wavenumbers because the subradical expression becomesnegative. The same takes place for the second wave when thevalues of the parameters satisfy the condition

« mt z2 Ž .tan u ) . 122 2j y « mt t

Ž . Ž .The conditions 11 and 12 correspond to the wave propaga-tion within certain angular domains in space defined by theangles u - u and u - u for the first and second waves,1 2respectively.

3. CONCLUSIONS

In this paper, we have deduced the eigenvalue equation forŽ .the media D C , solved this equation, and discussed some`h `¨

of the eigensolution properties.

ACKNOWLEDGMENT

This work was supported by the Brazilian agency CAPES.

REFERENCES

1. V. Dmitriev, Constitutive tensors and general properties of com-plex and bianisotropic media described by continuous groups of

Ž .symmetry, Electron Lett 34 1998 , 532]534.2. E.O. Kamenetskii, On the technology of making chiral and bian-

isotropic waveguides for microwave propagation, Microwave OptŽ .Technol Lett 11 1996 , 103]107.

3. H.C. Chen, Theory of electromagnetic waves. A coordinate-freeapproach, McGraw-Hill, New York, 1983.

4. I.V. Lindell and A.J. Viitanen, Plane wave propagation in uniaxialŽ .bianisotropic medium, Electron Lett 29 1993 , 150]152.

5. I.V. Lindell, S.A. Tretyakov, and A.J. Viitanen, Plane wave propa-gation in uniaxial chiro-omega medium, Microwave Opt Technol

Ž .Lett 6 1993 , 517]520.6. S.A. Tretyakov and A.A. Sochava, Proposed composite material

for nonreflecting shields and antenna radoms, Electron Lett 29Ž .1993 , 1048]1049.

7. M. Hamermesh, Group theory and its application to physicalproblems, Addison-Wesley, Reading, MA, 1962.

8. V. Dmitriev, Constitutive tensor nomenclature of Kamenetskii’sŽ .media, Microwave Opt Technol Lett July 1998 .

9. E. Wigner, Group theory and its application to quantum mechan-ics of atomic spectra, Academic Press, New York, 1959.

Q 1999 John Wiley & Sons, Inc.CCC 0895-2477r99

EFFICIENT ITERATIVE METHOD FORANALYSIS OF LARGE FINITEPHASED ARRAYSWei Chen1 and Lang Jen11 Institute of Electromagnetic Fields and Microwave TechniquesSouthwest Jiaotong UniversityChengdu, Sichuan 610031, P. R. China

Recei ed 20 August 1998; re¨ised 27 October 1998

ABSTRACT: A no¨el iterati e method for the solution of large systemsof linear equations in analyzing large finite phased arrays is proposed.A¨oiding the calculation of large matrices and gi ing a good approxima-tion of the currents initially this iterati e algorithm shows an excellentcon¨ergence. A computational example is presented to illustrate its highefficiency. Q 1999 John Wiley & Sons, Inc. Microwave Opt TechnolLett 21: 44]46, 1999.

Key words: antennas; phased antenna arrays; algorithms

1. INTRODUCTION

Phased arrays usually have a great number of elements, andthe classical moment method analysis of them will introducea tremendous computation proportional to the third power of

w xthe number of the expansion coefficients 1 . For analyzingphased arrays with over thousands of elements, approxima-tions or fewer than necessary expansions have to be usedw x2, 3 . Iterative methods for solving large systems of linearequations are also applied to the matrix equations in themoment method, and they achieve success in the analysis of

w xlarge bodies 4 . However, the number of iterations for con-vergence, which is, for example, equivalent to the number ofindependent eigenvalues in the conjugate gradient methodw x4 , generally increases with the number of antennas. Thegreat amount of computation also makes the iterative meth-ods ineffective for large arrays.

On the other hand, by assuming the array infinite andequal-amplitude excited, the infinite array theory solves theFloquet modes and gives good results for central elementsw x5 . The large array method, an improvement of the infinitearray theory, assumes Floquet currents on antennas in afinite array, and obtains more accurate scan element patterns

w xfor central elements 6 . However, both of them cannotpredict the properties of antennas close to an edge or acorner, which make up a considerable part of the array, andtheir properties are important for some radiation featuressuch as sidelobe levels and null positions. The accurateanalysis of large but finite phased arrays remains a problem.

