Electromagnetic coupling to conducting objects behind apertures in a

Radio Science,Volume 32, Number 3, Pages881-898, May-June 1997

Electromagneticcouplingto conductingobjectsbehind apertures in a conducting body KarimY. Kabalan,Ali EI-Hajj, Shahwan Khoury,andAsaadRayes ElectricalandComputerEngineering Department,ganericanUniversityof Beirut,Beirut,Lebanon

Abstract. A procedure for computing characteristic modesfor theproblemof electromagneticcouplingto arbitraryshapeconducting objectsbehindarbitraryshapeapertures in a conductingbodyis developed startingfromtheoperatorformulationof thecurrents.The mode currentsareobtainedfromtheachnittance andimpedance operators. This formulationis generalandcharacterized by its simplicityandefficiency.Thismethodis appliedto a finitethin wire backedby a rectangular aperturein a conducting planein an unbounded medium.The integralequationsin termsof the magneticandelectriccurrentdistributions are derivedfor the magneticandelectricfields.The momentmethodis usedto obtainmatrixequations approximatingtheintegralequations. The expansion functions arechosen to satisfythecontinuity equationsand to vanish' ' at the wire enupomts ---•--: .... Finm•y, ' " a, __,__,: ........... are • :•..• tratethe convergence of the solutionas the numberof modesincreases.

1. Theory

on a wire locatedbehinda circular aperturein a conducting screenbut alsoneglected the interaction of thewire onthe apertureelectricfiled. However, The electromagnetic couplingto conducting ob- theenergyscattered by the wire significantly influjectsbehindaperturesin a conducting bodyis an encesthe aperturefield. important problemin electromagnetic theory.It is of In analyzingthe subject,the completecoupling interestin manyapplications suchas electromag-between the conductors and the slots must be con1.1. Introduction

netic interference and radiation studies. A Iraowl-

edgeof theelectromagnetic fieldsin theaperturein the presenceof the conductoris essential.

The problemof couplingapertures of arbitrary size and shapeand a conducting body has been treatedby many researchers. King and Owyang [ 1960]treatedtheproblemof an arrayconsisting of two drivendipoleslocatedsymmetrically on either sideof theperforated screen. Lin et al. [1974]considered a wireexcitedthrougha slot,buttheycalculatedthe aperturefield in the absenceof the wire

by ignoring thescattering of thewire.Kaffkz[1974] derived theequivalent sources andtravelling current Copyright1997by theA•nericanGeophysical Union. Papernmnber96RS03689. 0048-6604/97/96RS-03689511.00 881

sidered.The interactionof the conductoron the aperturesfields can be significantand cannotbe ignored.Many papershaveconsidered this interaction in searchingfor a solutionto the problem. With the advances in numerical techniques, Butler and Umashankar[1976] formulatedintegro differentialequationsfor the generalproblemof a finite lengththin wire behinda perforatedconducting screen.Theseequationsare solvednumerically for specialcases,and resultsfor the wire current and aperture electric fields were presented. Umashankarand Wait [1978] treatedthe problem of electromagnetic couplingfrom an incidentelectromagneticfield to an infinitecablethrougha slotperforatedscreenby using a Fourier transform method.Naiheng and Harrington [1983] analyzed the problemof electromagnetic couplingto an infi-

