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1568 I l - E E TRANSACTIONS O N M I C K O WAV E T H E O K Y AN D TECHNIQUES, VOL. 38, N O . 11 , NOVEMBER 1990

Electrom agnetic Scattering and Radiationfrom Finite M icrostrip S tructures

TAPAN K. S A R K A R ,SENIOR MEMBER, IEEE, SADASIVA M.P o , ENIOR MEMBER, IEEE,AN D A N TO N I J E R . D J O R D J E V I C

Abstract -In this paper, two techniques are presented for the analysisof electromagnetic radiation and scattering from finite microstrip struc-tures. The two techniques are based on two different formulations, viz.the volume/surface and surface/surface formulations. In the volume/surface formulation the finite-sized dielectric is replaced by an equiva-lent volume polarization current whereas the conducting plates arereplaced by equivalent surface currents. For the surface/surface formu-lation the surface covering the dielectric volume i s replaced by equiva-lent electric and magnetic currents and the conducting plates by surfaceelectric currents. Both techniques can be utilized for the analysis ofarbitrarily shaped finite m icrostrip structures. The techniques are quiteaccurate and they are utilized to validate each other. Finally, typicalnumerical results are presented representing the agreement betweenthese two solution techniques.

I . INTRODUCTION

ANY PAPERS and textbooks are available for theM nalysis of microstrip antennas. However, most ofthe techniques utilize some form of approximation tocarry out the analysis. The most accurate of all thetechniques available uses the familiar Sommerfeld formu-lation. The drawback of this approach is the inherentinability to handle finite microstrip structures. In thepresent work, we overcome this inability and analyzefinite-sized microstrip structures. Moreover, the antennascan be of arbitrary shape.

In this paper, two different numerical procedures arepresented for the analysis of microstrip problems. Oneuses a volume/surface formulation and the oth er utilizesa surface/surface formulation. The techniques are usedto validate each other. In th e volume/surface formulationthe dielectric body and the conducting plates are approxi-mated by cubical cells and rectangular patches, re-spectively. Hence this procedure is most suited for rec-tangular/cubical geometries. In the surface/surfaceformulation both the dielectric surface and the conduct-ing plates are approx imated by triangular patches. Thu s,the surface/surface formulation can handle truly arbitrar-ily shaped microstrip structures.

Manuscript received January 11 , 1990; revised June 18, 1990.T. K. Sarkar is with the Department of Electrical Engineering, Syra-

S . M. Rao is with the Department of Electrical Engineering, Auburn

A. R. Djordjevid is with the Department of Electrical Engineering,

IEEE Log Number 9038473.

cuse University, Syracuse, NY 13244-1240.

University, Auburn, AL 36849-5201.

University of Belgrade, Belgrade, Yugoslavia.

11. V O L U M E / S U R FA C EORMULATION

In this section, we present the detailed mathematicalsteps to obtain a pair of coupled integral equations toanalyze the finite-sized microstrip problem using the vol-ume/surface formulation. For the sake of clarity, wepresent the formulation for a single conducting plate anda single dielectric region. Extending the formulation tohandle multiple conductors and dielectric regions isstraightforward.

A . Theory

Consider a perfectly conducting bodyS, and a dielec-tric body S, with constitutive parameters U , p 2 = p r p 0 ,and e2 = e r e Oplaced in an infinite homogeneous medium( p , , ~ ! ) s shown in Fig. 1. The composite structure isilluminated by an incident plane wave denoted byE ' . Itmay be noted that t he incident field is defined to be thatwhich would exist in space if the structure were notpresent. The incident field induces a surface current onthe conducting bodyJ , and p enetrates the dielectric bodyto produce the electric field inside the body given byE 2 .By invoking the equivalence principle[ l ] , we replace theconducting body by an equivalent surface currentJ , an dthe dielectric material by th e volume polarization currentJ d . These currents are situated in free space; hence thefree-space Green's function can b e utilized in th e compu-tation of the radiated or scattered fields. The presentformulation also holds for the more general case wherethe dielectric body may be inhomogeneous.

