Ee at 0 = 0” 0

0” *--3

-90” 90”

(b)

Figure 6 Theoretical and measured radiation patterns of (a) EH at 4 = 0” and (b) E, at (p = 90” of a DRA with er = 14.2, a = 9.5 mrn, and d = 3.0 mm. Solid lines, theoretical; dashed lines, measured

TABLE 2 Theoretical and Experimental Results of a DRA with cr = 14.2, a = 9.5 mm, and d = 3.0 mm

Parameter Theoretical Experimental ~ ~ -

Resonant frequency 7.074 GHz 6.700 GHz Antenna gain 3.6 3.4 Q factor 7.4 7.8 Bandwidth (VSWR S 2.5) 12.8% 12.2% Efficiency 100% 93 %

increases. The Q factor, however, decreases as the height decreases and as the radius increases.

Experimental results for two DRAs showed that the mag- netic wall approximation employed gives a good prediction in radiation pattern and directivity and a fairly good prediction in resonant frequency and Q factor. Better agreement be- tween theoretical and experimental results was observed for the short DRA, indicating that this model operates better for disc-type DRAs. More accurate prediction in these parame- ters would require the use of numerical methods. Efficiency measurements have shown that the DRA is a very efficient antenna, the wide bandwidth and high gain make it a pre- ferred choice in many antenna applications.

REFERENCES 1. S. J. Fiedziuszko, “Microwave Dielectric Resonators,” Microwaue

2. S. A. Long, M. W. McAllister, and L. C. Shen, “The Resonant J . , Sept. 1986, pp. 189-200.

Cylindrical Dielectric Cavity Antenna,” IEEE Trans. Antennas Propagat., Vol. A€-31, May 1983, pp. 406-412.

3. M. W. McAllister, S. A. Long, and G. L. Conway, “Rectangular Dielectric Resonator Antenna,” Electron. Lett., Vol. 19, March

4. M. W. McAllister and S. A. Long, “Resonant Hemispherical Dielectric Antenna,” Elecfron. Lett., Vol. 20, Aug. 1984, pp. 657-659.

5. R. K. Mongia, A. Ittipiboon, P. Bhartia, and M. Cuhaci, “Electric Monopole Antenna Using a Dielectric Ring Resonator,” Elec- Iron. Lett., Vol. 29, Aug. 1993, pp. 1530-1531.

6. R. K. Mongia and P. Bhartia, “Dielectric Resonator Antennas-A Review and General Design Relations for Reso- nant Frequency and Bandwidth,” Int. J. Microwaue Millimetre- Wace Cornput.-Aided Eng., Vol. 4, 1994, pp. 230-247.

7. J . T. H. St. Martin, Y. M. M. Antar, A. A. Kishk, A. Ittipiboon, and M. Cuhaci, “Dielectric Resonator Antenna Using Aperture Coupling,” Electron. Lett., Vol. 26, Nov. 1990, pp. 2015-2016.

8. R. A. Kranenburg and S. A. Long, “Microstrip Transmission Line Excitation of Dielectric Resonator Antennas,” Electron. Lett., Vol. 24, Sept. 1988, pp. 1156-1157.

9. R. A. Kranenburg, S. A. Long, and J. T. Williams, “Coplanar Waveguide Excitation of Dielectric Resonator Antennas,” IEEE Trans. Antennas Propagat., Vol. AP-39, Jan. 1991, pp. 119-122.

10. Y. Kobayashi and S. Tanaka, “Resonant Modes of a Dielectric Rod Resonator Short-circuited at Both Ends by Parallel Con- ducting Plates,” IEEE Trans. MicrowaL,e Theory Tech., Vol. MlT-

11. R. N. Simons and R. Q. Lee, “Effect of Parasitic Dielectric Resonators on CPW/Aperture-Coupled Dielectric Resonator Antennas,” IEE Roc. Pt. H , Vol. 140, Oct. 1993, pp. 336-338.

12. D. Kajfez and P. Guillon, Dielectric Resonators, Artech House, Dedham. MA, 1986.

13. R. F. Harrington, Time-Harmonic Electromagnetic Fields, Mc- Graw-Hill, New York. 1961.

14. J. D. Kraus, Antennas (2nd ed.), McGraw-Hill International Editions, New York. 1988.

15. C. A. Balanis, AdL>anced Engineering Electromagnetics, Wiley, New York, 1989.

16. 2. Wu and L. E. Davis, “Automation-Oriented Techniques for Quality-Factor Measurement of High-T, Superconducting Res- onators,” IEE Proc.-Sci. Meas. Technol., Vol. 141, Nov. 1994,

17. H. A. Wheeler, “The Radian Sphere around a Small Antenna,”

1983, pp. 218-219.

