptive integral method (AIM) is implementedin conjunction with the loop-tree (LT) decomposition of the elec-tric current density in the method of moments approximation ofthe electric field integral equation. The representation of the un-known currents in terms of its solenoidal and irrotational compo-nents allows for accurate, broadband electromagnetic (EM) simu-lation without low-frequency numerical instability problems, whilescaling of computational complexity and memory storage with thesize of the problem of the are of the same order as in the conven-tional AIM algorithm. The proposed algorithm is built as an ex-tension to the conventional AIM formulation that utilizes roof-topexpansion functions, thus providing direct and easy way for the de-velopment of the new stable formulation when the roof-top basedAIM is available. A new preconditioning strategy utilizing near in-teractions in the system which are typically available in the imple-mentation of fast solvers is proposed and tested. The discussed pre-conditioner can be used with both roof-top and LT formulations ofAIM and other fast algorithms. The resulting AIM implementationis validated through its application to the broadband, EM analysisof large microstrip antennas and planar interconnect structures.

Index Terms—Fast algorithms, full-wave electromagnetic (EM)CAD, loop-tree (LT) decomposition, low frequency, method of mo-ments (MoM).

I. INTRODUCTION

I T is a matter of common experience in the electromagnetic(EM) modeling community that the conventional integral

equation based method of moments (MoM) exhibits two majorshortcomings when used for broadband EM analysis of electri-cally large and/or geometrically complex structures. The firstshortcoming has to do with its numerical solution time andmemory requirements, both growing at least as a square of thenumber of unknowns involved in the MoM approximation ofthe EM boundary value problem. State-of-the-art applicationsof relevance to integrated microwave/RF, and mixed-signalelectronic devices and systems, call for EM models where the

Manuscript received June 20, 2003. This work was supported in part by theDefense Advanced Research Projects Agency (DARPA) NeoCAD programunder Grant N66001-01-1-8921 and in part by the Semiconductor ResearchCorporation.

V. I. Okhmatovski is with Cadence Design Systems, Incorporated, Tempe,AZ 85282 USA (e-mail: [email protected]).

J. D. Morsey was with the Center for Computational Electromagnetics, De-partment of Electrical and Computer Engineering, University of Illinois at Ur-bana-Champaign, Urbana, IL 61801 USA. He is now with the IBM T. J. WatsonResearch Center, Yorktown Heights, NY 10598 USA.

A. C. Cangellaris is with the Center for Computational Electromagnetics, De-partment of Electrical and Computer Engineering, University of Illinois at Ur-bana-Champaign, Urbana, IL 61801 USA.


number of unknowns is in the order of tens and even hun-dreds of thousands. This makes the direct implementation ofMoM-approximations of EM integral equations computation-ally prohibitive and thus impractical.

The second shortcoming of the MoM approximation of EMintegral equations is the so-called low frequency breakdown[1], [2] that occurs when the harmonic field wavelength be-comes substantially larger than the characteristic size of theMoM grid. In the application of MoM-based solvers for thenarrow band analysis of traditional EM devices of resonantlengths (e.g., antennas, waveguides, and various types ofRF/microwave passive components) this problem of geometryover-sampling tends to be the exception rather than the rule.However, over the last decade, the advent of miniaturizationand high-density integration of electronic devices has led tonew classes of RF/microwave passive components and as-sociated integrated waveguides and packaging structures ofincreasing geometric complexity, primarily due to the presenceof numerous minute features, the accurate modeling of whichresults in the aforementioned geometric over-sampling in thedevelopment of the MoM discrete model. Even in the absenceof minute features, the aggressive push toward the integrationof mixed-signal (e.g., high-speed digital and RF/microwave)functional blocks, calls for broadband EM modeling (from dcto multi-GHz frequencies) in support of computer-aided designof electromagnetically compatible, EMI-immune devices [3],[4]. While it is applications of this type that this paper isconcerned with, it is worth mentioning that another importantclass of applications where low-frequency breakdown of MoMsolvers is encountered is the kilohertz-range remote sensing ofthe buried objects [5], [6].

There are two major classes of methods that are capable oftackling effectively the computational complexity and largememory requirements of MoM approximations to EM integralequations. The first class includes the fast multipole method(FMM) [7] and its multilevel modifications (MLFMM) [8].The second class includes all the acceleration schemes thattake advantage of the convolution nature of the EM field in-tegral equation to expedite its calculation through the use offast Fourier transforms (FFT) algorithms [9]–[14]. For quasi-planar geometries both the FMM and FFT-based algorithmsreduce the required CPU time in the iterative solution of theMoM system from the aforementioned complexityto complexity. Also within the framework offast algorithms the memory consumption of MoM issubstantially reduced and scales proportionally to the problem

0018-926X/04$20.00 © 2004 IEEE

OKHMATOVSKI et al.: LT IMPLEMENTATION OF AIM FOR FEM MODELING 2131

size . Even though the FMM and FFT acceleration schemeslead to similar improvements in computational efficiency, theFFT-based schemes are much more simple to implement andmore flexible in usage of various types of Green’s functionsaccounting for the surrounding medium.

With regards to the theme of this paper, it is important to pointout that, irrespective of the fast solution methodology employed,the problem of low-frequency breakdown remains and has to beaddressed in the context of the specific fast solution scheme.Thus, if a robust and broadband EM simulator is to be devel-oped, it is necessary to introduce appropriate modifications inboth the development of the MoM approximation and the ac-celeration schemes to avoid the numerical instability at low fre-quencies. For the FMM class of fast solvers such a modificationwas presented in [16], [17]. As discussed in [18], the fact that inthe regular FMM the threshold distance separating the near andfar interactions in the system and/or the order of the multipoleexpansion are related to the wavelength makes necessary thereformulation of the entire FMM scheme in order to ensure nu-merical stability at low frequencies. For the class of FFT-basedfast solvers, the separation into the near and far interactions iseither not required at all (in the case when the MoM mesh isbound to the FFT grid [13]) or the threshold distance separatingfar and near interactions of the algorithm is not related to thewavelength (this is the case for the adaptive versions of suchsolvers where the MoM and FFT grids are independent [10],[12]).

In view of this property of the FFT-based fast solvers it wasproposed in [15] that the problem of low-frequency breakdownmay be circumvented through the use of a higher-order, lo-cally corrected Nyström method [19], combined with a nonuni-form grid FFT scheme [20]. To understand how the Nyströmscheme stabilizes the solution of the electric field integral equa-tion (EFIE) at low frequencies one must recall that the nature oflow-frequency breakdown is purely numerical [1]. It is causedby the loss due to round-off error of the small termcompared to much larger term in the scatteredfield representation

(1)

when . Since higher-order methods evaluate both ofthese terms with much higher accuracy, low-frequency numer-ical instability occurs at much lower frequencies compared tothe conventional MoM with Rao–Wilton–Glisson (RWG) ex-pansion functions [21].

In this paper, an alternative implementation of the adaptiveintegral method (AIM) [also referred to as the precorrectedFFT algorithm] is proposed for overcoming the low-frequencybreakdown. First, within the framework of the MoM approx-imation of the EFIE using roof-top expansion functions, theloop-tree (LT) decomposition of the unknown current is in-troduced as prescribed in [1], [2]. This change of expansionfunctions from roof-tops to loops and branches of the tree al-lows for the explicit separation of the irrotational and solenoidalcomponents of the current. Subsequently, the AIM solutionprocess is adjusted to accommodate the new representationsof the discretized current densities. As a result, the overall

solution complexity and memory usageof the conventional AIM are preserved, while the numericalstability of the solution is maintained down to very low fre-quencies.

The paper is organized as follows. In Section II, MoM ap-proximation in the standard and mixed-potential forms is out-lined, with emphasis on conversion from roof-top into the LTbasis. In Section III, the proposed alternative AIM process ispresented in standard and mixed potential forms, highlightingthe differences and similarities between the new and the con-ventional formulations. In the following, the proposed imple-mentation will be referred to as AIM-LT. Section IV discussesa new preconditioning strategy utilizing near interactions in thesystem. The numerical studies of Section V are used to demon-strate the validity of the AIM-LT methodology. Finally, SectionVI concludes the paper with a brief summary of the method anda few remarks about its attributes.

II. LOOP-TREE DECOMPOSITION OF THE MoM WITH

ROOF-TOP BASIS

The EFIE statement of the problem is usually obtainedthrough the application of Green’s theorem to the distinct vol-umes of the structure. Assuming perfect electrically conducting(PEC) surfaces, the EFIE forces the tangential electric field,produced by the current flowing on PEC surface of thecircuit , to cancel the applied tangential electric field onthe same surface . This results in standard [22]

(2)

or mixed potential formulation [12], [23] of EFIE

In (2), is the unit normal vector to ; and and are the po-sition vectors to the observation and source points, respectively.Expanding the unknown current over a set of roof-top ex-pansion functions with triangular [21] or rectangular support[22]

(3)

and testing the scattered field with the same basis functions, the integral equation (2) is reduced to a set of linear al-

gebraic equations

(4)

In (4), and contain, respectively, the coefficients of expan-sion (3) and the discrete form of the excitation. The impedance


matrix elements and excitation vector elements arerepresented by the following inner products

(5)

where and the Hilbert space innerproduct is defined as

(6)

In the form (4), the primary cause of the low-frequency break-down of the MoM approximation is easily recognized. As

, the first term in (4) drops below the numerical error levelof the second term due to the finite accuracy of computations.In order to avoid this problem, the representation of unknowncurrent in terms of loop currents (rotational component)and tree-branches currents (irrotational component) is in-troduced

(7)

The current flux through any cross section of the loop remainsconstant forcing the following conditions to be satisfied

(8)

where is a potential in the range of operator on the surface. Application of the Galerkin process with loops and the tree

branches as expansion and testing functions yields an alterna-tive form of the MoM matrix that is free of the low-frequencybreakdown problem

(9)

which in abbreviated form can be written as

(10)

The matrix elements in (9) are as follows:

(11)

In the process of evaluation of the matrix elements in (9) con-ditions (8) were taken into consideration. Also it was assumedthat the media and the PEC object are such that the transfer ofthe divergence operator to the current is allowed [23].

In practice submatrices , and areevaluated based on the matrices and resulting from theroof-top formulation of MoM. For this purpose basis conversionmatrices and are introduced as follow:s

(12)

where is the sparse transformation matrix from theloop-based expansion functions to roof-tops, is the sparse

transformation matrices from the branch-based ex-pansion functions to the roof-tops, and the symbol denotesmatrix transposition [1], [2]. The total number of loops andbranches in the tree, , is equal to the number of theroof-top expansion functions. Since the tree represents an undi-rected graph defined on the original MoM roof-top grid, the throw of the matrix contains only one nonzero element equal tounity in the th column corresponding to the th roof-top takenas the th branch in the tree. Each loop presents a directed graphon the roof-top mesh and the th row of matrix containsnonzero values in the columns corresponding to the roof-topscomposing the th loop. The sign of these elements is positive ifthe corresponding roof-top function is collinear with the orien-tation of the loop, and negative if it has the opposite direction.The absolute value of the nonzero elements in is equal to theinverse of the corresponding roof-top function width in orderto satisfy the condition of continuous current flux through anycross section of the loop. Using (12) we can represent matrix

in terms of the roof-top based impedance matricesand as

(13)

where the sparse matrices and

(14)

are the conversion matrices from the LT basis to the roof-topbasis. The zero in the matrix signifies the loop properties (8).As a result, the following formulas for the submatrices in (11)can be obtained in terms of and

(15)

Thus, the matrix equation (9) can be interpreted as the originalMoM matrix (4) with its inductive and capacitive partspreconditioned to the left and to the right with matrices and

, respectively, in order to improve spectral properties of theroof-top based impedance matrix at low fre-quencies and/or for electrically oversampled structures.

III. LOOP-TREE IMPLEMENTATION OF THE AIM ALGORITHM

In the iterative solution of (9) a repetitive computation of thematrix-vector product is required and can be acceler-


Fig. 1. Illustration of a pair of closely interacting loops and roof-top functionprojection on the FFT grid.

ated by means of the AIM algorithm. Namely, the matrix-vectorproduct is decomposed into near and far interactions in the fol-lowing manner

(16)

A. Near Interactions

The matrix of near interactions is sparse and the com-plexity of its product with is . In order to elaborate onthe definition of closely interacting loops let us consider loops

and shown in Fig. 1. The loop being a concatenationof roof-top functions , is considered closely inter-acting with the loop composed of the roof-tops ,if there exists at least one pair of roof-top functions

and , such that the distance between andis smaller than a certain preset threshold. The same rule is

applied when it is to be decided whether or not a tree branch in-teracts closely with a given loop. Alternatively, if the matrices ofnear interactions and defined in the roof-top basisare available from the conventional implementation of AIM, adifferent definition of the matrix of near interactions is al-lowed

(17)

Clearly, the more spatially localized the loops are the sparser thematrices and become. Whether the definitionor is adopted the matrix of near interactions is computedonce and stored, the storage requirement being of orderdue to its sparsity. The part of the matrix responsible for thefar interactions is neither computed directly nor stored.Instead the matrix-vector product is computed on thefly using FFT’s as described next.

B. Far Interactions

First, all the basis roof-top functions involved in the orig-inal MoM discretization of the object are projected onto the FFTgrid enclosing the object in the same manner as done in the con-ventional AIM [10]. If mixed-potential formulation is used, thencharge duplets are also projected on the same FFTgrid. Projection means that each basis function and is re-

placed by a set dipoles, as described by the following for-mulas

(18)

where for each basis function only of the dipoles havenonzero amplitudes, with being the total number of FFT gridpoints. The projection of the rectangular shaped roof-top func-tions on the FFT grid for the case of two-dimensional (2-D)structure is illustrated in Fig. 1. Various criteria can be devisedfor the evaluation of the dipole weights and , the most eco-nomical among them being the so-called multipole reproduc-tion criteria discussed in [10]. Substitution of the dipole repre-sentations (18) for the basis and testing functions into the innerproduct formulae for the impedance matrix elements (5) yieldsthe following expressions for the elements

(19)

In (19) multidimensional matrices are used. In order to distin-guish two groups of dimensions in them, both boldface typingand square brackets are utilized. The boldface characters implythe spatial vector and dyadic structure of the quantities whilethe square brackets emphasize their discrete nature due to theprojection on the FFT grid. In expanded form, for the case of a2-D object located in -plane, (19) becomes

where

(20)

In (20), and are the dimensions of the FFT grid alongand , respectively, while and are the corresponding

grid steps. Substitution of formula (19) into (15) leads to thefollowing expressions for the blocks of the impedance matrix

in LT basis

(21)

where subscript FFT is used to indicate that the matrices arecomputed using projections of the basis and testing functionson the FFT grid.


From (21) it is apparent that matrix products andstand for the projections of the testing and basis loops

on the FFT grid. Similarly, the products andcharacterize the projection of the testing and basis trees on theFFT grid. Also it is important to observe that these products arenothing else but the linear combinations of the projections fromthe individual basis/testing functions composing them. Thus, ina way analogous to the conventional AIM, the matrix-vectorproduct can be computed with flopsusing FFT due to the Toeplitz structure of the discretized dyadicGreen’s functions. The part of the matrix-vector multiply asso-ciated with inductive interactions in the system is evaluated asfollows:

(22)

The part corresponding to the capacitive coupling in the systemcan be evaluated in two different ways depending on whetherthe standard or mixed potential form of EFIE is considered

(23)

In (22) and (23), the operators and denoteforward and backward discrete FFT, respectively.

Clearly, if the AIM in the roof-top basis has been imple-mented and the fast computational kernel for evaluation of theproducts and is available, the matrix-vectorproduct can be easily computed as

(24)

which is equivalent to formulas (22) and (23).

C. Precorrection

The dipole representation (18) for basis and testing functionsprovides accurate approximation of the impedance matrix ele-ments only when the th and th elements are at a suffi-ciently large distance. Consequently, all elements of

containing contributions from that do not satisfy thisrequirement must be removed from the matrix-vector product

in order to obtain accurate approximation of the de-sired product in (16)

(25)

Since the branch-based expansion functions are nothing else butselected roof-top expansion functions, the precorrection proce-dure for the interactions between branch-based testing and ex-pansion functions is identical to that in the conventional AIM. Atthe same time, extraction of the near interactions correspondingto closely located loops and branches allows for two alternativeapproaches as has been discussed in the Subsection A. In thefirst approach, one may extract contributions of the closely in-teracting loops and branches by removing entire elements fromthe matrix pertinent to them. Alternatively, one hasthe option of extracting only the contributions to dueto the closely interacting roof-tops that contribute to the forma-tion of the pertinent loops or branches. Let us refer to these con-tributions as

(26)

Clearly, whichever of the two approaches, extraction ofor from in (25), is used the same

precorrection method must be applied when the elements ofthe matrix are evaluated in (16). From the point ofcomputational performance the two precorrection strategies areequivalent. However, the second approach, where isevaluated, may be easier to implement when the conventionalAIM algorithm is already in place. This is due to the fact that,if the near interactions for roof-top functions are available,there is no need for the computation of any additional elementsof matrices and in order to constructand . Irrespective of which approach is adopted, thematrix of near interactions is computed only once andthen stored; thus, both the computational complexity associatedwith its construction and the memory requirements for itsstorage scale as .

IV. PRECONDITIONER BASED ON NEAR INTERACTIONS

The matrix contains information about the strongestinteractions in the impedance matrix ; hence, its inverse

can serve as an effective preconditioner in suchcommonly used iterative solvers as the conjugate gradient (CG)method [24] or the generalized minimal residual (GMRES)algorithm [25]. Even though is sparse [Fig. 2(a)], itsdirect inversion exhibits computational complexity and

memory requirement since is in generala full matrix. The practiced solution to this problem is theclass of so-called incomplete factorization methods [26]. Thebasic idea behind these methods is to discard the elementshaving values below a certain threshold in the process of theLU factorization, thus obtaining a sparse approximation tothe originally full L and U matrices. Although these methodscan lead to reduction of both memory storage and CPU timecomplexity to , our numerical experiments with theirapplication to problems involving planar structures found themunable to provide acceptable iteration counts in certain cases.

To remedy the situation an alternative preconditioningstrategy is introduced. The key idea of this alternative approachis to achieve sparsity of through a reindexation of theunknowns (i.e., the coefficients in the loop- and branch-based


Fig. 2. Matrix of close interactions Z (a) Reindexed matrix z , (b) Land U matrices of z , and (c).

expansion functions) that would lead to a new matrixwith its elements clustered near the diagonal. Hence, afterreindexing, the new matrix has a banded structure, asdepicted in Fig. 2(b), and contains all the nonzero elements ofthe matrix . Subsequently, taking advantage of the bandedstructure of , its exact LU-factorization can be performed.This LU factorization is done only once and is of complexity

, where is the bandwidth of . The factorsand of are also banded with the same bandwidthas that of [Fig. 2(c)]. Thus, storage of the banded and

matrices of requires memory. The backwardsubstitution process, yielding the desired product of matrix

with the vector at each iteration, is ofcomplexity .

Clearly, performance of the preconditioner is strongly depen-dent on how the bandwidth changes with the problem size

. In order to explore further this dependence of on sizeit is necessary elaborate on the details of the reindexing proce-dure used. The algorithm discussed next is very similar to the re-verse Cuthill–McKee ordering [27], [28] which is enhanced by

considering the distance between two interacting elements as anadditional parameter for reordering. To band the sparse matrix

the process starts with an arbitrary element labeled R1.The testing function associated with it has expansion func-tions closely interacting with it, including the self-term. These

expansion functions are then reindexed and assigned indexesfrom 1 to in the order of increasing center-to-center dis-tance from the testing function. Clearly, expansion function 2,the closest to R1 is indexed as R2. Given its close proximity toR1, most of its near interactions have been assigned a new indexalready. Those few interaction that have not been reindexed arereindexed next, starting with and in order of increasingdistance from R2. This procedure continues until all elementshave been reindexed and the banded matrix has been generated.

The new preconditioner is sparse within the bandwidth, asdepicted in Fig. 2(b). From the description of the reindexing al-gorithm it can be deduced that for geometries with continuoussurfaces, such as a sphere or a square plate, the bandwidthis expected to grow with the size of the problem as .However, for the types of geometries encountered in planar in-tegrated circuits (see, for example, Fig. 5), the bandwidthdoes not change with the number of unknowns provided thatthe radius of the sphere that defines the range of near interac-tions is kept smaller than the pitch between the lines. This pointwill become clear in Section V through the numerical examplesused to investigate the performance of the proposed fast solver.

Another observation worth making is that although thepreconditioner is sparse within the bandwidth, the andmatrices are in general full within the same bandwidth. Sincememory usage is controlled by the bandwidth, one can fillin the bandwidth of the preconditioner prior to the LU fac-torization. The storage requirements of the LU factorizationremain unchanged; however, the quality of the preconditioneris improved significantly.

Finally, it is stressed that the proposed preconditioner can beapplied with both the LT and roof-top basis functions. Once thenear interactions and center-to-center distances between expan-sion functions have been defined, the sorting algorithm remainsessentially the same irrespective of the choice of the expan-sion functions. An exception is the case when large loops, re-ferred to as super loops for the purposes of this manuscript, arepresent in the LT implementation [1]. Such super loops containmore near interactions since they cover a larger area of the an-alyzed structure. Even though every effort should be made tocontain the size of the loops during the loop generation process,super loops are expected to occur, especially in conjunctionwith structures that include conducting portions forming closedloops (e.g., shorted sections of coplanar waveguides). Whensuch super-loops are naturally present, the near interactions be-tween these super loops and any elements except for other super-loops must be ignored, in order to prevent the formation of anexcessively oversized preconditioner.

V. NUMERICAL RESULTS AND DISCUSSION

The fast solution methodology developed in the previouschapters is suitable for the integral equation-based EM


Fig. 3. Input reflection coefficient of the 2� 2 microstrip patch array computed with regular AIM, and AIM in LT basis.

modeling of both 3-D and 2.5D structures. However, the primaryobjective of our research has been the development of a fast androbust solver for the broadband EM analysis of dense, complexpredominantly planar structures used in RF/microwave printedcircuits, planar antenna arrays, and the high-density intercon-nect circuits used at all levels of packaging of high-speed/high-frequency mixed-signal integrated systems. The examples pre-sented next involve representative members of the aforemen-tioned categories of structures.

A. Corporate-Fed Microstrip Antenna Array

In the first example, four different corporate-fed microstripantenna arrays are considered. The matrix of patch antennasconstituting these arrays had dimensions 2 2, 2 4, 4 4,and 8 4. The corresponding sizes of the MoM approxima-tions were , and , respectively.The geometry and dimensions of the 2 2 antenna array areshown in Fig. 3. The 8 4 array and the current distribution,

, at 20 GHz are depicted in Fig. 4. The air/dielectricinterface was taken to be mm above the ground plane.The current distribution depicted in Fig. 4 is calculated for thecase of a substrate with permittivity , whereas forthe remaining numerical results generated for the purposes ofdemonstrating the performance of the proposed fast solver thesurrounding medium of antennas was assumed to be air.

The 2 2 array was modeled using the original AIM withroof-top expansion functions, and the proposed LT formulationof AIM both implemented in the standard EFIE form (2). Themagnitude of the input reflection coefficient was computedusing standing wave characterization and is plotted versus fre-quency in Fig. 3. It can be seen that at higher frequencies the

Fig. 4. Geometry of 8� 4 array and current distribution log(jJJJ (rrr)j) at 20GHz.

methods are in very good agreement. However, at lower fre-quencies, the standard AIM exhibits numerical instability, whileits LT version remains robust and produces accurate results.

The parameters of the AIM implementation where chosen asfollows. For the projection of each roof-top function on the FFTgrid sixteen dipoles were used, as prescribed by for-mula (18). The area of near interactions for each basis/testingroof-top function, which establishes the threshold between nearand far interactions, was defined in terms of the FFT grid stepsand remained the same for all frequencies. For the specific ex-ample considered here, two functions were assumed closely in-teracting if any two dipoles from their projections were sepa-rated by less than 11 points in either or directions.


TABLE ICPU TIME AND MEMORY REQUIREMENTS ON A 1.4 GHz PENTIUM4

FOR CORPORATE-FED ARRAYS AT 20 GHz

Fig. 5. Geometry of the 16-wire microstrip interconnect structure. The circuitis printed on lossless grounded dielectric substrate of permittivity � = 4 andthickness d = 0:15 mm.

Table I summarizes all information necessary for quantifyingcomputational complexity of the proposed fast solver. Thediscussed AIM-LT implementation utilizing LT basis functionsexhibits the same computational complexity and

memory requirements as the conventional AIM imple-mented with roof-top functions. The low iteration count wasmade possible through the use of the preconditioner discussedin Section IV. The penalty paid for the effectiveness of thepreconditioner is the memory storage requirementand the complexity of its LU factorization. The lattercan definitely become a show-stopper in the use of such a pre-conditioner when is in the order of hundreds of thousands.Therefore, it is important that the numerous possibilities thatexist for the construction of more efficient preconditioners, inthe sense that they provide for lower computational complexitywithout jeopardizing the efficiency of the iterative solver, arethoroughly investigated. The choice made here reflects the situ-ation where it was preferable to trade-off up front CPU time for

the LU factorization of the preconditioner for the benefit of thesmall number of iterations in the solution. For example, sucha choice will be most appropriate when a multiport structurewith a large number of ports (hence, multiple right-hand sidevectors) is being modeled.

B. Multiconductor Interconnect

The second example considered is a multiconductor intercon-nect of enough nonuniformity along the direction of signal prop-agation to require full-wave EM modeling for the quantificationof its transmission and signal interference (crosstalk) proper-ties. For high-speed digital signal transmission as well as forinterference analysis pertinent to mixed-signal systems, the EMproperties of such interconnects must be computed from dc tomulti-GHz frequencies. Thus, in addition to the large compu-tational complexity of such structures, the need for broadbandEM analysis provides a significant challenge for conventionalfull-wave integral equation solvers aimed for primarily narrowband frequency modeling of microwave and RF waveguides andpassive components.

The top view of the geometry of a generic 16-line microstripinterconnect is depicted in Fig. 5. The sixteen wires are orga-nized into eight differential lines. It is stressed that the fine dis-cretization along the cross-section of the wires is needed for theaccurate prediction of both the speed of signal transmission andthe EM coupling between adjacent lines. For the purposes ofthis analysis the wires were assumed to be of zero thickness andperfectly conducting.

In order to provide a reference solution, a portion of this inter-connect structure was analyzed using the EM field solver fromSonnet [29]. The structure analyzed in Sonnet consisted only ofthe differential line that is depicted in Fig. 5 as having terminals1 through 4, and the two adjacent differential lines. Analysis of atruncated portion of the structure in Sonnet was necessitated bythe significant complexity of the 16-wire structure, the detailedmodeling of which makes the use of this direct EM solver verytime consuming. Magnitudes of some of the calculated -pa-rameters are presented in Fig. 6. Good agreement of the results isobserved over the entire bandwidth of analysis. The discrepan-cies between -parameters at higher frequencies, as well as forthose frequencies and ports for which coupling becomes week,are attributed to the difference of the geometries analyzed bythe two methods, namely, the entire 16-wire structure using theproposed method and the 8-wire portion of it using Sonnet. It isimportant to mention that no de-embedding of the port discon-tinuities was done in the generation of the presented -param-eters [30].

In order to demonstrate the robustness of proposed algorithmthe operating frequency of analysis was taken down to 0.001 Hz.The admittance matrix element versus frequency is plottedin Fig. 7 in the range from 0.001 Hz to 8 GHz. It can be seenthat while the Sonnet EM solver experiences the low-frequencybreakdown the LT implementation of the AIM algorithm pro-vides accurate results in the range of frequencies stretching fromdc to microwaves.

The numerical statistics of computations for the interconnectstructure of Fig. 5 is summarized in Table II. In order to demon-strate how the proposed algorithm scales with the size of the


Fig. 6. Magnitude of some of Y -parameters for 16-wire interconnect.

TABLE IICPU TIME AND MEMORY ON 1.4 GHz PENTIUM4 FOR FOUR-, EIGHT-,

AND 16-WIRE INTERCONNECTS AT 10 GHz

structure, portions of the structure including four, eight and 16wires were considered separately. Also, the numerical solutionwas carried out using two different preconditioners. For the firstpreconditioner only the matrix elements corresponding to in-teractions between testing/expansion functions separated by adistance of 0.15 mm or less were included. This means that thematrix elements of the basis and testing functions (loops andbranches) located on different wires were not utilized in the pre-conditioner. We refer to this preconditioner as P1 in Table II.For the second preconditioner, referred to as P2, the radius ofnear interactions was extended so that the coupling between ex-pansion and testing functions located on adjacent wires was in-cluded. In this case the penalty for the improved quality of pre-

conditioner and the faster convergence of the iterative solutionwas the substantial increase in memory usage. The number ofiterations taken by the GMRES algorithm [25] to converge to

outer residual error is plotted in Fig. 8 for the cases whenthe matrix equation was preconditioned with P1 and P2. Fig. 8demonstrates that while at low frequencies the choice of pre-conditioner is irrelevant, the iteration count may become exces-sively large at higher frequencies if the coupling between theneighboring lines is not included in the preconditioner. It is im-portant to point out that the total CPU time of the algorithm,presented in the last row of Table II, was calculated as the sumof the time required by the iterative solver and the time spent forthe LU-factorization. Other computations associated with taskssuch as the evaluation of the Green’s function and the computa-tion of the matrix elements for the near interactions , werenot included in the complexity evaluation. This is due to thesubjective nature of the time estimates for these computationsin view of the wide variety of methods that can be used for per-forming these calculations.

VI. CONCLUSION

This paper discusses a new fast algorithm for the iterativesolution of MoM approximations of EM field integral equa-tions pertinent to the analysis of primarily planar and/or layeredpassive structures. The methodology is based on an alterna-tive implementation of the AIM (pre-corrected-FFT algorithm),where the LT decomposition of unknown current is introducedto enhance the numerical stability of the iterative solution downto very low frequencies. Both mixed-potential and standard


Fig. 7. Magnitude of some of the Y -parameters for 16-wire interconnect at very low frequencies.

Fig. 8. Iteration count of GMRES solver versus frequency for 16-line interconnect depicted in Fig. 5.

formulations of the EFIE can be accommodated within theframework of the proposed method. For microstrip structures(2.5D geometries) the implementation of AIM utilizing LTbasis and testing functions exhibits complexityand requires memory, while for 3-D boundary elementstructures the CPU time and memory scale asand , respectively. In order to expedite the conver-gence of the iterative solver, a new preconditioning strategy thatutilizes near-field interactions of the MoM impedance matrixorganized in the LT basis was proposed and its efficiency

evaluated. Validation of the proposed fast solver was providedthrough its application to the numerical analysis of microstripantenna arrays and multiconductor interconnect structures.

REFERENCES

[1] W. Wu, A. W. Glisson, and D. Kajfez, “A comparison of two low-fre-quency formulations for the electric field integral equation,” in Proc.10th Annu. Review of Progress in Applied Computational Electromag-netics, vol. 2, Monterey, CA, Mar. 1994, pp. 484–491.


[2] M. Burton and S. Kashyap, “A study of a recent, moment-method al-gorithm that is accurate to very low frequencies,” Appl. ComputationalElectromagn. Soc. J., vol. 10, no. 3, pp. 58–68, Nov. 1995.

[3] B. Young, Digital Signal Integrity. New York: McGraw-Hill, 2001.[4] R. R. Tummala, Fundamentals of Microsystems Packaging. Engle-

wood Cliffs, NJ: Prentice-Hall, 2001.[5] T. J. Cui and W. C. Chew et al., “Fast forward and inverse methods

for buried objects,” in Fast and Efficient Algorithms in ComputationalElectromagnetics, W. C. Chew et al., Eds. Boston, MA: Artech House,2001, pp. 347–424.

[6] S. Chen and W. C. Chew, “Low-frequency scattering from penetrablebodies,” in Fast and Efficient Algorithms in Computational Electro-magnetics, W. C. Chew, Ed. Boston, MA: Artech House, 2001, pp.425–460.

[7] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole methodfor the wave equation: A pedestrian prescription,” IEEE Antennas Prop-agat. Mag., vol. 35, pp. 7–12, June 1993.

[8] J. M. Song, C. C. Lu, and W. C. Chew, “MLFMA for electromagneticscattering from large complex objects,” IEEE Trans. Antennas Prop-agat., vol. 45, pp. 1488–1493, Oct. 1997.

[9] J. R. Phillips and J. White, “A precorrected-FFT method for capacitanceextraction of complicated 3-D structures,” in Proc. Int. Conf. Computer-Aided Design, Santa Clara, CA, Nov. 1994.

[10] E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: Adaptive in-tegral method for solving large-scale electromagnetic scattering and ra-diation problems,” Radio Science, vol. 31, pp. 1255–1251, Sept.–Oct.1996.

[11] J. R. Phillips and J. K. White, “A precorrected-FFT method for electro-static analysis of complicated 3-D structures,” IEEE Trans. Computer-Aided Design, vol. 16, pp. 1059–1072, Oct. 1997.

[12] F. Ling and J.-M. Jin et al., “Full-wave analysis of multilayer microstripproblems,” in Fast and Efficient Algorithms in Computational Electro-magnetics, W. C. Chew et al., Eds. Boston, MA: Artech House, 2001,pp. 729–772.

[13] M. F. Catedra, R. P. Torres, J. Basterrechea, and E. Gago, The CG-FFTMethod—Application of Signal Processing Techniques to Electromag-netics. Norwood, MA: Artech House, 1995.

[14] N. N. Bojarski, “k-Space Formulation of the Electromagnetic ScatteringProblem,” Air Force Avionics Lab., Tech. Report, AFAL-TR-71-75,Mar. 1971.

