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LEAKYMODESANDHIGH-FREQUENCYEFFECTSINMICROWAVE INTEGRATEDCIRCUITSFRANCISCOMESAUniversity of SevilleSeville, SpainDAVIDR.JACKSONUniversity of HoustonHouston,Texas1. INTRODUCTIONThisarticleinvestigatesthehigh-frequencyeffectsthatoccur when sources or discontinuities are present on print-ed-circuittransmissionlines. Althoughmanyofthecon-clusions are general, the analysis andthe results willfocus onmicrostripandstripline types of transmissionlines, where there is a single-strip conductor that is withinalayeredstructurethat maycontaineitheroneortwoground planes. Examples include the microstrip structureshown in Fig. 1a, the covered microstrip shown in Fig. 1b,andthestriplinestructurethat hasanairgapoverthestripconductor,asshowninFig.1c.(Laterinthearticle,the analysis will be extended to include coupling betweentwo strip conductors, so that the crosstalk current inducedonapassivemicrostriplinefromcouplingtoanadjacentactivemicrostriplinecanbestudied.)Theanalysisandresultswillfocusonthecurrentandelds that are excited when a gap voltage source is locatedat theorigin(z =0) onaninnitestructure, wherethestrip conductor extends from N to N, as shown in Fig. 2.This source model allows for a semianalytical treatment ofthe problem in the spectral domain, which greatly simpli-esthesolution. Othertypesof sourcesmaybeconsid-ered, however. Many of the conclusions also apply todiscontinuities on the line (bends, junctions, etc.),althoughsuchdiscontinuityproblemsarenotspecificallytreatedhere.Oneofthemaingoalsofthearticleistoexaminethenatureofthecurrentonthetransmissionlinethatisex-cited by the gap voltage source, and explore what types ofspuriouseffectsoccurathighfrequency.Itiswellknownthat at lowfrequency, simple transmissionline theorymay be used to adequately predict the current on the stripconductor. In particular, the current I(z) predicted by sim-pletransmission-linetheoryisI(z) =12Z0ejkBMzz [[(1)whereZ0isthecharacteristicimpedanceofthelineandkBMzisthepropagationwavenumberofthetransmission-linemode, whichisalsocalledaboundmode (BM) ofpropagation, since the elds of this mode decaytrans-versely away from the structure (in the 7x direction), andhenceareboundtothe guidingstructure.Athighfrequency, simpletransmission-linetheoryisnolongeradequatetoaccuratelypredictthecurrentex-cited by the gap source, for various reasons. First, itshouldbenotedthatthedefinitionof thecharacteristicimpedance Z0 is not unique at high frequency [1]. Second,athighfrequency, thecurrent onthestripisnotsolelythatoftheboundtransmission-linemode(asassumedintheaboveequation), butitalsoconsistsofacontinuousspectrum (CS)part,whichisnotpredictedbytransmis-sion-linetheory, andwhichbecomesmoreimportant asthe frequency increases. The analysis presented hereallows for an accurate calculation of the bound-modeamplitudethat isexcitedbythegapsourceat anyfre-quency, evenwhenthe bound-mode current is not thedominant current ontheline. TheboundmodecurrenthastheformI(z) =ABMejkBMzz [[(2)wherethemodalamplitudeABMisdeterminedbyares-idueinthecomplexplane, asexplainedlater.The semianalytical treatment presented here allows foracalculationoftheCScurrent, whichmaybecomeveryLh(a)(b)(c)hhhchwcccc0coFigure1. Crosssectionsof (a) amicrostripline, (b) acoveredmicrostrip,and (c) stripline with anairgap.2268importantathighfrequencies.Themethodalsoprovidesa convenient description of the physics of the CS current.In particular, the CS current includes the effects ofleakymodes,whicharemodesthatradiate(leakenergy)astheypropagate. Theleakagemaybeof twotypes. Inonecasetheleakageoccursintothefundamental modeof the substrate, whichis either a surface-wave modeor a parallel-plate mode, depending on whether thereare one or two ground planes. (When only a single groundplane is present, the structure will be called open, sincethe structure is open to free space. When there aretwogroundplanes, thestructurewillbecalledclosed.)Assuming that only a single substrate mode is abovecutoff (the usual case), the leakage will occur into thefundamental TM0mode of the substrate. Inthe othercase, whichonlyoccursonopenstructures, theleakageoccurs into both the TM0substrate mode and intospace.Ineithercase,aleaky-mode(LM)currenthastheformI(z) =ALMejkLMzz [[(3)where the propagationwavenumber kLMz=b ja of theleakymodeiscomplexasaresultof radiationloss. Theexcitationofaleakymodemaycauseconsiderablespuri-ous effects, including power loss, interference with the de-siredboundmode, andcrosstalkwithneighboringcircuitcomponents. The semianalytical treatment presented hereallows for a calculationof the excitationof any leakymodes that are part of the CS current. In the representa-tionusedhere(andexplainedlater),leakymodesappearexplicitly as part of the continuous spectrum if the modesarephysical, meaningthat theleakagemechanismofthemode(eitherintothesubstrateorintoboththesub-strateandintospace) isconsistent withthephasecon-stantof theleakymode.The analysis also reveals that the CS current onthe strip conductor consists not only of leaky modesbut alsoof acomponent calledtheresidual-wavecur-rent. Theresidual-wave(RW)currentistheleftover, orresidual,partoftheCScurrentthatisnotrepresentableasasumofphysicalleaky-mode(LM)currents.Whennophysical leaky modes are present, the RWcurrent isidenticallythesameas theCScurrent. Whenphysicalleakymodesarepresent,theCScurrentconsistsofboththe LM and the RW currents.In either case, the RW cur-rentmaybecomeimportantathighfrequency.Theanal-ysis further shows that the RWcurrent consists, ingeneral, of two parts. One part is called the TM0 RW cur-rent, and exists for both open and closed structures. Thesecondtype, calledthek0RWcurrent, existsonlyforopenstructures. Anasymptotic analysisshowsthat forlargedistanceszfromthesource, theTM0RWcurrentbehavesasITM0RW (z) =ATM0RW1z [[3=2ejkTM0z [[(4)where kTM0is the wavenumber of the TM0 substrate mode,whilethek0RW currentbehavesasIk0RW(z) =Ak0RW1z [[2ejk0z [[(5)wherek0isthe wavenumber of freespace.Section2providesadetaileddescriptionof thesemi-analytical method of analysis that is used to calculate thestrip current and the consistent BM, LM, and RW compo-nents. Theanalysisshowshowthetotal stripcurrentisrepresentableasasumofBMandCScurrents,andalsohow the CS current can be represented as a sum of phys-icalLMcurrentstogetherwithRWcurrents.In Section 3, results will be presented to show the typesofspuriouseffectsthatcanoccurforthevarioustypesofprinted-circuittransmission-linestructuresin Fig. 1, duetotheLMcurrentsand theRW currents.Section4extendstheanalysisandtheresultsof theprevious sections by considering the calculationof thecrosstalk current that is excited on a passive (victim) line,when a gap voltage source is present on an excited line. Itis shown that simple transmission-line theory can predictthe crosstalk current accurately at low frequency, using aneven/odd-modeanalysis.However,athighfrequency,theCScurrentbecomes important,andthe crosstalkcurrentcannot beadequatelydescribedbysimpletransmission-linetheory.Section 5 examines the nature of the crosstalk eld thatsurroundsaprinted-circuittransmissionline, whenitisexcitedby agap voltagesource.It is seen thatat lowfre-quencytheelddecaysawayfromthelinefairlyrapidly,butathighfrequencythisisnotthecase. Whenphysicalleakymodes arepresent, verystrongfocusedradiationmayoccur inaspecific direction, leadingto veryhighcrosstalkelds.Section6 concludes the article by summarizing themost important observations and conclusions that are ob-tainedfromthestudyoftheprinted-circuittransmissionlineexcitedbyagapvoltage source.2. ANALYSISFORCURRENTEXCITEDBYAGAP SOURCEThissectiondescribesthetheoryunderlyingthecompu-tationof thecurrentexcitedonthestripconductorof ainnite printed-circuit line by a gap voltage source on thelineas showninFig.2. Thesubstrateisa losslessdielec-tric that can be layered and an optionalmetallic wall cancover the line (in which case the structure is referred to asaclosed structure, sinceitisclosedvertically,andradi-ationintospaceisprohibited).Thelineisassumedtobeinnitealongthelongitudinal direction(z) andthecon-ductorsareinnitesimallythinandperfectlyconducting.