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OBJECTS

nite wire througha slotin a conducting planeby the the apertureandfromthevanishingof thetangential method of moments.Hsi et al. [1985] used the componentof the electricfield on the conducting modelis then represented methodof momentsfor determiningthe currenton body.This mathematical an infinitewire, impedance terminatedfinitewire, or in the form of a coupledeigenvalue equationwhose unterminated finite lengthwire behindan aperture. solution determines the characteristic modes and Taylor et al, [1986] developeda formulationfor waves. The characteristicmode theory has been develdetermining the voltageand currenton an impedanceterminatedwire behindan aperturein a shield oped for conductingbodiesstartingfrom the opby includingthe loadingeffectson the apertureus- erator formulationfor the electriccurrentHarringing the receivingantennatheory.Butler [1980] pre- ton and Mautz [1971a]. The modecurrentsform a setoverthe conductorsurface, sentedan investigationof the generalproblemof weightedorthogonal determiningthe currentinducedon a conductorlo- andthe modefieldsform an orthogonalsetoverthe catedbehindan apertureperforatedscreencaused radiationsphereat infinity. In a similarway, this by an impressedsourceand solvedspecialcase theory has been used for aperturesperforatinga equations.Harrington [1982] analyzedthe general conductingscreenstartingfrom the operatorforproblem and discussedthe resonantbehaviorof a mulationfor the equivalentmagneticcurrentHarsmallaperturebackedby a conducting body.Davis ringtonand Mautz [1985]. The modecurrentsalso and Sistamzadeh[1982] considered the problemof form a weightedorthogonalset over the radiating coupling to a multiconductortransmissionline sphereat infinity.Thesetheorieshavebeenapplied throughapertureperforatedscreenand formulated to many situations:circular loop, elliptical loops, an upper boundfor the currentand voltageat the straightwire, helices,bodiesof revolution,wire obplane,and a slot in a line terminationsin frequencyand time domains. jects, a slot in a conducting Smha et al. [1986] considered the problemof ra- conductingcylinderGarbacz [1965]; Garbaczand diationfrom a rectangular waveguide-backed aper- Turp•n [ 1971]; Harrington and Mautz [ 1971b]; ture in an infinite conductingplane when an arbi- Kabalan et al. [1990]; El-Hajj et al. [1992]. This theorystemsfrom viewingan obstacleas a trarily orientedconducting plate is locatedin vicinity. In thiswork, boundaryequations at the aperture devicewhich transformsan incidentwave convergand the conductingbody are used. Butler et al. ing in uponan originwithin it into a perturbedwave [1991] considered the problemof a parallel-plate divergingaway from it. In the absenceof this obstawave incominguponthe origin guideopenedintoa semi-infinite half spacein which cle, any continuous residesa perfectlyconductingcylinderof arbitrary is transformedwithoutdisturbanceinto an outgoing crosssectionwhoseaxis is parallel to that of the wave. In the presenceof the obstacleat the origin, slot. Coupled integral equationsfor the aperture the incomingwave will be disturbedand will be electric field and the cylinder current are derived transformedinto a perturbedwave from the original from first principlesand are solvednumericallyus- outgoingwave.The purturbedoutgoingwaveis put ing the moment method.Manmkko et al. [1992] as a functionof the initial incidentwave using a considered the problemof a two-dimensional slot in perturbationoperator. one of the walls of the parallel-plateguideopened A set of patternfunctionsassociatedwith the ininto a half spacein which a two-dimensional con- comingwave is introduced.It is admittedthat these ductingcylinderresides. pattern functionsare transformedby the obstacle In this paper, the characteristic modetheory is into replicas,with at mosta magnitudeand phase usedto solve for the problemof electromagneticchange.This impliesa characteristicequationthat couplingto a conducting objectthroughan aperture is, an eigenvalueequation. in a conducting body.In this formulation,a matheThe set of patternfunctionscorresponding to the maticalmodelis derivedfrom the continuityof the set of eigenvalues forms an infinite completeortangentialcomponentof the magneticfield across thogonalseton the enclosedspherewith respectto

KABALAN ET AL.' ELECTROMAGNETIC COUPLING TO CONDUCTING OBJECTS

radiatedpower. The total power given as summation of the squaresof the magnitudesof the eigenvalues is found to be finite becauseorderingthe characteristic valuesin orderof decreasingmagnitude showsthat beyond a certain mode, little is contributedto the perturbedfunction. Therefore, only a finite numberof modesis significant,renderingthe problema finite dimensional one. Here, it is desirableto take advantageof the aperture in a screen (no scatterer) and scatterer (no aperture)formulationsusingthe characteristic mode theoryin treatingthepresentproblem.

883

'•

'"•'•..shorted //-• ,M• •. aperture '/ M ./

//2

'x• regiona

conductors

Figure 2a. Equivalentsituationof regiona.