28, Oct. 1980, pp. 1077-1085.

pp. 527-530.

IRE Proc., Aug. 1959, pp. 1325-1331.

Received 5-1 4- 96

Microwave and Optical Technology Letters, 13/3, 119-123 0 1996 John Wiley & Sons, Inc. CCC 0895-2477/96

ON THE USE OF SYMMETRY TO REDUCE THE COMPUTATIONAL REQUIREMENTS FOR FDTD ANALYSES OF FINITE PHASED ARRAYS David Crouch Hughes Missile Systems Company Advanced Electromagnetic Technologies Center Rancho Cucarnonga, California 91 729

KEY TERMS Phase arrays, symmetry, antennas, FDTD

MICROWAVE AND OPTICAL TECHNOLOGY LEnERS / Vol. 13, No. 3, October 20 1996 123

ABSTRACT S~inmet17e.s of jzriite phawd arrays (ire e.vploired to reduce the memor? and nit7 time reqirired for their atialy.sis nith the finite-difjerence rime- domaiti (FDTDI method. Resirltr arrpresetitedfor u 2 X 2 uray of n.ire dipoles arid are compared Lvith re.culis calcitlated wit17 the method of momerits arid the coni.etitioiinl FDTD triethod. 'Q 1YY6 Johri Wiky CC Soris, lnc.

1. INTRODUCTION

Historically, the analysis of finite phased arrays has been a difficult problem. Large arrays can with some justification be approximated by infinite arrays. which are amenable to n u - merical treatment when the periodicity of the array is ex- ploited. For small- and medium-sized arrays, however, edge effects become important, mutual coupling varies substan- tially from element to element. and the infinite-array approxi- mation breaks doun. Finite phased arrays can be treated numerically with the method of moments or the finitc- element method: these methods. however, are frequency-do- main methods, and separate analyses must be carried out at each lrequency of interest. The finite-difference-timc-do- main (FDTD) method, as its name implies. is a time-domain method and is not burdened by the same limitation: array performance data at multiple frequencies can be obtained from a single FDTD run . Although the FDTD method has been used to analyze individual antennas [IF41 and infinite phased arrays [S. 61. i t has rarely been applied to the analysih of tinite phased arrays. because of large memory and run-time requirements [7 ] . Because conventional FDTD analyses of small arrays of complex elements or of larger arrays of simple elements require large amounts of computer memory and CPU time, it is o f interest t o explore any means o f reducing the computational rcsources rcquired for their analysis.

In this article symmetry is used to reduce the size o f the computational grid required for the FDTD analysis of finite phased arrays having one or more orthogonal planes of symmctry. Section I1 presents the thcory for the symmetry- aided FDTD solutions for finite planar arrays, showing how the symmetry planes of a n array can be replaced with combi- nations of electric and magnetic walls. and how the radiated fields for arbitrary excitations of an array can be calculated through the use of superposition [8. Y]. The use of symmetry in the FDTD method is validated in Section 111. where the symmetiy-aided FDTD method is applied to a 2 X 2 array of wire dipoles. The results are conipared to those calculated with the method o f moments and with the conventional FDTD method; the latter comparison is used to determine the savings in memory and CPU timc achieved by exploiting the symmetry o f the array.

11. THEORY Symmetry is often used in FDTD analyses to reduce the size o f the computational grid. Consider. for example, a rectangu- lar waveguide excited in its TE, , , mode. This particular mode is symmetric about planes bisecting both the narrow and wide dimensions o f the guide. labeled AA' and BB', respectively. in Figure l(a). As pictured, AA' is a magnetic image plane. and BB' is an electric image plane. As long as no obstruc- tions capable of breaking the symmetry are introduced, one can replace AA' with a magnetic wall and BB' with an electric wall and directly calculate the fields in one quadrant only of the wavcguidc, thereby realizing a factor of 4 reduc- tion in the memory and run-time requirements [Figure l(b)].