(Althoughonlylosslessstriplikelineswill beconsideredhere, the present approach can be extended to lossy and/orslotlikelineswithsomemodications.)Thebasicideasofthetheorywill beexposedinthissectioninconnectionwithasinglemicrostripline.Section4willtreatthecur-rentexcitedonapairof coupledmicrostriplines.LEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2269Inordertocomputethesurfacecurrent onthestripconductor,Js=Jx ^ xxJz^ zz,thefollowingelectriceldinte-gralequation(EFIE)has tobesolved:Et[Js(x; z)] =Egapt(x; z) (6)where Egaptis the tangential electric eld impressed in thegap (where the t subscript denotes tangential) to modelthe gapvoltage source. This impressedelectric eldisequatedtotheeldthatiscomputedfromthestripcur-rent using the transverse part of the corresponding dyadicGreensfunctionofthelayeredstructureatthestripin-terface, Gt(x; z; x/; z/) =Gxx ^ xx^ xxGxz( ^ xx^ zz ^ zz^ xx) Gzz ^ zz^ zz, as thefollowingconvolutionproductEt(x; z) =_oo_w=2w=2Gt(x x/; z z/) Js(x/; z/) dx/dz/(7)wherethetranslational symmetryof thegroundedsub-strate in the tangential plane has been incorporated. Theimpressedeldisassumed tobeEgapt(x; z) = 1Drectxw=2_ _rectzD=2_ _^ zz (8)inordertogiveauniformgapeldthatcorrespondstohomogeneous voltage drop of 1 Valong the gap (D denotesthe longitudinal width of the gap, and rect(x) =1 if 7x7r1andiszerootherwise). Forfutureuseitisconvenienttonotethat thezFouriertransformof theimpressedgapeld,~EEgap(x; kz),of (8) isgiven by~E Egap(x; kz) = rectxw=2_ _sinckzD2(9)where sinc(x) =sin(x)/x. (In the present notation,~AA(ka) de-notestheFouriertransformofA(a)foranyfunctionA.)Next, the EFIE is solved by means of Galerkins method[2], for which the unknown induced current density on thestrip has to be expanded in basis functions. First, and forsimplicityofexplanation, thecurrentdensitywillbeex-pandedusingonlyonebasisfunction. Theextensiontoinclude multiple basis functionwill be discussedafter-ward.2.1. SingleBasisFunctionIfonlyasinglebasisfunctionisemployedtoexpandtheinduced current density on the strip, Js(x, z) will have onlyalongitudinalcomponent, Jsz, whichcanberepresentedbyJsz(x; z) =T(x)I(z) (10)whereT(x) denotesthetransverseprole, whichisnor-malizedto give_w=2w=2T(x) dx =1 (11)so that I(z) actually represents the longitudinal current onthelineatadistancezfromthesource. Specifically, thefollowingbasisfunctionhasbeenchosenT(x) =rectxw=2_ _pw2_ _2x2_ (12)whichwill giveagoodapproximationfornarrowstrips.TheEFIE (6)can bethenwrittenas_oo_w=2w=2Gzz(x x/; z z/)T(x/)I(z/) dx/dz/=Egapz(x; z)(13)The application of the longitudinal inverse Fourier trans-formI(z) =12p_oo~II(kz) ejkzzdkz(14)to the integral equation above (or alternatively, taking theFourier transform of both sides with respect to z) leads to~II(kz)_w=2w=2~GGzz(x x/; kz)T(x/) dx/=rectxw=2_ _ ~EEgapz(kz)(15)ThetildeabovetheGreensfunctionindicatesherea1DFouriertransformwithrespecttoz.IfGalerkinsmethodisnow applied,(15) canbenallyexpressedas~II(kz)_w=2w=2T(x)_w=2w=2~GGzz(x x/; kz)T(x/) dx/dx=_w=2w=2T(x) dx_ _ ~EEgapz(kz)(16)ywhxzc,j0Figure 2. Geometry of an innite microstrip line excited by a gapvoltage source atz =0.2270 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITSwhich, after takingintoaccount (11), allows this tobewrittenas~II(kz) =~EEgapz(kz)D(kz)(17)withD(kz) =_w=2w=2T(x)_w=2w=2~GGzz(x x/; kz)T(x/) dx/dx(18)ItshouldbenotedthatD(kz)actuallyrepresentsafunc-tion whose zeros are the modal propagation constants (kMz )oftheinnitemicrostripline.Acarefulstudyofthesin-gularitiesofthisfunctioninthecomplexkzplanewillbepresentedlaterinthis section.Twocommontechniqueshavebeencommonlyusedinthe literature to compute integrals such as D(kz): the spec-tral-domainapproach(SDA) [3,4] andthediscretecom-plex images technique (DCIT) [57]. Eachis discussedfurtherbelow.2.1.1. Spectral-DomainAnalysis. Asiswellknown, theuseof theSDAismotivatedbythefactthatthedyadicGreens functioncanbeobtainedinclosedform(or bymeansof certainalgorithms) inthespectral domain[810]. Thus, assumingthat~GGzz(kx; kz) is knowninclosedform, D(kz) canbeexpressedanalyticallyintermsof aspectral-domainintegralinvolvingthisGreensfunction.(ThetildeovertheGreensfunctionindicatesherea2DFourier transform of the Greens function.) This is accom-plished by rst expressing the inner integral in Eq. (18) asan inverse Fourier transform of the Fourier transform (inx), notingtheconvolutional natureof theintegral, andthen performing the outer integral in x in closed form, re-sultinginthe Fourier transformof thefunctionT(x).Theresultis (assumingthatT(x) is anevenfunction)D(kz) =12p_Ckx~GGzz(kx; kz) ~TT2(kx) dkx(19)where Ckxis anappropriate inverse Fourier transformpathfromminus innityto innity inthe complex kxplane[1113].Since the precedingintegralhas to be computedmanytimes to obtain the current prole given by (14), it is veryconvenient to minimize as much as possible the intensivenumerical effort required to compute (19). Different strat-egies have been proposed in the literature to speed up theoverall computation of similar integrals, and one that hasbeen found particularly efcient is the extraction of an as-ymptotic behavior that can be further expressed in closedform. This involves splittingthe integral (19) into twointegralsD(kz) =Ds(kz) Do(kz) (20)whereDs(kz) =_Ckx~GGzz(kx; kz) ~GGozz(kx; kz)_ _~TT2(kx) dkx(21)Do(kz) =_oo~GGozz(kx; kz) ~TT2(kx) dkx(22)sothat Ds(kz) canbecomputedwithlowcomputationaleffort. Anasymptoticexpressionforthespectral-domainGreenfunctionis~GGozz(kx; kz) = jom0212joemk2z_ _kx[ [k2xk20(23)(wherek0isthefree-spacewavenumber).Useofthisex-pression leads to a closed-formexpression for DN(kz)[14,15].2.1.2. DiscreteComplexImages Technique. Oneof themain numerical disadvantages of the spectral-domaintechniqueisthatobtainingthelongitudinalcurrentpro-lefromEq.(14)requiresnumericallyevaluatingthein-tegral (21) withinnitelimitsforeachvalueof kz. Thisdrawbackcanbeconsiderablyalleviatedif thediscretecomplex images technique (DCIT) is used to obtain aclosed-form expression for the Greens function in the spa-tial domain. This procedure has been employed elsewhere[7,16,17] todeal with2Dprintedlines whenEq. (6) isposed as a mixed potential integral equation (MPIE) [18].It is worth mentioning that the integrals to be repeatedlycomputedaredoubledefiniteintegralswithnonsingularintegrands (except at the strip edges) that can be very ef-cientlycomputedbyloworderGaussChebyshevquad-ratures [17]. Our experience shows that this schemereducestheCPUtimeatleast5timeswhencomparedtothatrequiredbytheSDA.2.1.3. Multiple Basis Functions. The procedure shown inthe previous section is numerically accurate only for thosesituations where the transverse component of the currentdensitycanbeneglected(whichisusuallyfoundatlowfrequencies and/or for narrow strips). In general, however,anexpansionwithmultiplebasisfunctionsisrequired.For this purpose, the current density on the strip isassumedtobe(withqeven andpodd)Jz(x; z) =

Nfq=0Tzq(x)Izq(z) (24)Jx(x; z) =

Nfp=1Txp(x)Ixp(z) (25)where Tzp(x) and Txq(x) are, respectively, the basis functionsusedtoexpandthetransverseproleofthezandxcom-ponentsofthecurrentdensity,andIzp(z)andIxq(z) arethecorrespondingunknowncoefcient functions accountingLEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2271for the longitudinal prole of the current density. ThetermNfdenotes the maximum order of the basis functions.The transverse prole is accounted for by the usualChebyshev polynomials of the rst and second kinds[Tn( ) andUm( ), respectively] weightedbytheproperedgecondition,namelyTzq(x) =(2 dq0)2wpTq(2x=w)1 2xw_ _2_ (26)Txp(x) =4wpjUp(2x=w)1 2xw_ _2(27)where dmnistheKroneckerdelta.Theuseofthisexpan-sionintheEFIE, afterusingGalerkinsmethodandap-plyingthe longitudinal Fourier transform, leads to thefollowinglinearequationsystemGxxpp/ (kz)_ _Gxzpq/ (kz)_ _Gzxqp/ (kz)_ _Gzzqq/ (kz)_ _____.~IIxp(kz)_ _~IIzq(kz)_ _____=0 [ ]~bbzq(kz)_ _____(28)whereGxxpp/ (kz) =_w=2w=2[Txp(x)]_w=2w=2~GGxx(x; x/; kz)Txp/ (x/) dx/dx(29a)Gxzpq/ (kz) =_w=2w=2[Txp(x)]_w=2w=2~GGxz(x; x/; kz)Tzq/ (x/) dx/dx(29b)Gzxqp/ (kz) =_w=2w=2[Tzq(x)]_w=2w=2~GGzx(x; x/; kz)Txp/ (x/) dx/dx(29c)Gzzqq/ (kz) =_w=2w=2[Tzq(x)]_w=2w=2~GGzz(x; x/; kz)Tzq/ (x/) dx/dx(29d)~bbzq(kz) =~EEgapz(kz)dq0(29e)Thecurrentalongtheline canbeobtainednallyasI(z) =_w=2w=2Jz(x; z) dx =

Nfq =0Izq(z)_w=2w=2Tzq(x) dx (30)whichcanberecognizedassimplyI(z) =Iz0(z)becauseofthenormalizationconditionimposedonthebasis func-tions.ThereforeI(z) =12p_Ckz~IIz0(kz) ejkzzdkz(31)ThecoefcientsoftheGalerkinmatrix(29a)(29d)canbenumericallyobtainedbyusingtheSDAortheDCITtechniquesdiscussedintheprevioussections, sincetheintroduction of multiple basis functions does not affect theasymptotic or singular behavior of the correspondingintegrals.2.2. TheComplexkzPlaneIt has been shown above that the computation of the cur-rentonthestripI(z) requiresaninverseFouriertrans-formgiven,ingeneral,byEq.