THE SIMPLEST ANTENNAS: RADIATION, RECEPTION AND SCATTERING BY AN ANTENNA IN 1-DIMENSION

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1 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) HE IMLE ANENNA: ADIAION, ECEION AND CAEING BY AN ANENNA IN -DIMENION teven J Weiss () nd Wlte K Khn () () U Amy esech Lbty, Adelphi, MD 783, UA Emil: stevenjweiss4civ@milmil () Deptment f Electicl nd Cmpute Engineeing, chl f Engineeing nd Applied cience, he Gege Wshingtn Univesity, Wshingtn, DC 5, UA Emil: wkkhn@gwuedu Abstct his ppe ims t pvide undestnding f sctteing nd e-ditin by eceiving ntenn thugh detiled nlysis f epesenttives fm the simplest clss f pssible ntenns, -dimensinl ntenns Cmplete, exct slutins f the electmgnetic fields sscited with such -dimensinl envinment nd ntenn stuctues cn be btined using nly elementy nlysis nd functins hese exct slutins my be mdeled by tnsmissin line equivlent cicuits he ppe cies this develpment futhe, ecnciling the tnsmissin line mdels with the tditinl ntenn equivlent cicuit fmultin nd the sctteing mtix epesenttin f ntenn they, in the pcess explining the vius spects f ntenns: ditin, eceptin, sctteing nd e-ditin Index ems Antenn, Antenn equivlent cicuits, Antenn sctteing, ctteed pwe, ctteing pmetes, nsmissin line mdels I INODUCION his ppe ims t pvide deepe undestnding f ntenns, especilly eceiving ntenns, thugh detiled nlysis f epesenttives f n especilly elementy ntenn type, -dimensinl ntenns [] Cmmnly, ntenns e cnsideed t dite fm sme cnfined vlume int (3-dimensinl) spce Nevetheless, thee e specil cicumstnces, eithe idelized infinite stuctues ntenns diting between pllel cnducting pltes, when the ntenn my be cnsideed s diting int - dimensinl spce his ppe nlyzes idelized infinite pln gemeties in which ntenns dite int - dimensinl spce Cnstuctin f diclly simplified stuctue which nevetheless stisfies ll pplicble physicl pinciples nd cnstints f the pupse f develping nd illustting the they is time-hned technique As exmples f such cnstuctins ne my cite the Cnt engine in hemdynmics [] nd the uing mchine in Cmpute cience [3] In the me mdest cntext f ntenns, the - dimensinl ntenn my spie t simil le While - dimensinl ntenns stisfy ll pplicble physicl ntenn pinciples nd cnstints, exct cmputtin f ditin, eceptin nd sctteing equies nly elementy lgeb nd elementy functins In cntst, nlyses f - nd 3- dimensinl ntenns invlve integl/diffeentil equtins nd cylindicl spheicl Bessel functins t yield even ppximte slutins Accdingly, ectin II intduces wht is pehps the simplest epesenttive f such -dimensinl ntenns cmpised f fee spce itself nd lded with n infinite plne, thin cnducting sheet, (hms/sque) ectin III develps the tnsmissin line equivlent cicuits f - dimensinl ntenns in genel nd the peceding simplest epesenttive in pticul [4, 5, 6] Cnsideing eceptin f plne wve by this ntenn, the exct eceived nd sctteed pwes e cmputed ectin IV eintepets these sme esults in tems f the tditinl ntenn equivlent cicuit fmultin [6 - ] ticul ttentin is given t the le f the ditin esistnce which ppes in the hevenin equivlent cicuit A nvel nlysis f the eceive cicuit demnsttes the specil cnditins unde which pwe in the ntenn impednce my be given the intepettin f sctteed pwe ectin V pesents sctteing mtix fmultin specilized f -dimensinl ntenns [ 4] Using this fmultin the univesl cnstnt elting the gin t the eceiving css-sectin f mtched -dimensinl ntenn system is estblished in n especilly tnspent clcultin Fequently, thughut this ppe, we will wite the fmuls f quntities in sevel diffeent nttins nd fms, nmliztins, etc; ne nttin dpted the ppch t hnd nd equivlents t fcilitte cmpisn with fm btined diectly vi diffeent tetment

2 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) In ectin VI, me cmplex nd inteesting clss fmily f ntenn stuctues is cmpised f dielectic slb tnsfme lded with thin esistive sheet [4 6] he dielectic slb tnsfme is pptined f ze eflectin f plne wve nmlly incident n the slb ntenn, ie, ze bcksctte hugh this cnditin the pmetes f the slb ntenn becme functins f the cnducting sheet ld esistnce, in effect defining diffeent ntenn f this clss f ech vlue f ld esistnce he eceived pwe, the pwe dissipted in the ntenn (hevenin) equivlent cicuit esistnce, nd the sctteed pwe e cmputed An e in pevius cmputtins f the sctteed pwe by Geen [6] nd uselves [] is cected F this clss f ntenns with ze bcksctte, the eceived pwe is gete thn the sctteed pwe he sctteed pwe is gete thn the pwe dissipted in the ntenn (hevenin) equivlent cicuit ditin esistnce II HE -DIMENIONAL ANENNA he essentil fetue tht distinguishes n ntenn mng scttees is the pesence f ne me lcl pts by vitue f which the sctteing my be mdified pwe bsbed by n ttched ld he (simplest) ntenn shwn in Fig is delimited, s t spek, by the infinite esistive film ( / sque ) which cts s ld n the lcl pt f this ntenn his the bstct cnceptin f the simplest ntenn will be mde cle nd cncete by the tnsmissin line mdel Fig elbted in the next sectin hee the ntenn is epesented by n idel, symmeticl 3-pt shunt- junctin, tnsmissin lines e ttched t the tw symmeticl pts, nd the esistive ld ttched t the thid pt Incident n the film (fm egin ) is plne wve In genel, eflectin cefficient bck int egin nd tnsmissin cefficient fm egin int egin, the shdw egin, my be defined Fig A -dimensinl ntenn teminted in plne, infinite esistive film ld At this pint we digess fm cnsidetins petining t the ntenn t inset bief side deling with the vect chcte f the incident plne wve It is ssumed thughut tht the vect pliztin f the incident plne wve which ppgtes n the sscited tnsmissin line is mtched t the eceiving ntenn While, n the ne hnd, cicuit epesenttin f n bitily plized incident plne wve des nt pesent ny fundmentl difficulty, it wuld detct fm the simplicity f pesenttin, dubling the size f the equivlent cicuit (tw tnsmissin lines) On the the hnd, the mplitude f the mtched pliztin is esily septed ut fm n bity incident wve s indicted next he electic field vect sscited with n bitily plized incident plne wve my be nlyzed s [7]: () whee E is the pliztin nmlly tnsmitted by the ntenn In the eceive mde, the ntenn is mtched t the vect plne wve with the sscited electic field vect E, the exct time evese f the field nmlly tnsmitted In view f the thgnlity eltin: E ( E zˆ ), he cmpnent E () is bviusly independent f the mgnitude f E Accdingly, we will del with nly this pliztin mtched cmpnent f ny bitily plized incident field in the subsequent nlysis III ANMIION LINE CICUI EEENAION AND ANALYI OF HE -DIMENIONAL ANENNA he -dimensinl ntenn is menble t simple fmultins using tnsmissin line they, mking use f the well-knwn cespndence f electic nd mgnetic field mplitudes f plne wves in spce nd the vltge nd cuent mplitudes n unifm tnsmissin lines [, 4] In view f this cespndence, we btin the cicuit epesenttin shwn in Fig f the physicl stuctue f Fig he pmetes f the equivlent tnsmissin lines e, f cuse, the ppgtin cnstnt k / f fee spce nd chcteistic impednce f fee spce he genet is ssumed t hve n intenl impednce equl t tht f fee-spce his chice mdels excittin f wve mplitude t n infinite emve he equivlent cicuit f this simple ntenn is n idelized shunt- junctin, Fig (nd ls with mtching tnsfme in Fig 8) he esistive sheet is epesented by the ld t the lcl pt f the ntenn, the esist he incident nd eflected wves fllw the cnventinl definitins in tems f ms phs quntities: E V I (3) g b V I (3b) e V I b (3c) Fig nsmissin line cicuit epesenttin, shwing n idel shunt- ld cnnectin