A I

B B'

B

Figure 1 (a) The symmetry planes of a rectangular waveguide escitcd in the TE,,, mode. AA' is a magnetic image plane, and BB' is a n electric image plane. (h) The size of the FDTD grid can he reduced by a factor of 4 by taking advantage of symmetry; only the lower left quadrant need be included in the grid

The same principle can be applied in the analysis of finite phased arrays possessing one or more planes of symmetry. Figure 2 illustrates a two-element array having a single plane of symmetry between the elements, labeled AA'. Notice that AA' must he a plane of symmetry with respect to the exeita- tions as well as with respect to the array itself. For purposes of illustration, suppose that the two elements are vertically polarized and that they are simultaneously excited by identi- cal wave forms. Under such conditions, AA' is a magnetic image plane: by replacing AA' with a magnetic wall, one can calculate the fields @ ( M ) radiated by the symmetrically ex- cited two-element array by including only one-half of the array in the FDTD grid [Figure 2(a)]. Now suppose that the simultaneous excitations of the two elements are equal and opposite, so that AA' is an electric image plane [Figure 2(b)]. By replacing AA' with an electric wall. one can in a similar manner calculate the fields @ ( E l radiated by the antisym- metrically excited array. Thc two field patterns @ ( E ) and @ ( M are linearly independent; the radiated fields for an arbitrary excitation of the array can be represented by a linear combination of @ ( E l and W M ) . For example, the fields radiated when element 1 only is excited are obtained by adding @,(El to @ ( M I : to obtain the fields radiated when element 2 only is excited, one simply subtracts Q ) ( E ) from W M ) . If the single-element field patterns radiated when elements 1 and 2 are individually excited are represented by @, and a?, respectively, then

Using the conventional FDTD method, one would include the entire array in the grid, and directly calculate the single-

124 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol 13. No 3 . October 20 1996

A I I I I I

1 / 2 I I I

A' In-Phase Excitation:

A I

' I 2 A'

180" Out-of-Phase Excitation: Figure 2 A two-element phased array having a single symmetry plane AA'. (a) In-phase excitation; for this case, AA' is a magnetic image plane. (b) 180" out-of-phase excitation; for this case AA' is an electric image plane

element patterns and Q2 [8, 91. If the symmetry of the array is exploited, howcvcr, only one half of the array nccd be accounted for in the computational grid; as a result, the grid size is reduced by a factor of 2, which in turn reduces the memory and run-time requirements by the same factor.

Now consider a 2 X 2 planar array of radiating elements like that shown in Figure 3. This array has two planes of symmetry. labeled AA' and BB'. If the excitations of the array elements are such that AA' is a magnetic (electric) image plane, then a magnetic (electric) wall can be placed at AA', and that part of the array opposite AA' can be elimi- nated from consideration. The same considerations apply to the plane BB'. By making full use of symmetry, only one

quarter of the array need be directly included in the FDTD grid.

There are then four possible combinations of boundary conditions by which the symmetry planes can be replaced. The field pattern produced by the array when AA' and BB' are replaced with a combination of electric and magnetic walls will be represented by WAA' = E, M , BB' = E, M I . For example, if AA' is replaced by an electric wall and BB' by a magnetic wall, the resulting field pattcrn is represented by W E , M ) . The four combinations of electric and magnetic wall boundary conditions and the resulting field patterns are summarized in Table 1.

The set of excitations described above is a complete set for describing radiation from the array, just as the singlc- elcment set is obtained by exciting each element individually in the presence of all the other elements. One set can easily be obtained from the other:

where are the single-element field patterns due to elements 1-4. Any time- or frequency-domain pattern can he constructed with the use of time- or phase-delayed versions of -4. In the time domain

where A I and T~

time delays, respectively. In the frequency domain arc an arbitrary set of amplitudes and

A

l

A'4' BB' Field Pattern

A' Figure 3 A 2 X 2 array having two planes of symmetry, AA' and BB'

where 4,Jw) is the Fourier transform of @ J t ) and (p_7-4 is an arbitrary set of phase shifts.