(31).Itiswellknownthatthe study of the singularities of the integrand of any com-plex spectral integral gives very valuable informationabout the integral itself [19]. In this way, referenceshaveshown[12,13,2022] thatastudyof theintegrandof(31)inthecomplexkzplaneprovidesadeepandveryuseful physical insight into the nature of the currentexcitedontheline, andfurthermore, howtodecomposethe current into its constituent parts. Since the exp( jkzz)functionisfullyanalyticintheentirekzplane, anysin-gularity of the integrand comes necessarily from~IIz0(kz) ~II(kz), andthereforethepresentstudywill focusonthepolesandbranchpointsofthis latterfunction.Looking at the form of~II(kz) given in (17), it can be rec-ognized that the numerator will not show any singularityin the kz plane since~EEgapz(kz) is the Fourier transform of abounded function. Thus, the pole singularities of~II(kz) willcome from the zeros of the denominator D(kz) (for the caseof multiplebasisfunctions, thisfunctionwill bethede-terminantoftheGalerkinmatrix),whichhavebeenpre-viously identied as the propagationwavenumbers formodes on the innite line. Also, the branch points of D(kz)willbethebranchpointsingularitiesof~II(kz).The discussion above shows that the nature of the com-plexkzplanedenedby~II(kz) isfullydeterminedbythefunctionD(kz).Thisfunctioncanberewritteningeneral(alsoformultiplebasis functions)asD(kz) =_CkxQ(kx; kz) dkx(32)whereQ(kx,kz)involvesthespectral-domainGreenfunc-tion(SDGF). TheexpressiongiveninEq. (19) forD(kz)makesapparentthatitssingularitiesarethemselvesde-terminedbyboththesingularitiesof theSDGFandthecorrespondingintegrationpathCkxinthecomplextrans-versewavenumberplane(kx).Therearetwotypesofsingularitiesinthecomplexkzplane: poles and branchpoints. Poles correspond to guidedmodesthatexistontheprinted-circuittransmissionline,andtheresiduesatthecorrespondingpolelocationsde-termine the amplitudes of the modes that get launched bythegapvoltagesource, includingbothboundandleaky2272 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITSmodes. During a path deformation to the steepest-descentpath(discussedlater), thebound-modepolesarealwayscaptured, and hence the bound-mode currents always ap-pear explicitly inthe nal expressionfor the current.Leaky-modepolesmayormaynotbecaptured,however.Leaky modes whose poles are capturedare saidto bephysical leaky modes, and these currents appear explic-itlyinthenalexpressionforthecurrent. Leakymodeswhosepolesarenotcaptureddonotappearexplicitlyinthenal expressionforthecurrent, andaresaidtobenonphysical. However, these nonphysical poles may stillhaveaninuenceonthetotalcurrent, byvirtueoftheirproximitytothesteepest-descentpath. Thiswillbepar-ticularlytruewhenanonphysical poleisclosetobeingphysical, sothatthepoleisclosetothesteepest-descentpath.Branchpointsinthekzplanearisebecauseof anon-uniquenessintheevaluationoftheintegral(32).Inpar-ticular, different pathsof integrationinthecomplexkxplane may be used for the path Ckx. These different pathsmay detouraroundthe pole and/orbranchpointsingular-itiesoftheSDGFindifferentways,resultingindifferentvaluesfortheintegral.ThiscorrespondstoD(kz)beingamultiple-valuedfunctionofkz. Mathematically, thismaybe convenientlyviewedas the complex kzplane corre-sponding to a Riemann surface. On each sheet of the sur-face a different path of integration is used in the kx planetoevaluate theintegralinEq.(32).The SDGF for an open structure such as microstrip hassingularitiesatparticularpointsinthetransversewave-numberktplane, wherek2t =k2xk2z. Thisresultsincor-respondingsingularitiesatthefollowinglocationsinthecomplexkxplane[13,2325]:1. Branchpointsatkx=7kxb,wherek2t =k2xbk2z=k202. Properpolesatkx=7kxp,wherek2t =k2xpk2z =k2sw,wherekswisthepropagationconstantforapropersurfacewave ofthe backgroundlayeredstructure3. Improper poles at kx=7kxi, where k2t =k2xik2z=k2imp,wherekimpisthepropagationconstantofanimpropermodeofthebackgroundlayeredstruc-tureTheimpropermodesofthebackgroundlayeredstruc-ture canbe classiedas either improper surface-wavemodes,ifthepropagationconstantkimpispurelyreal,orascomplexleakymodes,ifkimpiscomplex.(Animpropersurface-wave modeisneverregardedas aphysicalmode.Acomplex improper leaky mode is regarded as beingphysical if thephaseconstant of themodeislessthank0, sothattheradiatingleakymodeisafastwavewithrespecttofreespace[26,27].)A careful study of the integration paths in the kx plane,and their evolution as the point kz moves in the complex kzplane,revealsthatbranchpointsinthecomplexkzplaneoccuratthefollowinglocations:1. Branchpoints at kpzb= kTM0, where kTM0is thewavenumberof theTM0surface-wave(orparallel-plate)modeofthebackgroundlayeredstructure(itisassumedherethatonlytheTM0modeisabovecutoff).2. Branchpoints at kizb= kimp, where kimpis thewavenumberofanimproper(improperrealorcom-plexleaky) modeof thebackgroundlayeredstruc-ture.3. Branchpoints at k0zb= k0, where k0is the free-spacewavenumber.Thebranchpointsatkpzbandkizbgiverisetoadouble-val-uedfunctionofkz(correspondingtotwosheetsofthekzRiemann surface), since circling these branchpoints twicegives back the original function D(kz) at the starting pointinthekzplane. Thus, thepathof integrationinthekxplaneinintegral (32) evolvesbacktotheoriginal pathwhenthepoint kzmovestwicearoundthebranchpointbacktotheoriginalpoint.The branchpoints at k0zb= k0have a logarithmiccharacterinthesensethatthereareaninnitenumberof sheets associated with these branchpoints; that is, eachtime the branchpoint is circled in the kz plane, the path ofintegrationinthekxplanechanges,andneverreturnstothe original path after a nite number of encirclements ofthebranchpoint. Theoriginal pathof integrationliesinthe sheet that is denoted as the zero sheet, for which thepath of integration in the kx plane stays on the top (proper)sheetof thekxRiemannsurface.As shown in Ref. 13, the branchpoints corresponding tothe improper modes of the background structure atkizb=kimpdo not appear onthe zero sheet of the 7k0branchpoints onthe kzRiemannsurface (suchbranch-pointsappearonlyontheoddsheetsofthe 7k0branch-points).TheoriginalpathofintegrationCkz,whichstaysonthe zero sheet, thus sees the branchpoints only atk0zb= k0andkpzb= kTM0. Thebranchpointsat kTM0appearonlyonthe evensheetsof the 7k0branchpoints,includingthezerosheet.The complexkz plane for the case of an open structuresuchasmicrostriplineisshowninFig. 3. Thebranch-pointsat 7k0andat kTM0areshown.Toaidintheun-derstanding of the branchpoints and the kzRiemannsurface, anexampleis giventoconnect thelocationofxxxxkzplaneIm(kz)Re(kz)k0 kTM0CBAxxFigure 3. The kz plane showing two pairs of branchpoints at7k0and kTM0,apairofproperpolesat 7A,andtwopairsofleakypoles at 7B and 7C.LEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2273the kz point on the Riemann surface with the correspond-ingpathofintegrationinthekxplane. Threepointsareshown in the complex kz plane, labeled as A, B, and C. Thecorresponding paths of integrationinthe kxplane areshowninFigs.4a,4b, and 4c.Point A is located on the real axis of the kz plane, and isassumedtobeonthezerosheet of thebranchpointsat7k0andalsothetopsheetofthebranchpointsat kTM0.(The original path of integration Ckz is on this sheet.) Thispoint corresponds to a path of integration that stays alongthereal axisinthekxplane, showninFig. 4a, withoutdetouring around any pole or branchpoint singularities oftheSDGF.PointBisassumedtolieonthezerosheetofthe 7k0branchpoints,andthebottomsheetofthe kTM0branch-points.SuchapointisobtainedbystartingfrompointA,thenmovingintotherstquadrantofthekzplane, andthenmovingdown,crossingtherealaxisbetweenk0andkTM0. In doing so, the branchcut fromthe kTM0branchpointis crossed, to enter the lower sheet of this branchpoint onthe kz Riemann surface. The corresponding path in the kxplanedetoursaroundtheTM0polesoftheSDGFlocatedat 7kxp, but stays on the top sheet of the Riemann surfaceinthekxplane.Point C is assumed to be on the lower ( 1) sheet of thek0 branchpoints on the kz Riemann surface. Such a point isobtainedbystartingfrompointA, thenmovingintotherst quadrant of thekzplane, andthenmovingdown,crossing the real axis between the origin and k0. In doingso, thebranchcutfromthek0branchpointiscrossed, toenterthelowersheetofthisbranchpointonthekzRiem-annsurface. (As soonas this branchcut is crossed, thebranchpoints at kTM0no longer appear, since thesebranchpointsappearonlyontheevensheetsofthe 7k0branchpoints.) The corresponding path of integrationcrosses throughthebranchcuts of the SDGFinthe kxplane and lies partly on the improper lower sheet of the kxRiemannsurface.ThepathalsodetoursaroundtheTM0surface-wave polesofthe SDGF.Forcoveredstructuressuchascoveredmicrostrip,theonlybranchpointsthatoccurarethoseatthewavenum-bersoftheparallel-platemodesofthebackgroundstruc-ture.AssumingthatonlytheTM0parallel-platemodeisabove cutoff, there will be one pair of branchpoints on therealkzaxisatkzb= kTM0.Therewillalsobeaninnitenumber of branchpoints onthe imaginary axes, corre-spondingtothewavenumbersoftheevanescentparallel-plate modes. The complex kz plane for a covered structureisshowninFig.5.xxxxTM0Im(kx)Re(kx)Im(kx)Re(kx)xxIm(kx)Re(kx)(a)(c)(b)Figure 4. Integration paths in the transverse wavenumber com-plexplane (kx) correspondingto (a) point A, (b) point B, and(c) pointC inFig. 