3 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) Equtins 3 e specil cses f the genel definitin f the nmlized vltge sctteing pmetes futhe elbted in Appendix A Nte tht tnsmissin line cnventins (incidence fm left t ight) e used hee t ssign diectins t the wve mplitudes nd b, nd b he wve pmete n the equivlent tnsmissin line cicuit is pptinl t the electic field stength E, (), nmlized s tht: E ( Wtts / m ) (3) It my be seen fm (3) the V nd I cnsequently hve dimensins f Vlts/m nd Amps/m espectively F clcultin f pwes, Fig my be educed t the simple cicuit digm shwn in Fig 3 As bth ends f the tnsmissin line e teminted in the chcteistic impednce (ze eflectin cefficients) the tnsmissin line lengths ply n le nd e theefe left unspecified he film esistnce is, t gd ppximtin, independent f fequency Fig 3 educed cicuit epesenttin f Fig One my nw cmpute the eceived nd the sctteed pwe he pwe eceived in the sheet (pe sque mete css-sectin f the tnsvesely infinite stuctue) my be detemined fm E E ˆ ˆ g g I whee: / ˆ ˆ V Eg Eg ˆ (33) (33b) Accdingly, the eceived pwe is viusly given s: e V I E g ˆ V V ˆ 4 4ˆ ˆ (34) (34b) (34c) he -dimensinl gemety ls llws us t fmulte the sctteed field nd cmpute sctteed pwe in stightfwd mnne he sctteed pwe is meely definite mesue f the diffeence between the ttl field with the ntenn pesent nd the incident field [4,7,,,4,5] N cnsevtin eltin invlving eceived nd sctteed pwe exists nd nne is implied in this cmputtin he incident field is the field in the bsence f the ntenn nd its (eceive) ld (Fig with the ld emved) Distinguishing the incident nd eflected wve quntities f the incident fields within the tw egins by ze supescipts (35) b b (35b) In the pesence f the ntenn nd its eceive ld, the eflected nd tnsmitted wves e mdified s tht: ( ) (36) b b (36b) whee the eflectin nd tnsmissin cefficients (f Fig ) e given by / ( ), (37) / ( ) ˆ ˆ ( ), ˆ (37b) he ttl field in the pesence f the ntenn is the sum f the incident nd sctteed fields In egin, the sctteed field is theefe fund fm (35) nd (36) ttl field in egin = (38) + b = + sctteed field in egin (38b) sctteed field in egin sctteed field in egin (38c) ˆ sctteed field in egin (38d) In egin, similly, the sctteed field is deduced fm (35) nd (36) b sctteed field in egin (39) ( ) sctteed field in egin (39b) sctteed field in egin (39c) ˆ sctteed field in egin (39d) he esults f (38) nd (39) veify n intuitively imptnt spect he cuents induced in the esistive film dite (sctte) eqully in the tw diectins Adding pwes sctteed int the tw diectins (egins), the ttl sctteed pwe is: Eg s ˆ he eceived pwe nd the sctteed pwe e equl nly when ˆ / his cnditin cespnds t cnjugte mtch t the eceive ld pt ymmeticl sctteing nd mtch cnditins cmpt with identifictin f this simple ntenn s cnnicl minimum sctteing (CM) ntenn 3 (3)

4 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) [] Indeed, the idel shunt- equivlent cicuit, Figues nd 7, embdies the cicuit epesenttin f the CM ntenn cncept he identifictin is eviewed in the iginl sctteing mtix cntext in ectin V he eceived nd sctteed pwes my be witten in tems f the single vible,, s: ( ) (3) el pt cmpises cmpnents ccunting f (het) lsses nd ditin, s indicted in the figue Fig 5 Antenn s tnsmitte (excited by genet) s = ( - t ) (3) lts f the eceived nd sctteed pwes e shwn in Fig 4 Fig 4 eceived pwe, nd sctteed pwe, s vesus tnsmissin cefficient, t As peviusly stted, the sctteed pwe (pe sque mete) is meely ne definite mesue f the distubnce cused by the ntenn, nd this mesue cn esily exceed the pwe incident (pe sque mete) heefe, nt much shuld be mde f cmpisns f eceived nd sctteed pwe IV ANENNA CICUI FOMULAION In cntst with fithful mdel such s the tnsmissin line f the -dimensinl spce, nd the idel shunt- junctin f the specil simple ntenn, enginees egully descibe ny ntenn by stndd fm f equivlent cicuit F the ntenn s tnsmitte, this stndd equivlent cicuit is simply the input impednce F the ntenn s eceive, this stndd is hevenin ( Ntn) equivlent cicuit elevnt ppeties f the extenl electmgnetic fields e summized by the gin (diectivity) nd eceiving css-sectin pmetes [6, 8, 9] We begin with sme emks n these cicuit desciptins f genel ntenns befe plcing the specil simple - dimensinl ntenn f the peceding sectins int tht cnventinl cntext he stndd equivlent cicuit f the ntenn s tnsmitte is shwn in Fig 5 he -pt ntenn itself is epesented by its input impednce Z = + jx, whee the 4 Fig 6 hevenin nd Ntn equivlent eceive ntenn cicuits he stndd ntenn enginee s equivlent cicuit f the ntenn s eceive is shwn in Fig 6 he ntenn tgethe with the effect f the incident (plne wve) electmgnetic field is epesented by eithe the hevenin Ntn equivlent cicuits t the left f the figue As is well knwn, the pwe ppently dissipted in hevenin Ntn equivlent impednce, in genel, bes n necessy eltin whteve t ny ctul dissiptin mechnism within the physicl cicuit epesented by the hevenin Ntn cicuit Hweve, the f-field ssumptins which undelie the stndd ntenn fmultin esult in the cicumstnce, f ecipcl ntenns, tht the hevenin nd Ntn equivlent impednce equls the input impednce t the ntenn, Z g = Z his cincidence hs led t vius ttempts t link the pwe ppently dissipted in tht impednce with pwe e-dited nd/ sctteed by the ntenn Intuitively, it seems cle tht the mj sctteing f plne wve incident n, sy, the bck f pblic ntenn, cn hve little cnnectin with the input impednce t such n ntenn Indeed, this pint ws specificlly ecgnized by ilve in his celebted vlume f the MI ditin Lbty eies [6, p 48], In genel, the pwe dissiptin cmputed f the equivlent genet impednce is nt equl t the pwe dissipted in the netwk between V [ E ] nd the ld; hence, in genel G g it cnnt be intepeted s sctteed pwe his genel cicumstnce will be exemplified by the fmily f - dimensinl ntenns studied in sectin VI f which the detiled clcultins cn be cied ut, Fig 3 he usully inncuus qulifictin, in genel emplyed in the peceding cmments is pfundly significnt hee As will be descibed in the vey specil cse f cnnicl minimum-sctteing ntenn (the simple - dimensinl ntenn nlyzed in ectin II nd ectin III