On the conductor surfaceS,, the total tangential elec-tric field is zero, i.e.,

where

E ' ( r ) = incident field at locationr E S,

E"( r ) = scattered field at location r E S,.

The scattered field E S ( ( r ) onsists of two components .The first is the electric field produced by the electriccurrent J , induced on the conducting surface, and thesecond is the field produced by the volume polarizationcurrent J<,.Hence,

E ' ( r ) = L [J , ] + L [ d ] ( 2 )

0018-9480/90/ 1100-1568$01 OO 0 990 I E E E

-- _ _ - _ _ _ _ _ ~_ _ ~



S A R K A R ('I U / . : F I . I I ( 'T K O M , Z ( i N F T I( ' S ( ' A I T F K I N ( i A N I ) R A D I A T I O N 1569

The integrat ion in (6) is carried out over the volumeinstead of the surface used in (5). However, it is necessaryto enforce

(Ehc, HhC)

V . J o ( r ) = 0 for r E y /# 0 fo r r on the surface of V, / (7)

inside a hom ogeneous dielectric region. For an inho moge-

neous dielectric region, (7) need n ot be en forced. Equa-tion (7 ) implies that t he curren t flowing inside a homoge-neous dielectric region is divergenceless; that is, there areno charge densities associated with this current in theinterior of the homogeneous dielectric, excepton th esurface. Hence th e second integralin (6 ) ma y be reduced

How this is accomplished numerically is described in thenext section. Combining (1), ( 2 ) , ( 5 ) , and (6), the fieldintegral equation on the surface of the conductor is given

E', E'

Region # 1

S C

Fig. 1. Arbitrarily shaped conducting/dielectric body illuminated by a to a surface integral for a homogeneous dielectric region.plane wave.

where

and

w = angular frequency

e g = permittivity of free space

p 0 = permeability of free space

e I h

G( r , ' ) = free-space Green's function=R

R = [ ( x - ~ ' ) * + ( y - y ' ) ~ + ( z - - ~ ' ) ~ ]/ 2

To apply the electric field integral equation to the dielec-

tric region, we utilized the definition for the volumepolarization current:( 3 )

J [ / ( r )= j w E 0 ( i - 1 ) E t o t a ' ( r ) , r E L $ (9 )

where i is the relative complex permittivity of the dielec-tric mediu m, given by

U

E = E ,. - - -.

In (9), E")"" is the total electric field inside the dielectricregion, given by

( 10 )w e0

k = 2 r / A E t "t " ' (r ) E ' ( r ) + E ' ( r ) (11)

and by combining (9 ) and (1 11, we getI , y , z ' = source coordinates

x , y , z = field coordinates.r E V, / . (12)

Since the total scattered fields are computed, as before,from two electric currents, namely the surface currentonthe conductor and the volume polarization current in thedielectric region, we have

J J r )

J w 4 - 1 )he charge density pc on the conductor is related to th e E ' ( r ) = . - E \ ( r ) ,

electric current J , by the equation of continuity

( 4 )W P , .V ' . J = - j

By combining(4) an d (3)9 we have theJ , given by

U 1P

Produced

E ' ( r ) = L [ J , 1+ Wd1[k 1 , ( r ') G ( r , ') ds'~ ~ 4 7 r q ~, where L [J , ] an d L [ d ] are th e fields produ ced byJ , and

J d and are given by ( 5 ) and (6), respectively. Hence, in the+ v " J ( ( r ' ) G ( r 7 r ' ) ds' . ( 5 ) dielectric region,

J J r )E ' ( r ) = - L [ J [ - L [ J , / I , r E V, / . (13)

L< 1The scattered fields from the equivalent volume polar-

ization curren ts Jd flowing in the dielectric regionsV, ,ca nbe obtained in similar fashion:

m,1= 7 k ' j J < , ( r ' ) G ( r , r ' ) d i s '

j w c , , (i - 1)Thus, given an incident field E ' , one can utilize thecoupled integral equations( 8 ) and (13) to solve for J , andJ , ! . Once the currents are accurately computed, any other

1

J W ~ T ~ v,,7 quantity of interest may be easily obtained. In the follow-

r ' ) G ( , ' ) di,' (6) ing section , we discuss th e numerical solution pro ced ure1 of the coupled integral equations(8) and (13).