6.xxx xIm(kz)kz planeRe(kz)kTM0CkzFigure 5. The kz plane showing a pair of branchpoints at kTM0,apair(outofaninnitenumber)ofbranchpointsontheimagi-naryaxis, and twopairs ofpoles.2274 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITS2.3. DecompositionoftheTotalCurrentWhencomputingtheeldsof asimplesourcesuchasadipoleinthepresenceof agroundeddielectricslab, theexistenceofsquare-root-typebranchpointsat 7k0intheintegrand of the corresponding Fourier integral is relatedtoradiationinthefreespace, namely, toitscontinuousspectrum [23,24]. This principle holds for more complicat-ed systems, such as the gap source on the microstrip line,discussed here. The existence of branchpoints is related tomechanisms of radiation. Thus, for the microstrip line un-der study, the fact that~II(kz) has multiple pairs of branch-points indicates that radiation can take place by means ofdifferent mechanisms (or, in other words, that the contin-uous spectrumwill havedifferent components). Speci-cally, the 7k0branchpointsarerelatedtoradiationintofreespace,andthebranchpointsat kTM0arerelatedtoradiationintheformof surfacewavesorparallel-platemodesof thebackgroundwaveguide.Inadditiontobranchpoints,~II(kz)alsohaspolesingu-laritiesthat originatefromtherootsof D(kz). It iswellknownthatD(kz) has an innite number of roots:a niteset corresponding to bound (proper) modes and an inniteset of leaky(improper) modes [2835]. Specifically, thedifferentpolesof~II(kz)are locatedasfollows:*Boundmodesthepoleslieonthezerosheetwithrespect to the 7k0 branchpoints and on the top (prop-er)sheetwithrespecttothe kTM0branchpoints.*Surface-waveleakymodesthepoleslieonthezerosheetwithrespectto7k0branchpointsandonthebottom(improper) sheet withrespect tothe kTM0branchpoints.*Surface and space-wave leaky modesthe poles lie onthe 1 sheetof the 7k0branchpoints.Atypical diagramoftheintegrationpathCkztocom-pute I(z) together with some relevant singularities of~II(kz)isshowninFig. 6. Thepathisequivalenttoareal-axispaththat runs fromNtoN, detouringaroundthebranchpointsingularitiesandalsotheboundmodepolesontherealaxis.Withanunderstandingof thecomplexkzplane, itisnowpossible to discuss the physical significance of aparticular mode. This physical significance can be denedbythedegreeof excitationof thismodefromarealisticsource. A mode that is physically significant should be ex-citedbyasuitablenitesource, whichimpliesthat thecurrentI(z)onthelinewillthenhaveacomponentthatclosely resembles the current of the mode. (If a particularmodeisexcitedwithasufcient amplitude, thismodalcomponentmay dominatethe totalline current.)Onepossiblemeasure of therelativesignificanceofeachmicrostripmodecanbeobtainedbylookingatthedistance fromthecorrespondingpoleinthekzplanetotheintegrationcontourCkzontheRiemannsurface.Thetermdistance herereferstodistanceontheRiemannsurface. For example, two points on the opposite sides of abranchcut, bothonthesamesheet, willnotbeclose tooneanother, regardlessof howgeometricallyclosetheyare together. Two geometrically close points on either sideof a branchcut will be close on the Riemann surface if theyarelocatedondifferentsheets, suchthatthereisacon-tinuousconnectionbetweenthe twopoints.SincethekBMzpoleontherealaxiscorrespondingtoaboundmodeisonthesamesheetastheintegrationpathCkz, this pole is always close to the path of integration, andisthereforealwaysphysicallysignificant. Fortheleaky-mode poles, however, the poles may or may not be close totheintegrationpath,dependingonwhichsheettheyarelocated, andthevalueof thephaseconstant bof eachmode. Thephysical significanceof aleakymodecanbesummarizedconciselybysayingthat amodecanhavephysical significance only if the mode satises what is de-notedasthepathconsistencycondition(PCC). Thiscon-dition (which was called the condition of leakage in Ref.32) states that the path of integration in the kx plane usedtoobtainaparticularleaky-modesolutionshouldbecon-sistent with the phase constant b of the leaky mode that isobtained by using the given path. For example, if the pathinthekxplaneusedtoobtainaleakymodeistheoneshowninFig. 4b, whichcorrespondstoleakageintotheTM0modebutnot intospace, theresultingleakymodewillbephysicallysignificant(orphysical)providedthephase constant of the leaky mode is in the rangek0obokTM0.Ifthepathinthekxplaneusedtoobtainaleaky mode is the one shown in Fig. 4c, which correspondsto leakage into both the TM0 mode and into space, the re-sulting leaky mode will be physically significant (or phys-ical) provided the phaseconstant of the leaky modeis intherange bok0.Thisphysical significanceof aleakymodeisdirectlyrelatedtotheclosenessof thecorrespondingpoletothepath Ckz on the Riemann surface. For example, consider aleaky-modepolelocatedasshowninFig.6,totheleftofthe branchpoint k0; that is, +(kzp)ok0. Such a leaky modeis close to the path Ckz if it is on the 1 sheet of the 7k0branchpoints.Thismeansthatthepathofintegrationinthe kx plane is that shown in Fig. 4c. Similarly, if a leaky-modepoleislocatedintheregionk0o+(kzp)okTM0, thepole is close to the path Ckz if the pole is on the zero sheetof the 7k0branchpoints, andthe bottomsheet of thekTM0branchpoints.Amore mathematicalway of determining the physicalsignificance of the leaky modes, which also providekz planek0kTM0kzBMkzLMCkzIm(kz)Re(kz)xx xxFigure6. ThekzplaneshowingtheintegrationpathCkz, twopairs of branchpoints at7k0 and kTM0, a pair of proper poles atkBMz,and a pair ofleaky poles at kLMz.LEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2275additional insight into the nature of the current and eldsexcited by the source, is to deform the original integrationpath Ckz to a vertical set of steepest-descent paths (SDPs)[23,24,36]. Figure 7 illustrates this for the case of an openstructuresuchasmicrostrip(forz40).Figure 7a shows how the original Ckz path may be rstdeformed into a path CBC detouring around all the branch-cuts (theTM0andk0branchcuts) andaset of residuecontributions CBMfrompaths that encircle all of thebound-modespoleson thepositiverealaxis (inthegureonlyoneboundmodeisassumed).Thetotalcurrent(TC)onthestripcan bedecomposedasI(z) = j Res{~II(kBMz)]ejkBMzz12p_CBC~II(kz) ejkzzdkz(33)=IBM(z) ICS(z) (34)wheretheresiduecontributionaroundCBMaccountsforthe launching amplitude of the bound-mode (BM) currentfromthegapsourceandtheintegralaroundCBCdenesthecontinuous-spectrum(CS)currentonthelinethatisinducedbythegapsource. Itisinterestingtonotethatstandardtransmission-linetheoryonlypredictstheBMcurrent and not the CS current, which is a consequence ofthefull-wavemodelingofthecompletecurrentspectrumdue to the gap source. This latter current usually increas-eswithfrequencyandisresponsibleforspuriouseffectsobservedinthelinecurrent,as shownlater.TheCScurrentcanbefurtherdecomposedbydeform-ingtheCBCpathtoaset of vertical SDPsasshowninFig.7b.Inthisdeformationsomeleaky-mode(LM)polesmay be captured and therefore will appear explicitly in thenal decompositionof thestripcurrent. Inparticular, aleaky-modepolewill becapturedif theleakymodeisaphysical one, satisfying the path consistency condition de-scribedpreviously. Theresiduecontributionof thecap-tured LM poles will dene the launching amplitude of thephysicalleakymodes.Nonphysicalleaky-modepoleswillnot give an explicit contribution to the leaky-mode current(denedfromtheresiduesofthecapturedpoles),butthepossibleinuenceofsuchpolesisautomaticallyincludedinthecalculationof theresidual wave (RW) currents,whicharethecontributions fromthepathintegrationsalongthetwoverticalSDPpaths. TheRWcurrentsrep-resent that part of the CS current that is not representedexplicitlybythesumof thephysical LMcurrents. Thetotalstrip currentcanthereforebeexpressedasI(z) =IBM(z) ILM(z) ITM0RW (z)Ik0RW(z)

iIiRW(z)(35)whereIBM(z) = j Res~II(kBMz)_ _ejkBMzz(36a)ILM(z) = j Res~II(kLMz)_ _ejkLMzz(36b)ITM0RW (z) =12p_SDPTM0~II(kz) ejkzzdkz(36c)Ik0RW(z) =12p_SDPk0~II(kz) ejkzzdkz(36d)IiRW(z) =12p_SDPi~II(kz) ejkzzdkz(36e)The rst term represents the currents of the bound modesthat are launched by the gap source. Usually there is onlyone, andthisisthedesiredtransmission-linemode. Thesecond termrepresents the sumof all physical leakymodes,fromtheresiduesofthecapturedpoles.Ingeneral,there may be two types of physical leaky modes, those thatleak into only the TM0 mode, and those that leak into boththeTM0modesandintospace. Thenextterm istheTM0RWcurrent, arisingfromthepathintegrationalongthevertical SDP path that descends downward fromthebranchpoint at kzb=kTM0. Thenext termis thek0RWcurrent that arisesfromthepathintegrationalongthevertical SDP path that descends downward fromthebranchpoint at kzb=k0. Thelast seriestermisaset ofxx xxkz planeCkzCBCCBMkzLMkzBMxxx xkz planeCBCCLMCBMSDPTM0SDPk0(a)(b)Figure 7. (a) Complex kz plane showing the integration path Ckz,anintegrationpathCBCaroundthebranchcuts, andaresiduepathCBMaroundtheBMpole; (b)complexkzplane showingthepathdeformationofCkztoasetofverticalSDPs,aresiduepatharound the captured LM pole kLMz, and a residue path CBM aroundthe BMpole.2276 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITSRWcurrentsthatcomesfromSDPpaths(notshowninFig. 