There have been continuous efforts to search for moreefficient algorithms for solving the large systems of linearequations involved in analyzing large antenna arrays. Some

w xutilize the physical features of the problems 7]9 . It isnoticed in the synthesis of antenna arrays that the effects ofmutual couplings between antennas are of integration of cur-rents, and that small changes in the distribution laws of thecurrents can lead to few variations in the effects of mutual

w xcouplings 9 . Following that idea, and realizing that thedistribution of the excitations is useful prior informationabout the currents to be solved, this paper proposes a newiterative method for analyzing large but finite phased arrays.Physically, this method divides the solution of an array intotwo parts. The matrix equation for the current on one an-tenna is solved by a direct method as the property of oneantenna is sensitive to the distribution law of the current onit. On the other hand, the effects of the mutual couplings aresolved by iterations because it is insensitive to the distribu-tion laws of the currents. An example of a 49 = 55 paralleldipole array is presented as a computational test to illustratethe rapid convergence and high efficiency of this method.

2. FORMULATION

Let us begin with the system of linear equations for thecurrents on the antennas in an N-element phased array viathe electric field integral equations by the moment method:

tV VZ Z . . . Z I 1 111 12 1N 1tZ Z . . . Z I V V21 22 2 N 2 2 2 Ž .? s 1. . . .. . . . . . . .. . . .tZ Z . . . Z IN1 N 2 NN N V VN N

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 21, No. 1, April 5 199944

where Z is the self- or mutual-impedance matrix betweeni jthe ith and jth antennas. I is the column matrix composediof the expansion coefficients for the currents on the ithelement V is the column excitation matrix for the ith ele-iment excited by unit driving voltage, and V t represents theidriving voltage terminated at the ith antenna.

Consider only the ith antenna in the phased array; thematrix equation for the current on it can be adapted from

Ž .Eq. 1 as follows:

Nt 9 Ž .Z ? I s V V q Z ? I . 2Ýii i i i i j j

js1

Ž .Equation 2 divides the whole excitation upon the ith an-tenna into two parts. The first term represents the excitationproduced by the driving voltage, and the second is thatproduced by the currents on all other antennas, i.e., thecouplings of all other antennas on the ith element. Physically,the distribution law of the current on one antenna is predom-inantly determined by its geometrical structure. The relativeamplitude and phase of the current are mainly determined bythe distribution of the driving voltages in a phased array. Thedifferences of the effects of mutual coupling act as a distur-bance on the computed current I . Mathematically, Z oni ii

Ž .the left-hand side of Eq. 2 is a matrix with large valueelements, and Z on the right-hand side are composed ofi jsmall value elements. The errors of the computed currents onother antennas also act as a disturbance. Therefore, weconclude that, if an approximation of the currents exists,

Ž . Ž .Eq. 2 will give a better approximation of I . So Eq. 1 isiadapted in an iterative form:

1 2Žkq1. t Žkq1. kZ ? I s V V y Z ? I y Z ? I ,Ý Ýii i i i i j j i j jj j

Ž .i s 1,2,3, . . . , N. 3

There, I Žkq1. is the current on the ith antenna which is beingiŽ . 1computed in the k q 1 th iteration. Ý represents the sum-

mation of the elements which have been computed beforeŽ . 2the ith antenna in the k q 1 th iteration, and Ý represents

the summation of those which have been calculated in thek th iteration and will be solved after the ith element in

Ž .the k q 1 iteration. It is planned that every antenna in theŽ .array is calculated once by Eq. 3 in one step of the iteration.

In that way, the currents of antennas are solved one by one inone step of the iteration and are iterated in the next.

ŽAlthough the currents on different antennas not onlytheir magnitude, but also their distribution laws along the

.antennas are different, the distribution laws of the currentalong antennas are similar in a phased array of antennas ofthe same size, and the distribution of the magnitudes of thecurrents is similar to that of the excitations, except for someantennas near an edge or a corner. The initial currents ofthis iterative method are determined according to this phe-nomenon. Assuming that the distribution laws of the initialcurrents on all antennas are the same as that on a selectedtypical element, called the NCth antenna, and their ampli-tudes and phases between antennas are in the same distribu-tion as that of their excitation voltages, leads to

0 t t 0 Ž .I s V rV ? I , j s 1,2,3, . . . , N 4Ž .j j NC NC

where I 0 is the initial current on element NC and I 0 isNC jŽ . Ž .that on the jth. Substituting Eq. 4 into Eq. 1 yields the

equation for the initial current on the NCth antenna:

Nt t 0 t Ž .Z ? V rV ? I s V V . 5Ž .Ý NCj j NC NC NC NC½ 5

js1

Then the initial currents on other antennas are determinedŽ .by Eq. 4 . In that way, a good approximation of the currents

is set as the initial currents. The iteration method defined byŽ .Eq. 3 can begin.