The same principle can be applied to the analysis of larger arrays. For example, consider the 4 X 4 array shown in Fig- ure 4. The symmetry planes now subdivide the array into a set of four 2 X 2 subarrays. As before, there are four possible combinations of boundary conditions by which the symmetry planes can be replaced. Each subarray, however, cannot be

TABLE 1 Associated Field Patterns for a Two-by-Two Phased Array

Boundary Condition Combinations and Their

Magnetic wall Magnetic wall W M , M ) Magnetic wall Electric wall @( M , E ) Electric wall Magnetic wall @( E, M ) Electric wall Electric wall W E , E )

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 13, No. 3, October 20 1996 125

? A

A' A '

Figure 4 A 1 x 4 array having two planes of symmety. AA' and BB'

Figure 5 A 3 x 3 array having two planes of symmetry, AA' and BB'. each of which intersects three array elements

further subdivided. For each of the four combinations of boundary conditions, then. four FDTD runs must be carried out. each with only one of the four elements in thc included subarray excited. The resulting 16 field patterns are denoted by @,,(AA' = E , M. BB' = E. M ) , where I Z = 1 . . . . , 4 dc- notes the excited array element in the lower left subarray (as numbered in Figurc 4). For n = 1, the four field patterns QI(M, M ) . @ J M , E ) , @,( E. E l , and @ J E , M ) are sufficient to completely characterize radiation from the array when only the corner elements 1. 6, 11. and I6 are excited. As with the 2 X 2 array, the single-element patterns can be recovered from these symmetric patterns:

required to recover the nine single-element patterns needed to completely characterize the array. The first step is to replace planes AA' and BB' with thc same four combinations of electric and magnetic walls used in the analysis of the 3 x 2 array and excite element 1 only. The four resulting tield patterns are sufficient to characterize radiation from the array with only the corner elements 1. 3, 7, and 9 excited. Using the same notation used in the discussion of the 2 X 2 array, one obtains the following expressions for the single- element patterns for the corner elements:

where a,. a,,. and @,, are the single-element field patterns radiated by the uray with only elements 1, 6. 11, and 16 excited. respectively. The other 12 single-element field patterns can be calculated in a similar fashion by following the same procedure but instead carrying out the calculation with element 2, 3, or 1 excited. From the single-element patterns, the pattern for an arbitrary excitation of the array can be calculated by superposition.

The use of symmetry is not limited to arrays with even numbers of rows and columns, but can also be applied to arrays with odd numbers of rows and/or columns, so that one or more o f its symmetry planes bisect some of the array elements, as is true of the 3 X 3 array shown in Figure 5 . Like the 3 X 2 and 1 X 4 arrays considered previously, this array h a s two symmetry planes, labeled AA' and BB'. Each symmetry plane, however, bisects three of the array elements. Because this array has nine elements. nine FDTD runs are

Now assume for the purposes of illustration that the individual element excitations are such that a horizontal plane bisecting an clement can be replaced with an electric wall and a vertical plane with a magnetic wall, as is true for a TEI,, waveguide mode, for example. With only element 4 cxcitcd and with plane BB' replaced with an electric wall. two runs with plane AA' replaced in turn with an electric wall and then with a magnetic wall yield sufficient information to characterize radiation from the array with only elements 4 and 6 excited. Similarly, with only element 2 excited and with plane AA' replaced with a magnetic wall, two runs with plane BB' replaced by an electric wall and thcn by a magnetic wall are sufficient to characterize radiation from the array with only elements 2 and 8 excited. Finally, with plane AA' replaced by a magnetic wall and plane BB' replaccd by an electric wall. a single additional run with only element 5 excited is sufficient to characterize radiation from the array when element 5 only is excited. The remaining five single-

126 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol 13 No 3 October 20 1996

element field patterns are then given by lo-, lo,.,

Taken together, the nine patterns calculated are a complete set with which the field pattern for an arbitrary excitation of the 3 x 3 array can be calculated.

111. RESULTS

The principles described in the previous section will be demonstrated by applying them to the 2 X 2 array of thin-wire dipoles shown in Figure 6. Each wire element is 29 cm in length and 0.25 cm in radius; the width of the array is 20 cm, and the height measured from feed to feed is 50 cm. The symmetry planes AA' and BB' are replaced with electric and magnetic walls as discussed in the previous section. Because the array has front-to-back symmetry, the plane of the array is itself a symmetry plane and is replaced with a magnetic wall, which in combination with the other symmetry planes reduces the size of the computational grid by a factor of 8 compared to that required by the conventional FDTD method using the same cell size.

