158 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 41, NO. 2, MAY 1999

This means that the V-dipole antenna with the coaxial balun is usefulfor wide-band electromagnetic sensors.

VIII. C ONCLUSION

A V-dipole antenna with coaxial balun has been investigated. TheCAF was calculated from the effective length and input impedanceof antenna element and the electrical length of a coaxial feeder. Thecalculated and measured results were compared and discussed. As theresult, it has been shown that this V-dipole antenna can be used forwide-band electromagnetic sensors and the CAF is easily calculable.

ACKNOWLEDGMENT

The authors would like to thank K. Fujii for valuable discussions.

REFERENCES

[1] S. Ishigami, R. Gokita, I. Yokishima, and T. Iwasaki, “Measurement offast transient fields in the vicinity of short gap discharge,”IEICE Trans.Commun., vol. E78-B, no. 2, pp. 199–206, 1995.

[2] S. Ishigami, H. Iida, and T. Iwasaki, “Measurements of complex an-tenna factor of near-field 3-antenna method,”IEEE Trans. Electromagn.Compat., vol. 38, no. 3, pp. 424–432, 1996.

[3] M. Kanda, “Transients in a resistively loaded linear antenna comparedwith those in a conical antenna and a TEM horn,”IEEE Trans. AntennasPropagat., vol. AP-28, no. 1, pp. 132–136, Jan. 1980.

[4] K. P. Esselle and S. S. Stuchly, “A broad-band resistively loaded V-antenna: Experimental results,”IEEE Trans. Antennas Propagat., vol.39, pp. 1587–1591, Nov. 1991.

[5] T. Ikeda and T. Nakamura, private communication.[6] H. Hosoyama, T. Iwasaki, and S. Ishigami, “Evaluation of the complex

antenna factor of dipole antenna by measuring theS-parameter of thebalun,” Trans. Inst. Elect. Eng. Japan, vol. 117-A, no. 5, pp. 509–514,1997 (in Japanese).

[7] K. Fujii, S. Ishigami, and T. Iwasaki, “Evaluation of complex antennafactor of dipole antenna by the near-field 3-antenna method with themethod of moment,”Electron. Commun. Japan, vol. 80, pt. 1, no. 11,pp. 34–43, 1997.

[8] K. Fujii, S. Ishigami, T. Iwasaki, and S. Usuda, “Measurements of an-tenna factor of log-periodic dipole-array antenna by near-field referenceantenna method,”Trans. IEICE, vol. J80-B-II, no. 12, pp. 1091–1098,1997 (in Japanese).

Loaded TEM Cell Versus Free Space: Comparisonof the E-Fields Inside Dielectric Spherical Objects

Michele M. G. D’Amico

Abstract—A Crawford cell is a convenient tool for generating transverseelectromagnetic fields. In this work, a dielectric hemisphere is assumedto be placed on the central septum of a Crawford cell; the field inside thehemisphere is numerically evaluated and then compared with that insidethe corresponding sphere in free-space (illuminated by a plane wave) forwhich the analytical solution is known.

Index Terms—EMC measurements, TEM cell.

Manuscript received December 29, 1997; revised October 5, 1998.The author is with the Dipartimento di Elettronica e Informazione, Politec-

nico di Milano, Milano 20133, Italy.Publisher Item Identifier S 0018-9375(99)02068-2.

Fig. 1. Cross-sectional view of the Crawford cell.

Fig. 2. Longitudinal section of the Crawford cell with the DUT in place.

I. INTRODUCTION

Crawford cells (a particular kind of TEM cell) have been widelyused since their formal introduction in 1974 [1]; the electromagneticfield configuration inside the cell has been widely investigated forthe unloaded as well as for the loaded [2]–[4] TEM cell.

However, it is recognized that the proper question to ask is whetherthe test object is equivalently stressed when placed in a TEM cell asit would be in free-space [5]. In this work, we will address this point;i.e., our objective will be to investigate how close the E-fieldinsidean object when placed in a TEM cell resembles that inside the sameobject when in free-space.

For this purpose, we have designed a TEM cell using standardformulas found in the open literature. We have then assumed to placea dielectric object inside the cell on the central conducting septum.The shape of the object is chosen so that the field inside the objectcan be analytically computed, when the object is illuminated by aplane wave propagating in free-space.

The electric field inside the object is evaluated using a commercialthree-dimensional (3-D) electromagnetic wave simulator (HFSS)1;this simulator uses a finite-element technique to evaluateS-parameters and electromagnetic (EM) field distribution for passive3-D structures. This field is then quantitatively compared with thatobtained using the analytical solution for free-space.