7b)thatdescendfrombranchpointsassociatedwithimproper complex modes of the background structure.These RW currents are included whenever the phase con-stantsofthecorrespondingimpropermodesoftheback-ground structure are less than k0, so that thebranchpoints are intercepted by the pathdeformation.TheseRWcurrentsareusuallynegligiblebecauseofthelargeimaginarypartof thecorrespondingbranchpoints.AlthoughtheTM0andk0RWcurrents donot haveclosed-form expressions, it is possible to evaluate their as-ymptotic behaviors for large z [21,36]. The calculationshowsthattheTM0RWcurrentbehavesasITM0RW (z) ATM0RWejkTM0z(kTM0z)3=2(37)andthek0RW currentasIk0RW(z) Ak0RWejk0z(k0z)2(38)3. RESULTSFORTHECURRENTEXCITEDBYAGAPSOURCEInthis sectionresults will be presented for the threestructuresshowninFig.1,toillustratethetypesofspu-riouseffectsthatmaybeobservedinthecurrentexcitedbyagapvoltagesourceonaprinted-circuittransmissionline. Thethreestructuresaremicrostrip, coveredmicro-strip, andstriplinewithanairgap. Ineachcaseplotsofthetotal current arepresented, alongwithplotsof thebound mode and continuous-spectrumcurrents. Whenphysical leakymodes arepresent, thecontinuous-spec-trum(CS) current is decomposed into the leaky-mode(LM) and residual-wave (RW) parts, and these are shown.3.1. MicrostripFor the microstrip structure of Fig. 1a, the current excitedby thegapsourceconsistsofa BMcurrentand aCScur-rent. The CS current consists of any physical leaky modesthat exist at the frequency of interest, together with a RWcurrent,whichconsistsofboththeTM0RWcurrentandthek0RWcurrent.In all the cases presented, the only physical leaky modethatisfoundtoexistisonethatleaksintotheTM0sur-face-wave mode, and not into space. Such a leaky mode isphysical,andhenceisincludedinthecompositionoftheCS current, if k0obokTM0. Results are presented for botha moderate-permittivity substrate and a high-permittivitysubstrate[21].Resultsarerstpresentedforthecaseofamoderatepermittivity substrate, er=2.2, for several different valuesof w/h, corresponding to different characteristic impedances.The substrate thickness is h=1.0mm in all cases.Figure8showsthedispersionplotforanarrowline,withw/h=0.333. Forthisdimension, thelow-frequencycharacteristicimpedanceis 145.8 O.Aplot of the normal-izedphaseconstant of theboundmodeisshown, alongwith a plot for the leaky mode. Also, for reference, a plot ofthe normalized phase constants for the TM0 and TE1 sur-face-wavemodesofthegroundedsubstrateareincluded.Itisobservedthatleakagebeginsat30 GHz. Abovethisfrequency a leaky mode with a complex propagation wave-number exists onthe microstripline. The leaky modeleaksintotheTM0surface-wavemodeof thegroundedsubstrate. The leaky mode is an EH0 mode, meaning thatthe transverse prole of the current closely resembles theexpectedquasi-TEMshapefunction.Below30 GHzthereis no leaky mode, only apair of improper real modes(modes that areimproper, but withareal propagationwavenumber). The leakymode becomes physical aboveabout 35 GHz, where it crosses the TM0 dispersion line tobecome a fast wave with respect to the TM0 surface wave.Figure9showsthetotal current onthestripversusnormalized distance from the source, for various frequen-cies. Inthis gure, and inall subsequent results, thelengthofthegapsourceistakenasD=0.1l0. ItisseeninFig.9thatthetotalcurrentisincreasinglyoscillatoryas the frequency increases. This is due to the interferencebetweentheBMandCScurrents.Atlowfrequency,onlytheBMcurrentis significant,andthe currentlevel ises-sentiallyat withdistancefromthesource. Asthefre-quencyincreases, theamplitudeof theCScurrent alsoincreases,resultinginastrongerinterferenceeffect.Figure10showsthedispersionplotsforthecaseofamoderate strip width, w/h=1. For this case the low-frequencycharacteristicimpedanceis95.2 O. ComparingwithFig.8,itisseenthatleakagebeginsatalowerfre-quencyforthiswiderstrip,althoughtheloweringinfre-quency is not dramatic. The leakage starts at about27 GHz,withtheleakymodebecomingphysicalatabout33 GHz, wherethephaseconstant crossestheTM0dis-persioncurve. Theleakymodeis not physical betweenapproximately 47 and 70 GHz, where the dispersion curveNormalized phase constant1.41.351.31.251.21.151.11.05180 70 60 50 40 30 20 10 0Freq (GHz)RIMLMBMTM0TE1Figure 8. Normalized phase constants (normalized by k0) versusfrequency, foramicrostriplinewithamoderatesubstrateper-mittivity(er=2.2) andasmall stripwidth(w=0.333mm, h=1.0 mm). Theboundmode(BM) isshownalongwiththeleakymode(LM) andtheTM0andTE1surface-wavemodes. At lowfrequencythereisnoleakymode, onlyapairof improperrealmodes (RIM). (This gure is available infull color at http://www.mrw.interscience.wiley.com/erfme.)LEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2277isbelowtheunityline, sothatbok0. Inthisfrequencyrange a physical leaky mode would be one that leaks intoboth the TM0 surface-wave mode of the substrate and intospace. However,nosuchphysicalleakymodeisfound.Figure11shows thecurrent onthestripat variousfrequencies, for the case of moderate strip width. Compar-ingwithFig.9,itisseenthatthelevelofspuriousoscil-lationsisroughlythesame,althoughtheoveralllevelofthecurrentishigherforthewiderstripcase, becauseawider strip corresponds to a smaller characteristic imped-ance,andhencetheboundmodeisexcitedwithalargeramplitude. However, the CS current is also excited with alarger amplitude, giving roughly the same level of oscilla-tionsinthetotalcurrent.Figure 12 gives the dispersion behavior for a wide-stripcase, where w/h=3. For this case the low-frequency char-acteristicimpedanceis51.1 O. It isseenthat thetrendof a lower leakage frequency continues, with leakage nowbeginning at about 23 GHz, and the leaky mode becomingphysical atabout27 GHz. Theleakymodeisonceagainnonphysical inaparticularhigh-frequencyregion, fromabout33to64 GHz.The current plots in Fig. 13 for the wide-strip case showacontinuingincreaseintheoverallcurrentlevel, asex-pected.Aswiththeothercases,thelevelofthespuriousoscillation increases with frequency. However, for thiswider strip case an interesting new effect is also observed.Atahighfrequencyof70 GHz,theoscillationsdecayex-tremelyslowly with distancefrom the source.In fact, outto eight wavelengths from the source, there is no noticeabledecay in the oscillation amplitude of Fig. 13. This is becausetheCScurrent decaysveryslowlywithdistanceinthiscase. Theeffect hasbeenexaminedindetail inRef. 22,where it is shown that the effect is due to a leaky-mode poleNormalized phase constant1.451.41.351.31.251.21.151.11.0510.9580 70 60 50 40 30 20 10 0Freq (GHz)LMRIMTMOBMFigure10. Normalizedphaseconstants(normalizedbyk0)ver-susfrequency, foramicrostriplinewithamoderatesubstratepermittivity(er=2.2) andamoderatestripwidth(w=1.0 mm,h=1.0mm). The bound mode (BM) is shown along with the leakymode(LM) andtheTM0surface-wavemode. At lowfrequencythere is no leaky mode, only a pair of improper real modes(RIM). (Thisgureisavailableinfull colorathttp://www.mrw.interscience.wiley.com/erfme.)Amplitude (mA)5.85.65.45.254.84.64.44.243.83.68 7 6 5 4z/z03 2 1 01GHz5GHz10GHz20GHz40GHzFigure 11. The total strip current versus distance fromthesource,for thecaseofFig. 10.Results areshown forvarious fre-quencies. (This gure is available in full color at http://www.mrw.interscience.wiley.com/erfme.)Normalized phase constant1.51.41.31.21.110.980 70 60 50 40Freq (GHz)30 20 10 0RIMBMLMTM0TE1Figure12. Normalizedphaseconstants(normalizedbyk0)ver-susfrequency, foramicrostriplinewithamoderatesubstratepermittivity(er=2.2) andalargestripwidth(w=3.0mm, h=1.0 mm). Theboundmode(BM) isshownalongwiththeleakymode(LM) andtheTM0andTE1surface-wavemodes. At lowfrequencythereisnoleakymode, onlyapairof improperrealmodes (RIM). (This gure is available infull color at http://www.mrw.interscience.wiley.com/erfme.)Amplitude (mA)3.43.232.82.62.40 1 2 3 4 5 6 7 81GHz5GHz10GHz20GHz40GHzz/z0Figure 9. The total strip current versus distance from the source,forthe case of Fig. 8. Results are shown for various frequencies.2278 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITSapproaching the branchpoint at k0 in the complex kz plane.Thefrequencyatwhichthiseffectoccursisgivenapprox-imatelyforwidestripsbythefollowingempiricalformula[22]:k0er_w=2p (39)In Fig. 14, the current is investigated in more detail for thecaseof moderatestripwidth(w/h=1), atafrequencyof20GHz. Part (a) of this gure showsthe totalcurrentandits constituent parts, theBMandCScurrents; part (b)showstheCScurrentresolvedintoitsconstituentcompo-nents, the k0 RWcurrent and the TM0RWcurrent. There isnophysicalleakymodeatthisfrequency(seeFig.10),sotheCS currentis simplythesumof thetwoRWcurrents.TheTM0RWcurrentisthestrongerof thetworesidualwaves, although the amplitudes are somewhat comparable.At40GHzthesituationchangesinthataphysical leakymode now exists (see Fig. 10). At this higher frequency theoscillationsin the totalstripcurrentare larger,as seen inFig. 15a, because the magnitude of the CS current is largerat thishigher frequency. InFig. 