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IEEE TRANSACTIONS ONMICROWAVE THEORYAND TECHNIQUES,VOL. 44, NO. 5 , MAY 1996 65 1

A Robust Approach for the Derivationof Closed-Forrn Greens Functions

M. I. Aksun, Member, IEEE

Abstruct- Spatial-domain Greens functions for multilayer,planar geometries are cast into closed forms with two-level ap-proximation of the spectral-domain representation of the Greensfunctions. This approach is very robust and much faster com-pared to the original one-level approximation. Moreover, il doesnot require the investigation of the spectral-domain behavior ofthe Greens functions in advance to decide on the parametersof the approximation technique, and it can be applied to anycomponent of the dyadic Greens function with the same ease.

I. INTRODUCTION

UMERICAL modeling of printed structures used inNmonolithic millimeter and microwave integrated circuits(MMIC) can be efficiently and rigorously performed by em-ploying the method of moments (MOM). The MOM is basedupon the transformation of an operator equation, such asintegral, differential, or integro-differential operators, into amatrix equation [l]. Although the MOM is the most effi-cient numerical technique for moderate-size printed geometries(spanning several wavelengths in two dimensions), there isstill need for improvement, which could be accomplished inthe calculation of the matrix elements and in the solution ofthe matrix equation. For small geometries like those requiringcouple hundreds of unknowns, the matrix-fill time could be

the significant part of the overall solution time, however, forlarge geometries the matrix solution time will dominate theCPU time [2].

In the application of the spatial-domain MOM to the solutionof a mixed-potential integral equation (MPIE), one needsto calculate the Greens functions of the vector and scalarpotentials in the spatial domain where they are represented asoscillatory integrals, called Sommerfeld integrals. The eval-uation of these integrals is quite time consuming, thereforethe matrix-fill time would be significantly improved if theseintegrals can be evaluated efficiently. Recently, a techniquehas been proposed to approximate these integrals analyticallyfor a horizontal electric dipole over a thick substrate backedby a ground plane; this is called the closed-form Grleensfunctions method [3]. This technique was improved first fortwo layer geometries with arbitrary thicknesses [4], then formultilayer geometries with horizontal electric dipole (H.ED),

Manuscript received Setpember 28, 1994; revised January 17, 19961.Thiswork was supported in part by NATOs Scientific Affairs Division in theframeworkof the Science for StabilityProgramme.

The a uthor is with the Departmentof Electrical and Electronics Engineering,Bilkent University, 06533 Ankara, Turkey.

Publisher Item IdentifierS 0018-9480(96)03021-9.

horizontal magnetic dipole (HMD), vertical electric dipole(VE,D), and vertical magnetic dipole (VMD) sources [5] .However, a question remains to be answered on the robustnessand the efficiency of the technique, because some of theGreens functions are usually difficult to approximate and itis recommended that the function to be approximated needsto be examined in advance. The: source of difficulties inthis technique is the approximation of the spectral-domainGreens functions in terms of complex exponentials. Theoriginally proposed technique [3]/ uses the original Prony

metlhod which requires the same number of samples as thenumber of unknowns, that is, the number of samples mustbe twice as many as the number of complex exponentials(one: for the coefficient and one fo r the exponent). Therefore,it would be difficult to account for the fast changes in thespectral domain without using tens of complex exponentialsif not hundreds in certain cases, which is partly due to theuniform sampling required by the Prony method. The use ofthe least-square Prony method has improved the techniqueto account for the fast changes with a reasonable numberof exponentials [4], but due to the noise sensitivity of theProny methods [6], [7], it requires several trial and erroriterations which render the technique to be inefficient andnot robust. As a solution, another approximation technique,called the generalized pencil of function (GPOF) method [8],is employed in casting the Greens functions into closedfomis [5]. The GPOF method has turned out to be quiterobust and less noise sensitive when compared to the originaland least-square Pronys methods, and also provides a goodmeasure for choosing the number of exponentials used inthe approximation. However, it still requires one to study inadvance the spectral-domain behavior of the Greens functionin order to decide on the approximation parameters likethe number of sampling points and the maximum value ofthe sampling range. In addition, since the approximationtechniques, like the Prony and the GPOF methods, requirethe function to be sampled uniformly, one would need to

take hundreds of samples in order to be able to approxi-mate a slow converging function with rapid changes (evenif this were to occur in a small region), which is a typ-ical behavior of the spectral-domain Greens functions ofthe scalar potentials in a thin substrate. Because of thesedifficulties, the technique of deriving the closed-form Greensfunctions and subsequently using 1hem in MOM applicationsare considered to be not robust and could not be used muchfor the development of a general-purpose electromagnetic