A uniform Cartesian grid of size 30 X 30 X 60 is used, with Ax = Ay = Az = 1.0 cm and A t = 17.33 ps. A thin-wire subcell model [lo] is used to model the wire antenna. The free-space boundaries are terminated with the use of second-order Liao boundary conditions [ l l ] ; with the above grid dimensions, the Liao boundaries are 20 and 21 cells away from the array in the x and z directions, respectively. The elements of the array are excited by the time derivative of a

Z t

1/" Figure 6 A symmetric 2 x 2 array of wire dipoles. Here W = 20 cm, H = 50 cm, and L = 29 cm. Each wire has a radius of 0.25 crn

4 0 ' I -30; I 0 45 90 135 180 45 90 135 180

Theta (degrees) Theta (degrees)

10, I 10, I

Figure 7 E-plane gain patterns for a 2 x 2 array of wire dipoles obtained by superposition of the symmetric field patterns. Solid lines: symmetry-aided FDTD results. Dotted lines: method-of-moments results (NEC-2). (a) 0.5 GHz, (b) 1.0 GHz, (c) 1.5 GHz, (d) 2.0 GHz

Gaussian pulse;

where t, = 150 ps and the pulse is normalized to 1 V at its peak.

Four FDTD runs of 2048 time steps each were carried out to calculate the symmetric field patterns @(M, M ) , @ ( M , El , W E , M I , and W E , E); for each run, a frequency-domain near-to-far-field transformation was performed to calculate the E- and H-plane gain at 0.5, 1.0, 1.5, and 2.0 GHz, for which the corresponding FDTD grid resolutions are 60, 30, 20, and 15 cells per wavelength, respectively. The results were then compared to those calculated with method-of-moments (MOM) code NEC-2 [12]. Because the FDTD thin-wire ap- proximation does not accurately calculate the fields near the end of a thin wire. there is some ambiguity in the physical position of the wire endpoints. For purposes of comparison with the MOM results, the physical ends of the wires are assumed to lie beyond the last cell in which the thin-wire approximation is used [ 131. Excellent agreement between the two calculations is obtained when that length is taken to be half of a cell, which is used in the following results. The single-element field patterns obtained by exciting element 1 only, @,, are synthesized at each frequency through linear superposition of the four symmetric field patterns, using Eq. (2a). The single-element E- and H-plane patterns calculated with symmetry-aided FDTD and MOM are shown in Figures 7 and 8, respectively. One sees that there is excellent agree- ment between the FDTD- and MOM-calculated field patterns at all frequencies.

In order to further validate the use of symmetry and to gauge the memory and run-time savings, the array was also simulated with a conventional FDTD code in which symmetry is used only in the plane of the array. As with the symmetry- aided FDTD code, the plane of the array is replaced with a magnetic wall. The grid was expanded to 60 x 30 X 120 to maintain the same grid resolution as used in the symmetry- based code. This code was used to directly calculate the

MICROWAVE AND OPTICAL TECHNOLOGY LEnERS / Vol 13, No. 3, October 20 1996 127

lol I

-loo 45 90 135 180 -6A 45 90 135 1:O Phi (degrees) Phi (degrees)

1 0 45 90 135 180

2 -30’ 0 45 90 135 180

Phi (degrees) Phi (degrees)

Figure 8 H-plane gain patterns for a 2 x 2 array of wire dipoles ohtained by superpoxition of the symmetric field patterns. Solid lines: ymmrtn-aided FDTD rewlts. Dottcd lines: method-of-rnoments results (NEC-2). ( a ) 0.5 GHz. ( h ) 1.0 GHz. (c ) 1.5 GHz. ( d ) 2.0 GHz

single-element field patterns. The results are nearly identical for the two calculations: the real and imaginary parts of the single-element E, in the far field agree to at least three decimal places a t all frequencics. The conventional FDTD code requires 28.3 megabytes o f memory and 12.687 s of CPU time for a single run, whereas the symmetry-based code requires only 6.9 megabytes and 3,001 s of CPU time for a single run. These single-run results are summarized in Tablc 2. For either conventional or symmetry-aided FDTD. four runs arc required to calculate all four single-element field patterns. to which any desired set of amplitudes and phases can be applied. without the use of symmetry, 50,748 s of CPU are required for such a calculatiun, whereas only 12.004 seconds arc required when symmetry is used, yielding a memory and run-time savings of 3.1.