1570 IEEE TRANSACTIONS ON M I C R O WAV E THEORY AN D TECHNIQUES, VOL. 38, NO. 11, NOVEMBER 1990

B. Numerical Implementation

In the following analysis, we assume the conductingplate and dielectric region to be a rectangular plate and arectangular parallelepiped region, respectively. Further-more, we assume the conducting plate to be located in th ex - y plane of the rectangular coordinate system. Underthese assumptions, we obtain an approximate numerical

solution of the coupled integral equations described inthe previous section using the method of moments[2].To apply the method of moments, we have to assume

certain expansion and weighting functions. For the con-ducting plate, we assume pulse basis functions for the twocomponents of the currents J, and J , flowing on the

approach, however, we used both methods and obtainedthe average for better accuracy. The weighting functionschosen are d elta functions; hence the fields are computedat the cen ter of the patch.

Th e volume polarization curre nt densityJd is expandedin similar fashion:

where Q i j k is defined as

(20)i < x < x i + l ; Y j < Y < Y j + l ; z k < z < z k + l

0, otherwise

surface. These are not the shifted pulse functions that areconventionally used for the electromagnetic analysisofbent plates. The pulse expansions for both J , an d Jyoccupy the same physical space. Therefore,

ell,( 7 Y 9 2 ) =

_ _

d r J k 7 and are the unknown constants to besolved for. For the case of an inhomogeneous dielectricregion, we further assume that within each rectangularpulse expansion function cell, the complex dielectric con-

M< N< stant remains constant. Hence,

i w oL [ J J = - -< N , 4 a

J , r ( x , ~ ) = C C a i j p i j ( x , ~ ) (14)l = l I - 1

J ~ ( x ~ Y >C C ~ i j p i j ( x 9 ~ ) (15) ’

/ = I J = 1 / X C i j k + Y d i j k + z i , k ) Q 2 r j k G d U r

where the pulse functions a re defined as i J k v d

1 a ax- + y - - 2 -= ( x a y az

an d c y i j an d pi j are the unknowns to be solved for. Using(14) an d (151, ( 5 ) may be written as * [ i j k / Q i j k ( G,’ - G l ) dy’dz‘

However, for numerical coinputation the following inte-gro-differential form has been used:

a2 a 2 d 2

+ ( dx,+y- axay + % z ) L ‘,Ga!x’dy’

The partial derivatives in (18) may be computed by theseven-point difference formula described in[3]. In thisapproach, the function to be differentiated is evaluated atseven discrete points placed in the first and third quad-rants only. An alternative approach is to evaluate thefunction at mirror images of these points, i.e., locatingthem in the second and fourth quadrants. In the present

where G’ = G(x , , z lx *,y’, 2 ’ ) and x + = x i + l a nd x - =

xi. Similar remarks hold forG,’ , G , G,‘ , and G,-. Theexpression for th e last integral in (21) nee ds little explana-tion. Since the charge term for the x-directed currentexists only over two surfaces of the cube atx + and x-,the integration is performed overtwo surfaces. This is anapproximation to the original integral. For the originalintegral there is no surface charge contribution frominside a homog eneou s dielectric region. However, throughthis approximation, we have introduced an additionalsurface charge layer inside the homogeneous dielectricregion. It app ears that t he error introduced by this addi-tional surface charge layer is not appreciable in the nu-merical com putation.