1.2. Geometry and Formulation of the Problem.

The equivalenceprincipleHarrington [ 1961] is usedto dividethe originalproblemin Figure I into an equivalent model valid in region a and an equivalentmodelvalid in regionb, as shownin Figure 2. The apertureis closedwith perfectelectric conductors(short-circuiting the aperture)and provided with hypothesized attachedmagneticcurrent

The general problem of couplingto a scatterer throughan aperturein a conductingwall is depicted in Figure !. It consistsof two regionsboundedby a perfectelectricconductors andcoupledthroughan apertureof arbitrary size and shape.One region, called regiona, is consideredto be closed,i.e. of sheet on both sides of the conductor in order to have boundedextent.The otherregion,calledregionb, is the electric field originally presentat the aperture considered to be open,i.e. of unbounded extentopen unchanged.The first boundaryconditionenforces to infinity. The excitationis represented by imthe continuityof the tangentialelectricfieldsacross pressed sources (ji, Mi ) in region a. A perfectlythe slot in the originalproblem,thusthe equivalent electricconductingscattereris placedin regionb. magneticcurrentsheetsare of +M to the left of the The mediumin each regionis assumedto be loss slot and of-M to the fight of the slot. The equivafree, i.e. the radiatedpoweris theonly lostpower. lent surfacemagneticcurrentis givenby M=nxE

'/,• ,M /

,•/aperture

(1)

•..•/'/' ß • /'•. conducting shorted

aperture /

-M

•,....r. eglon a /f /

C

___•) .•r•'

scatterer

•-"'•

conductors

ionb

conducting

s•_s_s_s_s_s_s_s_s_••2 U • scatterer

conductors •'J•'I re,onb Figure1. A typicalproblemconsisting oœtwo regionsbounded by conductors and coupledby an aperture witha conducting bodyplacedbehindit.

Figure 2b. Equivalentsituationof regionb.

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whereE is the unknownelectricfield presentin the Thelinearity oftheoperator Htt'hasbeen used in aperturein the originalsituationand n is the nom•al (4) to replaceH,'(-M) by - H,•(M). The remaining unit vectorpointinginto regionb. boundaryconditionto be satisfiedis the vanishing Theimpressed sources (ji, M) in addition to the of the tangentialelectricfield acrossthe surfaceof surfacemagneticcurrent+M remainin regiona in the scatterer. The tangential componentsof the the equivalentsituation.The electromagnetic field in electricfields in regionb on the scatterersurface regiona is producedby the originalsourcesandthe Ep canbeputas equivalentsurfacemagneticcurrentin the presence of the short circuited aperturewall. The electroEtt'- Ett'(-M) + E,t'(J, ) (5) magneticfield in region b is producedby the onthescatterer equivalent surfacemagneticcurrent -M and the surface electric current inducedon the scattererJ.• in

the presenceof the shortcircuitedwall. The second boundaryconditionto be satisfiedis that the tangentialmagneticfields shouldbe continuousacross the aperturein the originalproblem. The tangentialcomponentof the magneticfield in a

regiona, H t , canbeputas

In (5), E,• (-M) is theelectricfielddueto surface magnetic currentandE,•(J,) is theelectricfielddue to the induced surface electric current on the scat-

terer. Both fieldsare evaluatedin the equivalentregion b with the apertureshort-circuited.The last boundaryconditioncan be expressedin equation

formbyequating Ef to zero.Theequality is rearrangedto

overtheapertureH• - Hj + Ht• (M)

(2) on the scatterer

Ett'(M) - E• (J, ) - 0

(6)

whereHi is the tangential component of the magneticfielddueto impressed sources and H•'(M) is Thelinearity oftheoperator Ett'hasbeen used in the tangentialcomponentof the magneticfield due

(6)to replace E• (-M) with- E• (M). Equations

to the magneticsurfacecurrent.Both Hi and (4) and (6) are the basicoperatorequationsfor deH•(M) are evaluatedin the equivalent region a terminingthe tangentialelectricfield overthe aperwith the apertureshorted.The tangentialcomponent ture (or equivalentsurfacemagneticcurrent)in the ofthemagnetic fieldinregion b, H f, canbeputas originalproblemand the surfaceelectriccurrenton the scatterer,whichare notknowna priori.