[15] S. Gedney, A. Zhu, W.-H. Tang, and P. Petre, “High-order pre-correctedFFT solution for electromagnetic scattering,” in Proc. IEEE Antennasand Propagation Int. Symp. Digest, vol. 3, San Antonio, TX, June 2002,pp. 566–569.

[16] J. Zhao and W. C. Chew et al., “Multilevel fast multipole algorithm atvery low frequencies,” in Fast and Efficient Algorithms in ComputationalElectromagnetics, W. C. Chew et al., Eds. Boston, MA: Artech House,2001, pp. 151–202.

[17] , “Integral equation solution of Maxwell’s equations from zero fre-quency to microwave frequencies,” IEEE Trans. Antennas Propagat.,vol. 48, pp. 1635–1645, Oct. 2000.

[18] S. Ohnuki and W. C. Chew, “Numerical accuracy of multipole expan-sion for 2-D MLFMA,” IEEE Trans. Antennas Propagat., vol. 51, pp.1883–1890, Aug. 2003.

[19] H. Contopanagos, B. Dembart, M. Epton, J. J. Ottusch, V. Rokhlin,J. L. Visher, and S. M. Wandzura, “Well-conditioned boundary inte-gral equations for three-dimensional electromagnetic scattering,” IEEETrans. Antennas Propagat., vol. 50, pp. 1824–1830, Dec. 2002.

[20] G. X. Fan and Q. H. Liu, “The CGFFT method with a discontinuous FFTalgorithm,” Microwave Opt. Technol. Lett., vol. 29, no. 1, pp. 47–49,2001.

[21] S. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scat-tering by surfaces of arbitrary shapes,” IEEE Trans. Antennas Propagat.,vol. 30, pp. 409–418, May 1982.

[22] G. V. Eleftheriades and R. Mosig, “On the network characterization ofplanar passive circuits using the method of moments,” IEEE Trans. Mi-crowave Theory Tech., vol. 44, pp. 438–445, Mar. 1996.

[23] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radi-ation by surfaces of arbitrary shape in layered media, part I: Theory,”IEEE Trans. Antennas Propagat., vol. 38, pp. 335–344, Mar. 1990.

[24] C. H. Smith, A. F. Peterson, and R. Mitra, “The biconjugate gradientmethod for electromagnetic scattering,” IEEE Trans. Antennas Prop-agat., vol. 38, pp. 938–940, June 1990.

[25] V. Fraysse, L. Giraud, and S. Gratton. A Set of GMRES Routines forReal and Complex Arithmetics. [Online]. Available: www.cerfacs.fr

[26] O. Axelsson, Iterative Solution Methods. Cambridge, MA: CambridgeUniv. Press, 1994.

[27] S. Pissanetsky, Sparse Matrix Technology, London, U.K.: AcademicPress, 1984.

[28] A. Georgem and J. W.-H. Liu, Computer Solution of Large Positive Def-inite Systems, London, U.K.: Prentice-Hall, 1981.

[29] The Sonnet User’s Manual, Apr. 1999.[30] V. I. Okhmatovski, J. Morsey, and A. C. Cangellaris, “On de-embedding

of port discontinuities in full-wave CAD models of multiport circuits,”IEEE Trans. Microwave Theory Tech., vol. 51, Dec. 2003. (in print).

Vladimir I. Okhmatovski (M’99) was born in Moscow, Russia, in 1974. Hereceived the M.S. (with distinction) and Candidate of Science (Ph.D.) degreesfrom the Moscow Power Engineering Institute, Moscow, Russia, in 1996 and1997, respectively.

In 1997, he joined the Radio Engineering Department, Moscow Power Engi-neering Institute, as an Assistant Professor. From 1998 to 1999, he was a Post-doctoral Fellow in the Microwave Laboratory, National Technical Universityof Athens, Greece. From 1999 to 2003, he was a Postdoctoral Research Asso-ciate with the Department of Electrical and Computer Engineering, Universityof Illinois at Urbana-Champaign. He is currently with the Department of CICAdvanced R&D, Cadence Design Systems, Tempe, AZ. He has authored andcoauthored over 30 papers in professional journals and conference proceedings.His research interests include modeling of high-speed interconnects, fast algo-rithms in computational electromagnetics, geometrical and physical theories ofdiffraction and conformal antennas and arrays.

Dr. Okhmatovski was the recipient of a 1995 scholarship of the Governmentof the Russian Federation, a 1996 Presidential Scholarship of the Russian Fed-eration, and a 1997–2000 scholarship of the Russian Academy of Science. In1996, he received Second Prize for the Best Young Scientist Report presented atthe VI International Conference on Mathematical Methods in ElectromagneticTheory (MMET’96). He was also the recipient of the Outstanding TechnicalPaper Award at the 3rd Electronics Packaging Technology Conference (EPTC2000).

Jason D. Morsey (S’01–M’03) received the B.S. (cum laude) and M.S. degreesin electrical engineering from Clemson University, Clemson, SC, in 1998 and2000, respectively, and the Ph.D. degree from the University of Illinois at Ur-bana-Champaign, in 2003.

He is currently with the IBM T. J. Watson Research Center, NY. His researchinterests include electromagnetic modeling of high-speed interconnects at alllevels of integration and their signal integrity analysis.

Andreas C. Cangellaris (M’86–SM’97–F’00) received the M.S. and Ph.D. de-grees in electrical and computer engineering from the University of California,Berkeley, in 1983 and 1985, respectively.

From 1985 to 1987, he was a Senior Research Engineer in the ElectronicsDepartment, General Motors Research Laboratories, Warren, MI. From 1987 to1992, he was an Assistant Professor on the faculty of Electrical and ComputerEngineering at the University of Arizona, Tuscon, and then an AssociateProfessor from 1992 to 1997. He is currently a Professor of Electrical andComputer Engineering at the University of Illinois at Urbana-Champaign(UIUC). Over the past 15 years, he has supervised the development of nu-merous electromagnetic modeling methodologies and computer-aided designtools for high-speed/high-frequency signal integrity-driven applications, whichhave been transferred successfully to industry. He has coauthored more than150 refereed papers and three book chapters on topics related to computationalelectromagnetics and interconnects and package modeling and simulation. Hisresearch work has been in the area of applied and computational electromag-netics with emphasis on their application to electrical modeling and simulationof RF/microwave components and systems, high-speed digital interconnects atthe board, package, and chip level, as well as the modeling and simulation ofelectromagnetic compatibility and electromagnetic interference.

Prof. Cangellaris is an active Member of the IEEE Microwave Theory andTechniques Society, the IEEE Components Packaging and Manufacturing Tech-nology Society, the IEEE Antennas and Propagation Society, and the IEEE Mag-netics Society, serving as a Member of technical program committees for majorconferences and symposia sponsored by these societies. He has served as As-sociate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,and is currently serving as Associate Editor of the IEEE TRANSACTIONS ON

ADVANCED PACKAGING and the IEEE Press Series on Electromagnetic Fieldsand Waves. He is the co-founder of the IEEE Topical Meeting on Electrical Per-formance of Electronic Packaging.


A Single-Level Low Rank IE-QR Algorithm for PECScattering Problems Using EFIE Formulation

Seung Mo Seo, Student Member, IEEE, and Jin-Fa Lee, Senior Member, IEEE

Abstract—This paper presents a single-level matrix compressionalgorithm, termed IE-QR, based on a low-rank approximation tospeed up the electric field integral equation (EFIE) formulation.It is shown, with the number of groups chosen to be proportionalto 1 2, where is the number of unknowns, the memory andCPU time for the resulting algorithm are both ( 1 5). Theunique features of the algorithm are: a. The IE-QR algorithm isbased on the near-rank-deficiency property for well-separatedgroups. This near-rank-deficiency assumption holds true formany integral equation methods such as Laplacian, radiation,and scattering problems in electromagnetics (EM). The samealgorithm can be adapted to other applications outside EM withfew or no modifications; and, b. The rank estimation is achieved bya dual-rank process, which ranks the transmitting and receivinggroups, respectively. Thus, the IE-QR algorithm can achievematrix compression without assembling the entire system matrix.Also, a “geometric-neighboring” preconditioner is presented inthis paper. This “geometric-neighboring” preconditioner whenused in conjunction with GMRES is proven to be both efficientand effective for solving the compressed matrix equations.

Index Terms—Electromagnetic (EM) scattering, low-rank QRfactorization, method of moments (MoM).

I. INTRODUCTION

METHOD of moments (MoM) using the electric field in-tegral equation (EFIE) formulation has been a very pop-

ular choice for solving electromagnetic scattering problems byperfect electric conducting (PEC) objects. However, it is wellknown that the traditional MoM suffers from the storage ofa dense impedance matrix and computational complexity forlarge-scale problems. However, significant progress has beenmade in using the fast multipole method (FMM) [1] to over-come these difficulties. The single-level FMM combined withthe iterative techniques has reduced the numerical complexityto to solve dense integral equation matrices that arisefrom the Helmholtz equation. One major drawback of this ap-proach is its dependence on the integral equation kernel. An-other approach is the so called ( ) algorithm presented in[2] and [3], which are based on low-rank representation of ma-trix blocks. The basic approach [2] is based on the FMMidea without the closed-form formula. In [2], the matrix com-pression is employed with the explicit formulation of the matrixblocks, which then results in an undesirable computa-tional complexity. Thus, the basic algorithm has been fine tunedin [3] through partial assembling of the matrix blocks. However,

Manuscript received March 17, 2003; revised September 16, 2003.The authors are with the ElectroScience Laboratory, Department of Electrical

and Computer Engineering, The Ohio State University, Columbus, OH 43212-1191 USA (e-mail: [email protected]).


the approach suggested in [3] was based on a statically-deter-mined map which was used to assemble a reduced impedancematrix through interpolation. The exact procedure of the inter-polation scheme adapted is not at all clear. In the present ap-proach proposed here in this paper, we have made two distinctdifferences: 1) Our QR decomposition procedure is based on adual-rank process which “ranks” both the transmitter and the re-ceivers and 2) the convergence of the QR process is determinedby both the orthogonal projection as well as the estimate errormatrix norm smaller than the specified tolerance.

This paper presents a low-rank IE-QR algorithm for ef-ficiently compress the MoM matrix to reduce the memoryrequirement, matrix filling time, and the time of the iterative so-lution to . The single-level low-rank IE-QR algorithmis based on the rank deficiency feature of the integral equationfor well-separated groups of basis functions. The algorithmforms a low-rank QR factorization of a matrix block, nonselfinteraction and nontouching groups, with only a portion of itbeing formed. Each group has a bounding box from binarypartition. The touching groups mean that the bounding boxes oftwo groups are overlapped. As a matter of fact, the total numberof entries being computed for a matrix block, assuming thematrix dimension , is , where is the numer-ically determined rank of the matrix block. The entire processcan be viewed as the classical rank-revealing QR factorizationusing modified Gram-Schmidt (MGS) with partial pivoting.

The rest of the paper is organized as follows: Section II givesa description of EFIE formulation for scattering problems; Thesingle-level low-rank IE-QR algorithm is given in Section III;Section IV then details the “geometric-neighboring” precondi-tioner; and one open cone plate example is shown in Section Vto validate and demonstrate the performance of the current ap-proach. Moreover, we investigate the performance of the single-level low-rank IE-QR algorithm; and finally, in Section VI, weprovide a brief summary.

II. EFIE FORMULATION FOR SCATTERING PROBLEMS

In this paper, we employ an integral equation for the surfacecurrent induced on a perfect conducting scatterer [4]. Applica-tion of the Galerkin method to the electric field integral equationresults in

(1)

0018-926X/04$20.00 © 2004 IEEE


where is the problem domain or the scatterer, is the freespace wavenumber, is the distance between observation andsource points, and is the characteristic wave impedance infree space. In the current paper, we have employed surface div-conformal vector basis functions for the surface current, namelythe Rao–Wilton–Glisson (RWG) basis functions.

III. SINGLE-LEVEL LOW-RANK IE-QR ALGORITHM

The single-level low-rank IE-QR algorithm is based on therank deficiency feature of the integral equation for well-sep-arated groups of basis functions. The algorithm factorizes thelocal impedance matrix due to the group and into

and matrices without a priori knowledge of ,where and are the number of receiving and transmittingbasis functions, respectively, and is the rank of interaction

...(2)

Here, is the column vector due to the th basis function inthe transmitting group (transmitter) and is the row vectordue to the th basis function in the receiving group (receiver).The information of the global matrix is stored in the ma-trices and where ; , and theself-impedance matrices , , which are com-puted directly from the integral equation formulation, whereis the number of groups of basis functions. Also, the geometricaltouching groups, in which the rank deficient aspect cannot bepreserved, are directly computed. The computing IE-QR of thetouching groups has expensive computational complexity com-pared with the direct computation.

The single-level low-rank IE-QR algorithm is used to gen-erate orthogonal columns of that approximately span thecolumn space of the original matrix and the columns of

that are the expansion coefficients of the correspondingcolumns of with respect to the column vectors in .

The rest columns of are obtained by solving theequations extracted from (2). The important advantage of the

single-level low-rank IE-QR algorithm is that it does not requirea priori knowledge of . Storing and using andinstead of helps both to save the required memory foreach group interaction and to reduce the numerical complexityof both the matrix assembly and matrix-vector multiplication inthe iterative solver. The detail of the algorithm can be shownbelow.

A. Definitions and Notations Needed for the IE-QR Algorithm

Definition 1: (MGS): Given linearly independent columnvectors, , we denote the orthonormal matrix

which is obtained by thesevectors through the MGS process. Namely

(3)

and

(4)

For the detail of the MGS process, see [9].Definition 2: (Orthogonal Component): Given an or-

thonormal matrix , and a columnvector , we define by

(5)

It is clear that .Definition 3: We say that a matrix is an approx-

imate of if

(6)

and we write .Definition 4: (Column Index Selection): Given a rect-

angular matrix , and assuming( ) exists, we define

as the minimum index , , such that

(7)

Definition 5: (Row Index Selection): Given a rectan-

gular matrix and its row partition ... , we define

according to

(8)

Definition 6: (Approximate Norm): Given a rect-angular matrix , and a series of indexes , wedefine an approximate norm of by

(9)

B. A Single-Level Low-Rank IE-QR Algorithm With MatrixBlock Partially Formed

The following is a straightforward theorem, so we just simplystate it.

Theorem 1: Given a matrix and its column partition, the matrix product

with (10), shown at the bottom of the next page, is anapproximate of , assuming is nonsingular.That is . Moreover, note that theerror for the approximate is

.... . .

......

. . ....

.... . .

...

(11)

SEO AND LEE: SINGLE-LEVEL LOW RANK IE-QR ALGORITHM FOR PEC SCATTERING 2143

Now we are ready to state our IE-QR algorithm with the ma-trix blocks partially formed. Suppose at the th step, we haveexplicitly formed a approximate of the permutedmatrix, namely

......

.... . .

......

. . ....

(12)

In (12), the entries marked by are available, i.e., havebeen computed, whereas the entries are not. Moreover, anorthonormal matrix , with dimensions ,has been constructed through previous steps. Also, in(12), and are row and column permutation matrices,respectively. The detail of these permutation matrices will bedescribed in the following algorithm.

Algorithm: IE-QR Algorithm with Matrix Block PartiallyFormed

Step 1) Let , we compute. Then the permutation matrix is an

identity with and columns interchanges.Step 2) Update to

......

.... . .

......

. . ....

(13)

Let , then compute

, , and .Update from to .Step 3) Compute , and the rowpermutation matrix is a identity with

and rows interchanged. Update to.

Step 4) If , where is a prescribed tolerance,then compute according to (10). Find in-dexes, , and compute

. If , stop theprocedure.

Otherwise, continue the IE-QR process.

Fig. 1. Geometry of the open cone scatterer, with the height of 20 cm and thebase diameter of 20 cm. (a) Surface triangulation and (b) mesh partitioning.

IV. GEOMETRIC-NEIGHBORING PRECONDITIONER

In this section, we consider the efficient solution of denselinear system by preconditioned iterativemethods, particularly GMRES method. An insightful discus-sion of three types of preconditioners, the operator splittingpreconditioner (OSP), the least squares approximate inversepreconditioner (LSAI), and the diagonal block approximateinverse preconditioner (DBAI) [5], for dense matrices arisingfrom the application of BIE is provided. Our approach containsthe idea of the mesh neighbor (MN) preconditioner in [6]and DBAI. The “mesh-neighboring” preconditioner proposedin the current paper, is based upon a two-step process [7].In the first step, we extract from the full impedance matrix,

, a sparse version, , which includes the nearrange interactions as well as a heuristic bias toward geometricalsingularities. Once the sparse matrix, , is obtained, the finalpreconditioner, , will be formed through an incompletefactorization with a heuristic dropping strategy [8]. The detailof the preconditioner can be found in [7].

V. NUMERICAL RESULTS

To demonstrate the efficient and validate the current single-level IE-QR approach, we have conducted studies on one nu-merical example. In the example, we employed constant (as con-stant as we possibly can) mesh density while increase the oper-ating frequency.

A. An Open Cone Plate

An open cone PEC scatterer is shown as inset in Fig. 1, whoseheight and diameter of the bottom are 20 cm. From Fig. 1(b), weclearly establish the open cone PEC scatterer is uniformly par-titioned using a simple mesh-partitioning algorithm. The parti-tioned groups are well-separated for the single-level low-rankIE-QR algorithm. The rank map for the open cone mesh at

(10)


Fig. 2. Rank map of the open cone example at 5 GHz (N = 5;280).

5 GHz is shown in Fig. 2. The number of unknowns is 5820 andthe maximum and minimum sizes of groups are 171 and 73, re-spectively. The black colored boxes represent the dense matrices(full MoM matrix assembling), which are for self-groups andtouching (or overlapping) groups. The maximum and minimumranks of all the coupling matrices of nonoverlapping groups are19 and 7, respectively. Therefore, the use of a low-rank repre-sentation results in significant CPU and memory reductions.

The radiation patterns computed by the single-level low-rankIE-QR algorithm are plotted in Fig. 3 along with the results fromfull EFIE code. The 41 % of memory compared to full EFIEcode is used. The approximation in the single-level low-rankIE-QR process, using a tolerance of 0.01, does not affect thesolution quality, as evidenced in Fig. 3. The radiation patternscomputed by the single-level IE-QR algorithm at 8 GHz areshown in Fig. 4. We see our results agree well with the resultsof full EFIE matrix.

B. Performance of the Single-Level IE-QR Algorithm

To study the memory and CPU time complexities of thesingle-level low-rank IE-QR algorithm, we increase the oper-ating frequencies and subsequently enlarge the problem size.

Fig. 3. Monostatic RCS patterns of the open cone example at 5 GHz.

The memory consumption and CPU time of the IE-QR processare shown in Figs. 5 and 6, respectively, and they both exhibitan complexity for small to moderate electrical size

SEO AND LEE: SINGLE-LEVEL LOW RANK IE-QR ALGORITHM FOR PEC SCATTERING 2145

Fig. 4. Monostatic RCS patterns of the open cone example at 8 GHz.

Fig. 5. Plot of the memory consumption for the open cone example. Note thatthe reference is the O(N ) line.

Fig. 6. Plot of the CPU time for the open cone example. Note that the referenceis the O(N ) line.

TABLE IPERFORMANCE OF SINGLE-LEVEL LOW-RANK IE-QR ALGORITHM WITH THE

OPEN CONE PLATE (tolerance = 10 )

TABLE IIPERFORMANCE OF SINGLE-LEVEL LOW-RANK IE-QR ALGORITHM WITH THE

OPEN CONE PLATE (tolerance = 10 )

problems. Finally, in Tables I and II, we summarize the compu-tational details of the application of the proposed single-levelIE-QR algorithm to the open cone example with the toleranceof and , respectively. The computations were doneon Pentium II 400 MHz.

VI. SUMMARY

This paper presents the novel single-level low-rank IE-QR al-gorithm. The algorithm proves memory consumption and CPUtime are reduced significantly.

REFERENCES

[1] R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole methodfor the wave equation: A Pedestrian prescription,” IEEE Trans. AntennasPropagat. Mag., vol. 35, pp. 7–12, June 1993.

[2] S. Kapur and J. Zhao, “A fast method of moments solver for efficientparameter extraction of MCMs,” in Proc. 34th Design Automation Conf.,vol. 39, June 1997, pp. 141–146.

[3] S. Kapur and D. E. Long, “IES : A fast integral equation solver forefficient 3-dimensional extraction,” in Proc. 37th Int. Conf. ComputerAided Design, Nov. 1997.


[5] K. Chen, “An analysis of sparse approximate inverse preconditioners forboundary integral equations,” SIAM J. Matrix Anal. Appl., vol. 22, pp.1058–1078, 2001.

[6] S. Vavasis, “Preconditioning for boundary integral equations,” SIAM J.Matrix Anal. Appl., vol. 13, pp. 905–925, 1992.

[7] J. F. Lee, R. Lee, and R. Burkholder, “Loop star basis functions and arobust preconditioner for EFIE scattering problems,” IEEE Trans. An-tennas Propagat., vol. 51, pp. 1855–1863, Aug. 2003.


[8] I. Gustafson, “Modified incomplete Choleski (MIC) methods,” inPreconditioning Methods: Analysis and Applications, D. J. Evans,Ed. New York: Gordon and Breach, 1983, pp. 265–293.

[9] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore andLondon: The Johns Hopkins Univ. Press, 1996, pp. 223–236.

Seung Mo Seo (S’00) was born in Seoul, Korea.He received the B.S. degree in electrical engineeringfrom Hong-Ik University, Seoul, in 1998 and theM.S. degree from The Ohio State University,Columbus, in 2001, where he is currently workingtoward the Ph.D. degree in electrical engineering.

From 1999 to the present, he has been a GraduateResearch Associate with the ElectroScience Labora-tory, Department of Electrical and Computer Engi-neering, The Ohio State University, where he focusedon development of fast integral equation methods.

Jin-Fa Lee (SM’99) received the B.S. degree from National Taiwan University,Taiwan, R.O.C., in 1982 and the M.S. and Ph.D. degrees from Carnegie-MellonUniversity, Pittsburg, PA, in 1986 and 1989, respectively, all in electrical engi-neering.

From 1988 to 1990, he was with ANSOFT Corporation, where he developedseveral CAD/CAE finite element programs for modeling three-dimensionalmicrowave and millimeter-wave circuits. His Ph.D studies resulted in the firstcommercial three-dimensional FEM package for modeling RF/Microwavecomponents, HFSS. From 1990 to 1991, he was a Postdoctoral Fellow at theUniversity of Illinois at Urbana-Champaign. From 1991 to 2000, he was withthe Department of Electrical and Computer Engineering, Worcester PolytechnicInstitute, Worcester, MA. Currently, he is an Associate Professor in the Electro-Science Laboratory, Department of Electrical and Computer Engineering, TheOhio State University, Columbus. His current research interests are analyzesof numerical methods, fast finite element methods, integral equation methods,hybrid methods, three-dimensional mesh generation, domain decompositionmethods, and mortar finite elements.


Accelerated Gradient Based Optimization UsingAdjoint Sensitivities

Natalia K. Nikolova, Member, IEEE, Reza Safian, Ezzeldin A. Soliman, Associate Member, IEEE,Mohamed H. Bakr, Member, IEEE, and John W. Bandler, Fellow, IEEE

Abstract—An electromagnetic feasible adjoint sensitivity tech-nique (EM-FAST) has been proposed recently for use with fre-quency-domain solvers . It makes the implementation of the ad-joint variable approach to design sensitivity analysis straightfor-ward while preserving the accuracy at a level comparable to that ofthe exact sensitivities. The overhead computations associated withthe estimation of the sensitivities in addition to the system analysisare due largely to the calculation of the derivatives of the systemmatrix. Here, we describe the integration of the EM-FAST withtwo methods for accelerated estimation of these derivatives: theboundary-layer concept and the Broyden update. We show that theBroyden update approach (Broyden-FAST) leads to an algorithmwhose efficiency is problem independent and allows the computa-tion of the response and its gradient through a single system anal-ysis with practically no overhead. Both approaches are illustratedthrough the design of simple antennas using method of momentssolvers.

Index Terms—Adjoint sensitivities, antenna design, Broyden up-date, design methodology, method of moments (MoM), optimiza-tion, sensitivity.

I. INTRODUCTION

T RADITIONAL full-wave electromagnetic (EM) solversdo not compute the gradient of the response (e.g., -pa-

rameters, input impedance or antenna gain) with respect tothe design parameters, which relate to the geometry and thematerials of the structure. Commercial high-frequency CADsoftware typically resorts to finite-difference approximationsof the response sensitivities, which are numerically inefficientbut simple to implement with existing EM solvers. To computethe response and its sensitivities, such an approach requiresa minimum of full-wave analyses, being the numberof the design parameters. This approach is also known as theperturbation approximate sensitivity technique (PAST) [1].Higher-order approximations may also be used at the expenseof an increased number of simulations. They are feasible when

Manuscript received December 12, 2002; revised September 30, 2003.This work was supported in part by the Natural Sciences and EngineeringResearch Council of Canada under Grants OGP0227660-03, OGP0007239,OGP0249780-02, STR234854-00, through the Micronet Network of Centresof Excellence and Bandler Corporation.

N. K. Nikolova, R. Safian, E. A. Soliman, and M. H. Bakr are with the Depart-ment of Electrical and Computer Engineering, McMaster University, Hamilton,ON L8S 4K1, Canada.

J. W. Bandler is with with the Department of Electrical and Computer Engi-neering, McMaster University, Hamilton, ON L8S 4K1, Canada and also withBandler Corporation, Dundas, ON L9H 5E7, Canada.


sufficient database for the system response in the design param-eter space becomes available. Such response data, for example,would gradually accumulate during optimization.

It is possible to derive exact sensitivity expressions forthe state variables of a system by directly differentiating itsequations with respect to the desired design parameters. Forexample, in [2], a mixed potential integral equation is developedfor the current density derivatives with the method of moments(MoM) applied to planar multilayer structures. This equation,when solved together with the original electric field integralequation, yields both the currents and their derivatives withrespect to the design parameters. Such an approach—generallyreferred to as sensitivity analysis via direct differentiation—canbe applied to both steady-state [3], [4] and dynamic [5] systems.For each design parameter, an additional linear system analysisis required to obtain the respective response derivative. Eachof these analyses is characterized by the same system matrix,which is also identical with the original system matrix.

A more efficient design sensitivity analysis is provided by theadjoint variable method [3]–[7]. It reduces the computationaloverhead of the sensitivity computation to just one additionallinear system analysis where the system matrix is the transposeof that of the original problem. Thus, its computational overheadis times smaller than that of the direct differentiation approachand is practically independent of the number of design param-eters . Adjoint-based design sensitivity analysis of microwavestructures has been first formulated in terms of circuit conceptsrather than field concepts, and it is referred to as the adjoint net-work method [7]–[12].

To obtain exact sensitivities, both the direct differentiationand the adjoint-variable techniques require the analytical deriva-tives of the system matrix with respect to the design parameters.This constitutes a major difficulty in applications with full-waveEM solvers for research or commercial design software. Re-cently, adjoint variable approaches were used with the finite-el-ement method (FEM); see, for example, [13]–[15]. The FEM iswell suited for exact sensitivity calculations because of the ana-lytical relation between the coefficients of the FEM matrix andthe coordinates of the vertices of the finite element grid. Thisanalytical relation, however, is not trivial. Its implementation inthe computation of the derivatives of the FEM system matrixwith respect to any geometrical or material design parameteris in practice difficult and, to our knowledge, has not been ex-ploited yet in commercial high-frequency CAD software.

A similar difficulty exists with the exact sensitivities for theMoM. The different varieties of MoM techniques rely on spe-cific Green’s functions, as well as different basis and weighting

0018-926X/04$20.00 © 2004 IEEE


functions. The dependence of the system matrix coefficients onpossible geometry perturbations is involved and case specific[2], [16].

In summary, exact sensitivities appear to be often impracticalin full-wave EM analysis for two reasons: (1) the analytical pre-processing is involved and solver specific; (2) the implementa-tion requires thorough reworking of the analysis engine. Thesecond requirement is especially unattractive in the develop-ment of commercial software.

Recently, a feasible adjoint-sensitivity technique (FAST)for applications with full-wave EM solvers (EM-FAST) hasbeen proposed [17]. It uses finite differences to approximatethe derivatives of the system matrix. Its implementation in aversatile CAD environment is straightforward since it requiresminor additions to existing frequency-domain computationalalgorithms. Its accuracy is comparable to that of the analyticalexact sensitivities. Its overhead is mostly due to the finite-differ-ence computation of the derivatives of the system matrix, and itis equivalent to that of the exact sensitivity calculations. Here,we propose the use of two techniques—the boundary-layerconcept and the Broyden update—to enhance the speed of theEM-FAST, which is crucial in gradient-based optimization.There is a certain loss of accuracy; however, the approximatedsensitivities are sufficiently accurate to efficiently guide the op-timization toward the optimal design. In applications requiringhigher accuracy of the response gradient such as tolerance oryield analysis, the original EM-FAST may be preferable.

We start with a brief outline of the EM-FAST [17] and itscomputational requirements. We then discuss ways to accelerateits performance through the boundary layer concept (BLC) andthe Broyden update. The resulting algorithms offer significantCPU time reduction in comparison with the original EM-FASTon the order of the number of design parameters . The savingsin comparison with the traditional finite-difference gradient ap-proximation applied directly to the set of responses (e.g., PAST)are drastic, especially in the case of multiple design variables.

II. FEASIBLE ADJOINT SENSITIVITY ANALYSIS

A. Definitions and Notations in Adjoint Sensitivity Analysis

Consider the system of complex-valued equations arisingform the discretization of a linear EM problem

(1)

where, is the vector of design parameters.These parameters typically have real values related to the ge-ometry and the materials of the structure. isthe state variable vector, e.g., complex-valued current distribu-tion in the MoM; is the excitation vector; is the systemmatrix whose complex coefficients depend on the geometry andmaterials.

The objective of sensitivity analysis is to determine the gra-dient of a properly defined response function at thecurrent solution of (1) with respect to the design parameters

(2)

We assume that the response is a scalar function, which is dif-ferentiable in and . We define the gradient operator as a rowoperator

(3)

When the gradient operator acts on a vector, e.g., , the resultis a matrix

...... (4)

The optimization problem is formulated as

(5)

where is the objective function to be minimized, and is thevector of optimal design parameters. Gradient-based optimizersrequire both the response of the current design and its sensi-tivity (3) in order to predict the next design iterate.

The sensitivities of the objective function are obtained as [18]

(6)

where is the solution of the complex adjoint problem

(7)

in which the adjoint excitation is defined by

... (8)

Here, and denote the real and imaginary parts, respectively,of a complex variable. The gradient reflects the explicitdependence of on . The matrix would typically be an-alytically available. In fact, the excitation is often insensitive tochanges in the design parameters, i.e., . For example,in a microstrip circuit, the excitation is defined at ports locatedat feed lines. If the design parameter variations affect neither thedielectric constant nor the height of the substrate, nor the widthof the feed line, the excitation remains unchanged.

In is a constant vector representing the solution atthe current design, i.e., (6) can be written explicitly as

(9)

The sensitivity expression (6) is a generalization of the well-known, linear, real-system sensitivity formula [3], [17].

As evident from (6) and (7), the adjoint approach provides thegradient of the response with respect to all design parameterswith just one additional system analysis (7) whose system ma-trix is simply related to that of the original problem (1). When

factorization of is used to solve (1), the factors of

NIKOLOVA et al.: ACCELERATED GRADIENT BASED OPTIMIZATION USING ADJOINT SENSITIVITIES 2149

are easily obtained by rearranging the factors of . Thus, theadditional system analysis (7) is practically avoided, the over-head being due only to the forward-backward substitutions. Inthe case of iterative solvers—often used when is large and/orsparse—a complete additional system analysis seems impera-tive.

B. The Feasible Technique

The matrices , which we refer to asderivative matrices, may be analytically available, as is the casewith the FEM. Then the sensitivities obtained with (9) are exact.The calculation of an analytically available matrix atthe current design is computationally equivalent to a -matrixfill; therefore, at each design iteration, the equivalent ofmatrix fills is needed. Thus, the advantage of analytically avail-able derivative matrices is in the accuracy of the derivative esti-mation rather than in its computational efficiency. When thefactors of the system matrix are available from the analysis of(1), the computation of the derivative matrices determines theoverhead associated with the sensitivity analysis since it is farmore computationally demanding than the forward-backwardsubstitutions when solving the adjoint problem (7). When thesystem equations of (1) and (7) are solved iteratively, the ad-ditional (adjoint) system analysis determines the computationaloverhead. Even in this case, the reduction of the time to estimatethe derivative matrices is desirable.

In full-wave EM analysis usually the derivative matrices, are not analytically available or they are

too complicated to obtain for the purposes of general and versa-tile design software. Then, we can resort to the finite-differenceapproximation [17], which requires

additional -matrix fills if forward (or backward) finitedifferences are used. The associated computational overhead isequivalent to that of the exact sensitivity estimation discussedabove. The important advantage here is that the implementa-tion with existing software is simple. The technique does notrequire any analytical preprocessing, which often restricts theversatility of the algorithm.

We have investigated the accuracy of the sensitivity estima-tion with the feasible adjoint technique [17] and we have foundthat it is excellent for relative perturbations between0.5% and 2%. The relative error in comparison with the exactsensitivities is well below 1% for a broad range of values of thedesign parameters, close to or far from the nulls of the sensi-tivity curves.

A detailed comparison between the computational require-ments of the EM-FAST and the commonly used finite differ-ences applied directly to the response is made in [17]. Here, weonly note that the EM-FAST reduces the number of requiredfull-wave analyses by a factor of being the number ofdesign parameters. However, there are overhead computationsassociated with the additional matrix fills in order to compute

. In certain cases, e.g., electrically smallproblems, the MoM matrix fill may account for a significant por-tion of the CPU time required by the overall analysis (matrixfill plus linear system solution). Such an overhead should notbe overlooked in a sequence of repetitive analyses performedduring optimization.