15b, theCScurrent isshown with its three components, the leaky-mode current,thek0RWcurrent,andtheTM0RWcurrent.Closeto thesource, the LM current is the strongest. However, becausethe LMcurrent decays exponentially with distance, the twoRWcurrentsbecomedominant forlargerdistances. TheTM0 RW current is the more dominant of the two, as for the20-GHz case. The presence of the physical leaky modecausesthelevelofspuriousoscillationtobequitesigni-cantclosetothesourceinthe40-GHzcase, muchlargerthan in the 20-GHz case. Farther away from the source thelevel of spurious oscillation is still slightly larger in the 40-GHz case, but the increase is not as dramatic.Figure16showsthedispersionbehaviorforamicro-stripwithahigh-permittivitysubstrate(er=10.2)andamoderatestripwidth(w/h=1).(Theleaky-modesolutionis plotted up to 37.5 GHz, which is already well within thenonphysical region.) ComparingwithFig. 10, it isseenthat the higher permittivity has lowered the frequency atwhichleakagebegins.Atabout24 GHzasecondsurface-wavemode, theTE1mode, begintopropagate, andthedispersioncurve for this mode is includedinthe plot.Above this frequency, there will be a third propagating RWcurrent, corresponding to the TE1 branchpoint on the realaxisinthecomplexkzplane.Figure17showsthecurrentversusdistancefromthesourceforthishigh-permittivitycase.Thelevelofspuri-ous oscillations inthestripcurrent increases withfre-quency, asforthemoderate-permittivitycaseof Fig. 11.Figure 18a shows the total strip current at 20 GHz for thecase of Fig. 16 along with its components, the BM and CScurrents. In Fig. 18b it is seen that a physical leaky modedominates the CS current near the source, although the k0RWandTM0RWcurrentsarenotnegligible, evennearthe source. Because of the exponential decay of this leakymode, theoscillationsinthetotal stripcurrent dieoutfairlyquicklywith distance.Amplitude (mA)1211109876548 7 6 5 4z/z03 2 1 01GHz10GHz20GHz40GHz60GHz 70GHzFigure 13. The total strip current versus distance fromthesource, forthecase ofFig. 12.Results areshown forvarious fre-quencies. (This gure is available in full color at http://www.mrw.interscience.wiley.com/erfme.)Amplitude (mA)Amplitude (mA)2345100.18 7 6 5 4z/z03 2 1 010.018 7 6 5 4z/z03 2 1 0TCBMCSCSK0 RWTM0 RW(a)(b)Figure14. Stripcurrentsforthecaseofamoderatesubstratepermittivity(er=2.2)andmoderatestripwidth(w=h=1.0 mm)at20GHz: (a) total current and its constituent parts,the bound-mode and continuous-spectrumcurrents; (b) continuous-spec-trumcurrentanditsconstituent parts, theTM0residual-waveandk0 residual-wave currents.LEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2279Amplitude (mA)10987654328 7 6 5 4z/z03 2 1 01GHz5GHz10GHz20GHz40GHzFigure 17. The total strip current versus distance fromthesource,for thecaseofFig. 16.Results areshown forvarious fre-quencies.Amplitude (mA)9876543Amplitude (mA)1E51E61E41E31E21E11E+01E+12108 7 6 5z/z0(a)(b)4 3 2 1 08 7 6 5 4z/z03 2 1 0TCBMCSCSTM0 RWK0 RWLMFigure 18. Strip currents for the high-permittivity case ofFig. 16 at20GHz: (a) the total current and its constituent parts,the bound-mode and continuous-spectrum currents; (b) thecontinuous-spectrumcurrentanditsconstituentpartsaleakymodecurrenttogetherwithTM0residual-waveandk0residual-wave currents.Normalized phase constants3.532.521.510.580 70 60 50 40Freq (GHz)30 20 10 0BMTM0TE1TM1LMRIMFigure16. Normalizedphaseconstants(normalizedbyk0)ver-sus frequency, for a microstrip line with a high substrate permit-tivity (er=10.2) and a moderate strip width (w=h=1.0mm). Theboundmode(BM)isshownalongwiththeleakymode(LM)andtheTM0andTE1surface-wavemodes.Atlowfrequencythereisnoleakymode, onlyapairofimproperrealmodes(RIM). (Thisgureisavailableinfull color at http://www.mrw.interscience.wiley.com/erfme.)Amplitude (mA)Amplitude (mA)23456100.010.118 7 6 5 4z/z03 2 1 0100.0018 7 6 5 4z/z03 2 1 0TCBMCSCSTM0 RWK0 RWLM RW(a)(b)Figure15. Thestripcurrentsforthecaseof amoderatesub-stratepermittivity(er=2.2) andmoderatestripwidth(w=h=1.0 mm) at 40GHz: (a) total current and its constituent parts, thebound-modeandcontinuous-spectrumcurrents; (b) continuous-spectrum current and its constituent partsthe leaky-mode cur-rent,the TM0 residual-wave, andk0 residual-wave currents.2280 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITSAt 40 GHz, the oscillations become stronger, as seen inFig. 19. Asbefore, theCScurrentisresponsiblefortheoscillations, asseeninFig. 19a. However, thestrongCScurrent is not due to the excitation of a leaky mode, as wasthe case for the moderate-permittivity substrate at40 GHz (Fig. 15). For the higher-permittivity substrateat 40 GHz, the leaky mode shown in Fig. 16 is not physical,sinceitsdispersioncurveislowerthanthat of theTE1surface-wave dispersion curve (while this leaky mode hasleakageonlyintotheTM0surface-wavemode). Further-more, no physical leaky mode that leaks into both the TM0andTE1surface-wavemodesisfoundatthisfrequency.Figure 19b shows that the strong CS current is due to theTE1RWcurrent. TheTE1RWcurrentisverystrongbe-cause the dispersioncurve for the leaky mode is onlyslightlybelowthatfortheTE1surface-wavemode. Theleaky mode is therefore only slightly into the nonphysicalregion.Mathematically,thismeansthateventhoughtheLMpoleisnotcapturedbythepathdeformationtothesteepest-descent paths fromthe branchpoints, the LMpoleisclosetotheTE1branchpoint,andthusmakesitsinuencefeltintheTE1RW current.3.2. CoveredMicrostripInpractice,microstriplinesareoftenusedinmicrowavecircuitsthatareplacedintopackagesofsometype. Itisthereforeimportanttodeterminetheinuenceofthetopand sidewalls of a package on the possible leakage behav-ior of various transmission lines. The effects of a top cover,as shown in Fig. 1b, is investigated here. The addition of atopcoverstill allowsthesamemethodof analysistobeusedas fortheopen microstripstructure, as discussedinSection2. Becauseof thecover, leakagemayonlyoccurintotheTM0parallel-platesubstratemode,andnotintofree space (it is assumed here that only the TM0 substratemode is above cutoff). Furthermore, there is only one typeofresidualcurrent,the TM0RW current.Oneofthemostimportantconclusions, demonstratedfrom the results below, is that the presence of the top coverlowersthefrequencyatwhichleakagebegins.Thisisbe-cause the top cover significantly raises the dispersion plotoftheTM0mode,whileinuencingthedispersionplotoftheguidedmodeto alesserdegree. Hence, physicalleak-agebeginsata lowerfrequencywiththetopcover.Furthermore, it is demonstrated that the top cover alsoincreases the overall amplitude of the CS current. Hence,spurious effects significantly increase due to the presenceofthetopcover.Figure20presents adispersionplot for amoderatepermittivity substrate (er=2.2) andastripwidth: sub-strate-height ratioof w/h=1, for uncoveredmicrostrip.ThisresultisthesameasthatshowninFig. 10, exceptthat the result is plotted in a normalized form, versus theelectrical thicknessof thesubstrateh/l0. (AlthoughtheresultinFig. 10isplottedagainstfrequencyusingxeddimensionsinmillimetersinsteadofplottingversusnor-malized electrical substrate thickness, the result is just asgeneral, since it can be directly converted into a result fornormalized substrate thickness, following the style ofRef. 37.) For the uncovered microstrip, the splittingpoint(the frequency at which leakage rst begins as a nonphys-ical leaky mode) is at about h/l0=0.09, and the frequencyat whichtheleakymodeenters thephysical fast-waveregion is at about h/l0=0.11. Clearly, physical leakage onAmplitude (mA)321076548 7 6 5 4z/z0(a)(b)3 2 1 0Amplitude (mA)0.0010.010.1110TCBMCSCSK0 RWTM0 RWTE1 RW8 7 6 5 4z/z03 2 1 0Figure19. Stripcurrentsforthehigh-permittivitycaseofFig.16at40GHz:(a)thetotalcurrentanditsconstituentparts,thebound-modeandcontinuous-spectrumcurrents;(b)thecontinu-ous-spectrumcurrent anditsconstituent partsaleakymodecurrent together with TM0 residual-wave, TE1 residual-wave, andk0residual-wave currents.Normalized phase constants11.051.11.151.21.251.31.351.40.12 0.1 0.08 0.06h/z00.04 0.02 0TM0 modeBound modeLeaky modeImproper real modesFigure20. Dispersionplotshowingthenormalizedphasecon-stant versusnormalizedfrequencyfor anopenmicrostripline(w/h=1.0, er=2.2).LEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2281typical microstrip lines occurs only for high frequencies orthickersubstrates.Figures21a21cshowthedispersioncurvesforcoverheightsof hc/h=2.0, 1.0, and0.5, respectively, showingthechanges inthedispersionplots as thetopcover isbroughtsuccessivelynearertothe strip.Foracoverheight hc/h=1.0, thefrequencyatwhichleakage begins is approximately h/l0=0.02. Whenthecoverheightisreducedstill furthertohc/h=0.05, leak-agenowoccursat all frequencies. Theseresultsclearlydemonstratethatforsmallercoverheightsthepotentialexists for spurious effects such as crosstalk and power lossduetotheleakymode.Aninvestigationintothecurrentexcitedonthestripbyagapvoltagesource, presentednext,willconrm this.Figure 22 shows a plot of the current for the uncoveredcaseofFig. 