For th e more general case of an inhomogeneous dielec-tric body, the analysis may be carried out by treating theregion as a combination of piecewise-homogeneous cubi-cal cells, and for each piecewise homogeneous cube(21)is enforced. For numerical evaluation of the second part



1571S A R K A R f U / . : FI.CC'TROMAGNETI<. S < ' AT T E K I N G A ND K A D l AT l O N

PI ? & I / - - ,

, ' \/ / I I

is not permitted from a strict mathematical point of view, ,' D,$ El , / II ' (0,O) D1' 5 /

'. _ _ _ / * '\produces reasonably accurate results [51. Finally, for ' - - - - -weighting functions, the point-matching solution is se-

of (21), we use the approach taken by Hohmann[4]. Thisis accomplished by first moving the derivatives appearing \

I \k - \ '\ '

E', H'.utside the integral to inside of the integral sign, thenanalytically performing the differentiation of the Green'sfunction, and finally integrating the result. Even though

_- - -\

I I1.

Jdd

, I / - - - -he interchange of the integration and the differentiation

since it produces a divergent integral, from a purely '\ @,O) ,' 2 \ //

//

num erical point of view, this "unco nvention al" appro ach

s c

lected. Fig. 2. Equivalent problem for region 1.

111. S U R FA C E / S U R FA C EORMULATION\

/ - - \

\n this section, we present a detailed derivation of thepair of coupled integral equations t o analyze a finite-sized

/1L - \

\ \, P29' \

/ - - - - - - IA. Theoly / 0 , 2' ]

microstrip struc ture using the surface/surface formula-tion. The formulation presented here holds only for ho-mogeneous dielectric regions.

IJ,,M,

0 0)

Again referring to Fig. 1 and invoking the equivalenceprinciple, the following two problems are formulated,each valid for regions external and internal to the dielec-tric body, in terms of conductor current J,. and theequivalent electric and magnetic currentsJ , and M , ,respectively.

For the problem valid external to the dielectric region,as shown in Fig.2, the conductor and d ielectr ic bodies arereplaced by fictitious mathematical surfaces and the en-tire region is filled with the homo geneous m aterial( p , , l )of medium 1. The cur ren t J , is allowed to flow on th emathematical surface S,. In addition, two equivalent cur-rents, J , an d M d ,whic h f-lowon the mathematical surfaceS, are introduced. Finally, the total electric fieldE l inmedium 1, which is the sum of the incident fieldE' andthe scattered field E s radiated by the currentsJ,., J , , andM , , is retained and th e field inside the surfaceS , is set tozero. By enforcing the continuity of the tangential electricfield at S, an d S i , where Sc- and S i are selected to beslightly interior to S,. and S d , the following equations arederived:

Fig. 3 . Equivalent problem for region 2.

be derived:

[E:,2( - a ) + EA2( - M,)Itd" = 0 ( 2 4 )

where th e superscr ipt s2 represents the medium in whichthe scattered electric field is evaluated.

Since we are using the electric field integral equation tosolve the microstrip problem, the electric fieldE (af terdropping the subscripts and superscripts representing themedium) due to the electr ic currentJ and the magnet iccur ren t M is given by

- E ' ( J , M ) = j w A + V @ + + E I V XF ( 2 5 )

where

where the superscr ipt sl represents the medium in which E e - I k R

the scattered field E s is computed and the subscriptsrepresent the surface on which the equation is enforced.

F = - / M - d s ' (28)455- s

For the problem valid for the interior region to thedielectric body, as shown Fig.3, the entire space is filledwith the material of the dielectric medium( p 2 , e 2 ) . Onthe mathematical surface S, , the equivalent currents J ,and M d are introduced. Th e electric field. insideS , is E 2and is zero outside S, . By enforcing the dnt inui ty of thetangential electric field on S: , where S: is the surfaceslightly exterior to s d , the following integral equation may

an dR = Ir - r'l. (29)

In (25)-(29), k is the wavenumber, and r and r' are theposition vectors of the observation and source points withrespect to a global coordinate origin 0. ote that thewavenumber k is either k , , o r k,, depending on themedium surrounding the currentJ an d M .



1572 I E E E TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 38, NO. 1 1 , N O V E M B E R 1990

- h + ._ t

Geometrical parameters associated with the nth ed ge.ig. 4.