over theaperture H) - Htt'(-M) + H) (J, ) (3)

Since theoperators H•'( ) andH,•( ) havethe

dimensions of an admittance,a linearoperatorY" which relates the tangentialcomponents of-H to whereH• (-M) isthetangential component of the the magneticcurrentM radiatingin the equivalent magneticfield due to the magneticsurfacecurrent

and Ht•(J•) is the tangentialcomponent of the magneticfield scattereddue to the inducedsurface

electriccurrenton the scatterer. Both H,•(-M)and H,•(J,)are evaluatedin the equivalentregion b with the apertureshortcircuited.The continuityof the tangential componentof the magneticfield acrosstheapertureyieldsovertheaperture,with the helpof (2) and(3), the followingequation:

- H•' (M) - Htt'(M) + Hf (J,) = H•

(4)

modelof region a, anda linearoperator YUwhich relatesthe tangentialcomponents of-H to the magneticcurrentM in the equivalentmodelof regionb are defined as follows:

Y" (M):

r(M) -

-H•' (M)

(7) (8)

In a dualmanner, sincetheoperator Ef ( ) has

KABALAN

ET AL.:

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the dimensions of an impedance, a linearoperator

Z• whichrelates thetangential components of theE to the electfiecurrentJ• in the equivalentmodel of regionb is definedas follows:

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885

II H,•(M,)ßM•ds=- lJH,•(M•)ßM,ds(16)

Apeft.

Apeft.

- II Ef(J,,).J•ds=-IIEf(J•,).J,, ds(17) Apeft.

Apert.

(9)

II Ef(M,).J,,ds=-IlHf(J,,).M,ds(18)

Furthermore,we define the linear operators C Apart. Apeft. whichrelatethe tangentialcomponents of H to the electriccurrentJ• radiatingin the equivalentmodel In (15)-(18), a andb referto regionand i andj refer

of region b, and C/ whichrelates thetangentialto fieldsandsources. Y•, Y•, andZ• arenotHercomponents of E to the magneticcurrentM radiatingin theequivalentmodelof regionb as follows:

C(J•) - H,• (J•)

mitian operators,that is, theseoperatorsare not self-adjointwith respectto innerproduct.However,

(15),(16),and(17) implythatoperators Y'•,Y•, and Z • are symmetric. Hence the admittance op(10) eratorY, whichis a linearcombination of Y• -and

C'(M) - E• (M)

operator Z, whichisequalto (11) Y•, andtheimpedance

Zu , are symmetric.Equation (18) indicates the

The total admittanceoperatorY whichrelatesthe reciprocitytheoremfor two sets of sources: .an tangentialcomponents of the magneticfieldsto the electric current source J,i onthescatterer where magneticcurrent,and the total impedance operator the magneticcurrentis zeroanda magneticcurrent Z which relatesthe tangentialcomponents of the sourceM• on the aperturewherethe electriccurelectric fields to the electric current are defined as rent is zero.The fieldsproducedby the two setsof follows: overthe aperture

sources are (E• (M,,O),(Hf (M•,O)) and

Y(M) - ya(M) + Yo(M)

(12)

(E)(O,J,i),(Hf(O,J,i))' respectively. The integrationis extendedoverall space.Nevertheless,

and on the conductor

(18) defines a relationbetween C and C':

Z(J,)-ZO(J•)

(13)

-"(35)

•-' I z +•-' V -0) is calledregionb. Regionb is alsounboundedopento positiveinfinity.The mediain both P• : - --+ --

regiona to region Linearity oftheoperator Ht• andofthesymmetric(At,•). Theapertureconnecting product

has

been

used

to

replace b isof lengthL• inthex direction andof widthW• in the z direction. It is centered in the y=0 pirate.A • M, Htt'* (-M*) >.-with- .-. wire of finitelengthLwis It is convenientto use (8), (10), (31), and (32) to- thinstraightconducting getherwiththe linearityof the symmetric productto placedin regionb parallelto the conducting screen