III. ACCELERATED OPTIMIZATION WITH APPROXIMATED

ADJOINT SENSITIVITIES

There are two techniques which can lead to faster calculationof the derivative matrices. The first one is the boundary-layerconcept (BLC) first proposed by Amari [19] in the sensitivityanalysis with the direct differentiation method. The accelera-tion offered by the BLC depends on the relation between the re-spective design parameter and the geometry of the structure aswe explain below. Its computational requirements are dependenton the number of design parameters . It requires modificationsof existing EM analysis software, which relate to meshing andmatrix building subroutines. Its advantage is that it yields sen-sitivity estimates of very good accuracy.

The second approach uses Broyden’s update to iterativelycompute approximate derivative matrices. This approachreduces the overhead drastically since its computational re-quirements—negligible compared to a matrix fill—practicallydo not depend on . The Broyden-update approach does notrequire any modifications of the EM analysis algorithms.

A. BLC With the EM-FAST

The BLC can be applied with solvers which allow nonuni-form discretization and/or unstructured grids, e.g., the FEMand the MoM. The idea is to perturb a certain geometricalparameter (the design parameter ) of a structure by respectivedeformations of as few grid elements as possible. This makesmost of the -matrix coefficients insensitive to the perturba-tion. Consequently, the matrix derivative is mostlysparse and only few nonzero coefficients need to be calculated.This is in contrast with the conventional EM-FAST where fullremeshing is applied to the perturbed structure, which resultsin a full matrix.

We present two examples, which illustrate the BLC. Throughthem, we investigate the accuracy of the modified EM-FASTalgorithm which exploits the BLC.

1) A Dipole of Finite Thickness: We analyze the sensitivityof the input impedance of a dipole with respect to the nor-malized length of the dipole . The dipole is dis-cretized into segments whose normalized length is uniform andequal to [see Fig. 1(a)]. Here, is the numberof segments. In this example, . This example is suit-able for design sensitivity tests because the input impedanceof a dipole is highly sensitive to its length, especially close toresonance. The thickness of the dipole is represented by theradius of its cross-section, which is constant and set to

. The derivatives and are calcu-lated, where and . We use the sym-metry of the structure and analyze half of it. The analysis algo-rithm is based on Pocklington’s equation, which is discretizedusing pulse basis functions and a point-matching technique [20].Magnetic frill excitation is applied.

Fig. 1(b) shows the perturbed geometry corresponding to achange of length at the th design iteration where onlythe boundary-layer (edge) segments are changed accordingly.The resulting derivative matrix has only one row andone column of nonzero elements. Fig. 1(c) shows the same pa-rameter perturbation this time realized with the conventional


Fig. 1. Perturbing the length of the dipole at the kth iteration with and withouta boundary layer.

EM-FAST approach. Since the centers of all segments in theperturbed structure change their mutual positions, thematrix is dense.

The input impedance sensitivities are calculated in four dif-ferent ways. First, the forward finite differences are applied di-rectly to the response

(10)

For each , the MoM solver is invoked twice to perform theanalysis at , and at , where .The sensitivity of is evaluated in the range from 0.3 to1.2 (see Figs. 2 and 3).

Second, the input impedance sensitivity is computed with theconventional EM-FAST [17]. The derivative matrixis dense and its coefficients are calculated using forward fi-nite differences applied to each matrix element

. This requires numerical integrations. The incre-ment is again set at . The derivativematrix is then used in (9) to compute and .The resulting sensitivity curves are used as a reference as theyare the closest to the exact sensitivities [17].

The third and the fourth derivative estimations use the ad-joint technique with the BLC. The matrix is verysparse and its computation is fast as it involves only numer-ical integrations. The perturbations are set so that the length ofthe edge elements is increased by

and for thethird and the fourth analysis, respectively. Some accuracy is sac-rificed as is clear from Figs. 2 and 3; however, it is sufficient forthe purposes of gradient-based optimization. The slight deteri-oration in accuracy is due to the nonuniformity of the segment

Fig. 2. Derivative of the input resistance of the dipole with respect to itsnormalized length.

Fig. 3. Derivative of the input reactance of the dipole with respect to itsnormalized length.

size introduced by the edge-only perturbation. We expect suchdeterioration to be less when higher-order basis and test func-tions are used such as triangular functions for wire antennas orrooftops for planar structures.

We now proceed with the optimization of the dipole for aninput impedance of . The objective function is de-fined as

(11)

We allow two geometrical parameters to vary: the normalizeddipole length and the normalized dipole diameter .The vector of design parameters is thus . Thefollowing constraints are imposed:

(12)

since this problem is known to be nonunique. The BLC is usedto compute the matrix derivative . Notice, however,that it cannot be exploited in the case of the design parameterbecause a change in the antenna diameter affects all -matrixcoefficients. The matrix is computed with the con-ventional EM-FAST technique, which requires a full matrix fill.


TABLE IOPTIMIZATION OF THE INPUT IMPEDANCE OF THE DIPOLE

Fig. 4. Geometry of the microstrip-fed patch antenna.

A similar situation would arise in the design of another wire an-tenna, a Yagi–Uda array. While the BLC is very useful whena design parameter represents the length of a wire, it can offerlittle or no computational savings if the design parameter is aseparation distance between wires. Its efficiency is case specific.

The initial design is where the ob-jective function is . After seven iterations, anoptimal solution is found at with

, which corresponds to an input impedance of. The progress of the optimization is

summarized in Table I. The gradient-based optimization routineof MATLAB1 fmincon is used.

2) A Microstrip-Fed Patch Antenna: The EM-FAST is alsointegrated with an in-house MoM solver, which performs anal-ysis of layered structures. The analysis technique is based onthe electric field integral equation. Here, we show an applicationwith the BLC to the optimization of a microstrip-fed rectangularpatch antenna. The length and the width of the patch areoptimized for a maximum real input impedance. The objectivefunction to be minimized is defined as

(13)

The geometry is shown in Fig. 4. The patch is printed on asubstrate of relative dielectric constant and height

mm. The initial design is given by mmand mm. The operating frequency is set at 2 GHz.

1MATLAB is a registered trademark of The MathWorks, Natick, MA.

Fig. 5. BLC and the perturbed mesh related to the design parameter x .

Fig. 6. Progress ofR = <Z of the patch antenna during the optimization.

TABLE IIDESIGN PARAMETERS OF THE PATCH AT EACH ITERATION

The BLC is applied as illustrated in Fig. 5. The calculationof the derivative matrices and is sig-nificantly faster than one matrix fill. A matrix fill is equiva-lent to integrations, where and showthe number of discrete steps along the length and the width ofthe patch, respectively. On the other hand, the estimations of

and with the BLC are equivalent toand numerical integrations, respec-

tively.The progress of the objective function is shown in Fig. 6 in

terms of . The changes of the design parameters with eachdesign iteration are listed in Table II.

B. The Broyden-Update Approach to Matrix DerivativeEstimation

The Broyden update is a classical rank-one formula proposedby Broyden [21] for the approximation of the Jacobianof a vector function . If the approximated Jacobianis denoted as at the th iteration, Broyden’s formula iswritten as

(14)


Fig. 7. Geometry of the Yagi–Uda array.

TABLE IIIOPTIMIZATION OF THE INPUT IMPEDANCE OF THE YAGI–UDA ARRAY WITH BROYDEN-FAST

where is the increment vector in the designparameter space. It has elements corresponding to the incre-ment of each design parameter. Broyden’s update has been usedin a number of applications such as gradient-based optimizationwhere analytical sensitivities are not available [22], the aggres-sive space mapping technique [23], etc.

We apply Broyden’s update to estimate iteratively the deriva-tive matrices , which are subsequentlyused in the sensitivity expression (9). We refer to this modifiedadjoint-based technique as Broyden-FAST. In the implementa-tion of (14), every complex-valued matrix coefficient

is a nonlinear function of the design parame-ters. We define as a vector which consists of the real and imag-inary parts of all elements of the matrix, and as a matrixwhich consists of their derivatives. To construct the vector , westack all the columns of in a vector followed by the vectorformed by all columns of . Thus, when is an ma-trix, is a vector with elements. A row of the matrixcontains the derivatives of the respective element of the vector

with respect to all design parameters. Therefore,is a matrix.

The approximate derivative matrices generated by theBroyden formula are typically less accurate [22] than thoseobtained by perturbations in the EM-FAST. Our experienceshows that as the optimization proceeds, the response sensitivityestimates produced by Broyden-FAST converge toward theexact sensitivities. As a precaution, in the case of a divergingobjective function, the algorithm defaults to the conventionalEM-FAST technique.

The advantage of the Broyden update is that it is problem-in-dependent and does not require any modifications of the analysisalgorithm. Moreover, its computational requirements are negli-gible in comparison with the EM-FAST. The response and itsgradient are obtained by a single system analysis with practi-cally no overhead regardless of the number of design parame-ters .

The potential of the Broyden update is demonstrated by twoexamples: the optimization of a Yagi–Uda array and the opti-mization of a microstrip-fed patch antenna.

1) Optimization of a Yagi–Uda Array: An initial design ofthe six-element Yagi–Uda antenna is given in Fig. 7. All dimen-sions are normalized with respect to the free-space wavelength

. We vary the normalized lengths of the reflector and the drivenelement, and , as well as the normalizedseparation distances and . Thus, thevector of design parameters is . The ob-jective function is set as in (11) with . The progress ofthe optimization is summarized in Table III where the changesof the design parameters, the input impedance and the objectivefunction are recorded at each iteration. An optimal solution isreached in nine iterations.

At the th design iteration, we update the four derivativematrices , with Broyden’s formula anduse them to compute the response sensitivities according to(9). The response sensitivities are then used by the optimiza-tion algorithm (fmincon) to produce the next design iterate.The Broyden-FAST sensitivities are then compared with thesensitivities calculated off-line where the derivative matrices


Fig. 8. Sensitivity of the objective function with respect to the length of thereflector during the optimization of Z of the Yagi–Uda antenna.

Fig. 9. Sensitivity of the objective function with respect to the length of thedriven element l during the optimization of Z of the Yagi–Uda antenna.

are obtained by the finite-difference approach of our originaltechnique, the EM-FAST [17]. The sensitivity curves areplotted in Figs. 8–11. At the first iteration only, we compute thederivative matrices using our original approach with forwardfinite differences and 1% perturbation over the initial designparameters and assign those to . That is why, at the firstiteration, the Broyden-FAST sensitivities and the EM-FASTsensitivities are identical. For all subsequent design iterations,Broyden-FAST uses (14). It is evident that our approach basedon the Broyden update produces sufficiently accurate sensi-tivity results that converge toward the exact sensitivities as theoptimization progresses.

To quantify the accuracy of the derivative matrices producedby the Broyden update in the FAST, we compute their globalrelative errors in norm

(15)

Fig. 10. Sensitivity of the objective function with respect to the separation sduring the optimization of Z of the Yagi–Uda antenna.

Fig. 11. Sensitivity of the objective function with respect to the separation sduring the optimization of Z of the Yagi–Uda antenna.

Fig. 12. Global error in the ZZZ-matrix derivative estimates of Broyden-FASTand EM-FAST.

Here, the exact derivative is computed using an analytical for-mula valid for this specific MoM solver [24]. Note that the errorestimate (15) operates on complex matrix elements. The errorsassociated with the normalized separation distances and


Fig. 13. Sensitivities of the objective function in the optimization of the Yagi–Uda antenna using the direct Broyden update: comparison with reference sensitivitiescomputed with the EM-FAST.

are plotted in Fig. 12. For comparison, we also plot the globalerrors of the matrix derivatives of the forward finite differencingin the original FAST computed for the same design iterates.As expected, the original FAST is robust while the accuracy ofthe Broyden estimates may vary throughout the optimization.These variations lead to the small (but observable) differencesbetween the EM-FAST response derivatives and the respectiveBroyden-FAST estimates (see Figs. 10 and 11). Notice that closeto the optimum solution where the optimizer takes very smallsteps (see Table III), the Broyden update may not perform verywell for all design parameters, e.g., , due to the nearly iden-tical -matrices of the consecutive design iterates; see (14). Ifnecessary, this can be avoided by defaulting to EM-FAST whensufficiently small value of the objective function is achieved.The improved accuracy of the sensitivity estimates may thus im-prove the convergence of the optimization at its final stages. Thehybrid approaches, however, are not a subject of our current dis-cussion.

The Broyden update, of course, can be applied directly tothe objective function . However, the objective function usu-ally exhibits strongly nonlinear behavior and sharp sensitivitieswith respect to the designable parameters. At the same time, theBroyden formula is based on a local linear approximation of thefunction and thus it performs better with only mildly nonlinearfunctions. The -matrix elements, on the other hand, are smoothfunctions of the shape or material parameters. In fact, the ma-jority of the MoM matrix elements are almost insensitive to

shape perturbations except for the diagonal (self-impedance) el-ements due to the intrinsic dependence on the distance betweenobservation and integration points. Thus, when the Broyden up-date is applied at the level of the system matrix, better conver-gence of the sensitivity estimates and of the overall optimizationprocess is expected.

In support of this observation, we repeat the Yagi–Udaantenna design, this time using Broyden’s update directly at thelevel of the objective function in order to estimate its derivatives(direct Broyden approach). We keep the optimization set-upidentical to that before: (1) the initial design is as shown inFig. 7; (2) the objective function is defined as in (11) with

; (3) the same optimization function fmincon ofMATLAB is used; (4) the stop criteriaand are the same; and (5) the value ofthe response sensitivity at the first iteration is supplied by theEM-FAST estimate. The direct-Broyden derivatives are usedby the optimizer to determine the subsequent design iterates.We also compute the objective function derivatives with theEM-FAST technique off-line in order to supply reference valuesfor comparison.

The direct-Broyden and the reference sensitivity curves areplotted in Fig. 13 for all four designable parameters. Theprogress of the optimization is summarized in Table IV. It isevident that the direct-Broyden derivatives do not convergewell toward the reference values, and the objective functionconverges to a different (worse) solution than that of the


TABLE IVOPTIMIZATION OF THE INPUT IMPEDANCE OF THE YAGI–UDA ARRAY WITH DIRECT BROYDEN UPDATE OF THE OBJECTIVE FUNCTION

Fig. 14. Progress of the objective function during the optimization of the microstrip-fed patch antenna.

Broyden-FAST optimization. This is due mostly to three inter-related factors: (1) the objective function is very sensitive to thedesignable parameters (especially and ); (2) the Broydensensitivity estimation can not track well such a rapidly changingfunction; (3) the incorrect sensitivity information misleads theoptimizer. Possible solutions to the problems encountered withthe direct Broyden sensitivity analysis are provided by the trustregion optimization approaches. This topic, however, is outsideof the scope of our work. With this example, we only illustratethe improved convergence of both the sensitivity analysis andthe optimization when the Broyden update is applied at the levelof the system matrix.

2) Optimization of a Microstrip-Fed Patch Antenna: Wenow apply the Broyden-FAST to the optimization of the patchantenna in Fig. 4. The length and the width of the patchare optimized for a minimum magnitude of the reflectioncoefficient. Thus, the objective function to be minimized is

(16)

Here, is the input impedance of the antenna, and is thecharacteristic impedance of the feeding microstrip line, which

approximately equals 50 in this example. The operating fre-quency is 2 GHz. The initial values of the designable parame-ters are mm and mm. The patch is meshedwith rectangular segments. The number of segments along thelength and the width of the patch are 11 and 17, respectively.One segment is used along the width of the feeding microstripline. The optimization is carried out using the fmincon functionof MATLAB.

We compute the sensitivities of the objective function withthree methods: 1) the direct-Broyden update; 2) the traditionalEM-FAST which employs finite differences to approximate thederivative matrices (without a boundary layer); and 3) the pro-posed Broyden-FAST which employs the Broyden update at thelevel of the system matrix. This time, we run three optimiza-tions, each being driven by the respective sensitivity analysistechnique. The three objective functions are plotted in Fig. 14versus the optimization iteration number. The design parametersversus the iteration number are plotted in Fig. 15. It is clear fromboth figures that the direct-Broyden method fails in meeting theoptimum design while the other two methods, EM-FAST andBroyden-FAST, are capable of achieving the optimum design.The obtained optimal patch dimensions are mmand mm.


Fig. 15. Progress of the design parameters during the optimization of the microstrip-fed patch antenna.

The reason that Broyden-FAST succeeded where the direct-Broyden approach failed is similar to the reason outlined inthe previous example with the Yagi–Uda antenna. The direct-Broyden approach is incapable to represent properly the fastvariations in the sensitivities especially close to an optimal so-lution, which is associated with a resonance. On the other hand,Broyden-FAST deals with the sensitivities of the -matrix ele-ments, which are slowly varying functions unaffected by reso-nance.

IV. CONCLUSION

We propose a novel technique for accelerated gradient basedoptimization. It integrates the recently developed feasibleadjoint sensitivity technique for full-wave EM analysis [17]with methods for accelerated estimation of the derivativesof the system matrix. We consider two such methods: theboundary-layer concept and the Broyden update, and investi-gate their accuracy and versatility. We show that the Broydenupdate of the matrix derivatives is efficient, problem-inde-pendent and sufficiently accurate for the purpose of gradientbased optimization. When integrated with our feasible ad-joint sensitivity technique, it allows the computation of thesystem response and its gradient in the design parameter spacethrough a single system analysis. The overhead associatedwith the gradient estimation is negligible in comparison withthe computational requirements of the full-wave analysis.Our Broyden-FAST technique combines the efficiency of theadjoint-variable sensitivity analysis with the simplicity of theBroyden update. Its implementation is straightforward and doesnot require any analytical preprocessing.

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[16] J. Ureel and D. De Zutter, “Shape sensitivities of capacitances of planarconducting surfaces using the method of moments,” IEEE Trans. Mi-crowave Theory Tech., vol. 44, pp. 198–207, Feb. 1996.

[17] N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasibleadjoint sensitivity technique for EM design optimization,” IEEE Trans.Microwave Theory Tech., vol. 50, pp. 51–2758, Dec. 2002.

[18] N. K. Nikolova, J. W. Bandler, and M. H. Bakr, “Adjoint techniquesfor sensitivity analysis in high-frequency structure CAD,” IEEE Trans.Microwave Theory Tech., Special Issue on Electromagnetics-Based Op-timization of Microwave Components and Circuits,, to be published.

[19] S. Amari, “Numerical cost of gradient computation within the method ofmoments and its reduction by means of a novel boundary-layer concept,”in Proc. IEEE MTT-S Int. Symp. Dig., vol. 3, 2001, pp. 1945–1948.


[20] C. A. Balanis, Antenna Theory, 2nd ed. New York: Wiley, 1997.[21] C. G. Broyden, “A class of methods for solving nonlinear simultaneous

equations,” Mathematics of Computation, vol. 19, pp. 577–593, 1965.[22] J. W. Bandler, S. H. Chen, S. Daijavad, and K. Madsen, “Efficient op-

timization with integrated gradient approximations,” IEEE Trans. Mi-crowave Theory Tech., vol. 36, pp. 444–455, Feb. 1988.

[23] J. W. Bandler, R. M. Biernacki, S. H. Chen, R. H. Hemmers, andK. Madsen, “Electromagnetic optimization exploiting aggressivespace mapping,” IEEE Trans. Microwave Theory Tech., vol. 43, pp.2874–2882, Dec. 1995.

[24] S. Glavic, “Electromagnetic Design Sensitivity Analysis of High-Fre-quency Structures,” M.A.Sc. thesis, McMaster University, May 2002.

Natalia K. Nikolova (formerly Georgieva)(S’93–M’97) received the Dipl.Eng. degree inradioelectronics from the Technical University ofVarna, Bulgaria, in 1989 and the Ph.D. degreein electrical engineering from the University ofElectro-Communications, Tokyo, Japan, in 1997.

From 1998 to 1999, she held a Postdoctoral Fellow-ship of the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC), during which time shewas initially with the Microwave and ElectromagneticsLaboratory, DalTech, Dalhousie University, Halifax,

Canada. For approximately one year, she was with the Simulation OptimizationSystems Research Laboratory, McMaster University, Hamilton, ON, Canada. InJuly 1999, she joined the Department of Electrical and Computer Engineering,McMaster University, where she is currently an Associate Professor. Her researchinterests include theoretical and computational electromagnetism, high-frequencyanalysis techniques, as well as CAD methods for high-frequency structures andantennas.

Dr. Nikolova currently holds a University Faculty Award of NSERC, which shereceived in 2000.

Reza Safian received the B.Sc. degree from theIsfahan University of Technology (IUT), Isfahan, Iran,in 1999 and the M.A.Sc. degree from McMaster Uni-versity, Hamilton, Canada, in 2003, both in electricalengineering. He is currently working toward the Ph.D.degree at the University of Toronto, Toronto, ON,Canada.

During 1999 to 2002, he was a Research Engineerin the Electrical and Computer Engineering ResearchCenter (ECERC), IUT. From 2002 to 2003, he was aResearch Assistant at McMaster University working

with the Computational Electromagnetics Laboratory and the Simulation Opti-mization Systems Research Laboratory. His research interests include theory ofelectromagnetism and computational electromagnetics.

Ezzeldin A. Soliman (S’97–A’99) was born in Cairo,Egypt, on May 18, 1970. He received the B.Sc. de-gree (distinction with honors) in electronics and com-munications engineering and the M.Sc. degree in en-gineering physics, both from Cairo University, Giza,Egypt, in June 1992 and Nov. 1995, respectively, andthe Ph.D. degree (summa cum laude) in electrical en-gineering from the University of Leuven, Leuven, Bel-gium, in February 2000.

From 1992 to 1996, he was a Research and aTeaching Assistant with the Department of Engi-

neering Physics, Faculty of Engineering, Cairo University. From 1996 to 2000,he has been a Research Assistant at both Interuniversity MicroElectronicsCenter (IMEC), Leuven, Belgium, and the Department of Electrical Engineering,University of Leuven. From April 2002 to July 2002, he was a Visiting Pro-fessor at IMEC. From October 2002 to September 2003, he was on a VisitingResearcher in the Department of Electrical and Computer Engineering, McMasterUniversity, Hamilton, ON, Canada. He is currently an Assistant Professor withthe Department of Engineering Physics, Faculty of Engineering, Cairo University.His research interests include computational electromagnetics, development andcharacterization of planar antennas in the multilayer thin film technology, neuralnetwork modeling of electromagnetic problems, and the EM-based optimizationtechniques.

Mohamed H. Bakr (S’98–M’00) received the B.Sc.degree in electronics and communications engineering(distinction with honors) and the Master’s degree inengineering mathematics from Cairo University, Giza,Egypt, in 1992 and 1996, respectively, and the Ph.D.degree from the Department of Electrical and ComputerEngineering, McMaster University, Hamilton, ON,Canada, in September 2000.

In 1997, he was a Student Intern with OptimizationSystems Associates (OSA), Inc. From 1998 to 2000, heworked as a Research Assistant with the Simulation Op-

timization Systems (SOS) Research Laboratory, McMaster University. In November2000, he joined the Computational Electromagnetics Research Laboratory (CERL),University of Victoria, Victoria, Canada as an NSERC Postdoctoral Fellow. He iscurrently an Assistant Professor with the Department of Electrical and Computer En-gineering, McMaster University. His research areas of interest include optimizationmethods, computer-aided design and modeling of microwave circuits, neural networkapplications, smart analysis of microwave circuits and efficient optimization usingtime/frequency domain methods.

John W. Bandler (S’66–M’66–SM’74–F’78) was bornin Jerusalem, on November 9, 1941. He studied at the Im-perial College of Science and Technology, London, U.K.,from 1960 to 1966, and received the B.Sc.(Eng.), Ph.D.,and D.Sc.(Eng.) degrees from the University of London,London, U.K., in 1963, 1967, and 1976, respectively.

In 1966, he joined Mullard Research Laboratories,Redhill, Surrey, U.K. From 1967 to 1969, he was aPostdoctorate Fellow and Sessional Lecturer at theUniversity of Manitoba, Winnipeg, Canada. He joinedMcMaster University, Hamilton, ON, Canada, in 1969,

where he has served as Chairman of the Department of Electrical Engineering andDean of the Faculty of Engineering, and is currently Professor Emeritus in theElectrical and Computer Engineering Department, directing research in the Simula-tion Optimization Systems Research Laboratory. He was President of OptimizationSystems Associates Inc. (OSA), which he founded in 1983, until November 20,1997, the date of acquisition of OSA by Hewlett-Packard Company (HP). OSAimplemented a first-generation yield-driven microwave CAD capability for Raytheonin 1985, followed by further innovations in linear and nonlinear microwave CADtechnology for the Raytheon/Texas Instruments Joint Venture MIMIC Program.OSA introduced the CAE systems RoMPE in 1988, HarPE in 1989, OSA90 andOSA90/hope in 1991, Empipe in 1992, Empipe3D and EmpipeExpress in 1996.OSA created the product empath in 1996 which was marketed by Sonnet Software,Inc., USA. Dr. Bandler is President of Bandler Corporation, which he founded in1997. He has published more than 350 papers from 1965 to 2003. He contributedto Modern Filter Theory and Design (Surrey, U.K.: Wiley-Interscience, 1973) andto Analog Methods for Computer-aided Analysis and Diagnosis Germany: MarcelDekker, Inc., 1988). Four of his papers have been reprinted in Computer-AidedFilter Design (New York: IEEE Press, 1973), one in each of Microwave IntegratedCircuits (Norwood, MA: Artech House, 1975), Low-Noise Microwave Transistorsand Amplifiers (New York: IEEE Press, 1981), Microwave Integrated Circuits, (Nor-wood, MA: Artech House, 1985, 2nd ed.), Statistical Design of Integrated Circuits(New York: IEEE Press, 1987)and Analog Fault Diagnosis (New York: IEEE Press,1987). He joined the Editorial Boards of the International Journal of NumericalModeling in 1987, the International Journal of Microwave and MillimeterwaveComputer-Aided Engineering in 1989, and Optimization and Engineering in 1998.He was Guest Editor of International Journal of Microwave and Millimeter-WaveComputer-Aided Engineering, Special Issue on Optimization-Oriented MicrowaveCAD (1997). He was Guest Coeditor Optimization and Engineering Special Issueon Surrogate Modeling and Space Mapping for Engineering Optimization (2001).

Dr. Bandler is a Fellow of the Canadian Academy of Engineering, the RoyalSociety of Canada, the Institution of Electrical Engineers (IEE), London, U.K.,anf the Engineering Institute of Canada. He is a Member of the Association ofProfessional Engineers of the Province of Ontario (Canada), the MIT Electro-magnetics Academy, and the Micronet Network of Centres of Excellence. Hereceived the Automatic Radio Frequency Techniques Group (ARFTG) AutomatedMeasurements Career Award in 1994. He was an Associate Editor of the IEEETRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (1969–1974), and hascontinued serving as a member of the Editorial Board. He was Guest Editor ofthe IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Special Issueon computer-oriented microwave practices (1974) and Guest Coeditor of the ofthe IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Special Issueon Process-Oriented Microwave CAD and Modeling (1992), and Guest Editor,IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, Special Issue onAutomated Circuit Design Using Electromagnetic Simulators (1997). He is GuestCoeditor, IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, SpecialIssue on Electromagnetics-Based Optimization of Microwave Components andCircuits (2004). He has served as Chair of the MTT-1 Technical Committee onComputer-Aided Design.


A Theoretical Study of the Stability Criteria forHybridized FDTD Algorithms for Multiscale

AnalysisM. Marrone, Student Member, IEEE, and R. Mittra, Life Fellow, IEEE

Abstract—In this paper, we propose two new hybrid, two-dimen-sional, generalized finite-difference time-domain algorithms—for-mulated by using the cell method—and designed to analyze objectswith fine details, without using very small time steps dictated by theCourant condition. A detailed analysis of the stability of the pro-posed algorithms is presented, with special attention devoted to thephenomenon of late time instabilities. Finally, some rules are pro-vided that would ensure the stability of the proposed algorithms.

Index Terms—Cell method (CM), generalized finite-differencetime-domain (FDTD) method, hybrid algorithms, stability, sub-gridding.

I. INTRODUCTION

WHEN analyzing complex and multiscale structures usingthe time domain methods, it often becomes necessary

to combine two algorithms, e.g., the finite-difference time-do-main (FDTD) and finite-element time-domain (FETD), to im-prove the accuracy of the solution without placing an inordi-nately heavy burden on the CPU. For instance, it is often desir-able to use the FDTD on a coarse, structured Cartesian mesh inthe bulk of the region, and to employ either a FDTD subgrid-ding scheme in a smaller region containing objects with finefeatures [1], [2], or to use the FETD on an unstructured meshto further improve the modeling accuracy in this region [3].Other alternative approaches that are still under investigation arethe hybrid alternating direction implicit (ADI)-FDTD schemeson subgridding [4] and the ADI-Multiresolution time domain(MRTD) methods [5]. These methods circumvent the problemof having to use a very small time step throughout the compu-tational domain to satisfy the Courant condition [6], which isassociated with the smallest length of the mesh edges in the en-tire computational domain. But they do introduce the burden oftemporal and spatial interpolations, and the distinct possibilityof instabilities introduced by the spurious reflections at the in-terface of the two domains with dissimilar meshes. What is evenmore disconcerting is that, although the two algorithms appliedin the different domains may be independently stable (in the ab-sence of the other domain), the combination may still be un-stable and produce the so-called late time instabilities. It hasbeen demonstrated that this kind of instabilities related to spa-tial interpolation schemes, such as those employed in the sub-

Manuscript received May 15, 2003; revised September 30, 2003.The authors are with the Electromagnetic Communication Laboratory,

Pennsylvania State University, University Park, PA 16802 USA (e-mail:[email protected]; [email protected]).


gridding schemes, are due to a lack of symmetry in some dis-crete operators, and are not dependent on the Courant condition[7], [8]. In this paper we propose two new hybrid generalizedFDTD algorithms where each of them is a combination of twoalgorithms that work on different grids—a coarse mesh FDTDin one domain, and either a fine mesh or a triangular one in theother—and with different time stepping schemes. To investigatetheir stability, we will need to account for the instabilities thatmay arise from the spatial as well as temporal interpolations.The two hybrid algorithms will be developed by using the re-cently-introduced cell method (CM) [[10]–[13]] that enables usto address both the problems of instability and connectivity be-tween the different algorithms.

This paper is organized as follows. Section II presents theCM and Section III lays the foundations of the main algorithmfor the analysis of the wave propagation in two dimen-sional (2-D) cases on a coarse mesh FDTD. Section IV presentstwo new algorithms, suitable for either a fine or an unstructuredmesh, which are later combined with the algorithm in Section IIIto set up two hybrid algorithms, whose stability analyzes arepresented in Section V, followed by a brief conclusion given inSection VI.

II. THE CM

A study of the mathematical structure common to many phys-ical theories [9] provides a discrete mathematical frameworkfor the electromagnetic field theory, which can be utilized todevelop a numerical method for solving electromagnetic staticand dynamic problems on unstructured grids. This approach isreferred to as the CM, which employs the global (integral) vari-ables instead of the local ones. To define the global variables,and to avoid any sign ambiguities in their values, we need toassociate them in a physically coherent way, with certain ori-ented space and time elements [9], [10]. Specifically, we needtwo types of oriented space elements: 1) inner-oriented space el-ements (points , lines , surfaces , volumes ) and 2) com-plementary outer-oriented space elements (points , lines ,surfaces , volumes ). In addition, we need two types of ori-ented time elements: i) inner oriented time elements (instants, intervals ) and ii) complementary outer oriented time ele-

ments (instants , intervals ). The principal global variables ofelectromagnetics are as follows:

• Electric voltage impulse ;• Magnetic flux ;• Magnetic voltage impulse ;

0018-926X/04$20.00 © 2004 IEEE

MARRONE AND MITTRA: THEORETICAL STUDY OF STABILITY CRITERIA FOR HYBRIDIZED FDTD ALGORITHMS 2159

Fig. 1. Primal and dual space cell complexes and the global variablesassociated with their primal and dual cells.

• Electric flux ;• Electric charge content ;• Electric current flow .

The above are associated, respectively, with the oriented spaceand time elements appearing inside the square brackets. A suit-able geometrical structure that supports the global variables inCM is comprised of two staggered and oriented space cell com-plexes. A space complex, which is synonymous with the 3-Dgrid, is a structured collection of space elements (cells), suchas nodes, edges, faces and volumes. In particular, we need twooriented space cell complexes: 1) a primal complex, made upof inner-oriented space elements, supporting the global vari-ables associated with the above elements and 2) a complemen-tary dual complex, comprising of outer-oriented space elements,supporting the global variables associated with the outer-ori-ented space elements.

Moreover, to avoid instability problems, we need to establisha relationship of duality between the two complexes. Specifi-cally, we need as many primal nodes as dual volumes ; andthe same is true for the primal edges and the dual faces ,the primal faces and the dual edges , the primal volumesand the dual nodes . In Fig. 1 we show an example of primaland dual cubical space cell complexes, and the association ofthe global variables with the space cells. The time structure uti-lized in the CM also involves two staggered and oriented timecell complexes, viz., a primal complex, comprised of inner-ori-ented time elements, and a dual complex made up of outer-ori-ented time elements. Moreover, in common with the space struc-ture, the two time complexes must be related by duality, i.e., weneed an equal number of primal instants and dual intervals ;the same is true for the primal intervals and the dual instants. For the sake of brevity, and taking into account the duality

between the primal and the dual time cell complexes, we willuse the following notation: primal instants , primal intervals

, dual instants and dual intervals . In Fig. 2 weshow an example of primal and dual time cell complexes withthe common choice of the primal and dual intervals equal to thetime step , and the association of the globalvariables with the time cells.