0018-9480/96$05.00 0 1996 IEEE

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652 IEEE TRANSACTIONS ON MICROWAVETHEORY AND TECHNIQUES, VOL. 44, NO, 5, MAY 1996

region -( i+l)

software. In this paper, a new approach based on a two-level approximation is proposed to overcome these difficulties,and demonstrated that it is very robust and computationallyefficient.

The procedure of the original one-level approximation isdescribed and difficulties associated with this approach are

demonstrated on some examples by using the GPOFmethod inSection I1 of this paper. This is followed in Section ID wherethe formulation of the new approach based on a two-levelapproximation and some numerical examples are included.Then, in Section IV, a discussion on the new technique andconclusions are provided.

Z =d i-hsource

11. DIFFICULTIESIN THE ORIGINALONE-LEVELAPPROXIMATION

Since the main goal of this paper is to introduce a robusttechnique to obtain the spatial-domain Greens functions inclosed forms for planarly-layered media, Fig. 1, it would beuseful to give the definition of the spatial-domain Greensfunctions

where, G and G are the Greens functions in the spatial andspectral domains, respectively, H i 2 )is the Hankel function ofthe second kind and S I P is the Sommerfeld integration pathdefined in Fig. 2. Note that this integral, called the Sommerfeldintegral, can not be evaluated analytically for the spectral-domain Greens functions G, which are obtained analyticallyfor planarly stratified media [ 5 ] , [9]. It was recognized byChow et al. [3] that if the spectral-domain Greens functionG is approximated by exponentials, the Sommerfeld inte-gral (1) can be evaluated analytically using the well-knownSommerfeld identity

Therefore, this places the emphasis of deriving the closed-form Greens functions on the exponential approximation.Since the approximation techniques used for this problem,namely the original Prony, the least square Prony and theGPOF methods, require uniform samples along a real variableof a complex-valued function, one might think of choosingthe integration path in (1) along the real IC, axis so that G can

be sampled along a real variable. However, one should noticethat k: = IC 2- IC; and sampling along real IC, results in anapproximation in terms of exponentials of IC, which cannotbe cast into a form of exponentials of IC, as required in theapplication of the Sommerfeld identity (2). Hence, a deformedpath on IC, plane, denoted by Ca p in Fig. 2, was defined as amapping of a real variable t onto the complex IC, plane by

Z= Z ,-hregion -(i+m) 1

region-(i-m)

Z=-Z -m -h

Fig. 1 . A typical planar geometry.

where IC ,and ICare defined in the source layer [3]. The Greens

functions are sampled uniformly on t E [0, To ] , which mapsonto the path Ca p with ICpmax= k [ l + T2]1/2 in the IC,-plane,and approximated in terms of exponentials of t which caneasily be transformed into a form of exponentials of IC,. Thisscheme is called the one-level approximation approach here inthis paper because the complex function to be approximated issampled between zero and To and is assumed to be negligiblebeyond To .

For a general-purpose algorithm, the spectral-domainGreens functions are obtained for a multilayer medium andneither surface-wave poles nor the real images are extracted.It is true that the extraction of the surface-wave poles (SWP)and the real images would have helped the exponentialapproximation techniques by making the Greens functions in

the spectral domain well-behaving (extraction of the SWPs)and fast-converging (extraction of the real images). However,since the contribution of the SWPs is small for geometrieson a thin substrate, and there is no analytical way of findingthe real images for multilayer planar structures except forsimple cases like single and double layers, the help gainedfor the approximation would be limited to a restricted class ofplanar geometries and would render the algorithm not generalpurpose and not robust.