20, atamoderatefrequency(h/l0=0.04). AtthisfrequencywemaynotefromFig.20thatthereisnoleaky mode, so that the continuous spectrum consists onlyoftheresidualwave.FromFig.22itisseenthattheCScurrentisverysmall, andthatthetotal stripcurrentisessentiallythesame asthatofonlytheboundmode. Theresidual waveisthereforeverysmall. At thismoderatefrequencythecontinuousspectrumissmallenoughthatno serious spurious effects are observed. Of course, signif-icantspuriouseffectswillbeobservedathigherfrequen-cies, as demonstrated in Section 3.1, but at this relativelylowfrequencythespuriouseffects arequitesmall.Figure 23shows aplot of the current for the samemoderate-permittivitycaseasinFig. 22, butforacoverheight of hc/h=1.0. Thenormalizedfrequencyish/l0=0.02(halfofthevalueusedinFig.22).Eventhoughthefrequencyis half that usedinthe uncoveredcase, thespuriousoscillationsareclearlymuchmoresevere.FromFig.21bitisseenthattheleakymodeisnotphysicalatthisfrequency, sotheCScurrentconsistsentirelyoftheRW current. Hence, one conclusion is that the presence ofatopcoverincreasestheamplitudeof theRW current.Normalized phase constants11.051.11.151.21.251.31.351.40.12 0.1 0.08 0.06h/z0(a)(b)(c)0.04 0.02 0Normalized phase constants1.11.151.21.251.31.41.350.08 0.06h/z00.04 0.02 0Normalized phase constants1.151.21.251.31.350.08 0.06h/z00.04 0.02 0TM0 modeBound modeLeaky modeImproper real modesFigure21. Dispersionplotshowingthenormalizedphasecon-stantversusnormalizedfrequencyforacoveredmicrostripline(w/h=1.0, er=2.2). The cover heights are (a) hc/h=2.0, (b) hc/h=1.0,and (c)hc/h=0.5.Amplitude (mA)65432101 0 2 3 4 5z/z06 7 8Total currentBound modeContinuous spectrumFigure22. Plotsofthetotalcurrent, BMcurrent, andCScur-rent onan uncovered microstrip line versus normalized distancefromthe delta gap voltage source at h/l0=0.04 (w/h=1.0,er=2.2).2282 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITSFigure24ashowsthecurrentforthecaseofasmallercover height, hc/h=0.5, at the same normalized frequencyof h/l0=0.02. All other parameters remain the same as inFig. 23. (See Fig. 21c for a dispersion plot.) This gure re-vealstwoimportantaspectsabouttheeffectsofreducingthecoverheight:1. Thelevel of oscillationinthetotal currenthasin-creasedsignificantly, comparedtothat inFig. 23,whichcorresponds tothesamefrequency. This ispartlybecause loweringthe cover height has de-creasedtheamplitudeoftheboundmode,butevenmore importantly, it has dramatically increased theamplitude of the CScurrent. Part (b) of Fig. 24showsthattheleakymodeisnowthemaincompo-nentof theCScurrent.2. Comparing Figs. 24b and 23, it is also seen that theRWcurrent is somewhat stronger for the lower coverheight. (In Fig. 23 there is no physical leaky mode atthis frequency, so the RW current is the same as theCScurrent.)However,themain reasonwhy theCScurrentis significantlylarger in Fig.24 is the pres-ence of the very strong leaky mode. Hence, loweringthe top cover both increases the amplitude of the RWcurrent and also results in the excitation of a strongLMcurrent.3.3. StriplinewithanAirgapThe stripline structure is one of the most well known andwidelyusedprinted-circuit transmission-linestructures.Ahomogeneous stripline structure will support only aTEMmodeof propagation, andnoleakymodes. Thisisbecausethesubstratebetweenthetwogroundplanesishomogeneous. However, duringmanufacture, anairgapmaybeinadvertentlyintroduced,resultingintheairgapstripline structure shown in Fig. 1c. It has been known forsometimethatairgapswithinstriplinepackagescande-gradetheperformanceofthestripline,althoughthefun-damental reasons were perhaps not clear. It is shown herethat the airgap lowers the phase constant of the dominantquasi-TEM stripline mode below that of the TM0 parallel-plate mode of the (now inhomogeneous) background struc-ture. Thiscausesthe striplinemodeto become aphysicalleakymode.Thismeansthatthestriplinemodeusedforsignal transmissionwill suffer frompower loss due toleakage, as well as crosstalk and other effects due to leak-ageintotheTM0mode.When a small airgap is present, it is shown that there isa bound mode present, as well as the leaky mode. Howev-er, the bound mode is one that has the eld characteristicsof aparallel-plate type of mode. Infact, as theairgapthicknesstendstozero, theboundmodeevolvesintoaTEMparallel-platemode. Aconventional striplinefeedAmplitude (mA)675432101 0 2 3 4 5z/z06 7 8Total currentBound modeContinuous spectrumFigure23. Plotsofthetotalcurrent, BMcurrent, andCScur-rent on a covered microstrip line versus normalized distance fromthedeltagapvoltagesourceath/l0=0.02(hc/h=1.0, w/h=1.0,er=2.2).Amplitude (mA)7865432101 0 2 3 4 5z/z0(a)(b)6 7 8Total currentBound modeContinuous spectrumAmplitude (mA)65432101 0 2 3 4 5z/z06 7 8Continuous spectrumLeaky modeResidual waveFigure 24. (a) Plot of the total current, BM current, and CS cur-rentonacoveredmicrostriplineversusdistancefromthedeltagapvoltagesource, forer=2.2, w/h=1.0, andacoverheightofhc/h=0.5,withnormalizedfrequency h/l0=0.02; (b) theCScur-rentisshownalongwithitsconstituentparts: theLMcurrentand the TM0 RW current.LEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2283wouldpredominantlyexcitetheleakymode,sincethisisthe mode that resembles theTEMstripline mode thatwould exist with no airgap. However, a practical feed willalso excite the bound-mode eld to some extent, more so asthe airgap thickness increases, since the elds of the leakyandboundmodestendtobecomelessdistinctastheair-gapthicknessincreases.Thefactthatbothaleakymodeandaboundmodeareexcitedmeansthatspuriousoscil-lationswilloccurinthestripcurrent,as was observedinthe results presented for the previous two structures, mi-crostrip and covered microstrip. This interference ex-plains the experimental observation in Ref. 11, whichwassubsequentlyinvestigatedmorethoroughlyin[20],whereaverysharpandpronouncedspuriousdipinthetransmission (S21) response was observed at a specific fre-quency,usingtheairgapstripline.Thespuriousdipthatwasobservedwasattributedtointerferencebetweenthebound mode and the leaky mode, and this is proved by anexamination of the currents excited by a gap voltagesource.The normalizedphase andattenuationconstants at30 GHz are plotted with respect to the airgap thickness inFigs. 25a and 25b, respectively. It is seen from part (a) thatthe LM will be a physical mode (a fast-wave solution withrespect to the TM0parallel-plate mode) for anairgapthicknesslessthan0.5 mm. Thephysical validityof theleaky mode is gradually lost as the airgap thicknessincreasesbeyondthisvalue.For an airgap thickness of d =0.3mm, plots of the totalcurrentaswellastheBMandCScurrentsareshowninFig. 26a. Figure26bshowstheCScurrent anditstwoconstituent parts, the LM current and the RWcurrent. Forthis airgap thickness, Fig. 26a shows that the CS currentis much more strongly excited than the BM current. More-over, thecurvesinFig. 26bshowthattheCScurrentisalmostentirelyaccountedforbytheLMcurrent. Thisisbecausetheleakymodeisphysicalforthisairgapthick-ness, andbecausetheeldof theleakymoderesemblesthat of the usual TEM stripline mode, and hence is excitedquitestronglybythegapvoltagesource. ThedipintheNormalized phase constants1.441.41.361.321.28Normalized atten. constant1.480.7o (mm)0.6 0.5 0.4 0.3(a)0.2 0.1 00.7o (mm)0.6 0.5 0.4 0.3(b)0.2 0.1 000.0020.0040.0060.0080.01TM0BoundLeakyFigure25. (a) The normalizedphase constants of the boundmode, leaky mode, andTM0mode of the parallel-plate back-ground structure, versus airgap thickness at 3.0 GHz; (b) the nor-malized attenuation constant of the leaky mode (h=1.0 mm,w=1.0 mm, er=2.2).Amplitude (mA)765432100 5 10 15z/z0(a)(b)20 25 30Total currentContinuous spectrumBound modeAmplitude (mA)65432100 5 10 15z/z020 25 30Continuous spectrumLeaky modeResidual waveFigure26. Plotsofthetotalcurrent, bound-modecurrent, andcontinuous-spectrumcurrentversusdistancefromthedeltagapsourceforthestructureof Fig. 25withanairgapthicknessof0.3 mm; (b) plots of thecontinuous-spectrum, leaky-mode, andresidual-wave currents.2284 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITSamplitudeof thetotal current at about 25wavelengthsfromthesourceisduetothedestructiveinterferencebe-tweentheBMandCScurrents, whichhaveanapproxi-mate1801phasedifferenceatthisdistance. BecausetheLM current dominates the CS current, it is concluded thatthe dip in the total current magnitude is due to a destruc-tive interference between the bound and leaky modes thatareexcitedbythe source.At the point of destructive interference, the amplitudesoftheleakyandboundmodesarenotnecessarilyequal,and this explains why the dip in the total current does notextend complexly to zero in Fig. 26a at z =25l0. However,bycorrect selectionof theairgapthickness, theampli-tudes of the two modes can be made equal, resulting in anessentiallycompletedestructiveinterference. Asmallerairgapthickness will causetheamplitude of theleakymode to dominate the amplitude of the bound mode at thepoint of destructive interference, while alarger airgapthicknesswillcausetheamplitudeoftheboundmodetodominate. Acritical airgap thickness thus exists thatequalizes thetwoamplitudes at theinterferencepoint.ThisisdemonstratedinFig. 