5 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) being n exmple f such n ntenn), the pwe dissipted in the ditin esistnce pwe cn be linked t the sctteed pwe A sht electic diple (when cnsideed in isltin fm extneus suppting stuctues) ppximtes theeticl cnnicl minimum-sctteing (CM) ntenn his pvides justifictin f clcultin f sctteed pwe fm the pwe dissipted in the ntenn hevenin equivlent impednce (ditin esistnce) ppeing in the eceive cicuit in the cse f this vey cmmn clss f ntenns [] We will nw plce the -dimensinl stuctue intduced in ectin II, Fig, nd mdeled s tnsmissin line cicuit in ectin III, Fig, int the pesent cntext As nted peviusly, the especilly simple ntenn is mdeled by n idel symmeticl shunt- junctin he ntenn s lcl pt is teminted by the ld esist epesenting the physicl esistive sheet he input impednce t this lcl pt (with genet sht-cicuited) is edily fund t be / Additin f n idel tnsfme, tuns ti : s in Fig 7, tnsfms this input impednce t pducing mtched ntenn he cnventinl tnsmit equivlent cicuit f this ntenn is shwn in Fig 8 Fig 7 he simple ntenn s tnsmitte (excited by genet) Fig 8 Cnventinl equivlent cicuit f simple ntenn s tnsmitte (excited by genet) he pmetes f the stndd hevenin eceive equivlent cicuit f this sme ntenn e nw ls esily clculted fm Fig he sht cicuit input impednce is, f cuse,, tht being the functin f the mtching tnsfme he pen cicuit vltge is E g Eg Open cicuit lcl pt vltge (4) he eceiving ntenn (hevenin) equivlent cicuit is shwn in Fig 9 whee E fm Fig nd g (3) Fig 9 Cnventinl hevenin equivlent cicuit f the simple ntenn s eceive (Fig 6) excited by n incident plne wve s in Fig Nte tht, due t the mtching tnsfme, the ld esistnce equivlent t the esistive sheet becmes he eceived pwe, edily clculted fm this cicuit, is identicl t tht given in (34b) I Eg I ( ) inc ( ) (46) which, n substituting (45b), is seen t gee with (43) 5 ˆ g E g E I = ˆ ubstituting f E, g 4 ˆ ˆ he pwe dissipted in the ntenn esistnce is: Eg nt s (4) (4b) (43) I (44) ˆ As pinted ut, in genel, the pwe ppently dissipted in the impednce f hevenin equivlent cicuit hs n physicl intepettin F the pesent simple mtched ntenn which disppes becmes invisible n pen cicuit, ie, s, the cse f cnnicl minimumsctteing ntenn, tht pwe equls the sctteed pwe, in geement with (34), (3), nd (3) F me genel ntenns, s f exmple the fmily f ntenns discussed in ectin VI f this ppe, n such equlity hlds etuning t the pwe eceived in the ld impednce s given by (4b), we nte tht this fm is edily tnsfmed in tems f the eflectin cefficient (ee Appendix A): ˆ ˆ 8ˆ ˆ (45) (45b) We nte tht the incident wve mplitude in the hevenin equivlent eceive cicuit Fig 9, is nt equl t the spce (tnsmissin line) incident wve mplitude ; / Accdingly, the eceived pwe my nw be expessed s

6 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) It is cle tht (43) nd (44) imply tht the sctteed pwe s is equl t the eceived pwe when the ld is mtched t the ntenn input impednce, ˆ, ( /, ) his is pecisely s wuld be expected f ny cnnicl minimum-sctteing ntenn [] nd gees with the tnsmissin line bsed clcultin f ectin III he pptinlity f ntenn gin (diectivity) t eceiving css-sectin f ny mtched, ecipcl ntenn system, the tw being elted by univesl cnstnt depending nly n the dimensinlity f the ntenn system, is n imptnt elementy esult [6, 8, 9] he specific vlue f the univesl cnstnt is genelly fund by evlutin f the cnstnt f pticul ntenn F 3- dimensins sht diple is cmmnly chsen In the tnsmit mde, the cicuit cnfigutin ssumes the fm shwn in Fig F this simple cse, the ntenn impednce is the sme in the equivlent (ctul) cicuit f the ntenn s tnsmitte nd the equivlent cicuit f the eceive ntenn s shwn in Fig Fig nsmit ntenn tnsmissin line mdel Fig Cicuit epesenttin f the tnsmit ntenn he -dimensinl ntenn dites in nly tw diectins: int egin int egin he gin f the ntenn diting int egin is witten s G() Gin is ( ) G( ) (47) in In view f the symmety f Fig nd the lssless chcte f the ntenn (tnsfme nd shunt- junctin), we hve () = () = in / Cnsequently, G () = G () = ; the ntenn dites istpiclly he ze subscipt ws dded t dente the specil simple ntenn Detiled cicuit clcultins f (), () nd in, f cuse, yield the sme esults Given n incming plne wve, the pwe eceived by n ntenn is cmmnly expessed in tems f its eceiving css-sectin, A whee A( ) nd inc A( ) A( ) ( ) nd A is the eceiving css-sectin f mtched ntenn system It is elementy t shw tht this eceiving csssectin f mtched system is pptinl t the gin f the ntenn with univesl cnstnt f pptinlity A [6, 8, 9] Me pecisely, thee is specific univesl cnstnt f ntenns diting nd eceiving int ech: 3- dimensinl, -dimensinl nd -dimensinl spce In summy, A G (48) A( ) A ( ) ( ) ( ) ( ) he fmuls f the eceiving css-sectin in tems f the gin f 3-, -, nd -dimensinl ntenns e listed in (48, b, c) he cnstnts f 3- nd -dimensinl ntenns e well knwn: the cnstnt A is /4 f 3-dimensinl ntenns nd /f -dimensinl ntennshe vlue f the univesl cnstnt f -dimensinl ntenns ½ is infeed fm (46) inc G( ) inc () ( ) 4 4 (48) ( istpic 3 - d) inc G( ) inc () ( ) (48b) ( istpic - d) inc G( ) inc () ( ) (48c) ( istpic - d) Equtin 48c cnfims tht the univesl cnstnt in the fmul f eceiving css-sectin in tems f the gin G f -dimensinl ntenn is / his is in geement with diect clcultin f the istpic mtched ntenn teminted in the esistive film (ie, / ) Cnsistent with this ppe inc is defined s pwe density W / m Accdingly, will hve units f W, W/m, nd W / m f the 3-, - nd -dimensinl cses, espectively F the simple mtched ntenn: E g (mtched) inc (49) 4 Identifictin f the univesl cnstnt f -dimensinl mtched ntenn systems is tken up gin in ectin V he evlutin will be seen t be pticully diect nd tnspent in tems f the sctteing fmultin f ntenns We emk tht -dimensinl ntenn evidently hs mximum chievble gin f, elized when ll input pwe is tnsmitted entiely in ne f the tw vilble diectins It fllws tht -dmensinl ntenn ls hs mximum chievble eceiving css-sectin, A G(mx), bsbing the entie incident pwe int the ntenn ld 6