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KABALAN ET AL.' ELECTROMAGNETIC COUPLING TO CONDUCTING OBJECTS

shortedand replacedby the equivalentmagnetic currentM. In Figure4b, the screenis removed,and by imagetheoryHarrington [1968] an imagemagnetic current is addedto the equivalentmagnetic region b

conducting screen

z

A

x

i

(E

a

i

,H a ) shorted

a p e rtu re

Figure3. A finitethinwireplacedbehinda rectangularaperture perforated infiniteconducting plane.

region

a

y=O

Y - Y•v,x - x•v, makingan angle(• withthez axis and centeredat the z=0 plane. The crosssectionof the wire is verysmallandhasa diameterof 2rw. The excitationof the apertureis a normaluniform

removed screen

planewaveincident at anincident angle• fromregiona andis represented by Harrington,[1968]

H:- exp(-jn'(zsin•9 i +ycos•9•))u. (42)

( F_• ,H=•

(Ei ,H i a

a

It is assumedthat no sourceexistin regionb. The conductingscreenis labelledS, the apertureis labelledA, andtheconducting wire is labelledW.

image current

2.2. Illustration

of the Procedure

A step-by-stepreductionof the originalproblem in Figure 1, with (•=0 degree,is shownin Figures4 and 5. Figure 4 showsschematicallyhow the left half spacein the originalproblemis transformedin the equivalentproblem.In Figure4a, the apertureis

region a

region a

y=0

Figure 4. Equivalence of regiona.

r

KABALAN ET AL.' ELECTROMAGNETIC COUPLING TO CONDUCTING OBJECTS

889

Figure5 illustrates theequivalence of theoriginal problem of therighthalf space.In Figure5a, the equivalence principle is usedto short-circuit theap-

conducting screen

,•,-h,,-•,•,•,4 ronlat-o it with _ns

conducting

•

[J

aperture •

e.n.•res the

continuity of theelectric fieldin theaperture andis the only sourceilluminating regionb. In Figure 5(b),thescreen is removed andby imagetheory,it encloses the imagemagnetic currentandtheimage

wire

shorted

-M

wire. The wire is consideredas a scattererthat inducessurfaceelectriccurrentdue to the magnetic currentsourceat the aperture.

The equivalent problemin regiona is that of the short-circuited fields(E '•, H '•) andthemagnetic currentM with its image radiatingin an infinite space.Similarly,theequivalent problemin regionb is that of the magneticcurrent-M with its image andtheinduced currentJ, withits imageradiating

y=y

region b y=O

in an infinite space. From the time harmonicMaxwell's equationsfor

removed

an e-• excitation, thefieldsproduced by M to the

screen

left of the screenare givenby Harrington [ 1968] conducting

/c

wire ,

,/

] V•/(o-) (44)

E(M) - -V x F(M)

(45)

where /c- co/zøø )!/2 is the wave nulnber,

r]-(At/00) ]/2isthe characteristic impedance, and image wire

F(M) istheelectric vectorpotential produced by

image

current

current

M in thepresence of theshorted screen:

y=y

y=-y

I regionb

region b

e-j••/(x-x /)2+y2 +(z_z /)2

F(M) =• JJ z')x/(x-x')• +Ya ^M(x', +(,z-z')

dx'dy'

(46)

y=O

Figure5. Equivalence of region b.

qJ/(or) isthemagnetic scalar potential produced by themagnetic charge density c•in thepresence of the

current,anda reflected waveis addedto the inci-

shorted screen:

dent wave.

The shortcircuitedmagneticfield at the aperture becomes

H '• - 2 exp(-js2sin0i)u•

(43)

(47)

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ELECTROMAGNETIC

- V-

(48)

jr0

Substituting(46), (47) and (48) into (44), the magneticfield tangentialto the aperturedue to M becomes

H'(M) --J • •AI1V•X"') V(X.,)2 +(Z.,)•

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whereA(J,) is themagnetic vectorpotential produced by J, inthepresence of theshorted screen:

A(J')-T••](x_x•)• +(y-y•)• +(z-Z) • e

-•-•

dz'u•

•J(x_x•)2 +(y+yw) 2+(zz'):