B. Numerical Implementation

In this work, the given conducting/dielectric structureis approximated by p lanar triangular patches. The trian-gular patches have the ability to conform to any geometri-

cal surface or boundary, permit easy descriptions of thepatch scheme to the computer, and may be used withgreater densities on those portions of the surface wheremore resolution is desired. Assuming a suitable triangula-tion for the scattering structure, two sets of basis func-tions are defined to approximateJ and M as follows.

Fig. 4 shows two triangles, T an d T,-, associated withthe nth edge of a triangulated surface modeling a scat-terer. Points in T may be designated either by theposition vector r , defined with respect to 0 , r by theposition vector p z , defined with free vertex T,'. Similarremarks apply to the position vectorp i , except that it isdirected toward the free vertex of T,-. It is assumed thatthe plus and minus designations of the triangles havebeen chosen in sucha way that the reference direction for

the positive current associated with the nth edge is fromT,: to T,;. Referring to Fig. 4, we define the two vectorbasis functions associated w ith the nth edge as

\ O , otherwise

an d

(31)( r ) = n x f,'

where h: an d n,' are the height of the edge from thefree vertex and the unit normal vector to the plane oftriangle T,:, respectively. Similar remarks apply to thequantities in triangle T,;.

The electric cu rrent J and the magnetic currentM onthe scattering structure may be approximated in terms ofthese two basis functions as

N< + Nd

J = c f f , I f , , + (32)n = l

T+m

Fig. 5. Testing direction for various field quantities.

an dNd

(33)= c P,sn = l

where N, and Nd are the numbers of edges on theconducting and dielectric surfaces in the triangulatedmodel, respectively. It may be noted that the functionsf,' are th e same functions described earlier by Raoet al.for the conducting body [6]. The functions f,* and g+are pointwise orthogonal in the triangle pair and thusform a unique pair to approximateJ an d M .

The next .step in the application of the m ethod ofmoments is to select a suitable testing procedure.Astesting functions, we chose the same expansion functionsdefined in (30) an d (31). With the symmetric product

definition( f ,g = / sds (34)

( E I ' ( J , ) + E , " ' ( J , ) + E , " ~ ( M , ) + E ~ , ~ , ' ) = O 35)

(E:2( - d ) + E:2( - M d ) , f,') = 0 . (37)

S

we test (221, (231, an d (24) with f,, yielding

( E : ' ( J,) + E:'( J d ) + E:'( M d )+ E:, f,' ) = 0 (36)

Referring to Fig. 5 , the integrals in ( 3 9 4 3 7 ) are evalu-ated by integrating from the centroid ofT to the m iddleof the edge 1, and thence to the centroid of T i . For allpath integrals, the incident field and the vector poten tialquantities A an d V X F are approximated along each

portion of the path by their values at the centroid. Theintegral V@ reduces to a difference of potentials at thecentroids. Substitution of the current expansion functionsof (32) an d (33) into (351437) yields a 2 N x 2 N system oflinear equations which may be written in the matrix form

Z I = v (38)



1573

where 2 is an N X N matrix and Z an d I/ are columnvectors of length N . In (38), N is the total number ofunknowns in the wlution procedure, equal to

N = N,. + 2 N , , ( 3 9 )where N , is the number of interior edg es (i.e., discountingthe edges on the boundary if the surface is open) for thetriangulated model of the conducting body, andN,, is thenumber of edges in the dielectric body model.

The matrix Z may be written as

z,, z,, z,,,

Z = ' h b ' b c ( 4 0 )

where Z,,, Zah, Z,, , Zha, Z,,, Zh,, Z,.,, ZCh, nd Z,., are

[ z c u zc b z<,c1

submatrices of dimension N,. X N,, N, X N,/, N,.X N(!, N,,

respectively.It may be noted that the matrix elements ofZ,, and

Z,, represent the tangential component of the scatteredelectric field produced by the currentsJ , and evaluatedon the conducting and dielectric surfaces, respectively.Th e evaluation of these elements is presentedin detail in[6] and hen ce will not be re peat ed h ere. Th e elements ofthe submatrix Z,, are zero since, in the equivalent modeldescribed in this work, the currents locatedo n the con-ductor do not interact with the inner medium of thedielectric object.