7 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) he eminde f this sectin dels with the field sctteed by eceiving ntenn In pticul, we cnside wht my be lened egding the sctteed field fm the pmetes f the cnventinl equivlent cicuit pmetes f the eceiving ntenn elicit tht infmtin fm the equivlent cicuit, we mke use f the cmpenstin theems nd the pinciple f supepsitin In bief, the cmpenstin theems stte tht, in cicuit ny element with knwn vltge dp ( cying knwn cuent vlue) cn be eplced by n idel, ze intenl impednce, cmpensting vltge suce ( n idel, infinite intenl impednce, cmpensting cuent suce) withut ffecting the vlidity f the Kichhff cicuit equtins [4] Fig epduces simplified vesin f the cnventinl equivlent cicuit f eceive ntenn, Fig 6 pecificlly, f CM ntenns, the ntenn becmes invisible, des nt sctte s, n pen cicuit when L Fig 3 Cmpenstin theem pplied within the hevenin equivlent cicuit f eceiving ntenn Fig hevenin equivlent cicuit f eceive ntenn shw tht in the specil cse f cnnicl minimum sctteing ntenn the pwe ppently dissipted in the (hevenin equivlent) esistnce f the ntenn eceive cicuit is equl t the sctteed pwe nd t indicte the fundmentl esn why this is nt equl t the sctteed pwe in genel, we emply the cmpenstin theem in the eceive cicuit Fig 3 [7,8,7,8] he ld element is eplced by the idel cuent genet hving mgnitude nd diectin equl t the knwn cuent s clculted fm the eceive cicuit hen, in ccdnce with the pinciple f supepsitin (ie, independent cmputtins f espnse f the cicuit t the tw diffeent genets nw pesent) yields tw septe slutins, cuents nd vltges he tw cicuits, ech excited by ne f the tw diffeent genets is shw in Fig 4 he sums (supepsitin) f these tw slutins shuld, in evey instnce, yield the ttl vlue knwn fm iginl eceive cicuit ince, due t the infinite seies esistnce sscited with the idel cuent genet, the fist cicuit cntibutes ze cuent, the sum cuent bviusly hs the cect vlue f the iginl cicuit by cnstuctin he sum f the tw vltges V nd V is shwn t yield the cect vlue V, Eq (4) Fig 4 inciple f supepsitin f excittins pplied t nlyze cicuit f Fig 3 int tw septe cicuits Fm Fig 4, by cnstuctin: I = I + I, I =, I = -I gc We then cmpute: Eg V V E g I E g Eg L E g V L L which veifies the cectness f the supepsitin L (4) (4b) Nw cnside the sctteed fields sscited with ech f the tw cicuits f Fig 4 he minimum sctteing ntenn disppes, becmes invisible, nd des nt sctte t ll n pen cicuit, Fig 4 he ntenn epesented in Fig 4b, the secnd supepsitin cicuit, is theefe nt suunded by ny sctteed field (due t 4) tht cn intefee with the field dited by the cicuit f the secnd supepsitin cicuit Als, tht secnd cicuit is identicl t the stndd equivlent cicuit f the CM ntenn s tnsmitte since the hevenin equivlent impednce f the ntenn cincides with the input impednce f the ntenn We my theefe identify the dited ( e-dited) fields with thse tht the ntenn nmlly dites In the bsence f ny the extenl field, the pwe dited in the cnditin f the secnd supepsitin equls the pwe dissipted in the ditin esistnce just s is the cse when the sme ntenn is emplyed s tnsmitte Cnvesely, given genel ntenn, such n ntenn pduces sctteed field even when the genel ntenn is teminted in n pen cicuit ht is, the genel ntenn (including the pcticlly necessy suppt stuctues) will pduce sctteed field, s-clled stuctul cmpnent, even when thee is n (pt) cuent in the eceive cicuit s in Fig 4 he field dited by the secnd cmpnent, Fig 4b, is supeimpsed n this stuctul cmpnent f the field he tw cmpnents intefee ince pwe depends n the sque f the ttl 7

8 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) field, thee cn be n diect eltin f ttl sctteed pwe t the pwe tht might be dited (e-dited) in the bsence f the stuctul cmpnent ee ectin VI in this cnnectin F the specil CM clss f ntenns, the eltinships f eceived pwe nd sctteed pwe given el ntenn impednce (ditin esistnce) nd ny cmplex eceive ld impednce my be cnveniently visulized n the ld eflectin cefficient plne (mith cht, impednce nmlized t the ditin esistnce) Fig 5 [4] Lci f cnstnt eceived pwe e cicles f cnstnt mgnitude f eflectin cefficient, eqs (46) nd (48) Nte gin tht these eflectin cefficients cespnd t incident nd eflected wve mplitudes in the cicuit f Fig; these e distinct fm the incident nd eflected wve mplitudes n the tnsmissin line equivlent cicuit Lci f cnstnt sctteed pwe e, s will be shwn, cicles (cs) centeed n the pint which ls cespnds t infinite ld impednce Indicted vlues f pwe, decibels, e eltive t the mximum eceived pwe when F the simplest CM ntenn descibed in ectin II nd mdeled in ectin III, pints n the el xis cespnd t the vlues gphed in Fig4 In pticul, t the mximum eceived pwe pint, the sctteed pwe equls the eceived pwe When the ld impednce is ze,, the sctteed pwe is twice the incident pwe eltive t the mximum eceived pwe (which is hlf the incident pwe) this equtes t eltive vlue f 6 db In genel hee, the cespndence f with vlues f is: L (4) whee ( L / ) equls the nmlized esistnce pmete in the mith cht he cnstuctin leding t the lci f cnstnt sctteed pwe is shwn in Fig 6 It fllws fm the cicuit f Fig I b ( ) ( ) (4) s I (4b) Fig 5 Ld eflectin cefficient plne: ed cicles eceived pwe eltive t mximum, Blck cicles ctteed pwe eltive t mximum eceived pwe Fig 6 Cnstuctin f sctteed pwe lci (cicul cs) n the cmplex ld eflectin cefficient plne V CAEING MAIX FOMULAION In genel, the sctteing mtix f cmpnent (ntenn) eltes cmplex wve mplitudes incident n (tveling twds the cmpnent) t cmplex wve mplitudes eflected (tveling wy fm) the cmpnent Cnside nw the sctteing mtix epesenttin f -pt, - dimensinl ntenn [,,3] he cnventinl nttin ssigns the lette with subscipts, etc f the wve mplitudes incident nt the ntenn (fm eithe side) nd the lette b with subscipt, etc, f wve mplitudes eflected fm the ntenn Hweve hee, bth t vid intducing new nttin nd t the sme time chieve desible cespndence with the definitin f the wve pmetes n the tnsmissin line equivlent cicuit, Fig, we mke use f the cespndence wve pmete sets whse diectins f incidence e defined with espect t eithe side f efeence plne (tnsmissin line pt) he incident wve mplitude, incident in the tnsmissin line sense f wve tveling twds the ight, (in this cse fm the ntenn n the left twds the ld t the ight) Fig, is intuitively equl t wve mplitude eflected fm the ntenn In the sme wy, the eflected wve mplitude b (eflected in the tnsmissin line sense f wve tveling t the left (in this cse fm the ld t the ight twds the 8

9 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) ntenn n the left) Fig, is intuitively equl t wve mplitude incident n the ntenn When we vil uselves f these stightfwd equivlences, the -dimensinl ntenn with ne lcl pt, designted (), nd tw- (tnsmissin line)-ditin-pts, designted () nd (3), is epesented s shwn in Fig 7 ditin pt () cespnds t tnsmissin line egin, nd ditin pt (3) cespnds t tnsmissin line egin Nte: the intuitive equivlences f the nmlized wve pmetes used hee e igusly demnstted in Appendix A he cespnding sctteing mtix epesenttin (5) eltes clumn mtices in which the peceding equivlences hve been substituted Fig 7 ctteing mtix f -dimensinl ntenn (5) If the ntenn stisfies the Lentz ecipcity cnstint, we hve mn = nm Accdingly, the gin f the ntenn (47) cn be expessed in tems f the tnsmissin nd eflectin pmetes, equivlently, in tems f sctteing pmetes: G() b b (5) (5b) F mtched system, nd the bve expessin educes t imilly, he cespnding diectivities e: D() G() (53) 3 G() (54) 3 (55) 3 D() 3 When the ntenn is lssless, hen, 3 (55b) (56) G() D() ; G() D() (57) Nw cnside this sme mtched ecipcl ntenn in the eceiving mde When wve is incident (in egin, pt ()) the pwe eceived is: inc () inc A (58) ee (48) In view f (53) nd (58), the eceiving csssectin f mtched system: A() G() G() A (59) since, due t the ecipcity cnstint, We cnclude tht the eceiving css-sectin f mtched, ecipcl, -dimensinl ntenn system hs, quite genelly, been shwn equl t the gin times univesl cnstnt, A In the pesent -dimensinl cse, A = ½ Indeed, the peceding nlysis demystifies nce nd f ll the fundmentl esn behind the existence f such univesl cnstnt vlid f ll mtched ecipcl ntenn systems f given dimensinlity, cf, equtins (48) We nte tht f 3-dimensinl ntenns the cespnding demnsttin ws given by Gtely, et l [, eq 7] Hweve, the necessy nttinl cmplexities invlving the spheicl mde functins bscue its diectness nd simplicity We nw develp the sctteing mtix f -dimensinl cnnicl minimum-sctteing (CM) ntenn uch lssless ecipcl ntenn is cmpletely defined by its (cmplex vltge) ditin ptten nd the ppety f becming invisible when the lcl eceive pt is pen cicuited, tht is, tets scttes n incident wve exctly s empty, fee spce In mny high fequency ntenns dissiptin within the ntenn stuctue plys negligible le nd the ntenn my be idelized s entiely lssless he nmlized sctteing mtix f lssless stuctue is unity (5) F mtched ntenn system,, this cnstint becmes (5) In the nttin f the clumn vects in (5), the sctteing mtix eltin descibing fee spce is evidently 9