(53)

-j v•lv.•[i,i)].•/(x.i)•+f+(z.t) • (49) •*(/9) is the electricscalarpotentialproducedby

the electricchargedensityp in the presenceof the

wherethenotation•/'A is usedto indicate theaperture tangentialcomponents of the gradient,and V' is usedto indicatethe divergenceof the magnetic currentin the aperture,that is, with respectto the primedcoordinates. Substituting(46) into (45), the electricfield tangentialto the wire dueto M becomes

e_j•: •](x_xw)2 +(y_yw)2 +(z_ •/)2 dz' Et(M )--•--• VwxIIM(x',Y')

shorted screen:

I/,øx/O:_ e-i'•/(x-x•)2+ - co +(yyw) dz/

p

+(y+y•)2 +(z_z/)2 (54)

1

A

x/(x_x,)2 +y2+(z_z/)2

at X=Xw,andy=yw

(50)

_ v, (j.(z•)u,)

p-

(55)

Substituting(53) and (54) into (51), the electric

tothewiredueto J, becomes In (50),thenotation V• isusedto indicate thewire fieldtangential tangentialcomponents of the curl.

Assumethat the wire is thin comparedto the wavelengthof the incidentfield. In this case,all currentsand fields within the wire are approximately axial with no circumferential variation around the perimeterof the wire cross section.

Hence,theinduced currentdensityis replaced by its

.:rl Ij•(z,)

E,(J.•) - -j•-•

w

e

-j•: x/(x-xw )2+(y-yw )2+(z-z')2

dz'uz

x/(x-Xw) 2+(y-yw) 2+(z-z/)2

axial component. From the time harmonicMax-

well'sequations, thefieldsproduced by J, to the rightof thescreen aregivenby Harrington[1961l

.:rl Ij ,)

+j•-• •(z w

E(J,)- -j•:rlA(J.)- rlV$'(o)

(51) e

-j•c4(x-xw):z +(Y+Yw):z +(z.z,):• dz'u

H(J,) - -VxA

(52)

x/(X-Xw) 2+(y+yw) 2+(z-z/)2

KABALAN

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ET AL.'

ELECTROMAGNETIC

Vw I[V' ßJ•(z')] w

e-j•J(x-x.) 2+(y-yw) 2+(Z-Z') 2

x/(X-Xw) • +(y-yw):+(z-z/)•

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891

2.3. Numerical Application Consider a bidirectional magnetic currentas follows'

dz'u zlX=Xw,y:yw

M(x, z) = M• (x,z)u• + M z(x,z)uz

(58)

and the unidirectionalelectric current as

• nv l[v'.J

+J4mcw

j• (z)_jz(Z)U z

(59)

w

e-j•J(•-•.)' +(y-y.)' +(z-z.)'

anddefine

dz'u[ J(X-Xw) 2+(y-yw) 2+(z-z/)2 zx=xw'Y=Yw e"j• cos(ru) sin(ru) (56)

u

-

-j•

u

u

= Cs(u)-jsn(u)

where thenotation V• isusedto indicate thecomponentsof the gradienttangentialto the wire, and Therefore, theoperators aregivenby

(6O)

V • isusedto indicate thedivergence of theelectric

current in the wire, that is, with respectto the primedcoordinates. Substituting(53) into (52), the magneticfield tan-

G(M): •-•ffM,(x',z')Sn(J(x-x/)'+ (z-z')' }dx/dz /u,

gentialto theaperture dueto J, becomes ß[M ,t'x' ,z" /u,,j +(z1Va.[iV, 'Sn(J(x-x'?

+ nKq

1V^ xIj•(z,)

at(J•)- 4•

+•

w

e

-jr 4(x-xw )2+(y-yw )2+(z-z')2

): + (z- z'): 4(x- Xw) 2+(Y-Yw

dz'uI

z y={}

^

KIA IMz(x''Z/)ßSn(J(x-x/) 2+(z-z/)• )dx/dz/Uz •:•:q

'

A

1 I,

4•:V^x •(z')

ßSn,J(x-x')• +(z- z/)2 dx/

w

(61) -jr x/(xx. )2+(y+y,,)2+(z-z'?