The elements of the submatrices Z u h , Z,,, and Z,,represent the tangential component of the scattered elec-tric field generated by the equivalent electric currentsJ ,and evaluated o n S,, S f , and S,, respectively. Again, thenumerical methods developed in [6] may be used tocompute these quantities. Note that while computing theelements of the matrix ZCh, he appropriate mater ialparameters should be included while evaluating the inte-grals.

Finally, the elements of the submatricesZ,,, Z,,, andZ,, may be evaluated by considering the general expres-sion

Z,,,, = f , : ( r >.V X [g,: ( r' > G ( r , ' ) d ~ ' s. (41)

For the elements of Z,, and Zhc, he field point r islocated on S,. nd S, respectively, and the Green's func-tion is calculated using k = k o . For the elements of Z,,,the Green's function is calculated using the dielectricmaterial parameters; i .e.,k = k , and the ob servation pointr is located on S,. The evaluation of these integrals maybe carried out by using the numerical methods specially

developed for triangular regions [71. However, note that( 4 1 ) s unbounded if r = r' . For this case, the integrals areevaluated in the Cauchy principal value sense.

X N,, N,! X NO, N,, X N(1 X N, > Nc/X NcI,and N,, X N,/ ,

1

IV . N U M E R I C A LESULTS

As a first example, we con sider th e electromagneticscattering from a 0 . l A dielectric cube of E, . = 4.0 coveredby two conducting plates of dimension0.1 A X 0.1 A . T h e

0 .

- i o .

-2 0

- - 3 0

-4 0

-50

-60

0 0

0 0

00

00

00

00

0 00 00 3 0 . 0 0 6 0 . 0 0 9 0 . 0 0 1 2 0 . 0 0 1 5 0 . 0 0 1 8 0 . 0 0

Fig. 6. Comparison for fa r fields: 2Olog,,, E,i versus 0 at = 0" for aO . l A l dielectric cube of E , = 4.0 with conducting plates on top andbottom illuminated by an x-polarized plane wave.

- olume Formulation_ . - urface Formulation

t: 00

5 00

0 00 30 00 60 00 90 00 120 0 0 1 5 0 00 18 00

Fig. 7. Comparison for far fields: 2010gl,, IE,l versus 0 at 4 = 90" fo r a0 . 1 A I dielectric cube of E , = 4.0 with conducting plates on top andbottom illuminated by an x-polarized plane wave.

two conducting plates are placedon the top and bot tomsurfaces of the dielectric cube. The composite struct ure is

illuminated by an x-polarized plane wave incident fromthe bot tom ( 0 = 180"). The scattered fields are plottedutilizing both th e surface/surface and the volume/surfaceformulation and are presented inFigs. 6 and 7. For thesurface/surface formulation, each of the conducting plateswas divided into 18 triangular patches. The dielectriccube was approximated by108 triangular patches. For thevolume/surface formulation, each of the conductingplates was divided into16 cells for J , and J , , for a total of32 unknowns. The dielectric cube is subdivided into64cubical cells representing 192 unknowns. As is evidentfrom the figures, the two solutions compare reasonablywell.

Finally, consider the case of a microstrip patch ante nna

as shown in Fig. 8. A square patch of 5 cm X5 cm isplaced over a dielectric slab( 20 cm x 15 cm X 0. 2 cm) ofE , = 2.56. The bottom plane of the dielectric slab is cov-ered by a conducting, finite ground plane. The antenna isexci ted by a s t r ipl ine at 1.875 G H z . F o r t h esurface/surface formulation, the feed line and the squarepatch are divided into41 t r iangular patches. Th e groundplane and the dielectric slab are approximated by 96 and



1574 IEEE TRANSACTIONS O N MICROWAVE THEORY AN D TECHNIOUES, VOL. 8, NO. 11 , NOVEMBER 1990

-60.00

-70.00

Feed Line Square Element\ / /

.