10 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) b, b (5) epesenting tnsmissin f wves fm egin int egin, bsent ny discntinuity t junctin f the tnsmissin line egins Undelining the bsence f n ntenn, the full sctteing mtix f fee spce my be witten s: (5b) he bve fm f the sctteing mtix epesenting - dimensinl fee spce s tnspent junctin seems intuitively me ppeling thn the spheicl mde epesenttin f fee spce f 3-dimensinl ntenn s pefect eflect [4,,] Hweve, the diffeence is me ppent thn el An ltentive epesenttin f - dimensinl fee spce, which pllels the ne necessy in the cse f spheicl mdes, ie, in tems f symmeticl wve mplitudes cnveging nd diveging fm the plne mking the junctin f the tnsmissin lines epesenting egin nd egin, is descibed in Appendix B A (mtched, lssless) cnnicl minimum-sctteing ntenn ws defined essentilly by the ppety tht such n ntenn becmes invisible emultes fee spce when the ccessible lcl pt is pen-cicuited [] usunt t this definitin, pen-cicuit cnditins ( ld with eflectin cefficient ) my be impsed t the lcl pt Cnsequently, b b, b being the wve incident n the ld Given the ld cnstint, the lcl pt my be eliminted fm the 33 ntenn sctteing mtix tht Equting this b (53) b (54) b sctteing mtix t tht f fee spce (54b) F (55) he eltin (53), emulting fee spce, fixes f given vltge ditin ptten : F (56) F (56b) F (lssless, unity cnditin) (56c) F F (56d) On the the hnd, f ecipcl ntenn, the nmlized sctteing mtix is symmeticl, the tnsmit nd eceive ditin pttens e cnnected s (57) he lssless cnstint (56c) nd ecipcity (57) must be ecnciled by F, in the pesent -dimensinl instnce (58) (59) 3 3 It fllws tht the pwe pttens f -dimensinl CM ntenns must be symmeticl, tht is: 3 s is ls the cse in thee dimensins (5) As n exmple illustting the peceding fmultin, cnside the cse f the simple cnnicl minimumsctteing ntenn, ledy fmili fm ectins II nd III, Fig 8, wheein the esistive film, Fig, the esistive ld, Fig, hs been eplced by n pen-cicuit (Nte tht the mtching tnsfme, shwn explicitly in Fig 8, meely mdifies the vlue f the esistive ld seen t the lcl pt f wht ws n entiely bity numeicl vlue in Fig ) Fig 8 nsfme necessy t yield mtched CM ntenn Finlly we cmpute the cmplete sctteing mtix f this ntenn nd then g n t evlute wht hppens t n incident wve = when the ntenn is pen-cicuited t the lcl pt s shwn Of cuse the ltte shuld shw tht the stuctue emultes fee spce (5), ie, = We cnstuct the sctteing mtix f this lssless ecipcl ntenn by (tivilly) evluting the vltge ditin ptten Input t pt () divides eqully between the left nd ight tnsmissin line egins, cicuit pts () nd (3) by symmety he mgnitudes fllw fm the mtch cnditin nd the cnsevtin f enegy (5) his ptten meets the equiements f CM ntenn (58), F Cnstucting the emining elements f the sctteing mtix f such CM ntenn, (5) (5b) F

11 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) Accdingly, the cmplete sctteing mtix f this simple ntenn is (53) An incident spce wve is nw ssumed in the left-hnd tnsmissin line (egin, cicuit pt ) his incident wve mtix nd the esulting eflected wve mtix b s cmputed fm this incident wve mtix vi the sctteing mtix (5) e: b (54) But the lcl pt eceiving the eflected wve mplitude b is pen-cicuited, Cnsequently, n dditinl incident wve cmpnent lc b is pduced t the lcl pt his dditinl incident cmpnent pduces n dditinl utging eflected wve cmpnent, gin cmputed fm the sctteing mtix (5) his is the secnd clumn mtix n the left side f (5) Adding the tw cmpnent eflected wve mtices yields the finl eflected wve It is seen tht the ditin cmpnents f the ttl eflected wve clumn mtix (tnsmissin line wve pmetes b ) e pecisely thse tht wuld hve been btined in the bsence f ny ntenn he pen cicuit cnditin hs endeed the cnnicl minimum sctteing ntenn invisible ht is, lc Fig 9 esistive film with qute-wve dielectic slb Equivlent tnsmissin line cicuit mdels f this fmily f -dimensinl ntenns with esistive sheet ld e shwn in Fig In pticul, the dielectic cnstnt f the slb, ie, the tnsfme tuns ti, will be chsen such tht the input impednce in equls [4, 5, 6] ht is, in n if n in ( tnsmissin line mtch ) (6) (6b) is the ttl eflected wve, demnstting = (55) VI ALICAION O A FAMILY OF ANENNA WIH ZEO BACKCAE Hving develped vius ppches nd cncepts illustted by nly the simplest exmple f n ntenn, we nw seek t pply these t me genel cse A fmily f -dimensinl ntenns chcteized by ze bcksctte pvides cse f inheent inteest; ne such fmily is cmpised f qute-wve dielectic slb (mtching tnsfme) gin teminted by esistive film ld he gemety f this cnfigutin is shwn in Fig 9 Fig Cicuit mdels f the ze bcksctte ntenns - qute-wve slb tnsfme with esistive sheet ld: ) Fithful tnsmissin line equivlent cicuit epesenttin, nd b) Idel tnsfme equivlent cicuit As indicted, this defines fmily f ntenn stuctues, ne membe ntenn with dielectic cnstnt nd slb thickness pppite f ech vlue f ld esistnce his cnditin lends pticul inteest t this exmple in tht it educes the ntenn eflectin cefficient (bcksctte) t ze Wning! Nte tht the pmetes f the bve cicuit the ld impednce (cespnding t the selectin f ne fixed membe fm the fmily f ze bcksctte ntenns) when ne investigtes, f exmple, the pwe eceived sctteed by this ne selected membe ntenn with sme new independently chsen nge f ld impednces In thse cicumstnces we shll etin the nttin f the ld defining the pticul membe ntenn (n is functin f ) nd emply the nttin L f the independently ssigned vlue f ld esistnce (n is nt functin f L) teminting the pticul selected ntenn

12 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) Fig tndd eceive ntenn hevenin equivlent cicuit f the ze bcksctte ntenn he stndd eceive ntenn hevenin equivlent cicuit f the fmily f ze bcksctte ntenns just descibed is edily btined fm the cicuit Fig nd is shwn in Fig Of cuse, the bcksctte is ze nly f the vlue f L = he intenl impednce f the genet is lbeled H t emphsize its sttus hee As peviusly nted f ecipcl ntenns, numeiclly, H = A, the ntenn input impednce in the tnsmit equivlent cicuit We nw cmpute pwe eceived in the ld L, L n (68) Using this fmultin, we check the eceived pwe when L = : ( ) ( L ) ( ) (69) Finlly, this eceived pwe f L = is ls esily checked diectly fm the tnsmissin line mdel ne 4 ( ) g L L n L L L n F the specil vlue L =, equtin 6 simplifies t (6) (63) Be in mind tht is the incident wve mplitude Fig nd nt the incident mplitude evluted f the eceive cicuit Fig ht mplitude is, fm (3): E g n EH = H n n (64) ubstituting the vlues f the hevenin equivlent pmetes nd slving f, ( ) Egn Afte sme lgeb, we btin the eltins E 4 g (65) (66) We emk tht ze bcksctte L = des nt imply impednce mtch (ze eflectin cefficient L ) in the stndd eceive equivlent cicuit, Fig Indeed, L L H L L H L which, when L = simplifies t (67) Fig Equivlent cicuit t the input t the tnsfme when the tnsfme tuns ti n is chsen t pduce mtched ld By cnstuctin, the simple equivlent cicuit Fig is pplicble when L = he vltge t the input t the ntenn (input t the tnsfme) is theefe E g / he vltge css the ld impednce epesenting the esistive sheet is nw fund simply by dividing by the tuns ti he pwe clcultin f the ld is then stightfwdly (vltge css ) V ( pwe int ) E g E g n V E g 4 ˆ (6) (6b) It is cle tht s ˆ, n pwe is deliveed t the ld In cntst, s ˆ, ll f the incident pwe is deliveed t the ld; this idelized cnditin cespnds t n As check f the ctul pwe blnce, we shuld be ble t shw tht u fmultin is cnsistent with the input pwe being equl t the pwe int the ld plus the pwe in egin (ie, the pwe int the chcteistic impednce epesented by the infinite tnsmissin line) identiclly in = + ˆ + ˆ ˆ (6) (6)