)• + (z- z'): J(x- xw)2+(y+yw

dz'uI

z y=0

0

C/" (M) - 2xOy IIM•(x/,z/)ß ß C Xw +(z-z) / Uz[Y=Yw X/ -x')2+ Y'' 2dx/dz A

(57)

where thenotation V• isused toindicate thecomponents of thecurltangential to theaperture.

I

(62)

892

KABALAN ET AL.: ELECTROMAGNETIC

_ •ctl

z/

R(J,)•-•fJz()'

COUPLING TO CONDUCTING

OBJECTS

N .P

P.N

M.(x,z)-

w

i=l

u.,f,(x,z)u+ j=•

(x,z)u,

ßSn(x/(xXw) 2+(y-yw) •+(z-z')• dz/ Uz +

wherei andj representthe subareasindicesand fi andgj are realfunctionsof x andz. In addition,the wire is subdividedinto small sublengths of length

J(). •ctl •w z/ AZw. Assume that there are Nw subdivisionsin the z 'Sn(4(x-Xw)' +(Y+Yw)' +(z-z/)')dz/Uz 4•:

•

direction.

The characteristic electric current is as-

sumed unidirectional and defined as Sw

+4•tl Vw[V/.[j w z(z/)u z].

J,.(z)- • D,•,h t(z)u: I=1

(66)

.Y=Yw ßSn(x/(x-Xw) •+(y-yw) •+(zz/)• Ix=xw 2.4. Reduction To Matrix Equations 4•c

w

z

(z',u z].

The methodof momentsHarrington [1982] is usedto reducethe operatorequationsinto matrix equations. To obtain symmetric matrices, Ix=xw ,Y=Yw Galerkin'smethodis usedand the testingfunctions over the apertureand wire surfacesare, respectively, the onesdefinedin (65) and(66). (63) Take the synm•etricproductof (26) with each magneticcurrenttestingfunctionand of (27) with eachelectriccurrenttestingfunction.The following matrix eigenvalueequationdue to n and m modes

ßSn(4(x-Xw) 2+(y+yw) 2+(z-z/) 2 cr(J•) - 4• I Oy 0•w Jz(z/).

ßCs(x/(x-Xw) 2+(y-yw) 2+(z-z/) 2dz/Uxly--0

are obtained as

_4•I Oy0w• Jz(z/). ßCs(X/(xxw)•+(y+ yw )•+(z-z/)• dz/ Uxly=0

&jLr, r,.],, ['X't' ][ol]m --•m[ 1•1 ][ol]m

(67)

(68)

B(M), C•(J,), C/•(J,), andX(J,) areobtained similar to G(M), C'(J,), C/'(J,), andR(J,), wheren=l, 2, ..., 2N.P, re=l, 2, ..., Nw. r,s,i,j=l, respectively by interchanging Sn with Cs and Cs with Sn. For a numerical solution of theproblem, the rectangularapertureis subdividedinto small subareasof lengthsAz in the z directionand Ax in the x direction. Assume that there are N subdivisions in the x direction and P subdivisions in the z

direction. Thus, there are N.P subareas.The characteristicmagneticcurrentsare bidirectional anddefined as

2, ..., N. P. andt,s= 1,2,...,Nw..The eigenvalues approximatethoseof the operatorequations(28) and (29). The eigenvectors definefunctions according to (31) and (32), which approximatethe eigenfunctionsof theoperatorequation. Therefore,substituting(61)-(64) and using the identityfor functionsF• andF2,

$$ . V,F2 )ds- -$$F2(V,. )ds (69)