In the CM, the electromagnetic laws can be formulated in analgebraic form on the cell complexes, in terms of the global vari-ables. In particular, these laws can be divided into the following

Fig. 2. Primal and dual time cell complexes and the global variables associatedwith their primal and dual cells.

two categories, depending upon the type of global variables thatare involved.

• Field equations (topological equations). These equationslink the global variables associated with space (time) cellsbelonging to the same type of complex (either the primalcell complex or the dual one). The field equations can beenforced on the cell complexes in an exact discrete form byusing appropriate incidence matrices. Let denotethe incidence matrices between the oriented primal ele-ments, such as edges and nodes, faces and edges, volumesand faces, respectively, and let

represent the corresponding incidence matrices be-tween the oriented dual elements, such as edges and nodes,faces and edges, volumes and faces, respectively [7], [9],[12], [13]. The above may be viewed as discrete counter-parts of the differential operators gradient, curl and di-vergence. Thus, the field equations can be expressed in aYee-like form [15] as follows:

— Faraday-Neumann law

(1)

which relates the global variables on the primal com-plex;

— Magnetic Gauss law

(2)

which relates the global variables on the primal com-plex;

— Ampère-Maxwell law

(3)

which relates the global variables on the dual complex;— Electric Gauss law

(4)

which relates the global variables on the dual complexwhere are arraysof scalars, the superscript indicates the association witheither a primal time instant or a dual time interval, whilethe superscript indicates the association with eithera primal time interval or a dual time instant.

• Space-time constitutive relations. These relations linkthe global variables associated with space and time cellsbelonging to different types of cell complexes. The electric


space-time constitutive relation links the variables and, whereas the magnetic space-time constitutive relation

provides a link between the variables and . These rela-tions employ the material parameters and can be enforcedon the cell complexes only in an approximate and discreteform by using suitable algebraic relationships. Next, byintroducing the following intermediate variables:

— electric voltage ;— magnetic voltage ;— electric current ;

that are global variables in space, but local variables intime, we can formulate the space-time constitutive rela-tions by combining two simpler relations. For example,to calculate the electric voltage impulse from the elec-tric flux , we can first calculate the electric voltagefrom the electric flux , by using the constitutive relation

, and we can then calculate the electric voltageimpulse from the electric voltage , by a time integralrelation . The time integration relations and theconstitutive relations we need are given below:

— Time integration relationships: These relationshipscan be extracted in an algebraic form from the fol-lowing time integral relationships:

In practice, we utilize the following two sets of timeintegration relations. The first set is

(5)

and it leads to the leap-frog time-stepping scheme em-ployed in conventional FDTD [15], [6]. The second setis

(6)

and leads to the Newmark time-stepping scheme em-ployed in FETD [16], which is an implicit scheme.

— Constitutive relations: These relations approximatethe following local constitutive relationships in an al-gebraic form:

Depending on the type of algorithms, we can utilizeeither one of the following two algebraic constitutiverelations:

(7)

or the:

(8)

where the matrices are referred toas constitutive matrices. These matrices can be gen-erated explicitly from the knowledge of the geomet-rical features of the cell complexes, and the features ofthe materials inside the computational domain. As wewill see in Section IV, the constitutive matrices must besymmetric and positive definite in order for the timeupdating scheme to be stable. These features can beobtained via the symmetrized microcell interpolationscheme (SMIS) [14], which also ensures that the con-stitutive matrices are physically consistent.

III. GENERALIZED FDTD 2-D- EXPLICIT ALGORITHM

FORMULATED BY THE CM

The formulation of the generalized FDTD algorithm on twostaggered cell complexes follows from the combination of thealgebraic electromagnetic laws of the CM. In particular, wefocus here only on the 2-D wave propagation, because the2-D wave propagation leads to similar theoretical results.Since the fields do not vary along the z-axis we can use a 2-Dcell complex obtained from a projection of a 3-D cell complexonto the xy-plane (Fig. 3).

The resulting 2-D cell complex is characterized by primalnodes ; edges ; faces ; and dual nodes ; edges and faces

. On this 2-D cell complex, the associations of the physicalvariables to the space elements are related to those in the 3-Dcase as follows:

• electric voltage ;• electric flux ;• magnetic voltage ;• magnetic flux ;• electric current ;

and we need to modify the field equations slightly from the 3-Dcase. In particular, for the FDTD algorithms that we need to setup, we are interested only in the Faraday–Neumann and Am-père–Maxwell laws. For the 2-D case, we can use the timeintegration relation in (5), to reduce these modified equations to

(9)

We point out that we have translated the quantities in the aboveequations by half a time-step, without loss of generality, in orderto simplify the analytical formalism of the stability criterion tobe discussed in Section V. Finally, by combining the field equa-tions given in (9) with the constitutive relations, we can derivean explicit time stepping algorithm (see Fig. 4), governing the

2-D wave propagation, which reads

(10)

In general, the incidence matrices that multiply the array vec-tors and in (10) must be adjoint one to the other, and theconstitutive matrices must be symmetric and positive definite,otherwise late time instabilities may arise. Moreover, a general-ized Courant condition must still be satisfied to avoid instabil-ities [7]. If the cell complexes we use are Cartesian orthogonal


Fig. 3. Associations of the physical variables with the elements of a 2-D cell complexes for the study of TM waves propagation.

Fig. 4. Graphical visualization of the CM-FDTD algorithm with the leap-frog time stepping.

grids, as shown in Fig. 3, the algorithm in (10) becomes iden-tical to the classical FDTD algorithm, once we replace the elec-tric and magnetic voltages with the electric and magnetic fieldcomponents. In this case, the conditions sufficient for avoidinglate time instabilities are satisfied automatically; hence, just asin the case of the conventional FDTD, the algorithm in (10) isstable if the time step satisfies the following Courant condi-tion [6]

(11)

where and are the grid step sizes and is the speed oflight. Since (10) can be utilized on arbitrary 2-D primal and dualcell complexes, we can derive a generalized FDTD algorithm forsuch grids, and we will refer to it as the CM-FDTD algorithm.

IV. HYBRID ALGORITHMS

When analyzing fine details in a small region via the FDTDmethod, it is often convenient to use subgridding in that region.Although this technique improves the accuracy of the results, it

places a heavy burden on the CPU, since it requires the use of avery small time step throughout the computational domain tosatisfy the Courant condition (11), which is associated with thesmallest length of the mesh edges in the entire computationaldomain. It is possible to overcome this problem by combiningtwo algorithms into a hybrid one, where we employ the FDTDalgorithm in the main part of the computational domain ,and an alternate one in a smaller region , which either usesthe subgridding (Fig. 5) or an unstructured mesh (Fig. 6). Thealternate algorithms should be chosen such that their stabilitycharacteristics are not so tied to the smallest length of the gridin the region as those ones of the FDTD algorithm.

The use of a complementary grid that is dual to the primalgrid throughout the entire computational domain, as for instancein the cases shown in Figs. 5 and 6, allows us to maintain theself-adjointeness between the matrices that multiply the arrayvectors V and F in (10). This approach is general and it is dif-ferent from that in [8] where one must replace the matricesand (or and in the 3-D cases) with some extendeddiscrete operators constructed via the use of some particular in-terpolation schemes.


Fig. 5. Example of two subdomains. The domain : square grid, the domain : square grid with subgridding.

Fig. 6. Example of two subdomains. The domain : square grid, the domain : square grid with a triangular grid.

To set up the hybrid algorithms for analyzing the wavepropagation in a 2-D domain , we consider splitting it into twosubdomains and , where we have a coarse mesh in the re-gion and a fine one in . When we discretize using theprimal and dual cell complexes, partitioning it into two subdo-mains creates a decomposition of the primal nodes set into twosets, viz., and . The primal edge set is also divided intotwo sets, namely and , where all the edges in connectonly nodes belonging to the set . Thus, in the subdomainwe have the primal nodes and the primal edges whereasin there are the primal nodes and the primal edgeswith their associated dual elements.

In addition, a decomposition of the set of the primal nodesas well as the edges into two introduces a partitioning of theoriginal incidence matrices and the constitutive matrices intofour blocks. These blocks are suitable not only for the derivationof the algorithms, but also for connecting them in a compactform that enables us to study the stability of the resulting hybridalgorithms more conveniently. In particular, for the incidencematrix , we have the following partition:

where is the incidence matrix related to the primal cellcomplex in ; is the incidence matrix related to the primalcell complex in ; and, is the incidence matrix related tothe part of the primal cell complex in connected to the primalcell complex in . Moreover, we assume that the common zonebetween the two cell complexes in and in , is connectedvia rectangular surfaces (see Figs. 5 and 6) in order for two of thefour blocks to be zero in the partitioned form of the constitutivematrices in (8). For instance

(12)

(13)

Finally, in practice, the only variables we are interested in calcu-lating are the electric and the magnetic voltages associated withthe elements of the two subdomains and . Specifically,they are

• electric voltages ;• magnetic voltages .

A. Algorithm for the Subdomain

In the subdomain we have a coarse Cartesian orthogonalmesh, and we can use the CM-FDTD algorithm given in (10).In this region, the blocks of the incidence matrix and of theconstitutive matrices (12), (13) can be reduced to yield:

•• .

For the stability analysis, discussed in Section V, it is convenientto rewrite the CM-FDTD algorithm as a two-step one as follows:

(14)

B. Algorithms for the Subdomain

In contrast to the subdomain contains objects with finefeatures that we want to model either by using a subgriddingscheme (see Fig. 5), or with an unstructured mesh (Fig. 6). Inorder to analyze the wave propagation in this region, wepropose two different algorithms for which the maximum al-lowable time step is not dependent on the smallest length of themesh edges in the same manner as it is in the FDTD algorithm.The first algorithm, which we will refer to as CM-NEW, is animplicit algorithm, whereas the second one, which we will namethe CM-TS-FDTD algorithm, is explicit in nature.

1) CM-NEW Algorithm: The CM-NEW algorithm is ob-tained in a way that is similar to the CM-FDTD algorithm, butit employs a Newmark time integration scheme. Specifically, itutilizes the time-stepping (6)

•

•


Fig. 7. Graphical visualization of the hybrid algorithm CM-FDTD (domain ) and CM-NEW (domain ).

Fig. 8. Graphical visualization of the hybrid algorithm CM-FDTD (domain ) and CM-TS-FDTD (domain ).

The above can be rewritten as the following two-step implicitalgorithm:

(15)

2) CM-TS-FDTD Algorithm: The CM-TS-FDTD algorithmis a type of CM-FDTD algorithm with a leap-frog time-steppingscheme, with a time subgridding (TS) of

••

••

•

•

C. The Hybrid Algorithms

The hybrid algorithms are derived by combining theCM-FDTD algorithm with either the CM-NEW or theCM-TS-FDTD. The are graphically represented in Figs. 7and 8, respectively.

V. STABILITY ANALYSIS

In order to study the stability of the hybrid algorithms, it isconvenient to restate them, without loss of generality, in terms ofthe new variables and , which are related to the variables

and as follows:

(16)

where the matrices and are real symmetric andpositive definite, provided that the matrix is symmetric andpositive definite. Moreover, let us define the following matrices,


that are useful for rewriting the hybrid algorithms in terms of thenew variables and

(17)

where .

A. Stability Analysis of the Hybrid Algorithm Set by CM-FDTD(Domain ) and CM-NEW (Domain )

To simplify the derivation of the stability criterion, we beginby assuming that there are no input sources. Then, by using (14),(15), (16), and (17), we can rewrite the hybrid time-steppingalgorithm as follows:

(18)

The system of equations in (18) has the following form:

(19)

where

and

Using the basic notions of discrete system theory (see Ap-pendix), it can be shown that the system in (19) is stable if twoof the following conditions are satisfied. The first conditionrequires that the matrix be symmetric, or that there exists anonsingular matrix such that

where is a symmetric matrix. One can show that late timeinstabilities would occur if the latter condition is not satisfied,and some of the eigenvalues of are complex [7]. Thesecond condition, a generalized Courant criterion, demands thatthe maximum eigenvalue in modulus max of must sat-isfy the bound given by

(20)

Fig. 9. Examples of domains . (a) Case SUB, (b) Case TRI-05.

If the constitutive matrices are symmetric and positive definite,then the are symmetric, and is symmetric positive def-inite. Then there exists the matrix

such that the matrix , given by

is symmetric. Therefore, the first condition for the stability ofthe hybrid algorithm is that the constitutive matrices be sym-metric and positive definite. Because the complexity of the com-bined algorithm, it is not possible to derive an explicit Courantcriterion, such as (11), which is employed in the FDTD algo-rithm to satisfy the second condition (20) for the stability. Forinstance, for the domains in Fig. 9, where the grid step sizes ofthe FDTD grid in the domain are m, it fol-lows from (10) that – s. For these domains, wehave calculated with varying from 0 to 2.5e–9 s andsteps of 0.05e–9 s, and have verified that (20) is satisfied when

– s. Moreover, the absence of instabilities has beenconfirmed by means of time domain simulations, on the samedomains, run with a – s, for 100 000 time steps.This leads us to conclude that this algorithm does not suffer forlate time instabilities, provided that the constitutive matrices aresymmetric and positive definite, and the Courant condition re-duce in practice to (10) regardless of the type of grid utilizedin the domain (either the subgridding or the triangular grid),due to the implicit nature of the CM-NEW algorithm.

B. Stability Analysis of the Hybrid Algorithm Set by CM-FDTD(Domain ) and CM-TS-FDTD (Domain )

Using (16), and (17), we can rewrite the CM-TS-FDTD algo-rithm, in the absence of input sources, as shown in (21) at thebottom of the next page. From (14), (16), and (17), and defining

we can write the hybrid algorithm as

(22)


The system in (22) has the same form as that in (19), providedwe define to be

and

If the constitutive matrices are symmetric and positive definite,then is symmetric, and if

(23)

then is symmetric and positive definite. In this caseare symmetric; hence, the matrix

is such that the matrix

is also symmetric. Unfortunately, it is not possible to find a morepractical criterion than the one given in (23), one which is validirrespective of the type of grid used in the domain .

Let us consider the following three cases: 1) the subdomainhas a subgridding [case SUB, Fig. 9(a)]; 2) a triangular grid

with a refinement factor of 0.5 [case TRI-05, Fig. 9(b)]; and 3)a triangular grid with refinement factor of 0.33 (case TRI-033).

For each of these cases the , which is the upper boundthat avoids late time instabilities, is presented in Table I. From

this Table we see that the depends on the refinement ofthe grid (cases TRI-05 and TRI-033). In summary, the stabilityconditions of the hybrid algorithm (22) are that the constitu-tive matrices be symmetric and positive definite, and that thetime-step has an upper bound given by (23). Turning now to theadditional condition for the stability, viz., (20), it is not possibleto find a criterion that is both practical and general. However,for the three cases analyzed above, we have calculated themax, with varying from 0 to 2.5e–9 s, and steps of 0.05e–9 s,to find the maximum sufficient to meet (20). The results are

TABLE I

TABLE II

presented in Table I. As we can see from this Table, with theexception of the triangular grid with a refinement factor of 0.33,the is equal or close to the , which ensures thestability of the FDTD on the subdomain .

Unfortunately, as we see from the Tables I and II, for eachcase the that avoids the late time instabilities is approx-imatively one half of the corresponding to the Courantlimit. However, since (23) represents a sufficiency condition, itdoes not necessarly follow that the late time instabilities wouldoccur if we use a time step such that .As a matter of fact, in the time domain simulations run for eachcase with there were no late time instabilities evenafter 100 000 time steps. This leads us to conclude that this algo-rithm is stable, provided that the constitutive matrices are sym-metric and positive definite, and the time step either satisfies

or, as in the cases analyzedhere, satisfies the criterion in (23). For the examples providedherein, allowed was always less than ; however,the algorithm is still less expensive then the one without the timesubgridding, so long as .

VI. CONCLUSION

In this paper, we have proposed two new hybrid 2-D time do-main algorithms, implemented by the CM, in order to analyzefine details without placing a heavy constraint on the maximumtime step imposed by the stability criterion. Additionally, thematrix notation of the CM has enabled us to carry out the sta-bility analysis of the hybrid algorithms, and to derive two condi-tions that guarantee stable numerical solutions. Finally, the the-oretically derived stability criteria have been tested via a nu-merical study. Although the analysis presented in this paper hasonly addressed the 2-D case, it can be readily extended to 3-Dgeometries, and the authors are currently in the process of pur-suing this effort.

(21)


APPENDIX

We wish to demonstrate that the discrete system

(24)

is stable if the matrix is symmetric, or there is a nonsingularmatrix such that

where is symmetric matrix, and if the maximum eigenvaluein modulus of satisfies the bound

(25)

The matrix can always be transferred into the Jordan canon-ical form through a nonsingular matrix as follows:

and, since is symmetric or it has the same eigenvalues asthose of a symmetric matrix, then is a diagonal matrix. Bydefining the new variable we can transform thesystem (24) into a new one which reads

(26)

The solution of the above equation is stable if and only if (24)is stable and if each of the following one component systems

are stable. The root conditions given in [18] states that in generalthe discrete system,

(27)

with , is stable, regardless of the nature of the inputsource, if the roots of the polynomial equation

(28)

satisfy . Using basic algebraic tools, we can demonstratethat the roots of the polynomial equation in (27) are ,if

(29)

Since the eigenvalues of the matrix are all real, thesystem (24) is stable if either , or, in general, themaximum eigenvalue in modulus of is such that

REFERENCES

[1] M. W. Chevalier, R. J. Luebbers, and V. P. Cable, “FDTD local gridwith material traverse,” IEEE Trans. Antennas Propagat., vol. 45, pp.411–421, Mar. 1997.

[2] M. Okoniewski, E. Okoniewska, and M. A. Stuchly, “Three dimensionalsubgridding algorithm for FDTD,” IEEE Trans. Antennas Propagat.,vol. 45, pp. 422–429, Mar. 1997.

[3] M. Feliziani and F. Maradei, “Mixed finite-difference/whitney-elementstime domain (FD/WE-TD) method,” IEEE Trans. Magn., vol. 34, pp.3222–3227, Mar. 1998.

[4] B. Z. Wang, Y. Wang, W. Yu, and R. Mittra, “A hybrid 2-D ADI-FDTDsubgridding scheme for modeling on-chip interconnects,” IEEE Trans.Adv. Packaging, vol. 24, pp. 528–533, Nov. 2001.

[5] Z. Chen and J. Zhang, “An unconditionally stable 3-D ADI-MRTDmethod free of the CFL stability condition,” IEEE Microwave WirelessComp. Lett., vol. 11, pp. 349–351, Aug. 2001.

[6] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Fi-nite-Difference Time-Domain Method. Norwood, MA: Artech House,2000.

[7] T. Weiland and R. Schuhmann, “Space and time stability of discrete timedomain algorithms,” in Proc. 4th Int. Workshop on Computational Elec-tromagnetics in the Time Domain, Nottingham, U.K., July 17–19, 2001,pp. 155–161.

[8] O. Podebrand, M. Clemens, and T. Weiland, “New flexible subgriddingscheme for the finite integration technique,” IEEE Trans. Magn., vol. 39,pp. 1662–1665, May 2003.

[9] E. Tonti, On the Formal Structure of Physical Theories, Italian NationalResearch Council, 1975. preprint.

[10] , “Finite formulation of the electromagnetic field,” IEEE Trans.Magn., vol. 38, pp. 333–336, Mar. 2002.

[11] M. Marrone, A. M. F. Frasson, and H. E. Hernández-Figueroa, “A novelnumerical approach for electromagnetic scattering: The cell method,” inProc. IEEE AP-S URSI, vol. 1, San Antonio, TX, June 16–21, 2002, pp.160–163.

[12] M. Marrone, V. F. Rodríguez-Esquerre, and H. E. Hernández-Figueroa,“Novel numerical method for the analysis of 2D photonic crystals: Thecell method,” Opt. Express, vol. 10, no. 22, pp. 1299–1304, Nov. 2002.

[13] M. Marrone, R. Mittra, and W. Yu, “A novel approach to deriving a stablehybridized FDTD algorithm using the cell method,” in Proc. IEEE AP-SURSI, Columbus, OH, June 22–27, 2003.

[14] M. Marrone, “A novel technique to build constitutive matrices for gen-eralized FDTD algorithms,” in Proc. IEEE AP-S URSI, Columbus, OH,June 22–27, 2003.

[15] K. S. Yee, “Numerical solution of initial boundary value problems in-volving Maxwell’s equations in isotropic media,” IEEE Trans. AntennasPropagat., vol. 14, pp. 302–307, Mar. 1966.

[16] S. D. Gedney and U. Navsariwala, “An unconditionally stable finite el-ement time-domain solution of the vector wave equation,” IEEE Trans.Microwave Guided Wave Lett., vol. 5, pp. 332–334, Oct. 1995.

[17] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore,MD: The Johns Hopkins Univ. Press, 1996.

[18] P. Henrici, Error Propagation for Difference Methods. New York:Wiley, 1963.

Massimiliano Marrone (S’02) received the Laureadegree (summa cum laude) in electrical engineeringand the Ph.D. degree in information technology fromthe University of Trieste, Italy, in 1999 and 2003,respectively.

From July 2001 to January 2002, he was a VisitingScholar at the Microwaves and Optics Department,Universidade Estadual de Campinas (UNICAMP),Sao Paulo, Brazil. From June 2002 to December2002, he was a Visiting scholar at the Electromag-netic Communication Laboratory, Pennsylvania

State University, University Park, where he is currently a Postdoctoral Re-searcher. His research interests include numerical techniques for modelingelectromagnetic fields in complex enviroments (FDTD, FEM, finite volumes),network theory, and foundations of electromagnetics.

Dr. Marrone is a Member of the International Compumag Society (ICS).


Raj Mittra (S’54–M’57–SM’69–F’71–LF’96) is aProfessor in the Electrical Engineering Department,Pennsylvania State University, University Park. He isalso Director of the Electromagnetic CommunicationLaboratory, which is affiliated with the Communica-tion and Space Sciences Laboratory of the ElectricalEngineering Department. Prior to joining Penn State,he was a Professor in the Electrical and ComputerEngineering Department, University of Illinois, Ur-bana-Champaign. He was a Visiting Professor at Ox-ford University, Oxford, U.K., and the Technical Uni-

versity of Denmark, Lyngby, Denmark. He is President of RM Associates, aconsulting organization that provides services to industrial and governmentalorganizations. He has published more than 600 technical papers and more than30 books or book chapters on various topics related to electromagnetics, an-tennas, microwaves, and electronic packaging. He has received three patents oncommunication antennas. Currently, he is the North American Editor of AEÜ.He has advised more than 80 Ph.D. students and about an equal number of M.S.students, and has mentored approximately 50 postdoctoral research associatesand visiting scholars in the EMC Laboratory. His professional interests includethe areas of communication antenna design, RF circuits, computational elec-tromagnetics, electromagnetic modeling and simulation of electronic packages,EMC analysis, radar scattering, frequency-selective surfaces, microwave andmillimeter-wave integrated circuits, and satellite antennas.

Prof. Mittra received the Guggenheim Fellowship Award in 1965, the IEEECentennial Medal in 1984, and the IEEE Millennium Medal in 2000. He is a pastPresident of AP-S and was Editor of the IEEE TRANSACTIONS ON ANTENNAS

AND PROPAGATION.


Communications______________________________________________________________________

Full-Wave Analysis of a Waveguide Printed Slot

Giorgio Montisci and Giuseppe Mazzarella

Abstract—In this paper, we propose a new waveguide slot array configu-ration, in which the slots are printed on a copper-clad substrate. This con-figuration can be obtained using a printed slot and therefore allows both asimpler realization of a slot array and permits its reuse when the patternrequirement changes. An innovative and highly effective method of mo-ments procedure has been devised. It employs a spectral domain approachfor the computation of the internal Green function. Our code has been vali-dated against a commercial finite element method code, and the agreementis highly satisfactory. The code has then been used to obtain the scatteringand radiating properties of the proposed printed slot. Such a slot behavesas a shunt admittance, so Elliott’s method can be used to design an arrayof printed slots.

Index Terms—Method of moments (MoM), slot antennas, waveguide ar-rays.

I. INTRODUCTION

Waveguide slot arrays have a number of advantages over other mi-crowave antenna systems. Among the others they have a very largeefficiency, due to the use of closed guiding structures for feeding, anda considerable mechanical strength. On the other hand, they have noflexibility since, once an array is realized, its electromagnetic behaviorcannot be changed. If the antenna requirements change, the array itselfmust be changed. Moreover the array realization requires, mainly inthe K-band, expensive procedures, such as electroerosion.

In order to get a structure mechanically flexible and easier to realize,but with all the other advantages of a waveguide slot array, it is pos-sible to replace the upper waveguide wall with a copper-clad laminate,with a single metal layer. The slots are etched on this metal layer usingone of the standard technologies for printed antennas. The electromag-netic behavior of the array then depends only on this slotted copper-cladlaminate, which can be easily replaced at only a small fraction of thetotal array cost. Moreover such an array can be realized in an easierway and with the same, or even higher, accuracy than usual waveguideslot arrays. The copper layer can be placed below or above the dielec-tric laminate, and a metallic frame is required to sustain the laminate(see Fig. 1). If the copper layer is below the laminate, this frame willcause almost unpredictable diffraction effects (due to both space andsurface waves). Therefore we propose here to insert the dielectric in-side the waveguide. In this way the slots radiate in an open half-space,exactly as in a standard array. Of course, the power handling capabilityis smaller than in a standard array, but this can be a problem only insome applications. The proposed structure (see Fig. 1) consists of ametallic “comb-like” structure, formed by the bottom and sidewalls ofthe waveguides, covered with the copper-clad laminate.

A standard slot array can be accurately designed using Elliott’s pro-cedure [1]. This procedure can be directly applied to the structure pro-posed here, as soon as the self-admittance of a single slot is known,since the mutual coupling is exactly equal to the standard case one. Ata first glance, another difference appears, namely the internal coupling

Manuscript received May 13, 2003; revised September 29, 2003.The authors are with the Dipartimento di Ingegneria Elettrica ed Elettronica,

Università di Cagliari, 09123 Cagliari, Italy (e-mail: [email protected]).Digital Object Identifier 10.1109/TAP.2004.832333

Fig. 1. Alternate phase feeding for the proposed slot array.

between waveguides. This coupling has been evaluated with HFSSusing two slots, each in one of two adjacent waveguides with a commoninternal dielectric layer: we consider a principal waveguide, which isfed with an incident TE10 mode and a parasite one. External boundaryconditions have been chosen so that only internal coupling is involvedin the analysis. We found that the voltage excitation of the slot in theparasite guide is about �30 dB with respect to that of the slot in theprincipal waveguide. Therefore this small coupling does not affect thearray behavior and can be neglected expecially if the waveguides arefed with alternating phases [2]. This case has been analyzed, too, andthe waveguide field distribution and the slots excitation are the same asin the case of two isolated waveguide.

To evaluate the slot self-admittance, we present here a specializedand very effective full-wave analysis, based on the method of moments(MoM) [3] which allows also to take exactly into account the thicknessof the metallic plate in which the slots are cut. An accurate and effectiveprocedure has been realized using entire domain basis functions [4].

The key point in every MoM procedure for waveguide slots is theevaluation of the waveguide magnetic field inside the equivalent mag-netic currents. As it is well known [5] the relevant Green function issingular and this can lead to numerical instabilities and poor conver-gence, both in the space domain (expressing the Green function as amodal series) and in the spectral domain. Since the F potential [6] isless singular in a magnetic current the elements of the MoM matrixhave been expressed in terms ofF . Previous papers on slots in inhomo-geneous waveguide [7] employ a modal series for the Green function.In this paper a spectral domain approach has been used since it is of farsimple use in presence of an air-dielectric boundary.

Our MoM code has been validated using a commercial finite elementmethod (FEM) code (Ansoft HFSS). The agreement is highly satisfac-tory, though our tailored code is significantly more effective than HFSS,which is a general-purpose commercial software and is not optimizedfor the proposed structure.

II. PROBLEM FORMULATION

Using the alternate phase feeding [2] for the array, we can assumethat short circuits exist at the upper corners of two adjacent waveguides(see Fig. 1). Therefore, the slot self-admittance is not affected by theadjacent waveguides, and, in order to evaluate it, we can consider alongitudinal slot in an isolated waveguide (Fig. 2). Let t be the dielectricsubstrate thickness and "r its permittivity. The fundamental mode of

0018-926X/04$20.00 © 2004 IEEE


Fig. 2. Geometry of the Slot. (a) Front view and (b) top view.

this waveguide impinges upon the slot and is scattered and radiated bythe slot itself.

In order to compute this interaction, and obtain the slot equivalentcircuit, both slot apertures (�i;�e) are replaced by equivalent surfacemagnetic currents, backed by perfect electric conductors. Accordingto the equivalence theorem, these currents M i;M e, are proportionalto the tangential electric field on the apertures, and are therefore un-knowns. In order to compute them, we enforce, at all interfaces �i;�e,the continuity of the tangential magnetic field

in �Hw[Mi] + in �H inc

= in �Hs[Mi;M

e] on �i

in �Hs[Mi;M

e] = in �He[Me] on �e (1)

wherein Hw; Hs; He are the waveguide region, slot region and ex-ternal region magnetic field, H inc is the incident magnetic field of thefundamental mode and in the normal to each surface. As usual forradiating slots [8], we assume that the slots radiate into a half-spacebounded by an infinite, perfectly conducting, ground plane. All the in-volved regions are regular and need not to be discretized in a MoMprocedure, since their Green function is known in (a simple) closedform.

In (1) we have also included explicitly the currents that produce eachfield. Since each magnetic field is an integral operator acting on itssource, (1) is already cast as a set of coupled integral equations. Theactual form of these operators is of course required, and their computa-tion will be discussed later in this section. The resulting integral equa-tions are then solved by the MoM, to obtain the unknown currents. Todo this, all unknowns are expressed as a linear combination of the samebasis functions. Usually the slots are narrow enough to neglect the lon-gitudinal component of the electric field on them [9]. Therefore only

the z-directed magnetic currents are used as unknowns, and each cur-rent (M i;Me) is expressed as truncated sinusoidal series with respectto z

M� =

N

p=1

a�p sin

p�

2L(z + L) iz =

N

n=1

a�pm

p(z): (2)

In (2) the apex � can assume the values i; e.Expansions (2) are then substituted into (1) and the resulting equa-

tions are scalar multiplied by mq and integrated over the apertures ofthe slots. Only a few terms in (2) are needed, at least for resonant or al-most-resonant slots, so that the resulting linear system in the unknownsaip; a

ep is small and quite well conditioned.

Both the internal and external magnetic fields in (1) are expressedusing the potential F

H = j!"F +1

j!�rr � F (3)

so the MoM matrix element is

Apq = j!"Slot

Fp�m

qds�

1

k20 slot

(r � Fp) (r �m

q) dS

(4)

wherein F p is the potential produced by mp.The external and slot region integrals are computed respectively using

a standard plane-wave spectrum representation of the external field andthe admittance matrix formalism, much as done in [6]. It is worth notingthatmp are the modes of the slot region (considered as a waveguide) andtherefore the admittance matrix allows to include exactly the wall thick-ness effect. The internal terms in (4) have been computed in the spec-tral domain expressing both the potentials and the currents as a Fouriertransform respect to the waveguide axis coordinate and as a (truncated)Fourier series respect to the transverse direction. In this latter case, theexpansion is actually an eigenmodes one.

Due to the waveguide inhomogeneity the z-directed magnetic cur-rent ~M produces both ~Fy and ~Fz components

~Mz(w) =

1

n=0

~Mzn(w) cosn�x

a(5)

~Fy(x; y; w) =

1

n=0

~Fyn(y; w) sinn�x

a

~Fz(x; y; w) =

1

n=0

~Fzn(y; w) cosn�x

a(6)

where

~F (x; y; w) =1

2�

+1

�1

F (x; y; z) exp(jwz)dz (7)

and likely for ~M . The use of expansions with different parity for ~Fyand ~Fz is dictated by the boundary conditions on the lateral walls ofthe waveguide.

Since the effect of the unknowns magnetic currents has been in-cluded in the continuity condition of the electric field at the sourcelocation, then we can compute the potential F by solving two equalhomogeneous Helmholtz equations, one for ~Fy and the other for ~Fz ,which become:

@ ~Fn@y2

+ �20n

~Fn = 0 0 � y � d

@ ~Fn@y2

+ �2n~Fn = 0 d � y � b

wherein

�20n = k

20 � w

2�

n�

a

2

�2n = "rk

20 � w

2�

n�

a

2

: (8)


Fig. 3. Comparison between our MoM analysis procedure and Ansoft HFSS(t = 0:3 mm, " = 2:2; slot length = 13:2 mm, slot width = 1 mm,slot o�set = 4 mm).

The general solution for (8) is

~Fn = A0n cos(�0ny) +B0n cos(�0ny) 0 � y � d

~Fn = An cos(�n(b� y))

+Bn cos(�n(b� y)) d � y � b (9)

wherein the coefficients A0n; B0n; An, and Bn are computed as afunction of the unknowns magnetic currents, by enforcing the conti-nuity and boundary conditions.

The MoM matrix elements are then computed in the spectral domainusing the Parseval identity in all the internal terms of (4). The integra-tion has been performed on the real axis, following the suggestions of[10].

III. RESULTS

The presented procedure has been applied to a number of differentcases. We discuss here only a typical set of results, using a WR90 wave-guide (A = 22:86 mm, B = 10:16 mm) partially filled with a dielec-tric slab. The ground plane thickness is equal to 50 �m and the losseshave been neglected. Actually, this is correct at the chosen X-band fre-quency but could be questionable, at least for the dielectric losses, athigher frequencies. In any case such losses can be taken into account, ifrequired, by using a complex permittivity in the Green function and inthe pole computation. Our pole-search procedure can be directly gen-eralized to look for complex poles.