The cell is designed to have a cutoff frequency of 200 MHz for thefirst non-TEM mode. Moreover, the cell has to be symmetrical and itscharacteristic impedance (when empty) is chosen to beZ0 � 50 .For the calculation of the characteristic impedanceZ0 we use theequations given in [6], while for the calculation of the higher ordermodes cutoff frequencies we refer to [7]. Using jointly the conditionsonZ0 and the cutoff frequency, we obtain the following dimensional

1HFSS (High-Frequency Structure Simulator) is a trademark of the Hewlett-Packard Company

0018–9375/99$10.00 1999 IEEE

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 41, NO. 2, MAY 1999 159

Fig. 3. E-field inside DUT D at 100 MHz, evaluated using a Mie theorysolution (continuous line) and an HFSS simulation (dashed line).

TABLE IGEOMETRICAL AND ELECTRICAL CHARACTERISTICS OF THETEST OBJECTS

values (see Figs. 1 and 2):2a = 48:2 cm; 2b = 43:4 cm; 2w = 39cm; H1 = 48:2 cm; andt = 1:5 mm. As far as the lengthH2 ofthe tapered transitions is concerned, we follow the design given in[8] and obtainH2 = H1=2 = 24:1 cm. The TEM cell is terminatedwith an ideal 50 load.

II. TEM CELL VERSUS FREE-SPACE

In the real world there is an almost unlimited choice for the shapeand dielectric characteristics of the DUT’s. We wanted to consider anontrivial shape and at the same time have an analytical solution forthe fields inside the object in free-space.

In this respect, the spherical shape seems to be a good choicesince a rigorous scattering theory exists [9]. When the DUT is placeddirectly on the central conducting septum, its free-space equivalent isfound by applying an image theorem. For our purposes, we placed adielectric hemisphere on the central conductor inside the cell whichcorresponds to a dielectric sphere in free-space.

We consider two different values for the diameterD—specifically9 and 16 cm. Choosing two different values for the dielectric constantwe get a total of four different objects (A to D; see Table I).Simulations and calculations are carried out at the frequencies of100, 200, and 400 MHz. For the first two frequencies, only the TEMmode can propagate, while at 400 MHz at least two higher ordermodes (TE01 and TE10) can exist.

To compare in a quantitative way the values of the electromagneticfields inside the object, it is necessary to establish a correspondencebetween the power injected into the TEM cell and the power densityassociated with a TEM wave propagating in free-space. Crawford andWorkman [10] propose the following very simple expression for theevaluation of the E-fieldE0 in an empty TEM cell:

E0 =P=Gc

b(V m�1) (1)

TABLE IIABSOLUTE VALUES OF THE E-FIELD, EVALUATED IN O FOR

THE DIFFERENT DUT’ S. INPUT POWER TO THE CELL IS 4 W

Fig. 4. As Fig. 3, but for DUT D at 200 MHz.

whereP is the net input power to the cell,Gc is the real part of thecharacteristic admittance of the cell, andb is the separation distancebetween the center conductor and the upper wall of the cell. Thisvalue ofE0 in the empty cell can be associated to the value of theE-field of a TEM wave propagating in free-space and illuminatingthe object.

All the simulations have been carried out assuming an input powerto the cellP = 4 W. Using this value in (1), we obtainE0 = 67:2V m�1. This is only an average value; in fact, the E-field is notconstant along thez axis, being higher on the central septum: usingHFSS we obtainE0 = 77:1 V m�1 for z = 0 andE0 = 57:5 Vm�1 for z = 21:7 cm. When using HFSS, the number of the nodesof the mesh has been adjusted to obtain an absolute accuracy betterthan 0.05 dB; the complete cell, including the tapered transitions, hasbeen considered in the simulation.

The E-field inside a DUT is evaluated along lineL (Fig. 2), veryclose to the central septum (1 mm). The center of the hemisphereis coincident with the originO of the reference frame. The absolutevalue of the E-fields inO for the different DUT’s at the variousfrequencies are collected in Table II.

Figs. 3–5 show the ratio (in decibels) between the E-field (alongline L) theoretically evaluated using Mie theory (solid lines) andobtained from 3-D simulations (dashed lines). The fields are referredto those calculated with Mie theory at the originO.

160 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 41, NO. 2, MAY 1999

Fig. 5. As Fg. 3, but for DUT D at 400 MHz.

From these figures it is clear that below 200 MHz the discrepancybetween the theoretical (free-space) and simulated (TEM cell) E-fields is mainly due to the underestimation ofE0 in the empty cellintroduced by (1). The actual value forE0 close to the septum is 77.1V m�1—about 1.2 dB higher than that obtained through (1). If wecompensate for it, the difference between free-space and TEM cellfields is smaller than 1 dB.

Things start to change at 200 MHz and, at 400 MHz, the discrep-ancies are no longer negligible: the general behavior of the E-field infree-space is similar to that of the field in the cell but differences ofseveral decibels can be observed. These differences tend to increasefor increasing values of the dielectric constant. For a large berillium("r = 6) sphere (DUTD) the maximum difference is about 3 dB(4.2 dB if the offset inE0 is taken into account); this differenceincreases to about 5 dB (6.2 dB) if silicon is considered("r = 12).