It would be instructive to consider the practical details ofthe implementation of the exponential approximation alongthe path defined in (3). It is of utmost importance to choosethe approximation parameters; T o , the number of exponen-tials to be used in the approximation, and the number ofsamples in t E [0, To ] , judiciously for the success of thisapproach. To illustrate the implementation of the one-levelexponential approximation and the difficulties involved, thespectral-domain Greens function for the scalar potential dueto an x-directed dipole, Gz, is given in Fig. 3, for a geometryof four layers at 30 GHz: First layer-PEC; second layer-E,Z = 12.5,dz = 0.03 cm; third la~er-6~3= 2.1,d3 = 0.07 cm;fourth layer-free-space, and the source and observation planesare chosen at the interface of the second and third layers. Sincethe expressions of the spectral-domain Greens functions in a



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AKSUN A ROBUST APPROACH FOR THE DERIVATION OF CLOSED-FORM GREE NS FUNCTIONS 653

kp -plane

Fig. 2. Definition of the Somm erfeld integration path and the path Cap used in one-level approximation.

4.0

Imaginary

4 . 00.0 0.2 0.4 0.6 0.8 1.0

tFig. 3. The magnitude of the spectral-domain Greens function G!: alongthe path Cap. First layer-PEC; second Iayer-e,z = 12.5, dz = 0.03 cin; thirdlayer-e,3 = 2.1, d3 = 0.07 cm; fourth layer-f ree-space, freq = 30 GHz.

multilayer medium are given in [5] for HED, VED, HMD,and VMD sources, they are not included in this paper. It is

evident from Fig. 3 that Greens functions can have sharppeaks and fast changes for small t , which maps to thie far-field region in the spatial domain. Therefore, one needs tosample the Greens function given in Fig. 3 at a peniod ofless than 0.05 along t so that the fine features of the functioncan be captured in the approximation. The choice of To isanother parameter that competes with the period of samplesbecause large To corresponds to large number of samplesand translates to a longer CPU time. Fortunately, for theexample given in Fig. 3, the Greens function decays quitefast in the spectral domain, therefore it would be enough tosample as far as To = 5 which requires 200 samples if A tis chosen to be 0.025. The spatial-domain Greens function isobtained via the GPOF method using the above approximationparameters (T o = 5, number of samples = 201, number ofexponentials = 13) and compared to the result obtained fromthe numerical integration, which are labeled as Apprx. andExact, respectively, in Fig. 4. Although, as it was mentionedabove, the SWPs are not extracted from the spectral-domainGreens function prior to the exponential approximation, thecontribution of the SWPs is also shown for the purpose ofcomparison and one can draw a conclusion that the exponentialapproximation algorithm (GPOF) works fine well withiin theinfluence range of the SWPs and beyond that an asymptotic

15.0

14.0

13.0

12.0

11.0. -.

-Approx.- ExactS W contribution -

a_ _ _ _

1.03 -2.0 -1.0 0.0

Fig. 4. The magnitude of the Greens function for the scalar po-tential and the surface wave contribution. First layer-PEC; secondlayer-e,z = 12.5,dz = 0.03 cm; third layer-c,~ = 2 . l , d 3 = 0.07 cm ;fourth layer-free-space, freq = 30 GHz.

approximation together with the surface-wave contribution canbe used to approximate the spatiad-domain Greens functions[1C% [Ill.

lJnfortunately, not all the Greens functions have fast de-caying spectral-domain behavior like the above example givenin Fig. 3. For example, the spectral-domain Greens functionfor the vertical component of the vector potential due toa 13ED [5], G&./jkz = Gfy/jky, does not decay as fastand moreover has a relatively isharp peak which requiressampling almost as frequently as that of the example givenin Fig. 3, as shown in Fig. 5. To demonstrate the effect of theapproximation parameters, the Greens function J Gf, dx (=~ - - ~ { G t . / j k ~ } )is given for thie same approximation pa-rameters as those of the above (example (T o = 5, numberof samples = 201) and compared to the results obtained bythe numerical integration of the spectral-domain representationof the Greens function and to thte results obtained by usingdifferent approximation parameters in Fig. 6. It is observedthat the approximated Greens fiinctions do not agree withthe exact solution for small values of p because the spectral-doinain Greens function is not sampled far enough to getan accurate near-field distribution. However, if the value ofTo is increased, the agreement between the approximated and



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AKSUN:A ROBUST APPROACHFOR THE DERIVATIONOF CLOSED-FORM GREENS FUNCTIONS 655

t ImLkplkp - plane

Cap2

Fig. 7. The paths C,,l and Capz used in two-level approximation.

to be decided by the user seems to have increased compared tothe one-level approximation, these parameters are determinedonly once for the class of geometries that are of interest; theyare used for the approximation of any component of the dyadicGreens function and for any geometrical constants. Moreover,the choice of these parameters do not require an investigationof the function to be approximated in advance because they canbe chosen for the possible limits of the geometrical constants.