V-Volume Fomu 1ation -_ _ _Surface Formulation

0.2 cm1

\

Fig. 8. Microstrip patch antenna over a finite ground plane (20 cm X 15cm). E,. = 2.56 and A = 16 cm.

0 . 0 0

-2.00

I Surface Formulation ,- - - .

-5 00 I I I

0.00 60.00 120.00 iEO.00 240.00 300.00 360.00

0.00

-10.00

-20.00

-30.00

-40.00

-50.00

-60.00

-70.00 I0 00 60.00 120.00 180.00 240.00 300.00 360.00

Fig. 10. Comparison of far fields: 2010g1,, EH lversus 0 at 4 = 0” forthe microstrip structure illuminated by a plane wave from the patchside and polarized along the feed line (for scattering problem).

574 triangles, respectively. For the volume/s urface for-mulation, th e square patch and the feed line are coveredby 30 subsections. The ground plane is divided into300subsections with 600 unknowns. The dielectric region issubdivided into 20 X 15 X 1 subsections, resulting in 900unknowns. Th is yields a total of1560 unknowns. In Fig. 9,we present a comparison of the far field (20log,,, E ,) fo rboth formulations. It is obvious that these two solutionsdo not compare well for this case. Although both solu-tions show somewhat similar behavior, the null at8 = 90”is much more pronounced in the volume/surface formu-lation. The problem here is that the excitation chosen

Fig. 11. Comparison of far field: 2010gI,,IE,J versus 0 at 4 = 90” forthe microstrip structure illuminated by a plane wave from th e patchside and polarized along the feed line (for scattering problem).

utilizing rectangu lar patches in the volume formulation isquite different from the excitation chosen using the trian-gular patches for the surface formulation. Hence it isclear that, unless the feed point of the antenna structureis characterized quite accurately, the radiated field, par-ticularly below the ground plane, will be quite differentfor the two approaches.

This statement is verified by solving the same problemas a scattering problem where the excitation is an electricfield incident from 8 = 0” from the patch side and polar-ized along the direction of the feed line. The far fields2010g,, IE,I at 4 = 0” and 2010g,, lEol a t 4 = 90” fo r 0 G8 G 360” computed with th e two techniques show reason-able agreement, as shown inFigs. 10 and 11.

V. C O N C L U S I O N S

In this work, two numerical techniques ar e developedfor the analysis of finite printed circuit antennas. The

techniques are based on volume and surface equivalenceprinciples. They are simple and efficient and solve themicrostrip problem accurately without assuming thegroun d pla nes to b e infinite, as is usually th e case. Typicalresults have been presented to validate the numericalprocedure.

R E F E R E N C E S[11 R. F. Harrington, Time-Harmonic Electromagnetic Fields. New

York: McGraw-Hill, 1961.[2] R. F. Harrington, Field Computation by Moment Methods. New

York: Macmillan, 1968.[3] M. Abramowitz and I. Stegun, Handbook of Mathematical Func-

tions. New York: Dover, 1965.[4] G. W. Hohmann, “Three dimensional induced polarization and

electromagnetic modelling,” Geophysics,vol. 40, pp. 309-324, 1975.[5 ] T. K. Sarkar, E. Arvas, and S . Ponnapalli, “Electromagnetic scatter-

ing from dielectric bodies,” Proc. IEEE, vol. 77: pp. 788-795, 1989.[6] S. M. Rao, D. . Wilton, and A. W. Glisson, “Electromagnetic

scattering by surfaces of arbitrary shape,” IEEE Trans. AntennasPropagat. , vol. AP-30, pp. 409-418, 1982.

[7] P. C. Hammer, 0 . P. Marlowe, and A. H. Stroud, “Numericalintegration over simplexes and cones,” Math. Tables Other AidsCompur., vol. 10, pp. 130-137, 1956.