13 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) F cnvenience nd ese f cmpisn with the wk f Geen [6], we intduce nttin f the (nw cmplex) vltge tnsmissin cefficient fm egin t egin in the pesence f the ntenn = t (63) the efeence plne f is tken t the input t the ntenn, ie, t the input t the qute wve tnsfme, t = t e - jp = n e - jp = - j n (64) Expessed in tems f this tnsmissin cefficient (65) Insmuch s the cnsidetins with the specil ld L = suffice f u pupses, the bvius lgebic simplifictin nd physicl tnspency cnsequent t tht chice, we will ssume L = in the eminde f this sectin he cmputtin f ntenn sctteing is bsed diectly n the tnsmissin line equivlent cicuits, Figs nd b he sctteed field is defined s the diffeence between the ttl field nd the field in the bsence f the ntenn Absent the ntenn is equivlent t, n (with the cnsequence nd ) in Fig Quntities epesenting this incident field will be distinguished with ze supescipt An input efeence plne is estblished t the input t the tnsmissin line sectin equivlent t the (physicl) length f the qute wve tnsfme ince the dielectic f the tnsfme hs been emved, the wvelength in this length f line evets t tht f fee spce with the cnsequence tht the phse length f this sectin is shtened fm the vlue in the dielectic (which ws, f cuse, ) t (66) n It is imptnt t undestnd tht in the bsence f the ntenn, the sptil eltins, ie, the physicl lengths the ntenn ccupied must be peseved his fundmentl pint, which my nt be bvius hee in the -dimensinl cse, becmes self-evident in - 3-dimensins Cnside 3-dimensinl spheicl dielectic shell s scttee If bsent the scttee wee intepeted s emvl f the shell withut peseving the physicl spce ccupied by the shell, the esulting gemety wuld invlve n bsud discntinuity in css-sectin with the dil cdinte ecisely becuse this pint is nt s fcefully evident in - dimensin, it ws missed by Geen [6] his ppe theefe pvides cected esults f the sctteed pwe 3 Absent the ntenn we hve b = b = (67) In the pesence f the ntenn nd its eceive ld, Fig, we hve b = = n e- jp (68) b b (68b) In egin, the sctteed field is ze s, b In egin, the ttl field is: sctteed field (69) j j n e e sctteed field (69b) n j j n sctteed field e e (69c) n he sctteed pwe my theefe (64) be witten s s s = t - e jp (-t ) = t - cs p (- t ) - jsin p (- t ) s = t - cs p (- t ) + sin p (- t ) s = + t (6) (6b) (6c) - t cs p (- t ) (6d) he incect expessin f the sctteed pwe peviusly given by Geen nd epeted by us in pevius pesenttin [], lcks the csine fct We nw clculte the pwe ppently dissipted in the hevenin equivlent esistnce in Fig, H (which is, f cuse, ls the input esistnce f the ntenn s dit H = A) nd cmpe with the ctul vlue f sctteed pwe just cmputed fm the fithful tnsmissin line mdel H H = E g n + n + n + Eg 4 ( ) ( ) + n (6) (6b)

14 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) In tems f the vible, we find H ; (6) H (63) We e nt supised by this cicumstnce tht is entiely in line with ilve s wning n this sce nd cited elie in this ppe As pinted ut in ectin III, geement f sctteed pwe with the pwe ppently dissipted in the hevenin equivlent impednce my be justified nly f CM ntenns Only f such ntenns tht d nt sctte t ll n pen cicuit, des the hevenin equivlent esistnce in the eceive cicuit pvide bsis f clculting the sctteed pwe On pen cicuit, the pesent ntenns educe t dielectic slbs he sctteing fm these dielectic slbs is evluted in Appendix C hee is, in genel, n physicl intepettin f the sctteed pwe clculted in (67) Nmlized plts f s,, nd H e pesented in Fig 3 It is seen tht the sctteed pwe is nt necessily gete equl t the eceived pwe Indeed, it is less thn the eceived pwe f the ze bcksctte cse N is eithe pwe (f these me genel, nt CM ntenns) equl t the pwe ppently dissipted in the hevenin equivlent ntenn impednce s Fig 5 tndd equivlent cicuit f the ze bcksctte ntenn tnsmit mde; nte A = H We wish t cmpute the gin f the ntenn his pmete is independent f the intenl impednce f the ntenn system g = A s indicted in Fig 5 he ttl dited pwe is E I g t A A 4 A E 4 4 g ( ) Eg ( n ) (64) As check, the sme ttl pwe is fund by dding up the pwe int ech egin f Fig 4 n Eg Eg () ; () (65) E g ( n ) () () (65b) 4 Hving these septe pwe clcultins pemits clcultin f the gin (equl t the diectivity f lssless ntenns) int ech egin () nd () : D() Eg n () Eg ( n ) 4 n ( ) n ( ) G () ; (66) (66b) Fig 3 lt f nmlized pwes f the fmily f eflectinless ntenns: eceived pwe, sctteed pwe s, nd pwe ppently dissipted in the hevenin equivlent impednce H = A f the stndd eceive cicuit H We nw cnside the pesent fmily f ntenns s tnsmittes he tnsmissin line gemety is given in Fig 4 while the stndd ntenn equivlent cicuit is given in Fig 5 D( ) Eg ( ) Eg ( n ) 4 = + n = ( + ) (67) = G() (67b) Fig 4 Idel tnsfme tnsmissin line epesenttin f the ze bcksctte ntenn in the tnsmit mde 4 hese vlues f gin diectivity cn nw be tuned t cmpute eceived pwe in the cnventinl wy fm the eceiving css-sectin he eceiving css-sectin f mtched ntenn system A is pptinl t the gin (ie, equl t the gin times univesl cnstnt A ) As peviusly indicted, equtins (48), the eceived pwe shuld equl the incident pwe multiplied by the eceiving