To demonstrate the robustness of the technique, the choiceof the parameters and the application of the above procedure,the Greens function J Gtx dx is obtained for the same ge-ometry given in Section 11. Its spectral-domain representationis given here as

to help explain the approximation procedure, where the layeri denotes the source layer, and A, B, C, and D are givenin [5]. It should also be noted that this expression is forthe case where the source and observation points are in thesame layer, i.e., layer i. If it is desired to find the Greensfunction for the observation layer different from the sourcelayer, then the coefficients of the up-going waves and clown-going waves must be carried to the observation layer with arecursive algorithm [5], [9]. Let us first give the parametricequations describing the paths C,,l and Cap2 for the first andsecond parts of the approximation, respectively

. For Cap1 ICzs= -jICi[T02 + t ] 0 5 t 5 To1 ( 5 )

For Cap, ICz,=kc;[- j t + (1-

-;2)] 0 5 t 5 T o 2

where t is the running variable sampled uniformly on thecorresponding range. Then, the above procedure is followedstep-by-step as:

1) T02 = 5 is chosen, for which ICfmaxz= k i [ l +To2]1/2 > 1, = Lo.

2) = 400 is chosen to ensure that the behavior ofGfx/jIC, for large 1, is captured. This choice is not

3)

4)

critical, 300 or 500 could have been chosen instead.Since there is no fine feature to pick up in this range,that is, the function is smooth, one can keep the rangelarge without having to use a large number of samples.Therefore, the number of samples is chosen to be 50.G:x/jICx is sampled along the path Cap, and the GPOF

method is applied

Ni Ni

n= l n = l

where b l , and P ln are coefficients and exponents ob-tained from the GPOF methlod, and Nl is the numberof exponentials used in this approximation. The choice

of the number of exponentials is based upon the numberof significant singular values obtained in an intermedi-ate step of the application of the GPOF method. Forthis specific problem, five axponentials are chosen toapproximate the Greens function on the range of IC, E[ICpmaxz , ICpmaxl ] .The transformation of the coefficientsbl n and the exponents Pln is necessary to cast theapproximating function into a form suitable for theapplication of the Sommerfeld identity (2) , that is, theapproximating function must be an exponential functionof k z * .Hence, a ln and a1, are obtained in terms of b l ,

The approximating function f ( k p ) is subtracted fromthe original function GtX/;iIC,, which guarantees theremaining function to be negligible beyond ICpmax2

and P l n in (8 )

dICpk,H~2)(k ,p)[ - f ( k , ) ]

Note that the first integral is evaluated along the pathCap, because the integrand is negligible on Cap,, but



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656 JEEE TRANSACTIONSON MICROWAVETHEORYAND I Q U E S , VOL. 44, NO . 5 , MAY 1996

ExactApprx. (two-level)Apprx. (one-level)

-10.0 I-3.0 -2.0 -1.0 0.0 1.0

1og10( IC0 P )Fig. 8.

layer-e,3 = 2.1, d3 = 0.07 cm; fourth layer-free-space, freq= 30 GHz.

The magnitude of the Greens function for thevector potentialG:= dx . First layer-PE C; second layer-e,z= 12.5, dz = 0.03 cm; third

the second integral is evaluated along Cup2 + CUp1.Therefore, the Sommerfeld identity (2) can be appliedto the integrals in (9).