To accelerate the convergence of the iteration, the typicalelement NC should be selected in the center of the array.The solution route in the first step of the iteration should bedesigned as from the central area of an array, where thedistribution laws of the currents along the antennas arealmost the same, to the edges and corners where greaterdifferences between elements appear.

3. NUMERICAL EXAMPLES

First, the reliability of this method is proved by analyzingmany types of arrays, even Yagi]Uda arrays. All of thosecomputations of small arrays show that the results of themethod proposed in this paper converge to the values ob-

Ž .tained via inverting Eq. 1 by Gaussian elimination.A 49 = 55 planar array of parallel dipoles placed 0.25l

high over an infinite conducting plane is analyzed. Here, l isthe wavelength. The elements are placed on a rectangulargrid. The full length of the dipole is 0.5l, and the radius ofthe dipole is 0.01l. The element spacing in the E-plane is0.55l and in the H-plane is 0.5l. The magnitudes of theexcitation voltages are in a 40 dB Taylor distribution inthe H-plane and a 35 dB one in the E-plane. The phasesof the voltages scan the main beam at 308 from broadsidein the H-plane. A system of simultaneous linear equations as

Ž .Eq. 1 is created from the electric field integral equationvia Galerkin’s procedure. Eleven sinusoidal expansions perelement are used for the relatively thick dipole. Thus, thereare 2695 dipoles in the array and 29,645 expansion coeffi-cients for solution.

All of the elements of the array are solved, and all of thecouplings between the antennas are taken into account.

Ž .Dipole 25, 28 , the center element, is chosen as the initialstandard element. The calculation route in iterations is de-signed as: first from the center of the array to the edges andcorners, second from the corners to the center, third from thecenter to the corners, and so on. Table 1 lists the computedinput impedances of the center element and some typicalones near one corner in the iterations. In its last line are theconvergent values. It is found that the calculated data arearound their convergent values after just a few iterations,especially those for antennas in the inner area of the array.The results after ten iterations are accurate enough for thisproblem with as many as 29,645 unknowns. The methodproposed is very efficient and reliable. As expected, theresults of the dipoles close to corners or edges have some-what worse convergence than the center ones. That is be-cause the corner elements are in rapidly varying electromag-netic circumstances and have much smaller currents. Theyare strongly influenced by the errors of the computed cur-rents on adjacent central dipoles.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 21, No. 1, April 5 1999 45

TABLE 1 Input Impedances of Dipoles in a 49 ===== 55 Scanned Array; L = 0.5l , a = 0.01l , d = 0.55l , d = 0.5l , u = 300e h ma x

Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .k Z 25, 28 V Z 3, 3 V Z 2, 2 V Z 1, 1 Vm m m m

0 204.57 y j22.731 194.55 y j14.55 246.81 y j6.50 135.83 q j57.582 204.54 y j23.00 185.03 y j2.77 205.60 y j64.453 196.70 y j1.90 205.69 y j72.37 137.04 q j58.804 204.48 y j22.85 194.76 y j0.64 204.96 y j68.505 197.76 y j0.15 202.39 y j64.09 137.57 q j58.796 204.53 y j22.86 197.73 y j1.59 203.43 y j64.657 197.32 y j1.72 203.81 y j64.22 137.69 q j58.698 204.51 y j22.89 197.21 y j1.80 203.95 y j64.269 196.87 y j1.66 203.96 y j64.51 137.68 q j58.68

10 204.50 y j22.89 196.93 y j1.59 203.95 y j64.5170r71 204.50 y j22.88 196.98 y j1.56 203.90 y j64.54 137.67 q j58.69

4. CONCLUSIONS AND DISCUSSIONS

An efficient iterative method for analyzing large but finitephased arrays is proposed. Compared with other iterativemethods solving the system of linear equations in analyzinglarge and finite phased arrays, the proposed method avoidscalculating the large matrices and requires considerably fewercomputations. The numbers of multiplications and additions

Ž .2in one step of the iteration are easy to estimate with MNas their highest order term, which is just half of the conjugate

w xgradient method 4 , where M is the number of expansionsper antenna. Moreover, this method chooses a close approxi-mation of the currents initially, makes use of the physicalfeatures of mutual couplings in arrays, and needs consider-ably fewer iterations for convergence.