KABALAN ET AL.' ELECTROMAGNETIC COUPLING TO CONDUCTING OBJECTS

893

Table1. TheConvergence of theCharacteristic Valuesfor a slotof Length 0.7)•in thex Direction and0.005)•inthez Direction N

bo

b•

5 7 9 11 13 19 29 41

3.191590 2.984210 2.889673 2.837438 2.804610 2.754427 2.726762 2.723427

-23.668320 -20.906797 -20.026301 -19.593253 -19.330403 -18.914394 -18.597807 -18.378583

b2

-1414.723925 -1024.089217 -913.299055 -863.978995 -863.782441 -799.809501 -778.389954 -767.710223

ba

-190340.575461 -77411.802836 -58906.104837 -51969.108950 -48487.376370 -44211.548658 -42073.503688 -41161.746163

in evaluating The elements of (67) and(68) areobtained together easwhichmay leadto discontinuities theintegrands, the integrals nearthediscontinuities withthematrixequivalents of (35) and(36). are evaluated analytically.The integrands are im--1 ....... ..1i:t• --- L1.....,.1.,• by 11•/I,111•.• ..... *"' *• lJIg;ll!g;l!tg;U OlUbl••-t..-.•tllgttar•,passed t.•J t.11•.• 2.5. Numerical Results

general function thatnumerically evaluates theintegrals. After evaluating the elements of matrices A generalcomputer program hasbeendeveloped subin C language to realizetheaboveequations.The [G],[B],[C•],[C•],[C/•],[X],and[R],special expansion andtestingfunctions are chosen for the routines such as NROOT and EIGEN from the magnetic currentto be composed of a rectangularIBM libraryareusedto solvefor thecorresponding andthe eigenfunctions of the problem. functionanda triangularfunctionandfor the elec- eigenvalues resultsareshown,theconvertric currentto be a triangularfunctionsuchthatthe Beforeanynumerical boundary conditions aresatisfied andthecontinuity genceof the algorithmis studied.This is donefor of themagnetic currentovera smallregionis as- severalaperturescoupledto finite size wiresand sured. The corresponding integralsare reduced to excitedwith a normalplanewave. The resultsof summationsbased on double-integralSimpsoh's the first four characteristicvaluesfor a narrow aprule. When the domains of a term have common atertureof length0.7)• in thex direction andof width

Table2. TheConvergence oftheCharacteristics Values fora WireofLength 0.5)• in the z Direction and 0.001)• Radius.

Nw

g0

g•

5 7 9 11 13 19 29 41

1.081039 1.102391 1.043151 0.99599 0.963480 0.915195 0.893420 0.890475

-1025.443340 -682.224020 -525.939422 -434.848293 -374.847516 -276.089249 -207.620080 -172.830739

7•2 -120519.268032 -63949.772917 -45573.304576 -36274.787085 -30589.716033 -21820.711019 -16058.297295 -13197.686195

g• -37867421.371541

-10933283.235639 -6584768.323019

-4860707.847839 -3932449.307015 -2652411.318687 -1889011.034108 -1526398.703122

894

KABALANET AL.' ELECTROMAGNETIC COUPLING TO CONDUCTING OBJECTS 0.4

1

0.2

0

/',,

-0.2

_." ,..

0.2

\\ __ __ __ REAL M

-0.4

,'

¾i

0

-o.2 j,,

",,,/

'ø'Slxx

?i

.-" n:0

\•'•\\N•. IMAGINARY -o.s I--\\'•\-••1' / '""• ' '•'"' ' -1

-0.6

-0.41

-0.8

-1

0

',./

0.1

0.2

0.3

,,'-- - n=l

,."...... n=2

0.4

0.5

0.6

07

-1.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 8. •e characteristic modesof the equivaFigure6. The equivalentmagneticcurrentfor an lentmagneticcurrentfor an ape•ure of len•h 0.7• x/%

apertureof length0.7X in thex direction andwidth

in the x direction and width 0.005•

in the z direc-

0.005X in the z directionand a wire of length0.5)• in the z directionplacedat (0,0.25)•,0)parallelto

tion and a wire of len•h 0.5• in the z direction placedat (0,0.25•,0) parallelto thez axis.

the z-axis.

[1985], the most importantmodesof the solution 0.005)• in the z directioncoupledwith a wire of arethose ofwhichb• and2'• aresmallest. Therelength0.5)• in the z directionandof radius0.001)• fore,themodes areordered n=1,2,3,...according to aregivenin Tables1 and2 for differentvaluesof N Ibll < lb21 < Ib•l