The results of our code have been validated through a comparisonwith the results of a commercial FEM code (HFSS 6.0). As it is wellknown a longitudinal slot in an empty waveguide behaves as a shuntadmittance and this behavior is retained also for a dielectric-backedprinted slot since our simulations show that the magnitude of j1+S11�S12j is very small [9]. Therefore only the normalized slot admittancehas been considered for the comparison. The agreement between ourcode and the commercial FEM one is very good (see Fig. 3). It is worthnoting that our code is about one order of magnitude faster than AnsoftHFSS and requires at least two orders of magnitude less memory thanHFSS.

In Fig. 4 the normalized resonant length of the slot has been shown,as a function of the slot offset, for different values of the substrate di-electric constant at the design frequency of 9.7 GHz. Finally in Fig. 5–7the normalized slot admittance is reported respectively as a function ofthe slot offset, the substrate dielectric constant and the dielectric sub-strate thickness.

Fig. 4. Resonant length of the slot as a function of the slot offset(frequency = 9:7 GHz, t = 0:3 mm, slot width = 1 mm).

Fig. 5. Normalized slot admittance at resonance as a function of the slot offset(t = 0:3 mm, " = 2:2; slot length = 13:2 mm, slot width = 1 mm).

Fig. 6. Normalized slot admittance at resonance as a function of " (t = 0:3mm, O�set = 4:0 mm, slot length = 13:2 mm, slot width = 1 mm).

In all the simulations we used seven expansion functions for each slot[N = 7 in (2)], since we found that this choice is enough to achieve agood convergence.


Fig. 7. Normalized slot admittance at resonance as a function of the dielectricsubstrate thickness t (O�set = 4:0 mm, " = 2:2; slot length = 13:2 mm,slot width = 1 mm).

IV. CONCLUSION

Afull-waveMoMprocedure forprintedwaveguideslotshasbeenpre-sented and validated against a commercial FEM code. From it both thetangential electric field and the scattering data of a printed slot can be ob-tained. From the latter we derive that a waveguide printed slot behavesas a shunt element, as a standard waveguide longitudinal slot does. TheMoMprocedurehasbeendevisedinorder toexploit thesimplegeometricform of a slot to realize an accurate and very effective code. As a result,an accurate electromagnetic characterization of this structure can be ob-tained, with the scattering data recast in a form suitable to be used in slotarray design procedures based on Elliot’s method. This allows to designslot arrays with the slots printed on the aperture plane, with the same ac-curacy of a standard waveguide slot array.

As a consequence, the new slot configuration proposed here can beeffectively used to allow an easier, less expensive array realization,which permits a simple reuse of the array itself when the pattern re-quirements changes.

REFERENCES

[1] R. S. Elliott, “An improved design procedure for small arrays of shuntslots,” IEEE Trans. Antennas Propagat., vol. AP-31, pp. 48–53, 1983.

[2] Y. Kimura, T. Hirano, J. Hirokawa, and M. Ando, “Chokes for alter-nating-phase fed single-layer slotted waveguide arrays,” Proc. IEEE Int.Antennas Propagat. Symp. Dig., vol. 38, pp. 82–85, July 2000.

[3] R. F. Harrington, Field Computation by Moment Methods. New York:IEEE Press, 1993.

[4] R. W. Lyon and A. J. Sangster, “Efficient moment method analysis of ra-diating slots in a thick-walled rectangular waveguide,” Proc. Inst. Elect.Eng., pt. H, vol. 128, no. 4, Aug. 1981.

[5] R. E. Collin, Field Theory of Guided Waves. New York, NY: IEEEPress, 1990.

[6] G. Mazzarella and G. Montisci, “A rigorous analysis of dielectric-cov-ered narrow longitudinal shunt slots with finite wall thickness,” Electro-magn., vol. 19, pp. 407–418, 1999.

[7] J. Joubert and D. A. McNamara, “Analysis of radiating slots in a rect-angular waveguide loaded with a dielectric slab,” IEEE Trans. AntennasPropagat., vol. 41, Sept. 1993.

[8] S. R. Rengarajan, “Compound radiating slot in a broad wall of a rect-angular waveguide,” IEEE Trans. Antennas Propagat., vol. AP-37, pp.1116–1124, 1989.

[9] T. Vu Khac and C. T. Carson, “Impedance properties of longitudinalslot antenna in the broad face of rectangular waveguide,” IEEE Trans.Antennas Propagat., vol. AP-21, pp. 708–710, 1973.

[10] J. R. Mosig, R. C. Hall, and F. E. Gardiol, Handbook of Microstrip An-tennas, J. R. James and P. S. Hall, Eds, London, U.K.: Peregrinus, 1993.

Dual Polarized Wide-Band Aperture Stacked PatchAntennas

K. Ghorbani and R. B. Waterhouse

Abstract—A wide-band, dual polarized printed antenna is designed anddeveloped in this paper. The antenna is based upon an aperture stackedpatch layout and incorporates a simple dual-layered feeding technique toachieve dual-polarized radiation. The printed antenna has a measured 10dB return loss bandwidth of 52% and an isolation between the excitationports of greater than 39 dB over this frequency range. The gain of the an-tenna is 7 4 dBi 0 4 dB and the typical issues associated with incor-porating an aperture excited solution are resolved by using a cross-shapedreflector patch to ensure the front-to-back ratio is greater than 20 dB.

Index Terms—Aperature antennas, microstrip antennas, polarization.

I. INTRODUCTION

As mobile communication services become more sophisticated, theneed for larger spectral bandwidth for delivery of these provisions isapparent. It was because of this trend the International Telecommunica-tions Union recommended that for IMT-2000 several frequency bandsbe utilized over almost a 50% range centered near 2 GHz [1]. Thus, basestation antennas must be able to operate efficiently over ever increasingfrequency ranges than were originally required. Also, to further en-hance the link performance between the base station and wireless user,diversity techniques have been proposed and subsequently used at thebase station. Polarization diversity is a useful technique to reduce thedetrimental effects of multipath fading and is a common procedure uti-lized at base stations of most mobile communication systems.

Printed antennas have many salient features that have made them pri-mary candidates for cellular base stations including their inherent easeof mass construction and their conformal nature. However, in their orig-inal form and subsequent bandwidth enhancement procedures, such asstacking patches [2], or for the case of an aperture excited patch, usinga large slot [3] it is difficult to achieve the previously mentioned band-width and therefore difficult to provide a single element solution. Asingle element solution allows for simple, low cost arrays to be de-veloped to provide the necessary sectoral radiation patterns typicallyrequired at a base station terminal.

Over past few years there have been several reported printed an-tennas that can achieve 50% bandwidth: Aperture stacked patches(ASPs) [4], Quasi Yagi–Uda printed antennas [5] and Suspendedpatches with three-dimensional feeds [6], to name a few. Each of theseprinted antennas has their relative figures of merit and can readilysatisfy the necessary bandwidth requirements for IMT-2000. However,of these solutions, the easiest to achieve good quality dual polarizationis the aperture-stacked patch, due to the inherent polarization purityof a thin slot excited patch antenna. There have been several dualpolarized slot coupled patch antennas investigated recently [7]–[11],however the antennas considered are relatively narrow band, withbandwidths less than or equal to 25%.

In this paper, we present the design and develop of a dual polarizedbroadband printed antenna capable of operation over a 50% impedance

Manuscript received February 13, 2003; revised September 17, 2003.K. Ghorbani is with the School of Electrical and Computer Engineering,

RMIT University, Melbourne, VIC 3000, Australia.R. B. Waterhouse was with Pharad Technologies, North Fitzroy, VIC 3068

Australia. He is now with Pharad Technologies, Baltimore, MD 21227 USA(e-mail: [email protected]).


0018-926X/04$20.00 © 2004 IEEE


Fig. 1. Geometry of dual polarized ASP antenna.

bandwidth, with polarization isolation greater than 35 dB and a gain of7 dBi � 0:4 dB over this band of frequencies. Our solution utilizes adual polar, broadband reflector patch below the feed/antenna ground-plane which improves the front-to-back ratio (FBR) of the element tomore than 20 dB across the band of interest, a key aspect for sectorizedcellular base stations and an issue with most aperture solutions.

II. ANTENNA CONFIGURATION AND DESIGN

The geometry of dual polarized ASP antenna is shown in Fig. 1. Ascan be seen from this diagram, 2 rectangular patches are mounted onmultiple dielectric layers (in this case 2 each) located above a feed layersituated above a ground-plane. Another feed layer used to provide theorthogonal polarization exists below the ground-plane. Power is cou-pled to the patch radiators from the feed networks via a cross-shapedslot in the common ground-plane. Beyond the lower feed substrateare a foam spacer and then a cross-shaped patch use to enhance thefront-to-back ratio.

The strategy outlined in [4] was used to design a linearly polar-ized aperture-stacked patch with the appropriate bandwidth (50%). En-semble 6.1, a planar method of moments (MoM) field simulator, wasutilized as the simulation tool. Symmetry is a very important aspect inthe design of the antenna to achieve a high degree of isolation betweenthe two polarization ports over all operation frequencies. By using acentered crossed slot to feed the patch element, a high degree of sym-metry can be maintained, however this necessitates a balanced feed net-work for each polarization excitation. Incorporating a single feed-lineon each polarization of the crossed slot would introduce asymmetry,which can degrade the isolation between the two ports. Using smallslots (as in [9], [10]) is not an option as the bandwidth requirement istoo large for such a solution. In the proposed configuration each feednetwork is placed on a separate layer on opposite sides of the groundplane of the antenna as can be seen in Fig. 1. This feed layout is based onthat presented in [8], [11] and avoids using an air-bridge (as in the casewhere both balanced feeds for each polarization are etched on the sameboard). Using separate feed layers improves the isolation between theports due to limiting spurious radiation from the air-bridge as well asreduces the construction cost of the antenna. The input impedance forthe two ports will differ slightly as the upper feed network can couple tothe patch elements. However this is typically negligible as electricallythick layers are used between the slot and the patches of the ASP andso direct coupling from the feed-line to the patches should be minimal.Having said that, the height of two feeding layers must be chosen sothat the mutual coupling between the slot and patch elements remainsthe same for both polarization ports. The proposed ASP consists of four

Fig. 2. Measured and Ensemble 6.1 return loss of dual polarized ASP antenna:(a) Port 1, (b) Port 2 (parameters: "r1 = 4:5; h1 = 1:58mm, "r2 = 1; h2 =35mm, "r3 = 2:2; h3 = 0:508mm, "r4 = 2:2; h4 = 0:508mm, "r5 =1:07;h5 = 9mm, "r6 = 2:2; h6 = 3:175mm, "r7 = 1;h7 = 13mm,"r4 = 2:2; h8 = 1:58mm, W1REF = 10mm, L1REF = 85mm,W2REF = 10mm, L2REF = 85mm, W1F = 0:4mm, L1STUB =12:3mm, W2F = 0:4mm, L2STUB = 13:1mm, W2SL = 10mm,L2SL = 46mm, W2SL = 10mm, L2SL = 46mm, W1 = 47mm,L1 = 47mm, W2 = 45mm, L2 = 45mm.).

Fig. 3. Measured and Ensemble 6.1 isolation between Port 1 and Port 2.

layers of Rogers Duroid 5880 laminate ("r3; "r4; "r6; "r8), one layer ofFR4 dielectric material ("r1) and three air/foam layers ("r2; "r5; "r7).


Fig. 4. Measured radiation patterns of dual polarized ASP antenna at 2.0 GHz: (a) H-plane Port 1, (b) H-plane Port 2, (c) E-plane Port 1, (d) E-plane Port 2.

ASP antennas normally produce a relatively low front to back ratio(typically 10–14 dB [4]) due to the resonant aperture used to enhancethe bandwidth of the aperture-coupled patches. In order to increase theFBR a microstrip antenna element as a reflector (based on the methoddescribed in [12]) was used. The length of the reflector is chosen so thephase of the radiated field due to the reflector is approximately oppositeto the radiated field from the aperture. Changing the spacing (betweenthe reflector and the aperture) and width of the reflector can control themagnitude of the radiated field. The total cancellation will occur whenthe magnitudes of the radiated field from the aperture and the reflectorare approximately equal while the phases are opposite. The spacingbetween the aperture and the reflector should also be sufficient so thereflector has negligible affect on the input impedance of the antenna.For the application considered here a cross-shaped element was chosento ensure minimal back radiation in both polarizations was achieved.

To reduce the cost of the antenna and to make it more rigid in con-struction, an FR4 substrate was used for the microstrip patch reflector.The spacing between the reflector patch and the lower feedline wasoptimized in order to achieve a FBR of greater than 20 dB without af-fecting the input impedance behavior of both polarization networks.This optimization was done by simulation. The procedure outlined in[12] was followed here. This procedure is pretty robust and so for thesake of brevity, we simply referenced it in this paper. The optimiza-tion procedure is conducted over the entire 10 dB return loss band-width. Plots of the affect of spacing on the FBR are given in [12]and these were consistent with the study conducted here. Typically for

the optimum design the FBR peaks at a frequency between the lowerband-edge and the center frequency. Slightly increasing the spacingfrom this value can enhance the peak FBR at the expense of the achiev-able 20 dB FBR bandwidth. The reflector patch enhanced the FBR bymore than 5 dB across the entire band to greater than 20 dB.

III. RESULTS AND DISCUSSION

Fig. 2(a) and (b) show the measured and calculated return loss fromPort 1 and Port 2 of the dual polarized ASP antenna, respectively (port1 corresponds to feed 1 in Fig. 1). As can be seen from these responses,there is good agreement between the theoretical and experimental re-sults. 10 dB return loss bandwidths of greater than 52% for both polar-izations were obtained. The lower edge of the matched band is slighterlower for Port 2 than Port 1. This can be attributed to the height of theupper feed substrate and so the coupling between the slot and the patchelement for this polarization is stronger than between the slot and thepatch element for the Port 1 excitation. The shift is less than 1%. Fig. 3shows the measured and calculated isolation between two ports. An iso-lation of more than 39 dB over the impedance-matched band has beenachieved. As mentioned before, the two feed networks are separatedby the ground plane, thereby eliminating any cross coupling betweenthe microstrip-lines in the two networks. The magnitude of isolationpresented here is comparable with the previous dual polarized aperturecoupled patches presented in the literature and is achieved over a con-siderably larger impedance bandwidth.


Fig. 5. Measured and Ensemble 6.1 gain of dual polarized ASP antenna.

A full radiation characterization of the antenna was conducted and theresultsat thebandedges (1.5and2.5GHz)aswell as thecenter frequency(2GHz).The results at 2GHzare shown inFig. 4. Thecross-polarizationlevels for this antenna are considerably low,with ahigher contribution intheE-planeofeachexcitedport.Thiscanbeattributedtotheinteractionofexcited surface waves, the slot and the edges of the truncated substrates.Such an interaction is always more prevalent in the E-plane. The cross-polar level for each excitation and in each plane is not less than 12 dBbelow the copolar value for scan angles up to �60�, angles consistentwith a three-sectored base station. Of course the worse cases are at theextremescanangles,wherethecopolarradiateddecaysmorerapidly.Thecross-polarization levels are the highest at 1.5 GHz, which is due to thesize of the ground-plane (here 105 mm� 105 mm). The finite size of theground-planewillhaveagreaterimpactatlowerfrequencies,asitsrelativesize issmaller.AmeasuredFBR(theratioofpoweratbroadside(� = 0�)and� = 180�)of approximately20dBwasobtainedfor theantennaoverthe entire impedance bandwidth.

The gain of the antenna was greater than 7 dBi over the entireVSWR < 2 : 1 band. A comparison between theory and experimentare shown in Fig. 5. As can be seen from this figure there is verygood agreement between the predicted and measured results. The gainof the dual-polarized ASP drops off at higher frequencies, beyondthe operation bandwidth of the antenna due to the slot becoming thedominant radiation mechanism [12].

There are many design parameters in the dual polarized ASP withreduced backward directed radiation. Because of this, it is important toensure that the overall performance of the antenna is not sensitive toparameter variations (tolerances in the dielectric constants of the mate-rials, or etching procedures). We conducted a parameter sensitivity in-vestigation via simulation (a three-sigma test) on the proposed antenna,allowing all the parameters associated with antenna (conductor/slot di-mensions (�0:02 mm, materials ("r � 0:02; h � 0:025 mm), to bevaried within the material tolerances and etching tolerances. The find-ings showed minimal impact on the return loss bandwidth and FBRperformance. There were slight increases in the peaks of the return losswithin the 10 dB return loss band by 1.5 dB when all the conductorswere over-etched by 0.02 mm, however, these peaks were still belowthe 10 dB requirement. These findings show that the antenna is verytolerant to moderate variations in all the parameters.

IV. CONCLUSION

In this paper, a dual-polarized wide-band patch antenna has beenpresented. The antenna has a VSWR < 2 : 1 bandwidth of greaterthan 50%, an isolation between ports of more than 39 dB and a gain of7:4 dBi � 0:4 dB over this frequency range. A cross-shaped reflector

patch ensures the FBR of the antenna is greater than 20 dB over theentire 50% matched-impedance bandwidth. The measured results arein very good agreement with the theoretical predictions. The antenna issimple to manufacture and is suited to being incorporated into a mobilecommunications base station array.

REFERENCES

[1] Handbook on Antennas in Wireless Communications, L. Godara, Ed.,CRC Press, New York, 2001.

[2] R. B. Waterhouse, “Design of probe-fed stacked patches,” IEEE Trans.Antennas Propagat., vol. 47, pp. 1780–1784, Dec. 1999.

[3] J. F. Zurcher, “The SSFIP: A global concept for high performance broad-band planar antennas,” Electron. Lett., vol. 24, pp. 1433–1435, Nov.1988.

[4] S. D. Targonski, R. B. Waterhouse, and D. M. Pozar, “A wideband aper-ture coupled stacked patch antenna using thick substrates,” Electron.Lett., vol. 32, pp. 1941–1942, Oct. 1996.

[5] N. Kaneda, W. R. Deal, Y. Qian, R. Waterhouse, and T. Itoh, “A broad-band planar quasiyagi antenna,” IEEE Trans. Antennas Propagat., vol.50, pp. 1158–1160, Aug. 2002.

[6] N. Herscovici, “A wide-band single-layer patch antenna,” IEEE Trans.Antennas Propagat., vol. 46, pp. 471–474, Apr. 1998.

[7] E. Aloni and R. Kastner, “Analysis of a dual circularly polarized mi-crostrip antenna fed by crossed slots,” IEEE Trans. Antennas Propagat.,vol. 42, pp. 1053–1058, Aug. 1994.

[8] M. Yamazaki, E. T. Rahardjo, and M. Haneishi, “Construction of a slot-coupled planar antenna for dual polarization,” Electron. Lett., vol. 30,pp. 1814–1815, 1994.

[9] S. Hienonen, A. Lehto, and A. V. Raisanen, “Simple broadband dual-po-larized aperture-coupled microstrip antenna,” in Proc. IEEE APS Symp.Digest, Orlando, FL, June 1999, pp. 1228–1231.

[10] S. C. Gao, L. W. Li, P. Gardner, and P. S. Hall, “Wideband dual-polarizedmicrostrip patch antenna,” Electron. Lett., vol. 37, pp. 1213–1214, Sept.2001.

[11] J. R. Sanford and A. Tengs, “A two substrate dual polarized aperture cou-pled patch,”Proc. IEEE Trans. Antennas and Propagation Symp. Digest,pp. 1544–1547, 1996.

[12] R. B. Waterhouse, D. Novak, A. Nirmalathas, and C. Lim, “Broadbandprinted sectoral coverage antennas for millimeter-wave wireless appli-cations,” IEEE Trans. Antennas Propagat., vol. 50, pp. 12–16, Jan. 2002.

Resonant Frequency of Equilateral Triangular MicrostripAntenna With and Without Air Gap

Debatosh Guha and Jawad Y. Siddiqui

Abstract—A tunable equilateral triangularmicrostrip patch (ETMP) an-tenna with a variable air gap between the substrate and the ground planehas been studied theoretically and experimentally. Calculated resonant fre-quencies for different air gap heights have been verifiedwithmeasurementsof a coax-fed antenna. The tunability of the antenna as a function of the airgap height is studied theoretically showing over 200% tunable range of anETMP with 50 mm side length printed on a substrate with = 10 5.The computed results for the antennas with zero air gap height are com-pared with a standard spectral domainmoment method analysis supportedby other previously reported experiments. Very close agreement is revealedin all comparisons.

Index Terms—Microstrip antenna, triangular microstrip patch, tunablemicrostrip antenna.

Manuscript received October 22, 2002; revised June 30, 2003. This work wassupported by the Center of Advanced Study in Radio Physics and Electronics,University of Calcutta.

The authors are with the Institute of Radio Physics and Electronics, Uni-versity of Calcutta, Calcutta 700 009, India (e-mail: [email protected]; [email protected]).


0018-926X/04$20.00 © 2004 IEEE


Fig. 5. Measured and Ensemble 6.1 gain of dual polarized ASP antenna.

A full radiation characterization of the antennawas conducted and theresultsat thebandedges (1.5and2.5GHz)aswell as thecenter frequency(2GHz).The results at 2GHzare shown inFig. 4.Thecross-polarizationlevels for this antenna are considerably low,with ahigher contribution intheE-planeofeachexcitedport.Thiscanbeattributedtotheinteractionofexcited surface waves, the slot and the edges of the truncated substrates.Such an interaction is always more prevalent in the E-plane. The cross-polar level for each excitation and in each plane is not less than 12 dBbelow the copolar value for scan angles up to �60�, angles consistentwith a three-sectored base station. Of course the worse cases are at theextremescanangles,wherethecopolarradiateddecaysmorerapidly.Thecross-polarization levels are the highest at 1.5 GHz, which is due to thesize of the ground-plane (here 105mm� 105mm). The finite size of theground-planewillhaveagreaterimpactatlowerfrequencies,asitsrelativesize issmaller.AmeasuredFBR(theratioofpoweratbroadside(� = 0�)and� = 180�)of approximately20dBwasobtainedfor theantennaoverthe entire impedance bandwidth.

The gain of the antenna was greater than 7 dBi over the entireVSWR < 2 : 1 band. A comparison between theory and experimentare shown in Fig. 5. As can be seen from this figure there is verygood agreement between the predicted and measured results. The gainof the dual-polarized ASP drops off at higher frequencies, beyondthe operation bandwidth of the antenna due to the slot becoming thedominant radiation mechanism [12].

There are many design parameters in the dual polarized ASP withreduced backward directed radiation. Because of this, it is important toensure that the overall performance of the antenna is not sensitive toparameter variations (tolerances in the dielectric constants of the mate-rials, or etching procedures). We conducted a parameter sensitivity in-vestigation via simulation (a three-sigma test) on the proposed antenna,allowing all the parameters associated with antenna (conductor/slot di-mensions (�0:02 mm, materials ("r � 0:02; h � 0:025 mm), to bevaried within the material tolerances and etching tolerances. The find-ings showed minimal impact on the return loss bandwidth and FBRperformance. There were slight increases in the peaks of the return losswithin the 10 dB return loss band by 1.5 dB when all the conductorswere over-etched by 0.02 mm, however, these peaks were still belowthe 10 dB requirement. These findings show that the antenna is verytolerant to moderate variations in all the parameters.

IV. CONCLUSION

In this paper, a dual-polarized wide-band patch antenna has beenpresented. The antenna has a VSWR < 2 : 1 bandwidth of greaterthan 50%, an isolation between ports of more than 39 dB and a gain of7:4 dBi � 0:4 dB over this frequency range. A cross-shaped reflector

patch ensures the FBR of the antenna is greater than 20 dB over theentire 50% matched-impedance bandwidth. The measured results arein very good agreement with the theoretical predictions. The antenna issimple to manufacture and is suited to being incorporated into a mobilecommunications base station array.

REFERENCES

[1] Handbook on Antennas in Wireless Communications, L. Godara, Ed.,CRC Press, New York, 2001.

[2] R. B. Waterhouse, “Design of probe-fed stacked patches,” IEEE Trans.Antennas Propagat., vol. 47, pp. 1780–1784, Dec. 1999.

[3] J. F. Zurcher, “The SSFIP: A global concept for high performance broad-band planar antennas,” Electron. Lett., vol. 24, pp. 1433–1435, Nov.1988.

[4] S. D. Targonski, R. B. Waterhouse, and D. M. Pozar, “A wideband aper-ture coupled stacked patch antenna using thick substrates,” Electron.Lett., vol. 32, pp. 1941–1942, Oct. 1996.

[5] N. Kaneda, W. R. Deal, Y. Qian, R. Waterhouse, and T. Itoh, “A broad-band planar quasiyagi antenna,” IEEE Trans. Antennas Propagat., vol.50, pp. 1158–1160, Aug. 2002.

[6] N. Herscovici, “A wide-band single-layer patch antenna,” IEEE Trans.Antennas Propagat., vol. 46, pp. 471–474, Apr. 1998.

[7] E. Aloni and R. Kastner, “Analysis of a dual circularly polarized mi-crostrip antenna fed by crossed slots,” IEEE Trans. Antennas Propagat.,vol. 42, pp. 1053–1058, Aug. 1994.

[8] M. Yamazaki, E. T. Rahardjo, and M. Haneishi, “Construction of a slot-coupled planar antenna for dual polarization,” Electron. Lett., vol. 30,pp. 1814–1815, 1994.

[9] S. Hienonen, A. Lehto, and A. V. Raisanen, “Simple broadband dual-po-larized aperture-coupled microstrip antenna,” in Proc. IEEE APS Symp.Digest, Orlando, FL, June 1999, pp. 1228–1231.

[10] S. C. Gao, L.W. Li, P. Gardner, and P. S. Hall, “Wideband dual-polarizedmicrostrip patch antenna,” Electron. Lett., vol. 37, pp. 1213–1214, Sept.2001.

[11] J. R. Sanford andA. Tengs, “A two substrate dual polarized aperture cou-pled patch,”Proc. IEEE Trans. Antennas and Propagation Symp. Digest,pp. 1544–1547, 1996.

[12] R. B. Waterhouse, D. Novak, A. Nirmalathas, and C. Lim, “Broadbandprinted sectoral coverage antennas for millimeter-wave wireless appli-cations,” IEEE Trans. Antennas Propagat., vol. 50, pp. 12–16, Jan. 2002.

Resonant Frequency of Equilateral Triangular MicrostripAntenna With and Without Air Gap


Abstract—A tunable equilateral triangular microstrip patch (ETMP) an-tenna with a variable air gap between the substrate and the ground planehas been studied theoretically and experimentally. Calculated resonant fre-quencies for different air gap heights have been verified with measurementsof a coax-fed antenna. The tunability of the antenna as a function of the airgap height is studied theoretically showing over 200% tunable range of anETMP with 50 mm side length printed on a substrate with = 10 5.The computed results for the antennas with zero air gap height are com-pared with a standard spectral domain moment method analysis supportedby other previously reported experiments. Very close agreement is revealedin all comparisons.

Index Terms—Microstrip antenna, triangular microstrip patch, tunablemicrostrip antenna.

Manuscript received October 22, 2002; revised June 30, 2003. This work wassupported by the Center of Advanced Study in Radio Physics and Electronics,University of Calcutta.



0018-926X/04$20.00 © 2004 IEEE


I. INTRODUCTION

Since the early phase of microstrip research the triangular geometryof microstrip patch is being investigated as planar circuit components[1]–[3] as well as microstrip antennas [4]–[17]. Many new configura-tions have also been studied in recent years to explore new character-istics [18]–[24]. Very recently, Gurel and Yazgan [25] have analyzed atunable equilateral triangular microstrip patch (ETMP) antenna. Theyhave introduced a variable air gap in between the ground plane andthe substrate like some previous investigations with other geometries[26]–[28].

The resonant frequency fr of the ETMP becomes a critical designparameter since it is inherently a narrow bandwidth structure. Startingfrom [1], most of the formulations for fr are reported on the basis ofthe cavity model analyzes (CMA) [1], [4], [7]–[17], [25], though a fewdeal with other techniques like spectral domain technique [3], [19], ge-ometrical theory [6] and spectral domain analysis with method of mo-ment solution (SDA MoM) [12]. One recent advance in this field [25]presents a new computation of the resonant frequency of an ETMPwithand without air gap employing Wolff’s cavity model [29]. In [25], abetter accuracy of their theory with respect to SDAMoM [12] and otheravailable cavity model results is also outlined. But the formulation [25]itself suffers from some limitations. The effective side length of theETMP derived in [25] is valid only for the substrates with "r < 10.

These shortcomings are addressed in this paper and a simple formulaof the resonant frequency of an ETMP with and without air gap is pro-posed on the basis of an improved cavity model recently proposed byone of the authors [30]. Like [30], the present model should be widelyapplicable, even to the MIC design on semiconductor materials with"r � 12. Furthermore, the variable air gap, as shown in Fig. 1, intro-duces tunability to the antenna and enhances its impedance bandwidth.Although Mirshekar-Syahkal and Hassani [20] investigated stackedETMP with dissimilar substrates, including air, no experimentalresult of an ETMP with an air gap is available in open literature. Thecalculated results are compared with measurements performed on HP8720C network analyzer. The tunability of the antenna as a functionof the variable air gap (h1 of Fig. 1) is presented as design data forvarious combinations of antenna parameters. Computed results for theETMPs without air gap are compared with a standard theory supportedby different experimental data.

II. THEORETICAL CALCULATIONS

Following the cavity model analysis by Helszajn [1], a simple andmore general expression for the resonant frequencies ofTMnml modesof an ETMP antenna with and with out air gaps can be given as [1]

fr;nm =2c

3se�p"r;e�

(n2 + nm+m2) (1)

where c is the velocity of light in free space, se� is the effective sidelength of an ETMP and "r;e� is the effective relative permittivity of themedium below the patch.

In the present formulation, se� and "r;e� are determined using theimproved cavity model [30]. The parameter "r;e� is derived in termsof "re and "r;dyn [30, eq. 3], where "re is the equivalent permittivityof the two-layer dielectric medium (Fig. 1) and "r;dyn is the dynamicdielectric constant of the medium below the patch. The quantity "r;dynof an ETMP can be obtained from the static main and static fringingcapacitances of an equivalent circular microstrip disk [30]. The equiv-alent circle of an equilateral triangular patch is determined on the basisof equal circumference of both the geometries keeping in view the samefringing field effects for the dominant modes. The radius of an equiva-lent circle thus can be equated as

a =3

2�s (2)

Fig. 1. Coax-fed ETMP antenna with an air gap between the substrate and theground plane.

The quantity "r;dyn for the TM modes under an ETMP now can becalculated from [30, eq.(4)–(16)] with n = 0:3525.The effective radius of the equivalent circle is calculated from [30]

which with the help of (2) determines the effective side length of theETMP as

se� =2�

3a (1 + q) (3)

where q is the parameter determining the fringing field effect derivedas [30, eq.(4)–(17)].

III. RESULTS

A tunable ETMP antenna with variable air gap analyzed inSection II and depicted in Fig. 1, reduces to the conventionalgeometry when h1 = 0. The computed results of such a structurehave been compared with some data reported earlier. The spectraldomain analyzes with method of moment solution (SDA MoM)is the most accurate method and hence the SDA MoM results[12] are taken as the reference to compare the computed valuesin Tables I and II. Table I deals with an antenna printed on alow dielectric constant substrate as measured by Dahele and Lee[7]. The present formulation is compared with another accuratecavity model analysis using GA optimization (CMA-GA) [17] withreference to the SDA MoM [12]. The comparison shows very closeapproximation of the calculated results to those of the SDA MoM[12] ones and even improvement over [17]. The measured valuesalso correspond to the theoretical values with close agreement.A similar comparison is presented in Table II for high dielectric

constant substrate. Here, the present theory rather shows a closerapproximation with the SDA MoM [12] results. The authors haveexperienced the same observation from the comparison with anotherSDA result [3] for a wide range of patches printed on a substratewith "r = 10:2.The dominant mode resonant frequency of the ETMP without

air gap is examined for different substrate heights and "r values inFig. 2. The effect of the substrate height h(h = h2) is predominantin the lower "r case where the fringing fields change considerablywith h [30, Fig. 2]. That effect is negligibly small in high dielectricconstant substrates. This is also supported by the SDA MoM [12]results, also depicted in Fig. 2 by discrete circles. Excellent agreementis revealed except for the low "r curve with h � 8 mm. This may


TABLE ICOMPARISON OF CALCULATED RESONANT FREQUENCIES OF AN ETMP AND

MEASURED VALUES (LOW DIELECTRIC PERMITTIVITY) "r = 2:32,s = 100 mm, h1 = 0, h2 = 1:59 mm, � = 3:0 mm

TABLE IICOMPARISON OF CALCULATED RESONANT FREQUENCIES OF AN ETMP AND

MEASURED VALUES (HIGH DIELECTRIC PERMITTIVITY) "r = 10:5,s = 41 mm, h1 = 0, h2 = 0:7 mm, � = 5:0 mm

be explained as an inherent limitation of the cavity model whichis best suitable for small h=�0 ratio. Similar disagreement withthe measurements of an identical patch is reported in [10] for lowdielectric constant substrate but this occurs at a much lower valueof h = 1:59 mm when compared with their integration averagemethod calculation.

No result of the ETMP antenna with air gap is available in open lit-erature and hence the calculated frequencies are compared with mea-surements of a coax-fed tunable antenna etched on RT-Duroid 5880and measured on HP 8720C network analyzer. The antenna was fedby a 50 SMA probe with � � 3:4 mm for all h1 values. The res-onant frequencies measured from the minima of the return loss tracesare compared with the calculated values in Table III. Very close agree-ment between the theory and experiment is observed for the first twooperating modes. However, the limited range of h1 values in the exper-iment could not give a clear view of tunability of the antenna.