15 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) css-sectin It must be eclled tht ze bck-sctte ntenn des nt fm mtched ntenn, is given by (63) We theefe hve, with sme lgeb, A ( ) A D() ( ) (68) inc n ( ) n (68b) Cmpisn with (48c) veifies tht A must hve the vlue / substituted in (68b) s fund in ectins III nd IV VII CONCLUION his ppe hs pvided new pth t deepe undestnding f eceive ntenns thugh detiled nlysis f n especilly elementy ntenn type, -dimensinl ntenn Anlyses f idelized infinite pln gemeties in which ntenns dite int -dimensinl spce wee pesented While cmputed esults, f cuse, d nt diectly cy ve t - nd 3-dimensinl ntenns, they cn pvide templte f cespnding clcultins in these functinlly me cmplicted dmins he ppe fcused n the detils f ditin, eceptin, sctteing nd editin in tems f vius ntenn (cicuit) fmultins stisfying ll pplicble physicl cnstints he undestnding gined in this wy is cnsequently pplicble t ntenns in genel Of pticul imptnce ws the nlysis f the eltin f sctteed pwe t the pwe int the ntenn impednce Becuse f the -dimensinl ditin ssumptin nd the inheent bility t clculte eceived nd sctteed pwe fm tnsmissin line pespective, it is esy t veify the specil cse, the clss f CM-like ntenns, in which intepeting the pwe int the ntenn s ditin esistnce cectly ( vey nely) yields the sctteed pwe, s well s t demnstte the fllcy f ssuming tht this clcultin lwys yields the cect vlue f sctteed pwe f ntenns in genel AENDIX A - HE CAEING AAMEE he definitin f tnsmissin nd eflectin cefficients used in this ppe fllws the tnsmissin line cnventin s shwn in Fig A he subscipts nd indicte pmete incidence fm the ight nd left, espectively Fig A nsmissin line cnventin V Z I (A) b V Z I (Ab) V Z ( I ) (A) If Z Z b V Z ( I ), then: (Ab) b b (A3) (A3b) Z nd Z e nmliztin numbes nd e nt necessily chcteistic impednces [] he pwe deliveed t the ight is fund s fllws: V Z I (A4) b V Z I (A4b) 4 V V Z I Z IV Z I (A5) 4 b V V Z I Z IV Z I (A5b) ubtcting (A5b) fm (A5) gives the pwe deliveed t the ight: 4 b ( Z Z ) V I ( Z Z ) IV 4 b V I IV (A6) (A6b) e (A6c) b V I imilly, the pwe deliveed in tems f left incidence pmetes: b e V ( I) e V I he cespnding eflectin cefficients becme: b V Z I Z Z V Z I Z Z b V Z ( I ) V Z I V Z I V Z I b l l ( ) ; ; (A7) (A8) (A8b) (A8c) he bve eflectin cefficient defined with efeence t the nmlized vltge sctteing mplitudes diffes theeticlly fm the eflectin cefficient cmmnly defined exclusively with efeence t tnsmissin line wve mplitudes In genel, the vltge nd cuents n unifm tnsmissin line my be witten [4] whee ( z z ) ( z z ) ( z z ) V ( z) Ae Be Ae ( z) (A9) Z I z Ae Be Ae z (A9b) ( ) ( z z ) ( z z ) ( z z ) ( ) ( z) ( z ) e, ( z ) A Dividing, we btin ( z z ) B V ( z) Z ( z) = = Z I( z) Z ( z) (A) (A) 5

16 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) lving this eltin f nd dpping the cdinte gument, Z Z Z Z (A) Fmlly, the expessins f the tw eflectin cefficients diffe nly in the cmplex cnjugte ppeing in the numet f the fmul f, (A8) F el vlues f Z =, Im{ Z } =, the tw eflectin cefficients e numeiclly equl Unde the sme cnstint, we hve t ny pint n tnsmissin line V = [ b] = A B (A3) s tht the nmlized vltge sctteing mplitudes nd tnsmissin line vltge wve mplitudes diffe nly by cnstnt fct f AENDIX B - HE CAEING AAMEE IN ALENAIVE COODINAE In the pesenttin f the sctteing mtix fmultin f -dimensinl ntenn, ectin V, we wee t pins t dpt fmultin which cnnected smthly with the tnsmissin line mdel nd cicuit fmultins f the peceding sectins Hweve, ne cnsequence f this ppch ws t bscue the equivlence f the - dimensinl esults with thse btined f 3-dimensinl ntenns emplying the spheicl vect mde bsis functins his ppendix clifies the eltinship between the peceding fms, in pticul the fm f the mtix epesenting fee spce, with tht fund f the cespnding sctteing mtix fmlism s develped f ntenns in 3-dimensins [,, 3] As indicted, the bsis f the sctteing epesenttin in 3-dimensin e incming (symmeticlly cnveging) nd utging (symmeticlly diveging) spheicl wve mdes hese wves ppgte in the ˆ diectin, functinlly s specified by the pppite spheicl Hnkel functin, in ll diectins fm the igin f cdintes In fee spce (n ntenn stuctue) the igin must be n diny nnsingul pint his egulity equiement fces the equlity f incming nd utging mplitudes, ie, the sctteing mtix is necessily the unit mtix imil cnsidetins gven in -dimensins Hweve, f plne wves, the bsence f singulities leds t flexibility in the sctteing epesenttin f fee spce nd in pticul t the fm shwn in (5) An ltentive bsis f the wves n u tnsmissin lines, ne tht mimics the symmety f wves incming nd utging fm centl pint, des indeed clsely pllel the 3- nd -dimensinl fms Cnside the ltentive cdintes induced by the el thgnl tnsfmtin in the mtix sub-spce f the ditin pts (tnsmissin line) wve pmetes: ; b b b b (B) (Bb) 6 he sctteing mtix f fee-spce becmes: (B) F F F F (Bb) he new cdinte system implies symmetic nd ntisymmetic excittins (ditin nd eflectin) fm plne in fee spce he esulting sctteing mtix epesenting fee spce is nw dignl with unit eflectin enties AENDIX C -DIMENIONAL CAEING BY A QUAE-WAVE DIELECIC LAB A qute-wve dielectic slb emins when the esistive film ld n membe f the fmily f ze-bcksctte ntenns f ectin VI is emved, L, effectively pen-cicuiting the ntenn, Fig C As we hve explined, the fields sctteed by the slb intefee with fields edited by the eceive ntenn equipped with its nml ld s tht the ttl sctteed pwe is genelly nt equl t the pwe ppently lst in the ditin esistnce f the hevenin equivlent eceive cicuit We shll pesent tw clcultins f this sctteed pwe he fist fllws the sme utline s used in ectin VI t cmpute the sctteed pwe fm the eceiving ntenn; the secnd will emply the ltentive symmeticl mde set f Appendix B As estblished in ectin VI, it is key t mintin the sme sptil eltins f efeence plnes f incident nd eflected wve pmetes in the bsence f the slb s exists when the slb is pesent hus, f ech membe f the fmily f ze bcksctte ntenns, fee-spce length equl t qute wvelength in the pppite tnsfme dielectic must be mintined when the slb is emved In egin tl Field Incident Field ctteed Field (C) b b ctteed Field (C) e ctteed Field (C3) j j e ctteed Field (C4) n j e ctteed Field (C5) n he sctteed pwe in egin is theefe n (C6) n Hee, the phse shift f the eflectin cefficient f the tnsfme thugh the qute wve dielectic slb t the estblished input efeence plne plyed n effective le, since b In egin tl Field Incident Field ctteed Field (C7) b b ctteed Field (C8)

17 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) j j n e e ctteed Field (C9) j j n e e ctteed Field (C) j n (C) e cs jsin ctteed Field n n n he sctteed pwe in egin is theefe wve pmetes (pptinl t tnsvese electic field), it fllws tht the symmeticl mde stisfies pen cicuit (pefect mgnetic cnduct) cnditins n the symmety plne while the ntisymmetic mde stisfies sht cicuit (pefect electic cnduct) cnditins n the symmety plne [4] Given these cnditins t the middle plne f the stuctue nd the symmeticl equivlent cicuit, Figs C, we edily infe the cespnding eflectin cefficients t the input f the slb n cs jsin n n n n 4n cs n n n (C) cs sin n n (C3) he ttl sctteed pwe fm the slb is theefe (C4) s s s s n n n 4 cs (C5) n n n n s n 4n cs (C6) n he lst simplifictin fllws fm the unity chcte f the sctteing mtix f the lssless idel tnsfme; the sum f the sques f eflectin nd tnsmissin cefficients is unity If, futheme, we mke use f the eltin between the tuns ti n nd the tnsmissin cefficient f the fmily f ze-bcksctte ntenns / n, we cn edily plt the sctteed pwe fm the slb n the sme bsciss used f the sctteed pwe fm the zebcksctte ntenns, Fig 3 ee Fig C3 We nw cmpute the sctteed pwe fm the qute wve slb emplying the ltentive symmeticl mdes, intduced in Appendix B, which me clsely esemble the incming nd utging wves ntully sscited with the centl lctin f - 3-dimensinl ntenn he symmeticl mde mplitudes cespnding t n incident plne wve mplitude fllw fm equtins (B) ince is el (unity) thgnl tnsfmtin f which, in pticul will be peseved;, ll pwe eltins (C7) As the dielectic slb (withut cnducting film ld) is symmeticl stuctue, we chse the middle f the slb s the plne f symmety f the symmeticl mdes, Fig C Futheme, becuse the pmetes e nmlized vltge Fig C Qute wve dielectic slb (emining when ze bcksctte ntenn is pen-cicuited, i e, esistive film ld emved) nd equivlent cicuit epesenttins: ) hysicl stuctue, b) Fithful tnsmissin line cicuit epesenttin, c) ymmeticl idel tnsfme equivlent cicuit, d) Idel tnsfme equivlent cicuit We nw cmpute the sctteed pwe in ech symmeticl mde F the fist mde tl FieldIncidentFieldctteed Field ' ' ' ' (C8) b b ctteed Field (C9) 7