III. CONCLUSIONS

In this work, we have compared the E-fieldinside a dielectrichemisphere placed on the central septum of a Crawford cell (evaluatedthrough a 3-D EM simulator) to that inside the equivalent object infree-space (for which an analytical solution is known).

Good agreement between theoretically derived and simulated fieldsis found when the cell is operated below the cutoff frequency of thefirst higher order mode. The discrepancy is less than 1 dB providedthat theE0 field in the empty cell is correctly evaluated. Beyondthis frequency limit the difference increases. The general behavior ofthe E-field in free-space is similar to that of the field in the cell butdiscrepancies of several decibels can be observed.

ACKNOWLEDGMENT

The authors are grateful to Prof. M. Politi for his valuable sugges-tions and help during the development of this work.

REFERENCES

[1] M. L. Crawford, “Generation of standard EM fields using TEM trans-mission cells,”IEEE Trans. Electromagn. Compat., vol. EMC-16, pp.189–195, Nov. 1974.

[2] M. Kanda, “Electromagnetic-field distrortion due to a conducting rect-angular cylinder in a transverse electromagnetic cell,”IEEE Trans.Electromagn. Compat., vol. EMC-24, pp. 294–301, Aug. 1982.

[3] A. Kucharski, “Electromagnetic field inside loaded TEM cells,” inProc.Int. Wroclaw Symp. Electromagn. Compat., Wroclaw, Poland, June 1990,pp. 345–349.

[4] P. Wilson and F. Gassmann, “Theoretical and practical investigation ofthe field inside a loaded/unloaded GTEM cell,” inProc. 10th Int. ZurichSymp. EMC, Zurich, Switzerland, Mar. 1993, pp. 595–598.

[5] P. Wilson, “A review of TEM cell development,” inProc. Int. Wro-claw Symp. Electromagn. Compat., Wroclaw, Poland, June 1994, pp.206–209.

[6] J. C. Tippet and D. C. Chang, “A new approximation for the capacitanceof a rectangular coaxial strip transmission line,”IEEE Trans. MicrowaveTheory Tech., vol. MTT-24, pp. 602–604, Sept. 1976.

[7] P. F. Wilson and M. T. Ma, “Simple approximate expressions for higherorder mode cutoff and resonant frequencies in TEM cells,”IEEE Trans.Electromagn. Compat., vol. EMC-28, pp. 125–130, Aug. 1986.

[8] W. T. Decker, M. L. Crawford, and W. A. Wilson, “Construction ofa transverse electromagnetic cell,” Tech. Rep., U.S. Dept. Commerce,Nov. 1978, NBS Tech. Note 1011.

[9] H. C. Van De Hulst,Light Scattering by Small Particles. New York:Wiley, 1957.

[10] M. L. Crawford and J. L. Workman, “Using a TEM cell for EMC mea-surements of electronic equipment,” Tech. Rep., U.S. Dept. Commerce,1981, NBS Tech. Note 1013.

A Higher Order (2,4) Scheme for ReducingDispersion in FDTD Algorithm

Kang Lan, Yaowu Liu, and Weigan Lin

Abstract—A finite-difference time-domain (FDTD) scheme with second-order accuracy in time and fourth-order in space is discussed for thesolution of Maxwell’s equations in the time domain. Compared withthe standard Yee FDTD algorithm, the higher order scheme reducesthe numerical dispersion and anisotropy and has improved stability.Dispersion analysis indicates that the frequency band in which the higherorder scheme yields an accurate solution is widened on the same grid. Thismeans a larger space increment can be chosen for the same excitation.Numerical results show the applications of the scheme in modeling wide-band electromagnetic phenomena on a coarse grid.

Index Terms—Absorbing boundary conditions, electromagnetic scatter-ing, FDTD method, numerical dispersion.

I. INTRODUCTION

The Yee’s finite-difference time-domain (FDTD) formulas [1] areconservative and dispersive with second-order accuracy both in timeand space. The numerical dispersion is the dominant limitation to theaccuracy of FDTD method. Generally, the dispersion is negligible ifwe choose the grid spacing finely enough. However, the applicablefrequency band is limited and large computer memory is needed,which limits the applications of the FDTD method in modelingelectrically large structures.

Manuscript received August 23, 1996; revised November 25, 1998.K. Lan and W. Lin are with the Institute of Applied Physics, University of

Electronic Science and Technology of China, Chengdu, 610054 China.Y. Liu is with the Department of Electronic Engineering, City University

of Hong Kong, Kowloon, Hong Kong.Publisher Item Identifier S 0018-9375(99)02069-4.

0018–9375/99$10.00 1999 IEEE