The tunability has been theoretically studied in Fig. 3. The designdata are presented for two different patches printed on two differentsubstrates having widely varying dielectric constant. The resonant fre-quency normalized with respect to that with h1 = 0 is plotted againsth1=h2 with h2 = 0:508 mm. The variation of f=f0 with h1=h2 isclearly observed for all s and "r values. Larger patches on high dielec-tric constant substrates should yield broader tunable frequency ranges.The curves for the small patch (s = 5 mm), unlike those for s =50 mm, show a blurred peak at some value near h1=h2 � 1 and thenfalls monotonically as h1=h2 increases. The decrease in f=f0 with theincrease in air gap height is due to the increase in the fringing factor qas thoroughly discussed in [30]. The increase in the air gap height h1keeping h2 � �0 increases the resultant thickness h of the mediumbelow the patch with the effect of lowering its equivalent dielectric con-stant to a significant fraction. This low permittivity value minimizesthe possibility of generating any surface wave even when the antennais electrically thick.

Fig. 2. Computed resonant frequency of TM10mode of the ETMP antennaversus thickness of the substrate h(h = h2). s = 100 mm, h1 = 0; ——Present Model, � SDA MoM [12].

TABLE IIICOMPARISON OF CALCULATED RESONANT FREQUENCIES OF AN ETMPWITH VARIABLE AIR GAP AND MEASUREMENTS s = 15:5 mm,

h2 = 0:508 mm, "r = 2:2, � = 3:4 mm

Fig. 3. Dominant Mode resonant frequency normalized with respect to thatfor h1 = 0Versus h1=h2. h2 = 0:508 mm; —— s = 50 mm, f0("r =2:32) = 2:59 GHz, f0("r = 10:5) = 1:24 GHz - - - s = 5 mm,f0("r = 2:32) = 23:43 GHz, f0("r = 10:5) = 12:09 GHz.

IV. CONCLUSION

A tunable equilateral triangular microstrip antenna with a variableair gap in between the substrate and the ground plane has been investi-gated both theoretically and experimentally. An improved formulationto compute accurate resonant frequencies of the operational modes inan ETMP is proposed on the basis of an improved cavity model [30].The computed results are compared with measurements of a proto-type tunable antenna printed on a RT-Duroid 5880 and measured onHP 8720C network analyzer. Very close agreement between the theoryand experiment is observed. Attractive tunability characteristics withthe change in air gap height are also studied theoretically. Over 200%


tunable ranges of an ETMP antenna on "r = 10:5 with s = 50 mmis demonstrated from theoretical data. Larger patches on higher dielec-tric constant substrates should offer larger tunable frequency ranges.The ETMP without air gap (Fig. 1, h1 = 0) reduces to the conven-tional geometry which was studied by many other researchers [1]–[17]earlier. The present model is compared with another accurate cavitymodel [17], taking SDA MoM [12] as the reference. Improvement ofthe present model is envisaged from the studies for different antennaparameters.

ACKNOWLEDGMENT

The authors would like to thank Dr. P. K. Saha and Dr. P. K. Basu,University of Calcutta, for their constant encouragement.

REFERENCES

[1] J. Helszajn and D. S. James, “Planar triangular resonators with magneticwalls,” IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 95–100,Feb. 1978.

[2] J. Helszajn, D. S. James, and W. T. Nisbet, “Circulators using planartriangular resonators,” IEEE Trans. Microwave Theory Tech., vol.MTT-27, pp. 188–193, Feb. 1979.

[3] A. K. Sharma and B. Bhat, “Analysis of triangular microstrip res-onators,” IEEE Trans. Microwave Theory Tech., vol. MTT-30, pp.2029–2031, Nov. 1982.

[4] Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and experiment onmicrostrip antennas,” IEEE Trans. Antennas Propagat., vol. AP-27, pp.137–145, 1979.

[5] I. J. Bahl and P. Bhartia, “Radiation characteristics of a triangular mi-crostrip antenna,” Arch. Elek. Ubertragungstech., vol. 35, pp. 214–219,1981.

[6] E. F. Kuester and D. C. Chang, “A geometrical theory for the resonantfrequencies and Q-factors of some triangular microstrip patch antennas,”IEEE Trans. Antennas Propagat., vol. AP-31, pp. 27–34, Jan. 1983.

[7] J. S. Dahele and K. F. Lee, “On the resonant frequencies of the trian-gular patch antenna,” IEEE Trans. Antennas Propagat., vol. AP-35, pp.100–101, Jan. 1987.

[8] R. Garg and S. A. Long, “An improved formula for the resonant fre-quency of the triangular microstrip patch antenna,” IEEE Trans. An-tennas Propagat., vol. AP-36, p. 570, Apr. 1988.

[9] K. F. Lee, K.M. Luk, and J. S. Dahele, “Characteristics of the equilateraltriangular patch antenna,” IEEE Trans. Antennas Propagat., vol. AP-36,pp. 1510–1518, Nov. 1988.

[10] X. Gang, “On the resonant frequencies of microstrip antennas,” IEEETrans. Antennas Propagat., vol. 37, pp. 245–247, Feb. 1989.

[11] R. Singh, A. De, and R. S. Yadava, “Comments on an improved formulafor the resonant frequency of the triangular microstrip patch antenna,”IEEE Trans. Antennas Propagat., vol. 39, pp. 1443–1444, Sept. 1991.

[12] W. Chen, K. F. Lee, and J. Dahele, “Theoretical and experimental studiesof the resonant frequencies of equilateral triangular microstrip antenna,”IEEE Trans. Antennas Propagat., vol. 40, pp. 1253–1256, Oct. 1992.

[13] N. Kumprasert and K. W. Kiranon, “Simple and accurate formula forthe resonant frequency of the equilateral triangular microstrip patch an-tenna,” IEEE Trans. Antennas Propagat., vol. 42, pp. 1178–1179, Aug.1994.

[14] K. Güney, “Resonant frequency of a triangular microstrip antenna,”Mi-crowave Opt. Technol. Lett., vol. 6, pp. 555–557, July 1993.

[15] , “Comments on ‘on the resonant frequencies of microstrip an-tennas’,” IEEE Trans. Antennas Propagat., vol. 42, pp. 1363–1365, Sept.1994.

[16] P. Mythili and A. Das, “Comments on ‘simple and accurate formula forthe resonant frequency of the equilateral triangular microstrip patch an-tenna’,” IEEE Trans. Antennas Propagat., vol. 48, p. 636, Jan. 2000.

[17] D. Karaboga, K. Güney, N. Karaboga, and A. Kaplan, “Simple and accu-rate effective side expression obtained by using a modified genetic algo-rithm for the resonant frequency of an equilateral triangular microstripantenna,” Int. J. Electron., vol. 83, pp. 99–108, Jan. 1997.

[18] J. P. Damiano et al., “Study of multilayer microstrip antennas with radi-ating elements of various geometry,” Proc. Inst. Elect. Eng., pt. H, vol.137, no. 3, pp. 163–170, 1990.

[19] H. R. Hassani and D. Mirshekar-Syahkal, “Analysis of triangular patchantennas including radome effects,” Proc. Inst. Elect. Eng., pt. H, vol.139, no. 3, pp. 251–256, 1992.

[20] D. Mirshekar-Syahkal and H. R. Hassani, “Characteristics of stackedrectangular and triangular patch antennas for dual band applications,” inProc. 8th Inst. Elect. Eng. Int. Conf. Antennas and Propagation, 1993,pp. 728–731.

[21] C. L. Tang and K. L. Wong, “A modified equilateral-triangular-ringmicrostrip antenna for circular polarization,” Microwave Opt. Technol.Lett., vol. 23, pp. 123–126, Oct. 1999.

[22] C. L. Tang, J. H. Lu, and K. L. Wong, “Novel dual-frequency and broad-band designs of single-feed slot-loaded equilateral-triangular microstripantennas,” IEEE Trans. Antennas Propagat., vol. 48, pp. 1048–1054,July 2000.

[23] J. H. Lu and K. L. Wong, “Single-feed circularly polarized equilateraltriangular microstrip antenna with a tuning stub,” IEEE Trans. AntennasPropagat., vol. 48, pp. 1869–1872, Dec. 2000.

[24] Y. W. Jang, “Characteristics of a large bandwidth rectangular mi-crostrip-fed inserted triangular patch in a circular slot antenna,”Microwave J., vol. 45, no. 5, pp. 288–298, May 2002.

[25] C. S. Gurel and E. Yazgan, “New computation of the resonant frequencyof a tunable equilateral triangular microstrip patch,” IEEE Trans. Mi-crowave Theory Tech., vol. 48, pp. 334–338, Mar. 2000.

[26] K. F. Lee, K. Y. Ho, and J. S. Dahele, “Circular disc microstrip antennawith an air gap,” IEEE Trans. Antennas Propagat., vol. 32, pp. 880–884,Aug. 1984.

[27] J. S. Dahele and K. F. Lee, “Theory and experiments on microstrip an-tennas with air gaps,” Proc. Inst. Elect. Eng., pt. H, vol. 132, no. 7, pp.455–460, Dec. 1985.

[28] F. Abboud, J. P. Damiano, and A. Papiernik, “A new model for cal-culating the input impedance of coax-fed circular microstrip antennaswith and without air gaps,” IEEE Trans. Antennas Propagat., vol. 38,pp. 1882–1885, Nov. 1990.

[29] I. Wolff and N. Knoppik, “Rectangular and circular microstrip disk ca-pacitors and resonators,” IEEE Trans. Microwave Theory Tech., vol. 22,pp. 857–864, Oct. 1974.

[30] D. Guha, “Resonant frequency of circular microstrip antennas with andwithout airgaps,” IEEE Trans. Antennas Propagat., vol. 49, pp. 55–59,Jan. 2001.

Effect of a Cavity Enclosure on the Resonant Frequency ofInverted Microstrip Circular Patch Antenna


Abstract—The inverted microstrip circular patch (IMCP) antenna hasbeen analyzed very recently by the present authors and the same microstripstructure enclosed in a cylindrical cavity has been employed by others todevelop various active integrated antennas. In this paper, the effect of thecylindrical enclosure in changing the resonant frequency of an IMCP hasbeen studied both theoretically and experimentally. The cavity-effect crit-ically depends on the enclosed patch and the cavity dimensions which arethoroughly investigated to help a designer in choosing the antenna parame-ters. An efficient analytical formulation is also proposed to predict accurateresonant frequencies of the cavity enclosed IMCPs and is verified with dif-ferent experimental data.

Index Terms—Cavity enclosed microstrip patch, inverted microstrippatch, microstrip antenna.

Manuscript received October 25, 2002; revised July 9, 2003. This work wassupported by the Center of Advanced Study in Radio Physics and Electronics,University of Calcutta.



0018-926X/04$20.00 © 2004 IEEE


tunable ranges of an ETMP antenna on "r = 10:5 with s = 50 mmis demonstrated from theoretical data. Larger patches on higher dielec-tric constant substrates should offer larger tunable frequency ranges.The ETMP without air gap (Fig. 1, h1 = 0) reduces to the conven-tional geometry which was studied by many other researchers [1]–[17]earlier. The present model is compared with another accurate cavitymodel [17], taking SDA MoM [12] as the reference. Improvement ofthe present model is envisaged from the studies for different antennaparameters.

ACKNOWLEDGMENT

The authors would like to thank Dr. P. K. Saha and Dr. P. K. Basu,University of Calcutta, for their constant encouragement.

REFERENCES

[1] J. Helszajn and D. S. James, “Planar triangular resonators with magneticwalls,” IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 95–100,Feb. 1978.

[2] J. Helszajn, D. S. James, and W. T. Nisbet, “Circulators using planartriangular resonators,” IEEE Trans. Microwave Theory Tech., vol.MTT-27, pp. 188–193, Feb. 1979.

[3] A. K. Sharma and B. Bhat, “Analysis of triangular microstrip res-onators,” IEEE Trans. Microwave Theory Tech., vol. MTT-30, pp.2029–2031, Nov. 1982.

[4] Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and experiment onmicrostrip antennas,” IEEE Trans. Antennas Propagat., vol. AP-27, pp.137–145, 1979.

[5] I. J. Bahl and P. Bhartia, “Radiation characteristics of a triangular mi-crostrip antenna,” Arch. Elek. Ubertragungstech., vol. 35, pp. 214–219,1981.

[6] E. F. Kuester and D. C. Chang, “A geometrical theory for the resonantfrequencies and Q-factors of some triangular microstrip patch antennas,”IEEE Trans. Antennas Propagat., vol. AP-31, pp. 27–34, Jan. 1983.

[7] J. S. Dahele and K. F. Lee, “On the resonant frequencies of the trian-gular patch antenna,” IEEE Trans. Antennas Propagat., vol. AP-35, pp.100–101, Jan. 1987.

[8] R. Garg and S. A. Long, “An improved formula for the resonant fre-quency of the triangular microstrip patch antenna,” IEEE Trans. An-tennas Propagat., vol. AP-36, p. 570, Apr. 1988.

[9] K. F. Lee, K. M. Luk, and J. S. Dahele, “Characteristics of the equilateraltriangular patch antenna,” IEEE Trans. Antennas Propagat., vol. AP-36,pp. 1510–1518, Nov. 1988.

[10] X. Gang, “On the resonant frequencies of microstrip antennas,” IEEETrans. Antennas Propagat., vol. 37, pp. 245–247, Feb. 1989.

[11] R. Singh, A. De, and R. S. Yadava, “Comments on an improved formulafor the resonant frequency of the triangular microstrip patch antenna,”IEEE Trans. Antennas Propagat., vol. 39, pp. 1443–1444, Sept. 1991.

[12] W. Chen, K. F. Lee, and J. Dahele, “Theoretical and experimental studiesof the resonant frequencies of equilateral triangular microstrip antenna,”IEEE Trans. Antennas Propagat., vol. 40, pp. 1253–1256, Oct. 1992.

[13] N. Kumprasert and K. W. Kiranon, “Simple and accurate formula forthe resonant frequency of the equilateral triangular microstrip patch an-tenna,” IEEE Trans. Antennas Propagat., vol. 42, pp. 1178–1179, Aug.1994.

[14] K. Güney, “Resonant frequency of a triangular microstrip antenna,” Mi-crowave Opt. Technol. Lett., vol. 6, pp. 555–557, July 1993.

[15] , “Comments on ‘on the resonant frequencies of microstrip an-tennas’,” IEEE Trans. Antennas Propagat., vol. 42, pp. 1363–1365, Sept.1994.

[16] P. Mythili and A. Das, “Comments on ‘simple and accurate formula forthe resonant frequency of the equilateral triangular microstrip patch an-tenna’,” IEEE Trans. Antennas Propagat., vol. 48, p. 636, Jan. 2000.

[17] D. Karaboga, K. Güney, N. Karaboga, and A. Kaplan, “Simple and accu-rate effective side expression obtained by using a modified genetic algo-rithm for the resonant frequency of an equilateral triangular microstripantenna,” Int. J. Electron., vol. 83, pp. 99–108, Jan. 1997.

[18] J. P. Damiano et al., “Study of multilayer microstrip antennas with radi-ating elements of various geometry,” Proc. Inst. Elect. Eng., pt. H, vol.137, no. 3, pp. 163–170, 1990.

[19] H. R. Hassani and D. Mirshekar-Syahkal, “Analysis of triangular patchantennas including radome effects,” Proc. Inst. Elect. Eng., pt. H, vol.139, no. 3, pp. 251–256, 1992.

[20] D. Mirshekar-Syahkal and H. R. Hassani, “Characteristics of stackedrectangular and triangular patch antennas for dual band applications,” inProc. 8th Inst. Elect. Eng. Int. Conf. Antennas and Propagation, 1993,pp. 728–731.

[21] C. L. Tang and K. L. Wong, “A modified equilateral-triangular-ringmicrostrip antenna for circular polarization,” Microwave Opt. Technol.Lett., vol. 23, pp. 123–126, Oct. 1999.

[22] C. L. Tang, J. H. Lu, and K. L. Wong, “Novel dual-frequency and broad-band designs of single-feed slot-loaded equilateral-triangular microstripantennas,” IEEE Trans. Antennas Propagat., vol. 48, pp. 1048–1054,July 2000.

[23] J. H. Lu and K. L. Wong, “Single-feed circularly polarized equilateraltriangular microstrip antenna with a tuning stub,” IEEE Trans. AntennasPropagat., vol. 48, pp. 1869–1872, Dec. 2000.

[24] Y. W. Jang, “Characteristics of a large bandwidth rectangular mi-crostrip-fed inserted triangular patch in a circular slot antenna,”Microwave J., vol. 45, no. 5, pp. 288–298, May 2002.

[25] C. S. Gurel and E. Yazgan, “New computation of the resonant frequencyof a tunable equilateral triangular microstrip patch,” IEEE Trans. Mi-crowave Theory Tech., vol. 48, pp. 334–338, Mar. 2000.

[26] K. F. Lee, K. Y. Ho, and J. S. Dahele, “Circular disc microstrip antennawith an air gap,” IEEE Trans. Antennas Propagat., vol. 32, pp. 880–884,Aug. 1984.

[27] J. S. Dahele and K. F. Lee, “Theory and experiments on microstrip an-tennas with air gaps,” Proc. Inst. Elect. Eng., pt. H, vol. 132, no. 7, pp.455–460, Dec. 1985.

[28] F. Abboud, J. P. Damiano, and A. Papiernik, “A new model for cal-culating the input impedance of coax-fed circular microstrip antennaswith and without air gaps,” IEEE Trans. Antennas Propagat., vol. 38,pp. 1882–1885, Nov. 1990.

[29] I. Wolff and N. Knoppik, “Rectangular and circular microstrip disk ca-pacitors and resonators,” IEEE Trans. Microwave Theory Tech., vol. 22,pp. 857–864, Oct. 1974.

[30] D. Guha, “Resonant frequency of circular microstrip antennas with andwithout airgaps,” IEEE Trans. Antennas Propagat., vol. 49, pp. 55–59,Jan. 2001.

Effect of a Cavity Enclosure on the Resonant Frequency ofInverted Microstrip Circular Patch Antenna


Abstract—The inverted microstrip circular patch (IMCP) antenna hasbeen analyzed very recently by the present authors and the samemicrostripstructure enclosed in a cylindrical cavity has been employed by others todevelop various active integrated antennas. In this paper, the effect of thecylindrical enclosure in changing the resonant frequency of an IMCP hasbeen studied both theoretically and experimentally. The cavity-effect crit-ically depends on the enclosed patch and the cavity dimensions which arethoroughly investigated to help a designer in choosing the antenna parame-ters. An efficient analytical formulation is also proposed to predict accurateresonant frequencies of the cavity enclosed IMCPs and is verified with dif-ferent experimental data.

Index Terms—Cavity enclosed microstrip patch, inverted microstrippatch, microstrip antenna.

Manuscript received October 25, 2002; revised July 9, 2003. This work wassupported by the Center of Advanced Study in Radio Physics and Electronics,University of Calcutta.



0018-926X/04$20.00 © 2004 IEEE


I. INTRODUCTION

The inverted microstrip circular patch (IMCP) in a cavity enclosedgeometry has been employed in designing several active antenna mod-ules in recent years [1]–[5]. The air dielectric below the patch in the in-verted microstrips is advantageous from various aspects [1] and henceshould be an attractive candidate in exploring new integrated antennas.The cavity enclosure, in addition, gives the antenna a compact, minia-ture and rigid shape without degrading its gain or radiation charac-teristics [1]. But a prior knowledge of the operating frequency of thecavity enclosed patch becomes important in all these designs. That wasdone experimentally [1], [3], [6] for some patch and cavity dimensions.Approximate formulas without considering the cavity effect were alsoused to compute some theoretical data.

Very recently, an IMCP in conventional configuration [Fig. 1(a)] hasbeen analyzed [7] to predict its accurate resonant frequency. The calcu-lated values due to this [7] was inquisitively compared with the exper-iments [1], [6] with cavity enclosed IMCPs. The comparison yields aninteresting observation that the theory for open IMCP [7] closely agreesthe experiments of the cavity enclosed IMCPs [6] and significantly dif-fers from another measurement [1]. No theoretical or experimental re-sults are available to understand or estimate the effect of the cavitywall on the resonant behavior of an IMCP. This has been addressed inthis paper both theoretically and experimentally. The effect of a cylin-drical cavity in changing the resonant frequency of an IMCP has beenanalytically modeled. Some measurements have been carried out withidentical open and cavity enclosed prototype antennas to understandthe cavity-effect. The computed results are compared with some pre-vious as well as new measurements.

II. THEORETICAL CALCULATIONS

A cavity enclosed IMCP geometry as shown in Fig. 1(b) maybelooked upon as a conventional IMCP shown in Fig. 1(a), surrounded bya metallic cylinder of radius r. So the cavity effect may be accountedfor in terms of the affected fringing fields caused by the closeness ofthe cavity wall to the patch edge.

Let us start with the cavity resonator model of a conventional IMCPgeometry [Fig. 1(a)] where the fringing fields under the influence of thedielectric medium on top of the patch of radius a is modeled in termsof the effective patch radius ae derived as [7]

ae = a(1 + q) (1)

where q is a factor representing the fringing field effects and is ex-pressed as [7]

q = 1+ "�1re

4�a

h

� 1 +(0:37 + 0:63"re)

�1 � 1

4(1+" )�

4ha

+ 2:6 +2:9h

a

+(0:37 + 0:63"re)

�1 � 1

4 + 2:6ah

+ 2:9ha

: (2)

The parameter q solely depends on the patch radius to air-gap heightratio (a=h1) and the effective relative permittivity "re of the mediumbelow the patch. The dielectric medium of height h2 [Fig. 1(a)] abovethe patch actually causes the change in the fringing electric fields. Thiseffect is accounted for in terms of the effective relative permittivity "rederived in [7, eqs.(4)–(11)] as the functions of a=h1 and h2=h1 ratios.

The resonant frequency of the TM11 mode in an IMCP can be de-termined as [7]

fr =�11 � c

2�ae(3)

Fig. 1. Circular microstrip patch antenna: (a) inverted microstrip geometry,(b) inverted microstrip patch enclosed in a cylindrical cavity, (c) equivalentmicrostrip patch printed on a substrate with relative permittivity "r;eq .

TABLE ICOMPUTED RESONANT FREQUENCIES OF THE TM11MODE OF CAVITY

ENCLOSED IMCPS COMPARED WITH THE MEASUREMENTS OF

NAVARRO et al. [1] r = 6:35 mm; "r = 2:3; h1 = 1:5 mm;h2 = 1:524 mm; � = 2 mm

where �11 is the first zero of the derivative of the Bessel function oforder one (�11 = 1:841) and c is the velocity of electromagnetic wavesin free space.

Now, if the IMCP is enclosed by a cylindrical enclosure of radius r[Fig. 1(b)] and ae � r or ae � r, under that condition one can surmisethat the cylindrical enclosure effectively turns to a circular cavity ofheight h1 resonating at a frequency close to the fr determined as (3).Standard h1 value of microstrip structure results in h1=r� 1 and thusit satisfies the condition of h1=r < 2:03 for exciting TMz

010 mode ina circular cavity [8] resonating at

f =2:4049� c

2�rp"r;eq

: (4)


TABLE IIRESONANT FREQUENCIES OF THE TM11MODE OF IMCP ANTENNAS WITH AND WITHOUT CAVITY ENCLOSURE: COMPUTED RESULTS

COMPARED WITH NEW MEASUREMENTS r = 6:35 mm; h1 = 1:6 mm; � = 2 mm

The cavity resonant frequency (4) equating to (3) under the condition ofae = r (for ae � r or ae � r) yields an equivalent dielectric constantof the medium filling the cavity as "r;eq � 1:706. The simple deriva-tion under the given conditions is significant one in incorporating all thefringing field effects caused by the cavity as well as by the inverted sub-strate of the IMCP. Since the cavity model of a circular microstrip diskrepresents a magnetic wall circular cavity of identical parameters [9],the newly modeled circular cavity filled with "r;eq can be representedby a simplified equivalent microstrip structure as shown in Fig. 1(c)where a patch of radius a is printed on a microstrip substrate with di-electric constant "r;eq and thickness h1.

The resonant frequency of the equivalent antenna configuration ofFig. 1(c) now can be readily determined from the improved formula-tions proposed by one of the authors in [10].

III. RESULTS AND DISCUSSIONS

The proposed model of the cavity enclosed IMCP is valid for thelimited patch dimensions expressed as ae � r or ae � r. The mea-surements of different sets of antennas [1], [6] are compared with thecomputed results to verify the present formulation as well as to examinethe extent of its validity with reference to the relative dimensions ofthe cavity and the IMCP. The closeness between the cavity wall andthe patch with respect to the patch-ground plane separation may be ex-pressed as (r � ae)=h1. The cavity wall affects the fringing fields inthe IMCP under limited condition. This may be judged in terms of acavity factor defined as g = f(r� ae)=h1g=(ae=a), where ae=a ratiorepresents the fringing factor.

The measurements of several IMCPs enclosed in a cavity [1] arecompared with the present theory in Table I where the smallest patchdimension corresponds to maximum value of g = 0:43. The presentmodel shows excellent agreement with all the measurements with anaverage error of 0.35% except for a particular one. The discrepancy fora = 4:5 mm may be due to any experimental error. The calculatedvalues for the open IMCPs [7] incorporated in Table I differ signifi-cantly from those for cavity enclosed antennas with an average relativedeviation of�22%. Thus, it predicts significant effect of the cylindricalenclosure in changing the resonant behavior of the IMCPs.

Similar investigations were done experimentally by the present au-thors by fabricating some open type as well as cavity enclosed iden-tical patches printed on different sets of substrate materials. The reso-nant frequencies of the probe-fed antennas were determined from theminima on the return loss traces measured on HP 8720C network an-alyzer. About 3-mm thick, 70 mm � 70 mm ground plane was usedfor the measurements of open type geometry [7]. For cavity enclosedantenna, it was of same thickness and other parameters are given inTable II. SMA probes with 0.254-mm diameter were used to excite thepatches and the feed locations were maintained at about 2 mm from the

Fig. 2. Resonant frequency of IMCP antenna with and without cavityenclosure and corresponding cavity factor g versus patch radius. "r = 2:3,r = 30 mm, h1 = 1:43 mm, h2 = 1:57 mm, � optimized in eachmeasurement [6].

patch center. Experimental results of the new measurements are com-pared with the computed data in Table II. Here also the effect of thecavity is equally pronounced from the measured data showing approx-imately 19% change in fr when g = 0:18 and 0.33. Comparatively asmaller patch in the same cavity resulting in g � 0:59 is almost freefrom the cavity-effect as revealed from the experiment.

From the critical studies of the data in Tables I and II, it is apparentthat the effect of the cavity is pronounced for those antenna dimensionswhich result in g � 0:43. But with a slightly larger value like g � 0:59,the patch resonance is hardly affected by the cavity enclosure. This hasbeen examined in Fig. 2 using the measured results of some IMCPs ina 60-mm diameter cavity [6]. Theoretical resonant frequency and cal-culated g values of the measured antennas are also plotted against thepatch dimension. The g values for the measured patches [6] are �0.5.All the experimental values except a single one are found to followthe theoretical curve for open type IMCP [7]. The measured point ata = 27:5 mm corresponding to g = 0:5 shows much deviation fromthe open IMCP curve indicating the effect of the cavity on it. This mea-sured value also shows closeness to the curve due to the present formu-lation for cavity enclosed IMCP. Thus it is apparent from the studiesthat the cavity enclosed IMCPs with g � 0:5 are under the influence ofthe cavity where the present formulas can be efficiently used to com-pute their dominant mode resonant frequencies.

Fig. 3 shows the effect of the height h1 in a cavity in changing thecavity factors g and r=ae and also the resonant frequency of a cavity en-closed IMCP on the basis of computed data. A theoretical curve for theresonant frequency of an identical IMCP without cavity is also included


Fig. 3. Resonant frequency of IMCP antenna with and without cavityenclosure and corresponding cavity factors g and r=aeversus h1. r =

6:35 mm, h2 = 1:524 mm, "r = 2:3, a = 4:0 mm.

for comparison. The increase in h1 causes decrease in both r=ae andg values revealing the increasing cavity-effect on the IMCP. The reso-nant frequency also decreases with the increase in h1 for both cavityenclosed and open type IMCPs. However, the effect of the cavity enclo-sure in diminishing the resonant frequency is significant over a consid-erable range of h1 values, particularly at the lower values of h1. Onlya single measured value is available [1] to compare with the theoreticalcurve in Fig. 3 and this value shown by a solid circle corresponds tog < 0:5. Excellent agreement between the experiment and the presenttheory is also revealed.

The present studies show that a cavity enclosure influences the res-onance of an IMCP under limited conditions resulting in lower reso-nant frequency. However, its effect on the radiation characteristics ofan IMCP is not significant to that extent as reported in [1]. Uniformradiation patterns with acceptable cross polarization level and 6.65 dBigain were reported [1] from the measurements of a typical cavity en-closed probe fed IMCP antenna.

IV. CONCLUSION

The inverted microstrip patches in open or cavity enclosed config-uration are attractive for integrating with active devices below it. Thechange in resonance condition under the influence of a cylindricalcavity enclosing an IMCP has been thoroughly investigated boththeoretically and experimentally. If different patches are examined ina same cylindrical enclosure, all are not affected by the cavity anda factor determining that dependence has been examined with thehelp of the new and previous data. A simple analytical formulationis proposed to predict the resonant frequency of cavity enclosedpatches so far those are under the influence of the cavity. The theoryis compared with different measurements resulting in very closeapproximation between them. The studies in this paper thus shouldguide a designer in choosing proper cavity and IMCP dimensions fora specified frequency within approximately 0.5% accuracy.

ACKNOWLEDGMENT

The authors would like to thank Dr. P. K. Saha and Dr. P. K. Basuof the University of Calcutta for their interest in the work and constantencouragement.

REFERENCES

[1] J. A. Navarro, L. Fan, and K. Chang, “Active inverted stripline circularpatch antennas for spatial power combining,” IEEE Trans MicrowaveTheory Tech., vol. 41, pp. 1856–1863, Oct. 1993.

[2] , “Novel FET integrated inverted stripline patch,” Electron. Lett.,vol. 30, no. 8, pp. 655–657, 1994.

[3] R. A. Flynt, L. Fan, J. A. Navarro, and K. Chang, “Low cost and com-pact active integrated antenna transceiver for system applications,” IEEETrans. Microwave Theory Tech., vol. 44, pp. 1642–1649, Oct. 1996.

[4] C. M. Montiel, L. Fan, and K. Chang, “A novel active antenna with self-mixing and wideband varactor-tuning capabilities for communicationand vehicle identification applications,” IEEE Trans Microwave TheoryTech., vol. 44, pp. 2421–2430, Dec. 1996.

[5] J. A. Navarro and K. Chang, “Active microstrip antenna,” in Advancesin Microstrip and Printed Antennas, K. F. Lee and W. Chen, Eds. NewYork: Wiley, 1997, ch. 8.

[6] J. A. Navarro, J. McSpadden, and K. Chang, “Experimental study ofinverted microstrip for integrated antennas applications,” in IEEE An-tennas Propagat. Int. Symp. Proc., Seattle, WA, 1994, pp. 920–923.

[7] D. Guha and J. Y. Siddiqui, “A new CAD model to calculate theresonant frequency of inverted microstrip circular patch antenna,”Microwave Opt. Technol. Lett., vol. 35, no. 6, Dec. 20, 2002.

[8] C. A. Balanis, Advanced Engineering Electromagnetics. New York:Wiley, 1989, ch. 9.

[9] I. Wolff and N. Knoppik, “Rectangular and circular microstrip disk ca-pacitors and resonators,” IEEE Trans Microwave Theory Tech, vol. 22,pp. 857–864, Oct. 1974.

[10] D. Guha, “Resonant frequency of circular microstrip antennas with andwithout air gaps,” IEEE Trans. Antennas Propaga., vol. 49, pp. 55–59,Jan. 2001.

Design and Development of Multiband Coaxial ContinuousTransverse Stub (CTS) Antenna Arrays

Robert Isom, Magdy F. Iskander, Zhengqing Yun, and Zhijun Zhang

Abstract—Continuous transverse stub (CTS) technology has beenadapted to use with coaxial lines to produce effective microwave an-tenna structures that radiate omnidirectionally, with high efficiency, lowreflection, and useful radiation patterns. In this paper, we describe thedesign, construction, and testing of a new type of antenna arrays, that is, asix-element multiband (4.2 and 19.4 GHz) CTS antenna array. The designof the CTS array was optimized through simulation using finite-differencetime-domain and then built and tested using both S-parameters andradiation pattern measurements. Simulation results agreed very well withmeasured data. These simple and low cost coaxial CTS structures couldbe adapted for base station applications in wireless communication, forsatellite communication systems, and Identification Friend-or-Foe systemsfor the military.

Index Terms—Antenna array, coaxial continuous transverse stub (CTS),continuous transverse stub (CTS), multiband array.

I. INTRODUCTION

Continuous transverse stub (CTS) technology developed in the early1990s at Hughes Aircraft Company [1] has attracted research attentionsrecently [2], [3]. This technology offers advantages over traditional ap-proaches to antenna design at microwave frequencies. Benefits of CTS

Manuscript received May 1, 2003; revised October 21, 2003.R. Isom is with the Antenna Department, Raytheon Systems, M/S 8019,

McKinney, TX 75071 USA.M. F. Iskander, Z. Yun, and Z. Zhang are with the Hawaii Center for Advanced

Communication, College of Engineering, University of Hawaii, Honolulu, HI96822 USA (e-mail: [email protected])


0018-926X/04$20.00 © 2004 IEEE


Fig. 3. Resonant frequency of IMCP antenna with and without cavityenclosure and corresponding cavity factors g and r=aeversus h1. r =

6:35 mm, h2 = 1:524 mm, "r = 2:3, a = 4:0 mm.

for comparison. The increase in h1 causes decrease in both r=ae andg values revealing the increasing cavity-effect on the IMCP. The reso-nant frequency also decreases with the increase in h1 for both cavityenclosed and open type IMCPs. However, the effect of the cavity enclo-sure in diminishing the resonant frequency is significant over a consid-erable range of h1 values, particularly at the lower values of h1. Onlya single measured value is available [1] to compare with the theoreticalcurve in Fig. 3 and this value shown by a solid circle corresponds tog < 0:5. Excellent agreement between the experiment and the presenttheory is also revealed.