IV. CONCLUSIONS

The computational resources-memory and CPU time- required for FDTD simulations o f finite phased arrays can be reduced if the array and its excitation possess one or more planes of symmetry. Each plane of symmetry can be replaced with a n electric or a magnetic wall. eliminating that part of the structure on the other side of the wall from consideration and reducing the meniory and run-time requirements by a factor of 2. For a planar array with two planes o f symmetry, a factor of 4 reduction in the memory and run-time require- ments is obtained. This technique can also be applied to threc-dimensional arrays with one or more planes of symme- try. The FDTD analysis of a three-dimensional array having three planes o f symmcty can be carried out with one-eighth

TABLE 2 Performance Comparison Data for the FDTD Analysis of a 2 x 2 Phased Array of Wire Dipoles With and Without the Use of Symmetry

No Symmetry With Symmety ~

Number ot unknown\ 1,290,000 324,000 Mcmory required (MB) 28 4 6 9 CPU time per run (s) 12.687 3,001

the computational resources required for a conventional FDTD analysis.

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Computation of the Radiation from Simple Antennas Using the Finite-Difference Time-Domain Method,” IEEE Trans. Anfennas Propaguf., Vol. AP-38, July 1990, pp. 1059-1068.

2. A. Reineix and B. Jecko, “Analysis of Microstrip Patch Antennas Using Finite Difference Time Domain Method,” IEEE Trans. Antennas Propagut., Vol. AP-37, Nov. 1989, pp. 1361-1369.

3. P. A. Tirkas and C. A. Balanis, “Finite-Difference Time-Domain Method for Antenna Radiation,” IEEE Trans. Antennas Propagat., Vol. AP-40. March 1992, pp. 334-340.

4. R. J. Luehbers and J. Beggs. “FDTD Calculation of Wide-Band Antenna Gain and Efficiency,” IEEE Trans. Antennas Propagaf., Vol. AP-40. NOV. 1992, pp, 1403-1407.

5. D. T. Prescott and N. V. Shuley, “Extensions to the FDTD Method for the Analysis of Infinitely Periodic Arrays,” IEEE Microwaiv Guided Ware Lett., Vol. MGW-4, Oct. 1994,

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7. E. Thiele and A. Taflove, “FD-TD Analysis of Vivaldi Flared Horn Antennas and Arrays,” IEEE Trans. Antennas Propagat., Vol. AP-42. May 1994, pp. 633-641.

8. D. M. Sullivan, “Mathematical Methods for Treatment Planning in Deep Regional Hyperthermia,” IEEE Trans. Microwate 7heoty Tech., Vol. MTT-39, pp. 864-872, May 1991.

9. C. E. Reuter, E. T. Thiele, A. Taflove, M. J . Piket-May, and A. J. Fenn. ”Linear Superposition of Phased Array Antenna Near Field Patterns Using the FD-TD Method,” in loth Annual Re- I ,ien of Progress in Applied Conlputational Electromagnetics, Naval Postgraduate School, Monterey, CA. March 1994, Vol. 1, pp. 459-466.

10. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Dornuin Method for E/ectromagne/ic.% CRC Press, Boca Raton, FL. 1993.

I t . Z. P. Liao. H. I*. Wong, B. P. Yang, and Y. F. Yuan, “A Transmitting Boundary for Transient Wave Analyses,’’ Sci Sin. Ser. .4, Vol. 27, Oct. 1984, pp. 1063-1076.

12. G. J. Burke and A. J. Poggio, “Numerical Electromagnetics Code (NEC)-Method of Moments,” NOSC Tech. Document No. 116, Vol. 2. 1981.

13. J . J . Boonzaaier. “The Radiation and Scattering by Thin Wires: A Finite-Difference Time-Domain Approach,” Ph.D. disserta- tion. University of Pretoria, Pretoria, South Africa, 1994.

pp. 352-354.

Receii3ed 5-21-96

Microwave and Optical Technology Letters, 13/3, 123-128 9 1996 John Wiley & Sons, Inc. CCC 0895-2477/96

MICROWAVE SCATTERING PARAMETERS OF LOSSY MATERIALS: EXPERIMENTS AND COMPUTATIONS T. Lasri, B. Bocquet, and Y. Leroy lnstitut d Electronique et de Microelectronique du Nord IEMN UMR CNRS 9929 Departernent Hyperfrequences et Semrconducteurs Domaine Universttarre et Scientifique de Villeneuve d’Ascq Avenue Poincare B P 69 59652 Villeneuve d’Ascq Cedex France

128 MICROWAVE AND OPTICAL TECHNOLOGY LEnERS / Vol 13. No 3, October 20 1996