The computation of this method can be further reducedby rearranging the number of iterations on different antennaelements. The antennas in the inner part of an array have amuch better convergence than those near a corner or anedge, and need considerably fewer iterations for the sameaccuracy, as shown in Table 1. Otherwise, this method has anexplicit physical meaning, and can easily be adapted to otherapplications such as analyzing nonlinearly spaced arrays orarrays of unequally sized antennas.

ACKNOWLEDGMENT

This work was supported by the National Natural ScienceFoundation of China.

REFERENCES

1. R.F. Harrington, Field computation by moment method, Macmil-lan, New York, 1968.

2. R.C. Hansen and D. Gammon, A Gibbsian model for finiteŽ .scanned arrays, IEEE Trans Antennas Propagat AP-44 1996 ,

243]248.3. D.M. Pozar, Finite phased arrays of rectangular microstrip patches,

Ž .IEEE Trans Antennas Propagat AP-34 1986 , 658]665.4. T.K, Sarkar and K.R. Siarkiewicz, Survey of numerical methods

for solution of large systems of linear equations for electromag-Ž .netic field problem, IEEE Trans Antennas Propagat AP-29 1981 ,

847]856.5. N. Amitay, V. Galindo, and C.P. Wu, Theory and analysis of

phased array antennas, Wiley, New York, 1972.6. R.C. Hansen, Evaluation of the large array method, Proc Inst

Ž .Elect Eng 137 1990 , 94]98.7. L. Gurel and W.C. Chew, A recursive algorithm for strips and

Ž .patches, Radio Sci 17 1992 , 387]401.8. A.K. Hamid, I.R. Ciric, and M. Hamid, Iterative solution of the

scattering by an arbitrary configuration of conducting or dielectricŽ .spheres, Proc Inst Elect Eng 148 1991 , 565]572.

9. W. Chen, C.F. Xie, and S.M. Zhang, Accurate pattern synthesis ofparallel dipole arrays and its application to unequal elements,

Ž .Electron Lett 24 1988 , 386]387.

Q 1999 John Wiley & Sons, Inc.CCC 0895-2477r99

MUTUAL COUPLING BETWEENRECTANGULAR MULTISEGMENTDIELECTRIC RESONATOR ANTENNASB. Henry,1 A. Petosa,2, Y. M. M. Antar,1 and G. A. Morin31 Department of Electrical and Computer EngineeringRoyal Military College of CanadaKingston, Ont. K7K 7B4, Canada2 Communications Research CentreOttawa, Ont. K2H 8S2, Canada3 Defence Research Establishment OttawaOttawa, Ont. K1H 0Z4, Canada

Recei ed 25 August 1998

ABSTRACT: Mutual coupling between two rectangular multisegmentdielectric resonator antennas is in¨estigated in both the E- and H-planes.Spacing distances up to 4.5l are considered, with comparisons madebetween numerical TLM simulations and experimentation. Good agree-ment is achie ed, with representati e plots presented. The in¨estigationis restricted to examining the dominant TE -mode, for which the111radiation pattern resembles that of a magnetic dipole. Q 1999 JohnWiley & Sons, Inc. Microwave Opt Technol Lett 21: 46]48, 1999.

Key words: dielectric resonator antennas; mutual coupling

INTRODUCTION

Ž .Dielectric resonator antennas DRAs have been suggestedas a potentially efficient antenna for use in personal commu-nication systems, global positioning systems, and wirelesscomputer networks. They offer several advantages, such ashigh efficiency, large bandwidth, low profile, small size, easeof fabrication, and low production cost. A good synopsis of

w xDRAs can be found in 1 .Ž .The multisegment dielectric resonator antenna MSDRA

has been found to offer very good coupling to microstriplines, and has been used as an element in various linear and

w xplanar arrays 5]7 . The MSDRA is an element consisting ofŽa low-permittivity dielectric positioned on top of a second or

. Ž .several insert s with higher permittivity. The higher permit-tivity dielectric insert provides a better impedance match to

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 21, No. 1, April 5 199946