The present studies show that a cavity enclosure influences the res-onance of an IMCP under limited conditions resulting in lower reso-nant frequency. However, its effect on the radiation characteristics ofan IMCP is not significant to that extent as reported in [1]. Uniformradiation patterns with acceptable cross polarization level and 6.65 dBigain were reported [1] from the measurements of a typical cavity en-closed probe fed IMCP antenna.

IV. CONCLUSION

The inverted microstrip patches in open or cavity enclosed config-uration are attractive for integrating with active devices below it. Thechange in resonance condition under the influence of a cylindricalcavity enclosing an IMCP has been thoroughly investigated boththeoretically and experimentally. If different patches are examined ina same cylindrical enclosure, all are not affected by the cavity anda factor determining that dependence has been examined with thehelp of the new and previous data. A simple analytical formulationis proposed to predict the resonant frequency of cavity enclosedpatches so far those are under the influence of the cavity. The theoryis compared with different measurements resulting in very closeapproximation between them. The studies in this paper thus shouldguide a designer in choosing proper cavity and IMCP dimensions fora specified frequency within approximately 0.5% accuracy.

ACKNOWLEDGMENT

The authors would like to thank Dr. P. K. Saha and Dr. P. K. Basuof the University of Calcutta for their interest in the work and constantencouragement.

REFERENCES

[1] J. A. Navarro, L. Fan, and K. Chang, “Active inverted stripline circularpatch antennas for spatial power combining,” IEEE Trans MicrowaveTheory Tech., vol. 41, pp. 1856–1863, Oct. 1993.

[2] , “Novel FET integrated inverted stripline patch,” Electron. Lett.,vol. 30, no. 8, pp. 655–657, 1994.

[3] R. A. Flynt, L. Fan, J. A. Navarro, and K. Chang, “Low cost and com-pact active integrated antenna transceiver for system applications,” IEEETrans. Microwave Theory Tech., vol. 44, pp. 1642–1649, Oct. 1996.

[4] C. M. Montiel, L. Fan, and K. Chang, “A novel active antenna with self-mixing and wideband varactor-tuning capabilities for communicationand vehicle identification applications,” IEEE Trans Microwave TheoryTech., vol. 44, pp. 2421–2430, Dec. 1996.

[5] J. A. Navarro and K. Chang, “Active microstrip antenna,” in Advancesin Microstrip and Printed Antennas, K. F. Lee andW. Chen, Eds. NewYork: Wiley, 1997, ch. 8.

[6] J. A. Navarro, J. McSpadden, and K. Chang, “Experimental study ofinverted microstrip for integrated antennas applications,” in IEEE An-tennas Propagat. Int. Symp. Proc., Seattle, WA, 1994, pp. 920–923.

[7] D. Guha and J. Y. Siddiqui, “A new CAD model to calculate theresonant frequency of inverted microstrip circular patch antenna,”Microwave Opt. Technol. Lett., vol. 35, no. 6, Dec. 20, 2002.

[8] C. A. Balanis, Advanced Engineering Electromagnetics. New York:Wiley, 1989, ch. 9.

[9] I. Wolff and N. Knoppik, “Rectangular and circular microstrip disk ca-pacitors and resonators,” IEEE Trans Microwave Theory Tech, vol. 22,pp. 857–864, Oct. 1974.

[10] D. Guha, “Resonant frequency of circular microstrip antennas with andwithout air gaps,” IEEE Trans. Antennas Propaga., vol. 49, pp. 55–59,Jan. 2001.

Design and Development of Multiband Coaxial ContinuousTransverse Stub (CTS) Antenna Arrays

Robert Isom, Magdy F. Iskander, Zhengqing Yun, and Zhijun Zhang

Abstract—Continuous transverse stub (CTS) technology has beenadapted to use with coaxial lines to produce effective microwave an-tenna structures that radiate omnidirectionally, with high efficiency, lowreflection, and useful radiation patterns. In this paper, we describe thedesign, construction, and testing of a new type of antenna arrays, that is, asix-element multiband (4.2 and 19.4 GHz) CTS antenna array. The designof the CTS array was optimized through simulation using finite-differencetime-domain and then built and tested using both S-parameters andradiation pattern measurements. Simulation results agreed very well withmeasured data. These simple and low cost coaxial CTS structures couldbe adapted for base station applications in wireless communication, forsatellite communication systems, and Identification Friend-or-Foe systemsfor the military.

Index Terms—Antenna array, coaxial continuous transverse stub (CTS),continuous transverse stub (CTS), multiband array.

I. INTRODUCTION

Continuous transverse stub (CTS) technology developed in the early1990s at Hughes Aircraft Company [1] has attracted research attentionsrecently [2], [3]. This technology offers advantages over traditional ap-proaches to antenna design at microwave frequencies. Benefits of CTS

Manuscript received May 1, 2003; revised October 21, 2003.R. Isom is with the Antenna Department, Raytheon Systems, M/S 8019,

McKinney, TX 75071 USA.M. F. Iskander, Z. Yun, and Z. Zhang are with the Hawaii Center for Advanced

Communication, College of Engineering, University of Hawaii, Honolulu, HI96822 USA (e-mail: [email protected])


0018-926X/04$20.00 © 2004 IEEE


Fig. 1. Schematic of a six-element multiband coaxial CTS antenna array. Firstset of stubs from input are high-frequency stubs. Second set of stubs from inputare low-frequency stubs.

structures include compact size, lightweight, low loss, and high direc-tivity. Parallel-plate CTS arrays have achieved average gains of 39.7 dBover a bandwidth of 37–40 GHz [2]. This high gain is achieved alongwith relative dimensional insensitivity, thus reducing fabrication costs.

CTS technology also offers greater tunable bandwidth than wave-guide or patch antennas, higher efficiencies, and polarization isolationof 25–50 dB. It is desired to apply the advantages of the presently avail-able planar CTS technology to a new design that incorporates coaxialgeometries. Such a process has been successfully implemented andsome of the obtained results for a new coaxial CTS antenna were re-ported elsewhere [4], [5].

Coaxial CTS structures offer several additional advantages overplanar CTS. They provide an omni-directional radiation pattern in theplane of the radiating stubs (perpendicular to the transmission line)as there is no azimuthal dependence in the designed cylindrical stubs.Secondly, coaxial structures are inherently easier to impedance match,thus providing higher efficiency and facilitate system integration withother coaxial structures.

In this paper, a two-band coaxial CTS is designed by simulating dif-ferent structures using the two-dimensional finite-difference time-do-main (2-D-FDTD) code. A prototype antenna is then constructed andtested. The measured S-parameters and radiation patterns agree wellwith the simulated results.

II. COAXIAL CTS DESIGN PROCEDURE

Design procedures for a coaxial CTS array include choosing the fol-lowing parameters for both of the high- and low-frequency sections (seeFig. 1): 1) width of stub segment: L1; 2) length of transmission line be-tween stubs: L2; 3) dielectric constant of fill material: "r; 4) diameter ofinner conductor: D1; 5) diameter of outer conductor: D2; and 6) diam-eter of radial stub: D3. To help characterize the antenna performanceas we vary some or all of these design parameters, a 2-D axially sym-metric cylindrical FDTD code was used to simulate the performanceand characterize the many tradeoffs involved in the design of these an-tennas. For the arrays we designed in this paper, we first designed twoseparated three-element arrays one at the lower frequency (4.2 GHz)while the other was designed at 19.4 GHz. The two sections were thenjoined and the entire six-element array was optimized to achieve thedesired S-parameters and radiation pattern characteristics. The designof each of the two sections follows a procedure similar to that describedin an earlier paper [5], and the discussion in this paper will focus on thedesign of the six-element multiband array.

III. DESIGN OF A MULTIBAND COAXIAL CTS ANTENNA ARRAY

The multiband array is a six-element coaxial CTS array with threeelements designed to operate at 4.2 GHz and the other three elements at

Fig. 2. Photograph of fabricated multiband six-element coaxial CTS antennaarray designed to operate at 4.2 and 19.4 GHz: angled view.

19.4 GHz. This array was designed to provide low reflection, high-ra-diation efficiency, and a broadside radiation pattern at two frequencies,4.2 GHz (C-band) and 19.4 GHz (K-band). In designing this array, weused dielectric loading (Teflon and polyethylene) to improve the per-formance and reduce the overall size of the array.

As mentioned earlier, two sets of stubs were designed: one that wasnonradiating (full transmission) at lower frequencies and radiates ef-fectively at higher frequencies (high-frequency stubs), and a second setof stubs that radiate at the lower frequency (low-frequency stubs). Thetwo sets need to be arranged in tandem with the high-frequency stubsnear the input. This way, high-frequency signals would radiate fromthe high-frequency stubs before reaching the low-frequency stubs atthe end, and the low-frequency signal, on the other hand, would effi-ciently pass through the high-frequency stubs connected to the input,and the input low-frequency signal will radiate when reaching the low-frequency stubs at the end of the array.

Toaccomplish this, itwasnecessary toknow theelectrical dimensionsthat would produce full transmission for the high-frequency stubs at thelower frequency. For this purpose, structures with narrow gaps and longstubdiameterswere simulated. Itwas found that narrowgap (� �) stubswith a stub diameter of approximately �=3 (12 mm at 8 GHz) or longerwillproduceclose to full transmissionat the lower frequencies.Athigherfrequencies, however, the same physical dimensions of the stubs werefoundtoradiateefficiently.It is thereforepossibletodesignacoaxialCTSstructure thatproducesefficient radiationatahighfrequencyandappearsalmost transparent at lower frequencies.

Using this approach, we simulated a coaxial CTS array with multi-band performance. The designed array was dielectrically loaded withTeflon and polyethylene to help improve impedance matching and re-duce the size. Further performance enhancements for the multibandarray were achieved by controlling design parameters such as the stubgap, stub height, and stub spacing of both the high-and low-frequencystubs. The entire multiband array was further optimized to account forthe mutual coupling effects.

Following a manual optimization using inhouse FDTD codes [5],a multiband coaxial CTS antenna array design was achieved thatproduced low reflection and good radiation characteristics at both theupper and lower frequency bands. The dimensions of the high-fre-quency stubs were L1 = 5:2 mm, L2 = 3:8 mm, D1 = 1:12 mm,D2 = 3:6 mm, and D3 = 44:4 mm. The dimensions of the low-fre-quency stubs were L1 = 18:9 mm, L2 = 29:6 mm, D1 = 1:12 mm,D2 = 3:6 mm, and D3 = 61:2 mm. The spacing between the lasthigh-frequency stub and the first low-frequency stub in the tandemconnection of the two arrays was 21.6 mm. The radial waveguidestubs were dielectrically loaded with Teflon rings. The coaxial line ofthe multiband array was filled with polyethylene. The Teflon (stubs)and polyethylene (coaxial line) were both simulated with "r = 2:2and tan � � 0 (negligible dielectric losses). Fig. 1 shows a schematicof the designed multiband six-element array, and Fig. 2 shows aphotograph of the fabricated and tested array.


Fig. 3. Comparison of simulated (solid with +) and measured (solid) S-parameter performance for six-element multiband coaxial CTS antenna array. Designfrequencies are 4.2 GHz (C-band) and 19.4 GHz (K-band).

Following fabrication of the six-element multiband coaxial CTS an-tenna array, the physical dimensions of the structure weremeasured andfound to be in accord with the design parameters. There was a slightair gap between the machined Teflon rings and the polyethylene-filledcoaxial transmission line at the base of the stub. A new FDTD modelthat matched these measured physical dimensions and included the de-viations from the initial design was then simulated. Simulations resultswere then compared with the measured S-parameters and radiation pat-terns for the fabricated prototype array.

Measurements of the coaxial CTS array were taken using the HP8510B Network Analyzer. The S-parameters were characterized across3–20 GHz band. These experimental results are shown along with thesimulation S-parameter results obtained using FDTD in Fig. 3. As maybe noted, there is a good agreement of resonant frequency performancebetweenmeasurement and simulations. Themultiband array was foundto possess S11 = �33 dB and 98% radiated power (S21 < �20 dB) inthe lower band at 4.2 GHz. The multiband array was found to possessS11 over �60 dB and 98% radiated power in the upper band at 19.4GHz. The 10-dB bandwidth (where S11 < �10 dB,VSWR < 2) wasfound to be 6% (� 250MHz) in lower band and 12% (� 2:2 GHz) inthe upper band.

The radiation pattern of the six-element multiband coaxial CTS an-tenna array was also measured at 4.2 GHz (see Fig. 4). The measuredradiation pattern produced broadside patterns, useful for many appli-cations. There was a slight rotation in the radiation pattern away fromthe source (� = 180

�). This deviation from the simulated results wassmall and did not change the broadside nature of the pattern or producesignificant changes in the side lobe levels. The radiation pattern at 19.4GHz was also measured in anechoic chamber (see Fig. 5) and foundto be in agreement with simulation results. Increased side lobe levelsin the 19.4 GHz pattern are seen toward endfire (at 180�) due to theeffect of the feed cable and connectors. This effect was not significantenough to change the main broadside nature of the pattern or the max-imum side lobe level as shown in Fig. 5.

Good agreement between simulated and measured results was seenfor both S-parameter and radiation performance. For both simulatedand measured results, the multiband coaxial CTS antenna array wasseen to produce low reflection (S11 < �20 dB) and good radiation

Fig. 4. Comparison of measured (+s) and simulated (solid) radiation patternat 4.2 GHz for the six-element multiband coaxial CTS antenna array. Feed isfrom 0 . Displayed on a dB-scale normalized to 0 dB.

characteristics (98% power radiated) at both 4.2 and 19.4 GHz. Thisvalidates the multiband design and illustrates the usefulness of the two-dimensional FDTD axially symmetric code as a design tool for coaxialCTS structures.

IV. TOLERANCE ANALYSIS

Planar CTS technology allows for relatively large tolerances in fab-rication without significant alteration in the antenna performance. Thisdimensional insensitivity is critical to achieving low cost fabrication.It was important to determine if coaxial CTS structures possessed thesame relative dimensional insensitivity.

Small changes were made in the simulation models, representing de-viations from the design dimensions, and FDTD simulations were runwith these changes included both as increases and decreases from thedesign dimensions. These changes were performed on all physical di-mensions at the same time. S-parameter results were compared to orig-inal design results. It was found that even by allowing for a large tol-erance as 5–10 mils, coaxial CTS structures maintained their antenna


Fig. 5. Comparison of measured (+s) and simulated (solid) radiation patternat 19.4 GHz for the six-element multiband coaxial CTS antenna array. Feed isfrom 180 . Displayed on a dB-scale normalized to 0 dB.

Fig. 6. Plot showing change in S11 for small changes in the physicaldimensions of the six-element multiband coaxial CTS array. The solid linerepresents the original simulation. The +s represent a decrease in size fromthe original dimensions. TheXs represent an increase in size from the originaldimensions.

performance and impedance matching characteristics at the design fre-quencies.

Specifically, FDTD simulation tests were run on three cases for thesix-element multiband array. These cases consisted of simulation at thedesign specifications, simulation at dimensions one cell size (.129 mmor 5 mils) larger than the design specifications, and simulation at di-mensions one cell size smaller than the design specifications. The re-sults of the simulations for S11 are shown in Fig. 6. These results showthat despite some small variation in the S-parameter performance ofthe array, its overall antenna performance was maintained in the re-gions around the design frequencies.

We also simulated the effect of small variations in the dielectric con-stant of the Teflon-filledmultiband antenna array on the antenna perfor-mance. A simulation was performed using the expected value of Teflon("r = 2:1), while two others were made using somewhat smaller andlarger values for Teflon ("r = 2:0 and 2.2, respectively). The vari-ations represent a change in the dielectric constant of approximately5%. The results of these simulations are shown in Fig. 7. As it may be

Fig. 7. Plot showing change in S11for small changes in the dielectric constantof the six-element coaxial CTS array. The solid line represents "r = 2:1. The+s represent "r = 2:0. The Xs represent "r = 2:2.

seen, coaxial CTS array is also robust and relatively insensitive to smallchanges in the dielectric constant of filling material.

V. CONCLUSION

Coaxial CTS technology provides low cost, high-efficiency antennaarrays and with excellent radiation and S-parameter characteristics.The radial stubs of coaxial CTS arrays provide an omni-directional pat-tern in the plane perpendicular to the coaxial line. Coaxial CTS struc-tures also possess the added advantages of ease of impedancematching,ease of feed, and maintains the insensibility to dimensional and dielec-tric constant tolerances.

In this paper, the design, construction and testing of a multiband an-tenna array was described. S-parameter performance from 3–20 GHzwas measured using the HP8510BNetwork Analyzer. Good agreementwas achieved between simulated S-parameter results obtained usingFDTD and measured results, particularly at the desired multiband fre-quencies of 4.2 and 19.4 GHz. Themeasured and simulated results con-firmed that it was possible to obtain low reflection, high-radiation effi-ciency, and good radiation pattern at frequencies in two different bands(C-band and K-band).

Radiation pattern measurements were also taken at 4.2 and 19.4 GHzand the broadside radiation pattern characteristic was verified. Severalsimulations were also performed to confirm the dimensional tolerance,and hence emphasize the low cost fabrication advantage of the coaxialCTS antenna design.

The excellent radiation performance shown by the coaxial CTS an-tenna arrays would enable many high-frequency communication ap-plications for both military and commercial use. In particular, suchhigh-efficiency antenna arrays would be useful for close range wire-less connectivity (e.g., Bluetooth). Additionally, this technology haspotential military application (e.g., Identification Friend-or-Foe (IFF)systems). They are compact in size and lightweight. Potentially, beam-steering capability can be accomplished in the same manner as planarCTS designs, either mechanically or using Ferroelectric materials.

REFERENCES

[1] W. W. Milroy, “The continuous transverse stub (CTS) array: Basictheory, experiment and application,” in Proc. Antenna ApplicationsSymp., Allerton Park, IL, Sept. 25-27, 1991.


[2] , “Continuous transverse stub element devices and methods ofmaking same,” U.S. Patent 5 266 961, Aug. 29, 1991.

[3] M. F. Iskander, Z. Yun, Z. Zhang, R. Jensen, and S. Redd, “Design ofa low-cost 2-D beem-steering antenna using ferroelectric material andthe CTS technology,” IEEE Trans. Microwave Theory Tech., vol. 49, pp.1000–1003, May 2001.

[4] Z. Zhang, M. F. Iskander, and Z. Yun, “Coaxial continuous transversestub element device antenna array and filter,” U.S. Patent 6 201 509, Nov.5, 1999.

[5] M. F. Iskander, Z. Zhang, Z. Yun, and R. Isom, “Coaxial continuoustransverse stub (CTS) array,” IEEE Microwave Wireless ComponentLett., pp. 489–491, Dec. 2001.

Near-Field, Spherical-Scanning Antenna MeasurementsWith Nonideal Probe Locations

Ronald C. Wittmann, Bradley K. Alpert, and Michael H. Francis

Abstract—We introduce a near-field, spherical-scanning algorithm forantenna measurements that relaxes the usual condition requiring datapoints to be on a regular spherical grid. Computational complexity is ofthe same order as for the standard (ideal-positioning) spherical-scanningtechnique. The new procedure has been tested extensively.

Index Terms—Near-field measurements, probe-position correction,spherical scanning.

I. INTRODUCTION

As frequency increases and wavelength decreases, it becomesdifficult to maintain mechanical tolerances in near-field scanningantenna measurements. Therefore, the paradigm shifts from takingmeasurements at predefined locations to accurately determining thepositions where measurements are actually made. Standard algo-rithms for transformation from near-field to far-field require thatdata points lie on a regular grid. Our goal is to relax this conditionwithout increasing computational complexity or sacrificing accuracy.Previously, we have dealt with planar near-field scanning [1]. Here,we turn our attention to spherical near-field scanning [2] . Althoughthe details are different, the basic approach is the same: The techniquerelies on efficient linear transformation between spherical-modecoefficients and probe response at actual measurement locations. Theconjugate-gradient method is applied to determine the coefficientsthat produce a weighted-least-squares match to the measured proberesponse. In the following, we sketch the theory and demonstrate thealgorithm through numerical simulation.

II. THEORY

The electric field of an antenna, operating at frequency f = !=2�,may be expressed as an expansion of spherical waves

E(r) =

N

n=1

n

m=�n

b1nmmnm(r) + b2nmnnm(r) (1)

where the vector modal (Hansen) functions mnm and nnm are de-scribed in [3, Ch. 7], for example, and the coefficients b1nm and b2nmcompletely characterize the radiated electromagnetic fields of the test

Manuscript received August 19, 2003.The authors are with the National Institute of Standards and Technology,

Boulder, CO 80305 USA (e-mail: [email protected]).Digital Object Identifier 10.1109/TAP.2004.832316

Fig. 1. Spherical scanning geometry.

antenna. The time-dependent factor exp(�i!t) has been suppressed.Equation (1) is valid in free space outside the minimum sphere; that is,outside the smallest sphere centered on the coordinate origin that en-closes the radiating structure. The summation over nmust be truncatedfor practical reasons. Normally, it suffices to chooseN � ka, where ais the radius of the minimum sphere and k = 2�=� = !=c.

In spherical scanning, the probe is effectively moved over the sur-face of a sphere of radius r so that it always “points” in the radial di-rection �r. As shown in Fig. 1, the probe’s position and orientation isdescribed by the Euler angles ('; �; �), where � and ' are the usualspherical-coordinate angles that define the location of the probe. Theangle � measures rotation of the probe about r. We assume that thereceiving pattern is broad enough that small pointing errors are unim-portant. Although possible in principle, correcting for probe wobblewould be costly in terms of measurement and processing time.

In order to simplify the collection and processing of measurementdata, we follow common practice [4] and restrict our attention to specialprobes that have a response w(r; '; �; �) with a simple � dependence

w(r; '; �; �) = �w(r; '; �;��=2) sin�+ w(r; '; �; 0) cos�:

(2)

Such probes are not difficult to construct [5, Ch. 1]. They are called� = �1 probes for reasons that may not be clear in this context. Anexample is a probe that measures transverse components of the electricfield with, say, E' = w(r; '; �; 0) and E� = w(r; '; �;��=2). Anypractical probe will approach a � = �1 probe as r is increased.

The Jensen transmission formula [4] expresses the measurementvector w(r) as an expansion in spherical harmonics

w(r) � w(r; '; �;��=2)�� + w(r; '; �; 0)'''

=nm

[B1

nm(kr)Xnm(r) +B2

nm(kr)Ynm(r) ] (3)

where

B1

nm

B2

nm

=Mn

b1nmb2nm

: (4)

In (4), Mn is a known 2� 2 matrix depending on r and the probereceiving function. The vector spherical harmonicsXnm andYnm =ir �Xnm are defined by Jackson [6].

Using (2), (3), and (4), we write

W = Ab: (5)

U.S. Government work not protected by U.S. copyright.


Fig. 2. Simulated far-field pattern of a maximum directivity antenna forN =128 (broken line) and the far-field pattern obtained ignoring probe-positionerrors, Ir = I� = I� = 3.

The vectorW contains the probe responsew(ri; 'i; �i; �i) at the mea-surement points, b is a vector of (unknown) expansion coefficients,and A is a (known) linear operator. Equation (5) is generally overde-termined (with more measurements than unknowns), so we solve thenormal equations

A��W =Mb (6)

where

M = A��A (7)

to obtain a least-squares estimate for b. HereA� is the Hermitian ad-joint (conjugate transpose) ofA, and �� is a diagonal matrix of positiveweights to be associated with the data points. When measurements arenominally located on an equispaced grid we choose �ii = j sin �ij tocompensate for oversampling near the poles. (The choice of �� can sig-nificantly affect the condition number of the operatorM.)

We solve the system (6) using a conjugate-gradient algorithm. Thisrequires one application of the operatorM per iteration.Our implemen-tation involvesanunequally spaced fastFourier transformation [7], [8] in� and' and interpolation in r. Because the expansion is bandlimited, theaccuracyof this interpolationcanbecontrolled.Thecomputationalcom-plexityof each iteration isO(N3),whichagreeswith thestandardspher-ical near-field scanning technique. The procedure is completely analo-gous to the planar near-field scanning case (see [1] for more detail).

III. VERIFICATION

To test the effectiveness of the position-correction algorithm, weconsider the transmitting pattern of a maximum-directivity antennawith a directivity of about 42 dB, corresponding to N = 128 ([4, p.55]). The amplitude of this pattern, as a function of �, is shown in Fig.2. Standard spherical near-field measurement techniques require mea-surements on a regular grid in � and ' with

��;�' �360�

2N + 1= �: (8)

Position errors are simulated by evaluating the probe response at thepoints

(ri; �i; 'i) = (rp; �pi; 'pi) + ("ri�; "�i�; "'i�) (9)where (rp; �pi; 'pi) is a nominal measurement point and "xi isa sample from a uniform probability distribution on the interval[�Ix; Ix].

Fig. 3. Original pattern (broken line) and the difference between the computedand original patterns after 1 iteration, Ir = I� = I� = 3.

Fig. 4. Original pattern (broken line) and the difference between the computedand original patterns after 15 iterations, Ir = I� = I� = 3.

Fig. 5. Original pattern (broken line) and the difference between the computedand original patterns after 56 iterations, Ir = I� = I� = 3.

First, consider the case Ir = I� = I' = 3. In practice, we striveto limit probe-position errors to �=50 or less, so allowing radial errors


Fig. 6. Residual as a function of iteration number, Ir = I� = I� = 3.

up to 3� may be regarded as extreme. Fig. 2 shows the far-field patternobtained if the data are processed without position correction; that is,assuming that the measurement points lie on the ideal grid. This patternbears no resemblance to the true pattern. Figs. 3 –5 show the differencebetween the computed pattern and the original after 1, 15, and 56 it-erations. (Execution time was about 3 min per iteration on a 2 GHzpersonal computer.) At 15 iterations (Fig. 4), errors are about 1 dB at asidelobe level 40 dB below peak, which is usually adequate in practice.After 56 iterations (Fig. 5) the residual

" = kA��W�Mbk=kA��Wk (10)

is less than 10�5. Fig. 6 shows the residual as a function of iterationnumber.

With less severe probe-position errors, fewer iterations are required.For example, when Ir = I� = I� = 0:25 (still extreme relative todesired tolerances), only 13 iterations are required to reach " = 10�5.

IV. SUMMARY

We have developed a spherical near-field scanning algorithm thatdoes not require data to be measured on a regular grid. Computationalcomplexity for probe-position correction is of the same order as for thestandard (ideal-positioning) spherical-scanning algorithm. The new al-gorithm is robust: we have successfully tested it for transverse probe-position deviations of up to 3 maximum sample intervals (�), and forradial deviations of up to three wavelengths (�). The software is avail-able from the authors.

REFERENCES

[1] R. C. Wittmann, B. K. Alpert, and M. H. Francis, “Near-field antennameasurements using nonideal measurement locations,” IEEE Trans. An-tennas Propagat., vol. 46, pp. 716–722, May 1998.

[2] , “Spherical near-field antenna measurements using nonidealmeasurement locations,” in Proc. Antenna Measurements Tech. Assoc.,Cleveland, OH, Nov. 3–8, 2002, pp. 43–48.

[3] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill,1941.

[4] J. E. Hansen, Ed., Spherical Near-Field Antenna Measurements,London, U.K.: Peregrinus, 1988.

[5] R. C. Wittmann and C. F. Stubenrauch, Spherical Near-Field Scanning:Experimental and Theoretical Studies: National Institute of Standardsand Technology NISTIR 3955, July 1990.

[6] J. D. Jackson, Classical Electrodynamics, 2nd ed. New York: Wiley,1975.

[7] A. Dutt and V. Rohklin, “Fast Fourier transforms for nonequispaceddata,” in Proc. SIAM J. Scientific Comput., vol. 14, Nov. 1993, pp.1369–1393.

[8] G. Beylkin, “On the fast fourier transform of functions with singulari-ties,” Appl. Computat. Harmonic Anal., vol. 2, pp. 363–381, 1995.

Resonance Series Representation of the Early-Time FieldScattered by a Coated Cylinder

Heike Vollmer and Edward J. Rothwell

Abstract—The transient response of a coated cylinder to a plane-waveexcitation is examined. It is shown that the early-time response has a com-ponent very close to the resonance representation for a planar structure,which may be isolated by subtracting the response of an uncoated cylinder.This component can be used with the E-Pulse technique to diagnosechanges in coating parameters.

Index Terms—Electromagnetic transient scattering, nondestructivetesting, dielectric materials.

I. INTRODUCTION

The conducting surfaces of air vehicles are often coated with ab-sorbingmaterial to reduce their radar cross section. Since the propertiesof the materials may degrade due to environmental exposure, detectingchanges in the material parameters is important. It has been shown thatthe pulse response of a planar coated conductor consists of a reflectionfrom the air-coating interface, followed by a superposition of dampedsinusoids representing the multiple reflections within the coating layer[1]. This natural resonance series allows the E-pulse technique, origi-nally developed for radar target discrimination, to be used for detectingmaterial changes [2].

The surfaces of actual air vehicles are often curved rather than planar.To determine if the temporal response of a coated curved surface canalso be written as a resonance series, we examine the transient fieldreflected by a coated circular cylinder. We show that under certaincircumstances the early-time portion of the cylinder response is verynearly a resonance series, with frequencies close to those of a planarstructure.

II. CALCULATION OF THE SCATTERED FIELD

Consider a circular conducting cylinder of a radius aligned alongthe z-axis, coated with a dielectric material of radius b such that thethickness of the coating is d = b � a. The coating has a permittivity" = "r"0 and a permeability � = �0. A plane wave is incident alongthe x-axis as shown in Fig. 1, with the electric field oriented in eithertransverse-magnetic (TM) or transverse-electric (TE) polarization.

In the case of TM polarization, the incident electric field is writtenas ~Ei = zE0e

�jk x where k0 = !=c. The far-zone scattered electricfield is given by

Esz � 2j

�k0

e�jk �

p�

1

n=0

Anjn cosn�: (1)

Manuscript received May 19, 2003; revised September 10, 2003.The authors are with the Department of Electrical and Computer Engi-

neering, Michigan State University, East Lansing, MI 48824 USA (e-mail:[email protected]).


0018-926X/04$20.00 © 2004 IEEE


Fig. 1. Geometry of a TE-polarized plane wave incident on a dielectric-coatedcylinder.

The coefficients An may be found in the literature (for example, [3]).For computational purposes we choose to write them as ratios of Besselfunctions:

An = �E0"nj�n

(an � bn)fnangn � bn

where

an =Jn(kb)

Jn(k0b)�

Jn(ka)

Jn(k0b)

Yn(kb)

Yn(ka);

bn =p"r

J 0n(kb)

J 0n(k0b)� Jn(ka)

J 0n(k0b)

Y 0n(kb)

Yn(ka)

fn =Jn(k0b)

H(2)n (k0b)

; gn = fnH

(2)n (k0b)

J 0n(k0b)

"n =1; n = 0

0; n > 0

and where k = k0p"r . While the Bessel functions may overflow or

underflow for large values of n, their ratios remain computable.For the case of TE polarization, the incident magnetic field is ~Hi =

z(E0=�0)e�jk x where �0 = �0="0. The far-zone scattered electric

field is given by

Es� �

2j

�k0

e�jk �

p�

1

n=0

Bnjn cosn� (2)

with the coefficients

Bn = �E0"nj�n (cn � dn)fn

cngn � dn

where

cn =Jn(kb)

Jn(k0b)� J 0n(ka)

Jn(k0b)

Yn(kb)

Y 0n(ka);

dn =1p"r

J 0n(kb)

J 0n(k0b)� J 0n(ka)

J 0n(k0b)

Y 0n(kb)

Y 0n(ka):

III. THE TRANSIENT SCATTERED FIELD

The transient field scattered by the coated cylinder may be found bycomputing the inverse transform of (1) or (2). The resulting impulseresponse has an early-time component consisting of multiple reflec-tions within the dielectric layer, followed by a series of creeping waves[4]–[6]. It is found that the multiple reflections do not comprise a purenatural mode series, but are superimposed with a term that extends intothe early time due to interaction with the curved surface (due in part,perhaps, to the presence of an early-time branch cut contribution). It

is possible to remove much of the extended interaction term by sub-tracting the field reflected by an uncoated conducting cylinder of radiusa. When a resonance series is fit to the resulting waveform, the reso-nance frequencies are found to be nearly those of a planar conductorwith a coating of identical permittivity and thickness.

As an example, consider a coated cylinder with d = 0:1 m and"r = 9. Equations (1) and (2) are evaluated for � = � (backscatteredfield) at 2500 frequency points in the range 0–2.5 GHz using 200 termsin each series (giving an accuracy of seven decimal places). Note thatthe coating is 2:5� thick at the highest frequency, and that the terme�jk �=

p� is suppressed. The frequency-domain data is windowed

using a Gaussian function and transformed into the time domain usingan inverse FFT. The results for a = 0:7 m and a = 1:9 m are shownin Figs. 2–5. It is clearly seen in each figure that the early time consistsof a number of multiple reflections. In Fig. 4 the late-time creepingwave appears around t = 17 nsec. The creeping wave is also presentin Fig. 2, but is not as pronounced.

Also shown in these figures is the temporal field reflected by a planarcoated conductor with a coating of thickness d = 0:1m and a dielectricconstant of "r = 9. The planar response is found by computing thereflection coefficient of a normally-incident plane wave on a planarcoated conductor [7]

R =�� P 2

1� �P 2

where P = exp(�jkd);� = (� � �0)=(� + �0), and � = �0=p"r .