5) The remaining function is sampled along the path Cu,2with 100 samples. Since the maximum range for thesampling ( k p m a x 2 )is rather small, compared to that ofthe one-level approximation scheme, the frequency ofsampling can be made quite high without substantiallyincreasing the number of samples. For all practicalpurposes (including the worst case situation) the choice

of 200 as the number of samples would be more thanenough to get a good approximation

where b2 , and t32n are the coefficients and exponents ofthe exponentials of t obtained from the application ofthe GPOF method, and and azn are the coefficientsand exponents of the exponentials of k z t . The numberof exponentials N2 in this part of the approximation

is chosen to be 8, again by the number of significantsingular values.To summarize, the approximation parameters as chosen

here are as follows: for the first part of the approximation;TOl = 400, To 2 = 5, number of samples = 50, and number ofexponentials = 5, for the second part of the approximation;number of samples = 100, number of exponentials = 8. Notethat the total number of exponentials used in this approxi-mation is 13. The Greens function obtained by employingthe above procedure is given in Fig. 8 along with the dataobtained from direct numerical evaluation of the Sommerfeld-

4.0

0 Apprx.(GAxx

2.0

0.0

-2.01.0-3.0 -2.0 -1.0 0 .0

oil10 ( L O P1

0 Apprx. (GAzz)

0 Apprx. (Gqz)- Exac t tGqz)

-3.0 -2.0 -1.0 0. 0 1 .o

(b)

Fig. 9. (a) The magnitude of the normalized Greensfunctions 4aG &,/ p3,4 ae3 GZ . First layer-PEC; secondlayer-c,z = 12.5,dz = 0.03 cm; third layer-e,s = 2. l ,d3 = 0.07 cm ;fourth layer-free-space, freq= 30 GHz. (b) The magnitude of thenormalized Greens functions 4nG tz, /p3 ,4ne3G;. First layer-PEC; secondlayer-e,Z = 12.5,dz = 0.03 cm; third layer-e,3 = 2.1, ds = 0.07 cm ;fourth layer-freespace, freq= 30 GHz.

type integral (exact), and from the one-level approximationapproach with the parameters of approximation To = 200,number of samples = 400 and the number of exponentials= 13. Note that the values of the parameters used in theone-level approximation are chosen to make the computationtime minimum with a reasonable agreement. However, thoseof the two-level approximation are typical values and thenumber of samples for the second part of the approximationcan even be reduced to 50 with no change in the results.The two approximation techniques for the above example arecompared for the CPU time on a SPARCstation 10/41, using



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AKSUN: A ROBUST APPROACH FOR THE DERIVATION OF CLOSED-FORM GREENS FUNCTIONS 651

0.0

-2.0

-4.0

-6.0

4.0 :--,--e.-&

- - - _ _........_,

4.-.-.

.-..__m....-.2.0

-

-

0 Exact

(a)

4.0

2. 0

0.0

-6.0 . I-3.0 -2.0 -1.0 0.0 1 .o

(b)

Fig. 10. (a) The magnitude of the normalized Greens function47rG&/p3.First layer-PEC; second layer-e,z = 12.5,dz = 0.03 cm; thirdlayer-e,3 = 2. l ,d3 = 0.07 cm; fourth layer-free-space.(b) The magnitudeof the normalized Greens function4 a e 3 G g . First layer-PEC; secondlayer-e,z = 12.5,dz = 0.03 cm; third layer-e,3 = 2.1, d3 = 0.07 cm;fourth layer-free-space.

the same number of total exponentials (= 13) for differentapproximation parameters, and presented in the table format

below:Approximation CPU time

Approximation Parameters (set)

one-level To = 200, N , = 400 198.0one-level To = 200, N , = 500 382.0two-level

two-level

TOl = 400, N,, = 50

Tol = 400, N , , = 50Toz = 5 ,N ,, = 50

T02 = 5, N ,, = 100

1 .2

3 .5

where N , is the number of samples in one-level approximationscheine while N sl and Ns 2 are the number of samples of thefirst and second parts of the approximation, respectively, inthe two-level approximation approach. It is obvious that thetwo- level approximation approach improves the computationalefficiency significantly.

The robustness of the two-level approach can be demon-strated by casting the other Greens functions into closedforms with the use of the same approximation parameters asthose: used for 1 G& dx , namely T0l = 400,N,, = 50To2 := 5 , N ,, = 100. The normalized Greens functions of thevector and scalar potentials due to HED and VED sourcesare obtained (47rG&/p3,47r~G$, 47rGfZ/p3 and 4m3G9following the two-level approach (Apprx.) and evaluatingthe !$ommerfeld integrals numerically (Exact), and given inFig. 9(a) and (b). This test shows thlat the same set of approx-imat,ion parameters can be used for any Greens function, thatis, there is no need for an advance investigation of the Greensfunction and no need for any trial steps. The assessmentof the robustness of the proposed approach also requires a

stud!{ of the sensitivity of the approximation parameters tothe geometrical constants and the frequency. Therefore, theGreens functions for the vector and the scalar potentials areob ta in closed forms for the same geometrical constantsand for the same approximation piirameters used above, butthe frequency of operation is changed to 1 GHz, 10 GHz and100 GHz, which is equivalent, in effect, to a change of thegeometrical constants, Fig. 10(a) and (b). It is observed thatthe agreements between the exact and approximate sets ofdata are still perfect and hence it is safe to conclude that thetwo-level approach proposed in this paper is very robust.