18 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) j ' ' ' ' n in c e ctteed Field (C) he sctteed pwe in mde is theefe ' j j s e j (C3) Fig C Cicuits f ltentive mdes eq C7 incidence clcultin: ) ymmeticl idel fme equivlent cicuit f qute wve dielectic slb, b) ' mde incidence pen cicuit bisectin, c) ' mde incidence sht cicuit bisectin Of cuse, the eflectin cefficient cespnding t n pen cicuit, c In de t cmpute the pen in cicuit must be efeed bck thugh hlf the qute wve tnsfme, phse length f he pen cicuit j / e eflectin cefficient is theefe multiplied by his eflectin cefficient cespnds t nmlized impednce j in the dielectic he idel tnsfme f the symmeticl equivlent cicuit with ti : n cespnding t the tnsitin fm dielectic t fee spce chcteistic impednce pduces nmlized input impednce j / n It fllws tht the input eflectin cefficient j n j in j j n (C) ubstituting in the bve pesciptin f the sctteed field in mde, we btin A pecisely pllel clcultin f the ntisymmetic mde, sc, pduces j n in j n j j nd the sctteed pwe in mde ' j j s e j (C4) (C5) In view f the thgnlity f the symmetic nd ntisymmetic mdes, the ttl sctteed pwe is the sum f the sctteed pwes in ech mde, (C6) ' ' ' s s s Althugh the tw clcultins f sctteed pwe esult in quite diffeent lgebic fms, they pduce identicl numeicl esults f the ttl sctteed pwe, Fig C3 In me detil, ' s mkes by f the lge cntibutin t the sctteed pwe his is t be expected s the electic field is stng ne the symmety plne f this mde nd theefe stngly influenced by the dielectic he electic field is cespndingly wek ne the symmety plne f the ntisymmetic mde nd theefe is nly wekly influenced by the dielectic Fig C3 Dielectic slb, nmlized sctteed pwe s j j e ctteed Field j (C) 8

19 Fum f Electmgnetic esech Methds nd Applictin echnlgies (FEMA) EFEENCE [] Weiss nd W K Khn, eview f ctteing nd e-ditin by eceiving Antenn, ceedings f the Antenn Applictin ympsium, Alletn k, Mnticell, Illinis [] Cnt, éflexins su l puissnce mtice du feu et su les mchines ppes à dévelppe cette puissnce, is, Bchelie, 84 [3] A M uing, (937) [Deliveed t the ciety Nvembe 936], "On Cmputble Numbes, with n Applictin t the Entscheidungspblem," ceedings f the Lndn Mthemticl ciety, 4: 3 65 [4] C G Mntgmey, H Dicke, nd E M ucell, inciples f Micwve Cicuits, ditin Lbty eies, vl 8, New Yk, McGw-Hill, 948 [5] H teyskl, On the we Absbed nd ctteed by n Antenn, IEEE Antenns nd pgtin Mgzine, Vl 5, N 6, pp 4 45, Dec [6] ilve, Micwve Antenn hey nd Design, ditin Lbty eies, vl, ectins -8, McGw-Hill Bk Cmpny, Inc, New Yk, 949 [7] E Cllin, Limittins f the hevenin nd Ntn Equivlent Cicuits f eceiving Antenn, IEEE Antenns nd pgtin Mgzine, Vl45, N pp9-4, Apil 3 [8] J Ahni, Antenne, Oxfd t the Clendn ess, 946 [9] C A Blnis, Antenn hey, nd Ed, Jhn Wiley & ns, Inc, New Yk, 997 [] Best nd B C Knt, A util n the eceiving nd ctteing peties f Antenns, IEEE Antenns nd pgtin Mgzine, Vl 5, N 5, pp Oct 9 [] W K Khn nd H Kuss, Minimum ctteing Antenns, IEEE ns n Antenns nd p, Vl A-3, pp , eptembe 965 [] A C Gtely, J, D J tck nd B u-h Che, A Netwk Desciptin f Antenn blems, ceedings f the IEEE, Vl 56, N 7, pp 8-93, July 968 [3] W Wsylkiwskyj nd W K Khn, ctteing peties nd Mutul Cupling f Antenns with escibed ditin ttens, IEEE nsctins n Antenns nd pgtin, Vl A-8, N 6, pp 74 75, Nvembe 97 [4] W K Khn (invited ppe), "Inteeltin f ditin nd ctteing by Antenns," ceedings f the Eleventh Ntinl di cience Cnfeence, Ci, Egypt, Mch -4, 994 [5] B Geen, ctteing fm Cnjugte Mtched Antenns, IEEE ns n Antenns nd p, Vl A-4, N, pp 7-, Jnuy 966 [6] B Geen, he Genel hey f Antenn ctteing, he Ohi tte Univesity esech Fundtin ept 3-7, 3 Nvembe 963 (hd dissettin) [7] W K Khn, "nsmit-eceive Chcteistics f n Antenn nd ime evesibility f the Mxwell Equtins," Millennium Cnfeence n Antenns nd pgtin, Dvs, witzelnd, Apil 9-4, EA ublictins Divisin, EEC, AG Ndwijk, he Nethelnds [8] J D yde, Netwks, Lines nd Fields, entice- Hll, New Yk, 949 (he cmpenstin theem, -, p ) [9] A D Besle, Equivlent Cicuit Desciptins f Antenn ctteing, ivte cmmunictin, Octbe [] D C Yul, "On ctteing Mtices Nmlized t Cmplex t Numbes," ceedings IE, Vl 49, N 7, p, July 96 teven J Weiss btined his Bchel's degee in Electicl Engineeing fm the cheste Institute f echnlgy in 985 (BEE) nd gdute degees fm he Gege Wshingtn Univesity in 989 (M) nd 995 (Dc) bth with cncenttins in Electphysics He is the tem lede f the Antenn em t the Amy esech Lb nd hs wked with the Amy since 989 In this cpcity, he hs been instumentl in the develpment f numeus specilized ntenns f mility pplictins D Weiss hs tught t he Jhns Hpkins Univesity since, teching cuses in Antenn ystems, Advnced Antenn ystems, nd Intemedite Electmgnetics D Weiss is fellw f the Wshingtn Acdemy f ciences nd is n the bd f diects f the Applied Cmputtinl Electmgnetics ciety (ACE) He is eni Membe f the IEEE nd hs seved in ll f the ffice psitins f the Wshingtn ectin f the IEEE He is membe f UI Cmmissins A nd B nd seves s the vice-chi f Cmmissin A He is egisteed pfessinl enginee in the sttes f Mylnd nd Delwe Wlte K Khn eceived the BEE degee fm Cpe Unin, New Yk, NY, in 95 He eceived MEE nd DEE degees fm New Yk Univesity lytechnic chl f Engineeing (fmely lytechnic Institute f Bklyn) in 954 nd 96, espectively Fm 95 t 954 he ws engged in mnpulse d system develpment t Wheele Lbties, Get Neck, NY In