This reflection coefficient is computed at 2500 frequency points, win-dowed and transformed into the time domain. We expect that in thelimit a ! 1 the reflections from the coated cylinder will be iden-tical to those from the planar conductor. For finite values of a thereflections are similar, but are not identical. Improved results can beobtained by subtracting the backscattered field from an uncoated con-ducting cylinder, found by substituting [8]

An = �E0"nj�n Jn(k0a)

H(2)n (k0a)

;

Bn = �E0"nj�n J 0n(k0a)

H(2)n (k0a)

into (1) and (2). The resulting fields are much closer to the planar case,as seen in Figs. 2–5. In all cases, the TM fields are a better match forthe planar fields than are the TE fields.

IV. DETECTION OF CHANGES IN COATING PARAMETERS USING THE

E-PULSE TECHNIQUE

Since the late-time field reflected from a planar coated conductor isa pure resonance series, an E pulse can be constructed that producesa null result when convolved with the reflected field. If this E pulse isconvolved with the field reflected by a conductor with different coatingparameters (thickness or dielectric constant) the result will be nonzero.This allows a detection of changes in the coating parameters. To quan-tify the change we compute the E-pulse discrimination number (EDNa)[2]

EDNa =

T +W

T

c2e(t)dt

T

0

e2(t)dtT +W

T

r2(t)dt

:

Here, r(t) is the reflected field, e(t) is the E-pulse waveform, and ce(t)is the convolution of the E pulse with the reflected field. Also, Te is theduration of the E pulse, TL is the time at which the response beginsto be a natural mode series, and W is the width of the natural mode


Fig. 2. Field reflected by a coated circular cylinder with parameters a = 0:7m, d = 0:1m, "r = 9. TM excitation. E Pulse is that for the planar surface.

Fig. 3. Field reflected by a coated circular cylinder with parameters a = 1:9m, d = 0:1m, "r = 9. TM excitation.

Fig. 4. Field reflected by a coated circular cylinder with parameters a = 0:7m, d = 0:1m, "r = 9. TE excitation.


Fig. 5. Field reflected by a coated circular cylinder with parameters a = 1:9m, d = 0:1m, "r = 9. TE excitation.

Fig. 6. Normalized energy in the convolution of the planar-surface E Pulse with the field reflected from a coated cylinder.

component. In the case of the coated cylinder, TL +W is the time atwhich the creeping wave first occurs.

Ideally, we would like to use the E pulse computed for the planarreflected field to detect changes in the cylinder coatings. If we do,there will be an error introduced since the coated cylinder responseis not a pure resonance series. However, if the EDNa due to this erroris smaller than the EDNa due to a change in coating parameters, thechange should be detectable. Fig. 6 shows the EDNa computed usingthe planar E Pulse (shown in Fig. 2) with the coated cylinder response,for various values of a. If the planar E Pulsematches the coated cylinderresponse, the result is an EDNa of 0. As expected, the EDNa is nonzero,but as a increases the coated cylinder more closely resembles the planarsurface and the EDNA decreases. Even so, we see that for both TE andTM responses, the EDNa is relatively large, ranging from 0.01 to 0.1. Incontrast, when we subtract the uncoated cylinder response, the EDNAis significantly reduced. As a numerical example, we find in [2] that anincrease in "r of 5% produces an EDNA of 0.0001. Thus for coatedcylinders of radius a = 0:9 (nine times the thickness) and larger thechange should be detectable with TM polarization as long as the con-ducting cylinder response is subtracted. Similarly, an increase in d of5% produces an EDNa of 0.0004, which should be detectable for coatedcylinders of radius a = 0:4 and larger.

V. CONCLUSION

We have shown that the early-time temporal response of a coatedconducting cylinder consists of a series of multiple reflections. Thesereflections do not form a pure resonance series, but are very close toa resonance series when the response of an uncoated cylinder is sub-tracted. In this case, the E-Pulse technique can be used to detect changesin the permittivity or thickness of the coating.

REFERENCES

[1] J. Oh, E. Rothwell, D. Nyquist, andM. Havrilla, “Natural resonance rep-resentation of the transient field reflected by a conductor-backed lossylayer,” J. Electromagn. Waves Applicat., vol. 17, no. 5, pp. 673–694,2003.

[2] G. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L.Frasch, “E-pulse diagnostics of simple layered materials,” IEEE Trans.Antennas Propagat., vol. 51, pp. 3221–3227, Dec. 2003.

[3] C. C. H. Tang, “Backscattering from dielectric-coated infinite cylindricalobstacles,” J. Appl. Phys., vol. 28, no. 5, pp. 628–633, May 1957.

[4] N. Wang, “Electromagnetic scattering from a dielectric-coated circularcylinder,” IEEE Trans. Antenntas Propagat., vol. AP-33, pp. 960–963,Sept. 1985.


[5] R. Paknys and N. Wang, “Creeping wave propagation constants andmodal impedance for a dielectric coated cylinder,” IEEE Trans. An-tennas Propagat., vol. AP-34, pp. 674–680, May 1986.

[6] , “Excitation of creeping waves on a circular cylinder with a thickdielectric coating,” IEEE Trans. Antennas Propagat., vol. AP-35, pp.1487–1489, Dec. 1987.

[7] E. J. Rothwell and M. J. Cloud, Electromagnetics. Boca Raton, FL:CRC Press, 2001.

[8] C. A. Balanis, Advanced Engineering Electromagnetics. New York:Wiley, 1989.

High-Order Symplectic Integration Methods forFinite Element Solutions to Time Dependent

Maxwell Equations

R. Rieben, D. White, and G. Rodrigue

Abstract—In this paper, we motivate the use of high-order integrationmethods for finite element solutions of the time dependent Maxwell equa-tions. In particular, we present a symplectic algorithm for the integration ofthe coupled first-order Maxwell equations for computing the time depen-dent electric and magnetic fields. Symplectic methods have the benefit ofconserving total electromagnetic field energy and are, therefore, preferredover dissipative methods (such as traditional Runge–Kutta) in applicationsthat require high-accuracy and energy conservation over long periods oftime integration. We show that in the context of symplectic methods, sev-eral popular schemes can be elegantly cast in a single algorithm. We con-clude with some numerical examples which demonstrate the superior per-formance of high-order time integration methods.

Index Terms—Finite element methods, high-order methods, Maxwellequations, symplectic methods, time domain analysis.

I. INTRODUCTION

We are concerned with the finite element solution of the time de-pendent Maxwell equations on unstructured grids using a combina-tion of both high-order spatial and high-order temporal discretizationmethods. In this paper we focus our attention on the high-order tem-poral discretization process, and we investigate the use of symplecticintegration methods. Such methods were originally developed to solvenumerical systems derived from a Hamiltonian formulation and havebeen successfully used in the fields of astronomy and molecular dy-namics where numerical accuracy and energy conservation are very im-portant over large time integration periods [1]. Recently, these methodshave been adapted for use in computational electromagnetics (CEM)in conjunction with the finite difference method. In [2] and [3] a sym-plectic finite-difference time-domain (FDTD) algorithm is presented

Manuscript received April 23, 2003; revised September 22, 2003. This workwas performed under the auspices of the U.S. Department of Energy by the Uni-versity of California, Lawrence Livermore National Laboratory under contractW-7405-Eng-48 and the U.S. Air Force under Contract F49620-01-1-0327.

R. Rieben and G. Rodrigue are with the University of California Davis and In-stitute for Scientific Computing Research, Lawrence Livermore National Labo-ratory, Livermore, CA 94551 USA (e-mail: [email protected]; [email protected]).

D. White was with the Center for Applied Scientific Computing Research,Lawrence Livermore National Laboratory, Livermore, CA 94551 USA (e-mail:[email protected]). He is nowwith the Defense Sciences Engineering Division,Lawrence Livermore National Laboratory, Livermore CA, 94551 USA (e-mail:[email protected]).


that is implicit, fourth order accurate and valid for orthogonal three-di-mensional grids . In [4] and [5], the authors present a modified sym-plectic FDTD method that is up to fourth-order accurate in space andtime. A variation using the linear “serendipity” finite elements of [6] isalso mentioned. Here, we proceed in a similar manner using high-ordersymplectic integration methods in conjunction with a high-order vectorfinite element method for use in nonorthogonal, unstructured grids. Thespatial discretization is handled by the use of Nédeléc [7] basis func-tions of arbitrary order which are based on the properties of differen-tial forms [8], [9]. For the Galerkin procedure applied to either the fre-quency domain or time dependent Maxwell equations [10], there aresignificant advantages to both one-form and two-form finite elementbasis functions [11]; including the proper modeling of the jump discon-tinuity of field intensities and flux densities across material interfaces,the elimination of spurious modes in eigenvalue computations and theconservation of charge in time-dependent simulations [11]. These prop-erties are crucial for the elimination of late time instabilities caused byimproper spatial discretization as investigated by [12]–[14].

We begin with a method of lines approach to the discretization ofthe time dependent Maxwell equations. We approximate the coupledpartial differential equations using a high-order vector finite elementscheme which yields a linear system of ordinary differential equations(ODEs). This system is then be discretized in time via a finite differencemethod to produce a series of update steps which propagate the solu-tions forward in time. However, most high-order numerical integrationmethods (e.g., Runge–Kutta, Adams–Bashforth) are dissipative. Thiscan lead to misleading results for systems that need to be iterated forlong time intervals [15], [16]. A solution is to use a symplectic timeintegration method that conserves energy. Therefore, in this paper weinvestigate and promote the use of symplectic methods for the time in-tegration of Maxwell’s equations.

II. AMPERE-FARADAY SYSTEM

We begin with the coupled first-order time dependentMaxwell equa-tions

�@

@tE =r� (��1B)� J(t)

@

@tB = �r�E (1)

where � and � are (possibly tensor valued) functions representing thematerial properties of the system and J(t) is a time dependent currentsource. Using a Galerkin finite element procedure with one-form (orCurl-conforming) vector basis functions to discretize the electric fieldintensity and two-form (or Div-conforming) vector basis functions todiscretize the magnetic flux density yields the following linear systemof ODEs

A@

@te =K

TD b� A j

@

@tb = �K e (2)

where e and b represent the discrete differential one-form and two-form electric and magnetic fields, respectively, K represents the dis-crete Curl operator (i.e., the topological derivative matrix), A is theone-form mass matrix computed using the material property function� to represent the dielectric properties, D is the two-form mass ma-trix computed using the material property function ��1 to representthe magnetic permeability and j is the discrete two-form time depen-dent current source. Note that the vectors e and b will have differentdimensions and that the matrix K will be rectangular. This is due tothe dimensions of the Nédeléc polynomial spaces from which they are

0018-926X/04$20.00 © 2004 IEEE


[5] R. Paknys and N. Wang, “Creeping wave propagation constants andmodal impedance for a dielectric coated cylinder,” IEEE Trans. An-tennas Propagat., vol. AP-34, pp. 674–680, May 1986.

[6] , “Excitation of creeping waves on a circular cylinder with a thickdielectric coating,” IEEE Trans. Antennas Propagat., vol. AP-35, pp.1487–1489, Dec. 1987.

[7] E. J. Rothwell and M. J. Cloud, Electromagnetics. Boca Raton, FL:CRC Press, 2001.

[8] C. A. Balanis, Advanced Engineering Electromagnetics. New York:Wiley, 1989.

High-Order Symplectic Integration Methods forFinite Element Solutions to Time Dependent

Maxwell Equations

R. Rieben, D. White, and G. Rodrigue

Abstract—In this paper, we motivate the use of high-order integrationmethods for finite element solutions of the time dependent Maxwell equa-tions. In particular, we present a symplectic algorithm for the integration ofthe coupled first-order Maxwell equations for computing the time depen-dent electric and magnetic fields. Symplectic methods have the benefit ofconserving total electromagnetic field energy and are, therefore, preferredover dissipative methods (such as traditional Runge–Kutta) in applicationsthat require high-accuracy and energy conservation over long periods oftime integration. We show that in the context of symplectic methods, sev-eral popular schemes can be elegantly cast in a single algorithm. We con-clude with some numerical examples which demonstrate the superior per-formance of high-order time integration methods.

Index Terms—Finite element methods, high-order methods, Maxwellequations, symplectic methods, time domain analysis.

I. INTRODUCTION

We are concerned with the finite element solution of the time de-pendent Maxwell equations on unstructured grids using a combina-tion of both high-order spatial and high-order temporal discretizationmethods. In this paper we focus our attention on the high-order tem-poral discretization process, and we investigate the use of symplecticintegration methods. Such methods were originally developed to solvenumerical systems derived from a Hamiltonian formulation and havebeen successfully used in the fields of astronomy and molecular dy-namics where numerical accuracy and energy conservation are very im-portant over large time integration periods [1]. Recently, these methodshave been adapted for use in computational electromagnetics (CEM)in conjunction with the finite difference method. In [2] and [3] a sym-plectic finite-difference time-domain (FDTD) algorithm is presented

Manuscript received April 23, 2003; revised September 22, 2003. This workwas performed under the auspices of the U.S. Department of Energy by the Uni-versity of California, Lawrence Livermore National Laboratory under contractW-7405-Eng-48 and the U.S. Air Force under Contract F49620-01-1-0327.

R. Rieben and G. Rodrigue are with the University of California Davis and In-stitute for Scientific Computing Research, Lawrence Livermore National Labo-ratory, Livermore, CA 94551 USA (e-mail: [email protected]; [email protected]).

D. White was with the Center for Applied Scientific Computing Research,Lawrence Livermore National Laboratory, Livermore, CA 94551 USA (e-mail:[email protected]). He is nowwith the Defense Sciences Engineering Division,Lawrence Livermore National Laboratory, Livermore CA, 94551 USA (e-mail:[email protected]).


that is implicit, fourth order accurate and valid for orthogonal three-di-mensional grids . In [4] and [5], the authors present a modified sym-plectic FDTD method that is up to fourth-order accurate in space andtime. A variation using the linear “serendipity” finite elements of [6] isalso mentioned. Here, we proceed in a similar manner using high-ordersymplectic integration methods in conjunction with a high-order vectorfinite element method for use in nonorthogonal, unstructured grids. Thespatial discretization is handled by the use of Nédeléc [7] basis func-tions of arbitrary order which are based on the properties of differen-tial forms [8], [9]. For the Galerkin procedure applied to either the fre-quency domain or time dependent Maxwell equations [10], there aresignificant advantages to both one-form and two-form finite elementbasis functions [11]; including the proper modeling of the jump discon-tinuity of field intensities and flux densities across material interfaces,the elimination of spurious modes in eigenvalue computations and theconservation of charge in time-dependent simulations [11]. These prop-erties are crucial for the elimination of late time instabilities caused byimproper spatial discretization as investigated by [12]–[14].We begin with a method of lines approach to the discretization of

the time dependent Maxwell equations. We approximate the coupledpartial differential equations using a high-order vector finite elementscheme which yields a linear system of ordinary differential equations(ODEs). This system is then be discretized in time via a finite differencemethod to produce a series of update steps which propagate the solu-tions forward in time. However, most high-order numerical integrationmethods (e.g., Runge–Kutta, Adams–Bashforth) are dissipative. Thiscan lead to misleading results for systems that need to be iterated forlong time intervals [15], [16]. A solution is to use a symplectic timeintegration method that conserves energy. Therefore, in this paper weinvestigate and promote the use of symplectic methods for the time in-tegration of Maxwell’s equations.

II. AMPERE-FARADAY SYSTEM

We begin with the coupled first-order time dependentMaxwell equa-tions

�@

@tE =r� (��1B)� J(t)

@

@tB = �r�E (1)

where � and � are (possibly tensor valued) functions representing thematerial properties of the system and J(t) is a time dependent currentsource. Using a Galerkin finite element procedure with one-form (orCurl-conforming) vector basis functions to discretize the electric fieldintensity and two-form (or Div-conforming) vector basis functions todiscretize the magnetic flux density yields the following linear systemof ODEs

A@

@te =K

TD b� A j

@

@tb = �K e (2)

where e and b represent the discrete differential one-form and two-form electric and magnetic fields, respectively, K represents the dis-crete Curl operator (i.e., the topological derivative matrix), A is theone-form mass matrix computed using the material property function� to represent the dielectric properties, D is the two-form mass ma-trix computed using the material property function ��1 to representthe magnetic permeability and j is the discrete two-form time depen-dent current source. Note that the vectors e and b will have differentdimensions and that the matrix K will be rectangular. This is due tothe dimensions of the Nédeléc polynomial spaces from which they are

0018-926X/04$20.00 © 2004 IEEE


derived [7]. For an electromagnetic problem with no physical dissi-pation due to conductivity or absorbing boundary conditions the totalelectromagnetic energy should remain constant. In this particular finiteelement method the instantaneous energy is the numerical version ofthe total energy given by

E = eTA e+ b

TD b: (3)

Many time integration methods such a forward Euler, backward Euler,Runge–Kutta, Adams–Bashforth, etc. are inherently dissipative and theenergy as measured by (3) is not conserved; given an initial conditionthe electromagnetic energy will decay exponentially.

The very popular second-order central difference (also known as a“leap frog”) method applied to system (2) can be written as

en = en�1 +�t(A�1KTD bn�1=2 � j)

bn+1=2 = bn�1=2 +�t(�K en): (4)

It is well known that this particular method is both conditionallystable and nondissipative; the energy as measured by (3) is conserved.Our goal is to apply higher order energy conserving time integrationmethods to system (2). This is required to take full advantage of thehigher order finite element basis functions. The resulting method ishigher order in both space and time and will have significantly lessnumerical dispersion than low-order FDTD type methods, which isimportant for electrically large problems.

III. CONSERVATIVE TIME INTEGRATION

Consider a general system of ODEs, with field values p and q and anindependent variable t, that is of the specific form

@

@tp =F (q; t)

@

@tq =G(p): (5)

Systems of this form have the property of being nondissipative, i.e., thesystem does not lose energy as it evolves in time. Numerical integra-tion methods for solving system (5) should likewise be nondissipative.For linear equations, such methods are typically written as an updatescheme of the form

pn+1

qn+1=M

pn

qn(6)

where the field values at a new state are expressed in terms of values atprevious states. There are three specific cases of interest based on thematrix norm ofM, given by

jMj

> 1; unstable= 1; neutrally stable (nondissipative)< 1; stable, dissipative.

(7)

When the eigenvalues of the update matrix all lie within the unit circlein the complex plane, themethodwill be stable and dissipative. Nondis-sipative methods have the additional property that the eigenvalues ofthe update matrix all lie on the unit circle in the complex plane, withadditional constraints on the eignevectors for stability [11]. The map-ping is said to be symplectic if the following relation holds [17]

@MTS @M = S (8)

where

@M =@p ~F @q ~F

@p ~G @q ~G; S =

0 I

�I 0

TABLE ICOEFFICIENTS FOR METHODS OF ORDER ONE THROUGH FOUR

where ~F and ~G represent discretized versions of the original functionsF andG. ThematrixS is referred to as the symplectic matrix, where theword symplectic literally means “intertwined.” Note that this definitiononly makes sense if the vectors of unknowns p and q are of the samedimension, as in the case of a Hamiltonian system where q denotes thegeneralized coordinates and p the generalized momenta.As a specific example, consider the simple harmonic oscillator

(SHO) where F (q; t) = q and G(p) = �p. An exact solution to thissimple problem is given by p(t) = sin(t) and q(t) = cos(t). Wecan quantify the energy of this system (i.e., a conserved or constantquantity) by the value

E = p2(t) + q

2(t)

which for this specific example is equal to 1. Applying the leap frogmethod to the SHO yields the following update scheme

pn

qn+1=2=

1 �t

��t (1��t2)

pn�1

qn�1=2:

It is a straightforward calculation to show that this update scheme sat-isfies (8) and is therefore symplectic. However, it is also straightfor-ward to show that this mapping does not conserve the exact value ofE under iteration. This is due to the fact that symplectic maps solvesomeHamiltonian exactly, but not the exact one of the system [1], [17].However, as shown by Yoshida [18], the numerical value of the inexactconserved quantity ~E oscillates about the exact value E and the ampli-tude of this oscillation is reduced as the order of the symplectic methodis increased.To demonstrate the properties of symplectic integrators for conserva-

tive systems, we proceed to solve the SHO system numerically usingboth a symplectic method (the order three case from Table I) and anonsymplectic fourth-order Runge–Kutta method. In both cases, thesystem is propagated from t = 0 to t = 250 using a time step of�t = 0:8 and the computed maximum global phase error will growlinearly at each time step. Where the two cases differ is in the compu-tation of the energy of the system. Fig. 1 shows the computed numericalenergy of the system at each time step for both methods. For the sym-plectic method, the numerical energy is of the form

~E = �1 cos( 1 t) E

while for the nonsymplectic method the energy is of the form

~E = �2 exp(� 2 t) E :

Fig. 2 shows a parametric plot of the conjugate variables as a functionof time. The numerical energy for the symplectic method oscillates at afixed amplitude around the exact value, and is therefore conserved (in


Fig. 1. Numerical energy at each time step using a symplectic method and a nonsymplectic Runge–Kutta method.

(a) (b)

Fig. 2. Parametric phase plots of the conjugate variables of a simple harmonic oscillator using (a) a symplectic method and (b) a nonsymplectic Runge–Kuttamethod.

a time averaged sense). The energy for the nonsymplectic method dis-sipates exponentially from the exact value, indicating spurious damp-ening of the system.

Such behavior is typical of symplectic methods when applied to con-servative systems, and has therefore motivated us to apply them to theparticular system of (2). It should be noted that when a symplecticmethod is applied to the Maxwell system of ODEs (2) the result doesnot satisfy the symplectic property of (8). This is due to the fact (asmentioned previously) that the vectors e and b are not of the same di-mension and that the matrix K is rectangular. Nevertheless this doesnot preclude the method from being used, in fact it has been success-fully used in FDTD schemes where the dimension of e (the number ofmesh edges) is different than the dimension of b (the number of meshfaces) [2], [3]. We demonstrate through computational experiments inSection V that high-order symplectic methods do work when applied tosystem (2) and correctly reproduce the previously mentioned featuresof stability, high accuracy, and no nonphysical dissipation.

IV. GENERAL SYMPLECTIC ALGORITHM

We now present the general symplectic integration algorithm used inour experiments. The algorithm is valid for ODE systems of the form(5), such as (2). The inputs, procedure and outputs of the method arepresented in Algorithm 1. Numerical methods for the integration of a

set of differential equations are typically characterized by the accuracyof a single step in time (the independent variable). If for some smalltime step�t the integration is performed so that it is accurate throughorder �tk , then the method is of kth order. In general, a method oforder k will require k evaluations of the functions F andG. Therefore,as the order of the method is increased the overall computational costswill increase likewise. However, as we will show in the next section,for higher order methods, it is possible to increase the size of the timestep�t (while still maintaining numerical stability), thereby reducingthe overall number of time steps. The order of the method can be ad-justed simply by providing the algorithm with a corresponding set ofcoefficients, a and b, each of length order. Table I lists exact values ofthe sets of coefficients a and b for methods of order one through four,as originally computed by Ruth [15] and Candy [19].

Algorithm 1: General SymplecticIntegration Algorithminput : , the order of the method

and , two functions and ,two sets of coefficients and , theinitial conditions and , initial andfinal time , the time step to useoutput : and , the fields at time


Compute the number of time steps:

nstep =t�n � t0

�t

Set initial conditions:

p1 F0

q1 G0

Begin loop over time steps:for to do

Begin integration method update :

pin pi

qin qi

for j = 1 tototo order do

Compute the update time for this step :

tj = i ��t+j�1

k=1

ak ��t

Update the �eld values :

pout = pin + bj ��t � F (qin; tj)

qout = qin + aj ��t �G(pout)

pin pout

qin qout

end

Update �eld values for this time step :

pi+1 pout

qi+1 qout

end


We now present some computational examples using the symplecticintegration algorithm in conjunction with high-order finite element ma-trices for the spatial discretization of Maxwell’s equations. The com-putational domain for these examples is a unit cube subject to either aPEC (Dirichlet) or a natural zero flux (Neumann) boundary condition.The Ampere-Faraday system is discretized in space using a very coarseeight element hexahedral mesh in conjunction with high-order vectorbasis functions of polynomial degree p = 4

In each of the following examples, the time integration schemes aresubject to a stability condition. This stability condition is based onthe spectral radius of the amplification matrix which is applied to thesystem at every time step in an update method of the form (6). For thediscrete Maxwell equations of system (4), there exists an upper boundon the largest stable time step given by [11]

�t �2

MaxEig(A�1KTDK): (9)

We have found that about 0.95 times the upper bound of this constraintis sufficient for symplectic methods of order one through three; higherorder methods require a smaller time step to remain stable. For ex-ample, we have found that for the fourth-order method from Table I,about 0.70 times the upper bound is sufficient for stability.

In addition, for each of the following examples, evaluation of thefunction F during the update phase requires that a linear system in-volving the matrix A must be solved. To simplify this process, we per-form the linear solve using a diagonally scaled Conjugate Gradient al-gorithm. However, this process could be made more efficient by usinga sparse direct solver.

Fig. 3. Global phase error at each time step for the first-order symplecticintegration method.

Fig. 4. Global phase error at each time step for the third-order symplecticintegration method.

A. Example 1

In this example we demonstrate the growth of global phase error forthe time integration of (2) using two different methods. We begin bysolving the general eigenvalue problem

Sx = �Ax (10)

subject to a zero flux boundary condition [20]. Here, S is the one-formstiffnessmatrix (i.e theCurl –Curlmatrix) and this system representsthe resonant modes of the unit cube. We locate the first nonzero eigen-value of this system (representing the first resonant mode of the cavity)and its corresponding eigenvector. Using basis functions of polynomialdegree p = 4 on a coarse eight element mesh, the first resonant modeis computed to an accuracy of 10�4. We then use the computed eigen-vector as the initial condition for the electric field in (2), the magneticfield will have a zero value initial condition. System (2) is then propa-gated forward in time for a total of 300 s (using a value of unity for thespeed of light). The resulting computed electric field will be an oscil-latory cosine wave with a frequency equal to the first resonant mode ofthe cube. We compare the global phase error in the computed solutionagainst the exact value using both a first and third-order symplecticintegration method. The first-order method is integrated using a timestep of �t = 0:005 s yielding a total of 60,000 time steps while thethird-order method is integrated using a time step of �t = 0:015 syielding a total of 20 000 time steps. The resulting computations there-fore require the same total amount of CPU time to complete. The re-sulting global phase errors are shown in Figs. 3 and 4. Note that in bothcases, the maximum global phase error grows linearly at each time step,but the third-order method yields a much slower rate of growth with a


Fig. 5. Numerical energy at each time step for the first-order method.

Fig. 6. Numerical energy at each time step for the third-order method.

maximum global phase error two orders of magnitude smaller than thefirst-order method for roughly the same computational cost.

Figs. 5 and 6 show the computed value of the numerical energy from(3) at each time step for both the first and third-order methods (forvisual clarity, only values for the last 50 s are shown). Note that for bothcases the numerical energy oscillates around the exact value, but forthe third-order case, the amplitude of this oscillation is several ordersof magnitude smaller than for first-order method, again for roughly thesame computational cost.

B. Example 2

In this example we compute the resonant modes of the cubic cavitysubject to a PEC boundary condition using two different integrationmethods. We do this by creating an oscillating electromagnetic field in-side the cavity by applying a time dependent current source to a randomsampling of the interior degrees of freedom. The simple current sourcehas a temporal profile equal to the second derivative of a Gaussianpulse. Setting the speed of light equal to unity, we let the simulation runfor a physical time of 300 s, then Fourier transform the resulting fieldamplitude to obtain both the transverse electric and transverse mag-netic resonant modes of the cavity [21]. The errors for the first fiveexcited modes of the cavity are computed using both a first-order anda third-order symplectic integration method. The exact values and thecomputed Fourier spectrum for the case of the third-order method areshown in Fig. 7. The results for both calculations are summarized inTable II. Again, note that for roughly the same computational cost, thethird order method gives results that are more accurate than the firstorder method. We know from eigenvalue computation of Example 1that the high-order spatial discretization is capable of computing the

Fig. 7. Computed resonant modes of cubic cavity using a third ordersymplectic method. Vertical lines represent exact values.

TABLE IICOMPARISON OF RESULTS FOR TWO INTEGRATION METHODS

modes to an accuracy of 10�4, and the data in Table II clearly showsthis same accuracy can be achieved in the time domain only if a higherorder time integration is used.

VI. CONCLUSION

The results of this paper are twofold. First, we have demonstratedthat high-order time integrationmethods used in conjunctionwith high-order spatial discretizations can yield more accurate numerical resultsfor roughly the same computational cost as a low-order method. Sec-ondly, we have presented a general symplectic method for the integra-tion of the time dependent Maxwell equations. Symplectic time inte-gration methods have been developed for Hamiltonian systems such asthose that arise in astrophysics and molecular dynamics, where verylong time integration is required. We show that these methods can besuccessfully applied to a finite element discretization of Maxwell’sequations, resulting in higher order and energy conserving integration.The higher order symplectic methods used in this paper are no morecomplicated or expensive than traditional Runge–Kutta methods.

REFERENCES

[1] J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Prob-lems. London, U.K.: Chapman and Hall, 1994.

[2] I. Saitoh, Y. Suzuki, and N. Takahashi, “The symplectic finite differencetime domainmethod,” IEEE Trans. Magn., vol. 37, pp. 3251–3254, Sept.2001.

[3] I. Saitoh and N. Takahashi, “Stability of symplectic finite-differencetime-domain methods,” IEEE Trans. Magn., vol. 38, pp. 665–668, Mar.2002.

[4] T. Hirono, W. W. Lui, and K. Yokoyama, “Time-domain simulation ofelectromagnetic field using a symplectic integrator,” IEEE MicrowaveGuided Wave Lett., vol. 7, no. 9, pp. 279–281, 1997.


[5] T. Hirono, W. W. Lui, K. Yokoyama, and S. Seki, “Stability andnumerical dispersion of symplectic fourth-order time-domain schemesfor optical field simulation,” J. Lightwave Tech., vol. 16, no. 10, pp.1915–1920, 1998.

[6] O. C. Zienkiewicz, The Finite Element Method in Engineering Sci-ence.. London, UK.: McGraw-Hill, 1971.

[7] J. C. Nédélec, “Mixed finite elements in R3,” Numer. Math., vol. 35, pp.315–341, 1980.

[8] D. A. White, “Numerical modeling of optical gradient traps using thevector finite element method,” J. Comput. Phys., vol. 159, no. 1, pp.13–37, 2000.

[9] R. Rieben, D. White, and G. Rodrigue, “Improved conditioning of fi-nite element matrices using new high-order interpolatory bases,” IEEETrans. Antennas Propagat., 2004, to be published.

[10] J. Dao and J. Jin, “A general approach for the stability analysis of thetime domain finite element method for electromagnetic simulations,”IEEE Trans. Antennas Propagat., vol. 50, pp. 1624–1632, Nov. 2002.

[11] G. Rodrigue and D. White, “A vector finite element time-domainmethod for solving Maxwell’s equations on unstructured hexahedralgrids,” SIAM J. Sci. Comp., vol. 23, no. 3, pp. 683–706, 2001.

[12] P. Thoma, “Numerical stability of finite difference time domianmethods,” IEEE Trans. Magn., vol. 34, pp. 2740–2743, Sept. 1998.

[13] F. L. Teixeira and W. C. Chew, “Lattice electromagnetic theory from atopological viewpoint,” J.Math. Phys., vol. 40, no. 1, pp. 169–187, 1999.

[14] S. D. Gedney and J. A. Roden, “Numerical stability of nonorthog-onal FDTD methods,” IEEE Trans. Antennas Propagat., vol. 48, pp.231–239, Feb. 2000.

[15] D. Ruth, “A canonical integration technique,” IEEE Trans. Nucl. Sci.,vol. NS-30, pp. 2669–2671, Aug. 1983.

[16] E. Forest and D. Ruth, “Fourth-order symplectic integration,” PhysicaD, vol. 43, no. 1, pp. 105–117, 1990.

[17] A. M. Stewart and A. R. Humphries,Dynamical Systems and NumericalAnalysis: Cambridge University Press, 1996.

[18] H. Yoshida, “Symplectic integrators for Hamiltonian-systems – Basictheory,” in Proc. IAU Symposia (152), The Netherlands, 1992, pp.407–411.

[19] J. Candy andW. Rozmus, “A symplectic integration algorithm for seper-able Hamiltonian functions,” J. Comput. Phys., vol. 92, pp. 230–256,1991.

[20] D. A. White and J. M. Koning, “Computing solenoidal eigenmodes ofthe vector Helmholtz equation: A novel approach,” IEEE Trans. Magn.,vol. 38, pp. 3420–3425, Sept. 2002.

[21] C. Balanis, Advanced Engineering Electromagnetics. New York:Wiley, 1989, vol. 75.


Corrections_________________________________________________________________________________

Corrections to “Phased Arrays Based on OscillatorsCoupled on Triangular and Hexagonal Lattices”

Ronald J. Pogorzelski

In [1], three lines above (11), the symbol I should be i. Also, thesymbol A appearing in (A3), (A4), and (A6) of the Appendix should

Manuscript received April 29, 2004.The author is with the Jet Propulsion Laboratory, California Institute of Tech-

nology, Pasadena, CA 91109 USA (e-mail: [email protected]).Digital Object Identifier 10.1109/TAP.2004.832315

be S and the symbol S appearing in Section C of the Appendix shouldbeA. Lastly, in equation (15), the symbol � in the argument of the sinefunctions and the 1p

3appearing in front of each sine function should

be deleted.

REFERENCES

[1] R. J. Pogorzelski, “Phased arrays based on oscillators coupled on trian-gular and hexagonal lattices,” IEEE Trans. Antennas Propagat., vol. 52,pp. 790–800, Mar. 2004.

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