IV. CONCLUSIONThe closed-form Greens functions developed previously

suffer from the difficulties of choosing approximation param-eters, for the exponential approximation techniques used in thederivation, thereby rendering the technique to be inefficientand not robust. Moreover, the extraction of the SWPs andreal images may not be possible or efficient for multilayergeornetries when the original approach is used. Here, a newapproach based on a two-level approximation is proposed toovercome these difficulties and to make the use of closed-formGreens functions attractive for those developing EM softwareand for researchers in the field. The major advantages of thisapproach are its robustness and the: computational efficiency,both of which are demonstrated in the text.

REFERENCIES

[ l ] R. F. Harrington,Field Computation by Moment Methods. Malabar,n: Krieger, 1983.

[2] A. Taflove, Sta te of theart and future directions in finite-differenceandrelated techniques in supercomputing computational electromagnetics,in Direc tions in Electromagnetic WaveModeling, H. L. Bertoni, andL.B. Felsen, Eds. New York Plenum, 1991.

[3] Y. L. Chow,J. J. Yang, D. F. Fang, and G.E. Howard, A closed-formspatial Greens function for the thick m icrostrip substrate,IEEE Trans.Microwave Theory Tech., vol. 39, pp. 588-592, Mar. 1991.

[4] M. I. Aksun andR. Mittra, Derivation of closed-form Greens functionsfor a general microstrip geometries,IEEE Trans. Microwave TheoryTech., vol. 40, pp. 2055-2062, Nov. 1992.



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[71

[81

[91

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G . Dural and M.I. Aksun, C losed-form Greens functions for generalsources and stratified media,IEEE Trans. Microwave Theory Tech.,vol. 43, pp. 1545-1552, July 1995.A. J. Mackay and A. McCowan, An improved pencil-of-functionmethod for extrac ting poles of an EM sys tem from its transien tre-sponse, IEEE Trans. Antennas Propagat., vol. AP-35, pp. 4 3 5 4 1 ,Apr. 1987.D. G. D udley, Parametric modeling of transient electromagnetic sys-tems, Radio Sci.,, vol. 14, pp. 387-396, 1979.Y. Hua and T. K. Sarkar, Generalized pencil-of-function method forextracting poles of an EM system from its transient response,IEEETrans. Antennas Propagat., vol. AP-37, pp. 229-234, Feb., 1989.W. C. Chew,Waves and Fields in Inhomogeneous M edia. New YorkVan Nostrand, 1990.Sina Barkeshli, P. H. Path& and Miguel Marin, An asy mptotic closed-form microstriu surface Greens function forthe efficient momentmethod analysis of mutual coupling in microstrip antennas,IEEE Trans.Antennas Propagat., vol. 38, pp. 1374-1383, Sept. 199 0.

[ l l ] L. B. Felsen and N. Marcuvitz,Radiation and Scattering of Waves.Englewood Cliffs,NJ: Prentice-Hall, 1973.

M. I. Aksun (M92) received the B.S. and M.S.degrees in electrical and electronics engineenngfrom the Middle East Technical University, Ankara,Turkey, in 1981 and 1983, respectively, and thePh.D. degree in electrical and com puter engineeringfrom the University of Illinois at Urbana-Champaignin 1990.

From 1 990 to 1992,he was a post doctoral fellowat the Electromagnetic Communication L aboratory,University of Illinois at Urbana-Champaign. Since1992 , he has been on the fa culty of the Department

of Electrical and Electronics Engineering at Bilkent University, Ankara,Turkey, where he is currently an Associate Professor. His research interestsincludethe numerical methods for electromagnetics, microstrip antennas, andmicrowave and millimeter-wave integrated circuits.

robust approach for

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