SARKAR et al. : ELECTROMAGNETIC SCATTERING AND RADIATION 1575

Tapan K. Sarkar (S’69-M’76-SM’81) was bornin Calcutta, India, on August 2, 1948. He re-ceived the B.Tech. degree from the Indian Insti-tute of Technology, Kharagpur, India, in 1969,the M.Sc.E. degree from the University of NewBrunswick. Fredericton. Canada. in 1971. andthe M.S. and Ph.D. degrees from’ Syracuse Uni-versity, Syracuse, NY, in 1975.

From 1975 to 1976 he was with the TACODivision of General Instruments Corporation.

He was with the Rochester Institute of ‘Technol-ogy, Rochester, NY, from 1976 to 1985. He was a Research Fellow atthe Gordon Mckay Laboratory, Harvard University, Cambridge, MA,from 1977 to 1978. He is now a Professor in the Department ofElectrical and Computer Engineering at Syracuse University. His cur-rent research interests deal with numerical solution of operator equa-tions arising in electromagnetics and signal processing with applicationto system design. He has authored or coauthored over 154 journalarticles and conference papers and has written chapters in eight books.

Dr. Sarkar is a registered professional engineer in the state of NewYork. He received the Best Paper Award of the IEEE TRANSACTIONSO N ELEcTmMmNEric COMPATIBILITY n 1979. He also received one ofthe “best solution” awards in May 1977 at the Rome Air DevelopmentCenter (RADC) Spectral Estimation Workshop. He was an AssociateEditor for feature articles of the IEEE Antennas and Propagation SocietyNewsletter and the IEEE TRANSACTIONS N ELECTROMAGNETICOM-PATIBILITY. He was the Technical Program Chairman for the 1988 IE EEAntennas and Propagation Society International Symposium and URSIRadio Science Meeting. Dr. Sarkar is an Associate Editor for the

Journal of Electromagnetic Wai,es and Applications and is on the editorialboard of the International Journal of Microwace and Millimeter-WareComputer Aided Engineering. He has been appointed U.S. ResearchCouncil Representative to many URSI General Assemblies. He is alsothe Vice Chairman of the Intercommission Working Group of Interna-tional URSI on Time Domain Metrology. He is a member of Sigma Xiand the International Union of Radio Science Commissions A and B.

Sadasiva M. Ra o (M88-SM’90) received thebachelor’s degree in electrical communicationengineering from Osmania University in 1974,the master’s degree in microwave engineeringfrom the Indian Institute of Sciences in 1976,and the Ph.D. degree with specialization in elec-tromagnetic theory from the University of Mis-sissippi in 1980.

He served as a Research Assistant at theUniversities of Mississippi and Syracuse from1976 to 1980 and was an Assistant Professor in

the Department of Electiical Engineering, RocheQer Institute of Tech-nology, Rochester, NY, from 1980 to 1985. He was a Senior Scientist atOsmania University from 1985 to 1987 From 1987 to 1988, he was aVisiting Associate Professor in the Department of Electrical Engineer-ing, University of Howton, Houston, TX. Currently, he is an AssociateProfessor in the Electrical Engineering Department, Auburn University,Auburn, AL. His areas of interest include acoustic and electromagneticscattering, antenna analysis, and numerical methods applied to radarcross-section studies.

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Antonije R. Djordjevic was born in Belgrade,Yugoslavia, in 1952. He received the B Sc.,M.Sc., and D Sc. degrees from the University ofBelgrade in 1975, 1977, and 1979, respectively .

In 1975 he joined the Department of Electri-cal Engineering, University of Belgrade, whereat present he is an Associate Professor in Mi-

crowaves. From FebruaIy 1983 until February1984 he was with the Department of ElectricalEngineering, Rochester Institute of Technology,Rochester, NY, ds d Visiting Associate Profes-

sor. His research interests are numerical problems in electromagnetics,especially those applied to antennas and microwave passive components.He is a coauthor of a monograph on wire antennas dnd of two textbooks

moment with equivalence principle of micro strip

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