27forairgapthicknessofd =0.353 mm. For this case there is an almost perfect can-cellation of the total current on the strip at z =32l0, point-ing out how the simultaneous presence of the LM and theBMcurrentscansignificantlydistorttheexpectedtrans-missionbehaviorofthe line.4. CROSSTALKBETWEENTWOMICROSTRIPLINESThissectioninvestigatesthecurrentexcitedonapairofcoupled microstrip lines when one of the lines (the sourceline, i.e., line 1) is excited by a gap voltage source (see Fig.28) [38]. Thecrosstalkcurrent isdenedasthecurrentinduced on line 2 (the victim line, which is passive). At lowfrequency the current on the two lines is well predicted bystandard transmission-line theory, which gives the follow-ingexpressionsforthecurrenton eachstripITLT1(z) =141Ze0ejkezz1Zo0ejkozz_ _(40)ITLT2(z) =141Ze0ejkezz1Zo0ejkozz_ _(41)whereZe0andZo0are the frequency-dependent even-modeand odd-mode characteristic impedances and kez and koz arethecorrespondingwavenumbers.However,simpletrans-mission-linetheoryfailstopredicttheeffectsofthecon-tinuous spectrum, and ignores all radiation effects. It willbeseenthattheCScurrentcanbepredominantathighfrequency,andcanthusbeasourceofundesirablespuri-ouseffectsthat canruintheperformanceof microwavecircuitryinvolvingneighboringprinted-circuitlines.4.1. FormulationfortheCurrentsThesameconceptspresentedinSection2applytothepresentcase,withsomeextensiontoaccountforthefactthatcurrentsnowexistontwolines.Oneimportantfea-tureofthepresentstructureisthatthecurrentsdonotshowanysymmetryaboutthecenterof eitherline, andthusamultiple-basisfunctionexpansionisnecessaryinpractice to model these currents. Thus, the surface currentcanbeexpanded,similarlyto (24)and(25),asJ(l)z (x; z) =

Nfq =0Tzl;q(x)Izl;q(z) (42)J(l)x (x; z) =

Nfp=1Txl;p(x)Ixl;p(z) (43)wherethesubscript l =1, 2isnowaddedtodenotethestripconductor(theintegersqandparenolongerevenand odd, respectively, as they were for the case of a singlestrip).However, withtheaimof keepingtheexplanationofthe problem as simple as possible, only a single longitudi-nal basis function will be used on each line in the followingexplanation. This implies that the surface current on eachline is expanded as J(1)z(x; z) =T1(x)I1(z) andJ(2)z(x; z) =T2(x)I2(z) andthusthe EFIEcanbewrittenasE(1)z[T1(x)I1(z)] E(1)z[T2(x)I2(z)] =Egapz(z) (44)E(2)z[T1(x)I1(z)] E(2)z[T2(x)I2(z)] =0 (45)Amplitude (mA)765432100 5 10 15z/z020 25 35 30Total currentBound modeBound mode+Leaky modeContinuous spectrumFigure 27. The same type of plot as in Fig. 26a but for an airgapthicknessof0.353mm, whichresultsinanalmostaperfectde-structiveinterferencebetweenthebound-modeandcontinuous-spectrum currentsatzE32l0.1VI1(z)zw1w2sI2(z)Figure 28. Top view of a pair of coupled microstrip lines. Line 1is fed by a gap voltage source, causing a crosstalk current in line 2.LEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2285TheGalerkintestingprocedurecombinedwiththeappli-cation of the inverse longitudinal Fourier transform leadstothe followingsystemof equations~II1(kz)_w1T1(x)E(1)z[T1(x)ejkzz] dx ~II2(kz)_w1T1(x)E(1)z[T2(x)ejkzz] dx =~EEgapz(kz)ejkzz(46)~II1(kz)_w2T2(x)E(2)z[T1(x)ejkzz] dx ~II2(kz)_w2T2(x)E(2)z[T2(x)ejkzz] dx =0(47)whichcanberewrittenasZ11(kz) Z12(kz)Z21(kz) Z22(kz)_ _~II1(kz)~II2(kz)_ _=~EEgapz(kz)0_ _(48)where the Zij(kz) coefcients (i, j =1, 2) have the followingform:*SpaceDomain:Zij(kz) =_wi_wjTi(x)~GGzz(x x/; kz)Tj(x/) dx/dx (49)*SpectralDomain:Zij(kz) =12p_Ckx~TTi(kx)~GGzz(kx; kz) ~TTj(kx)ejkxs(1dij)dkx(50)The form given in (49) will be used when the DCIT is usedwiththecorrespondingMPIEinthespacedomain, andexpression (50) is used when the EFIE is solved using theSDAapproach.FollowingthesamerationaleasinSection2,thetotalcurrent on the lines is given by I1(z) =Iz1;0(z) andI2(z) =Iz2;0(z), which are given in terms of their correspond-inglongitudinalFouriertransformsasIl(z) =12p_Ckz~IIzl;0(kz)ejkzzdkz; l =1; 2 (51)4.2. ResultsThetheorypresentedabove(usingmultiplebasisfunc-tions)hasbeenimplementedinacomputercodetostudythecrosstalkcurrentinline2,I2(z),forapairofcoupledmicrostrip lines when line 1 is excited by a 1-V gap voltagesource.Figure29showsthecrosstalkcurrentfortwomicro-striplinesatvariousfrequencies(low,medium,andhighfrequencies) and a wide separation between the lines(s/h=10). The crosstalk current is plotted versus distancez from the origin (the source is at z =0 on line 1). Both thetotal current (TC) numerically obtained from (51) and thetransmission-linetheory(TLT)currentgivenby(41)areshown. The curves showthat, at the lowfrequencyof1 GHz (h/l0E0.003), transmission-line theory predictsvery accurately the crosstalk current. At 20 GHz (h/l0E0.065) there is overall agreement, but the exact cross-talk current shows a noticeable oscillation, which is due tothe presence of the CS current. At 40GHz (h/l0E0.13) theCScurrent has becomestrongenoughthat thereis noagreement at all between the exact and the TLT currents.It is interesting to note that at low frequency, for a largelineseparation, thecrosstalkcurrentexhibitsanalmostperfectly linear growth with distance z. This is explainablefromEq.(41),which mayberewrittenasITLT2(z) =141Ze01Zo0_ _ejkezzj2Zo0ejkazzsin(Dkzz=2)(52)wherekaz=(kezkoz)=2is theaverageof thetwobound-modewavenumbersandDkz=kez kozistheirdifference.Asthelineseparationincreases,theeven-andodd-modecharacteristic impedances and wavenumbers approacheachother, makingtherst terminEq. (52) negligibleandthesecondterm approximatelylinear.To explore the nature of the spurious behavior of I2(z) athigh frequency (40 GHz), Fig. 30 shows the decompositionof thiscurrentintermsof itsconstituentparts, theBMand the CS currents. The TLT current predicted byEq.(52)isalsoshownforcomparison.ItisseenthattheCS current is significantly stronger than the BM current,conrming that the CS current is responsible for the spu-riouseffects. It isalsointerestingtonotethat theTLTcurrentpredictsfairlyaccuratelytheactual BMcurrentAmplitude (mA)2.521.510.5020 18 16 14 12 10z/z08 6 4 2 0 1 GHz20 GHz40 GHzTLT:1GHzTLT:20GHzTLT:40GHZFigure29. Crosstalk currentI2(z)versusz/l0for er=2.2,s/h=10.0,w1=w2=h=1.0mm. Both the totalcurrentand thetrans-mission-linetheory (TLT) currentsare plotted forthree differentfrequencies.2286 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITSontheline.Thus,transmission-linetheorypredictsaccu-rately the amplitude of the BM currents on the line, whichin the exact theory comes from the residues at the two BMpolesinthecomplexkzplane. Transmission-linetheorydoesnotaccountfortheCScurrent,however,sotheTLTcurrentis far fromthe actualcurrentontheline.It is also interesting to note from Fig. 29 that for low tomoderatefrequencies, thecrosstalkcurrentdecreasesasthefrequencyincreases. This is becausetheeven- andodd-mode wavenumbers become more nearly equal as thefrequency increases. As seen from Eq. (52), this results inthe crosstalkcurrent I2(z) becoming smaller. However,when the frequency becomes very high, the crosstalk cur-rentincreaseswithfrequency, duetotheincreaseoftheCScurrent. ThisisillustratedbyFig. 31, whichshowsthatat80 GHzthecrosstalkcurrenthassignificantlyin-creasedfrom40 GHz.For large line separations, the CScurrent stronglydominates the crosstalk current at high frequencies.This is illustrated by Fig. 32, which shows that at80 GHz, the total crosstalk current and the CS componentofthecrosstalkcurrentareessentiallyindistinguishable.Physically, thisisbecauseforlargelineseparationandhighfrequency,the crosstalk currentis due mainly to ra-diationfromthesourcelineintotheTM0parallel-platemode,asopposedtoquasistaticcoupling.Radiationfromthe source line mainly induces a CS current on the victimline, instead of a bound mode, since the bound modes havewavenumbers that are larger than that of the TM0 parallel-platemode,andhencecannotcoupletotheTM0parallel-plate radiation.Thebehaviorof thehigh-frequencycrosstalkcurrentfor various lineseparations is showninFig. 33. For asmall line separation (s/h=2), the crosstalk current is wellaccountedfor by TLTmodel, but this good agreementworsens as the line separation increases. This fact can beadduced to the predominant quasistatic nature of thecrosstalk coupling for small separation between the lines,which is accounted for by the TLT model. As the frequencyAmplitude (mA)0.350.30.250.20.150.10.05020 18 16 14 12 10z/z08 6 4 2 0Freq=40GHzTCBMCSTLTFigure 30. Total crosstalk current I2 (z) and its constituent partsversus z/l0 for er=2.2, s/h=10.0, w1=w2=h=1.0mm at 40GHz.Amplitude (mA) 0.70.50.40.30.20.1010.90.80.620 18 16 14 12 10z/z08 6 4 2 040 GHz80 GHzFigure 31. Comparison of the total crosstalk current I2 (z) versusz/l0 for er=2.2, s/h=10.0, w1=w2=h=1.0mm at 40 and 80GHz.Amplitude (mA) 0.70.50.40.30.20.1010.90.80.620 18 16 14 12 10z/z08 6 4 2 0TCBMCSFigure 32. Total crosstalk current I2 (z) and its constituent partsversus z/l0 for er=2.2, s/h=10.0, w1=w2=h=1.0mm at 80GHz.The TC and CS curves are indistinguishable, while the BM curveis essentially zero.Crosstalk current amplitude (mA)3.52.521.510.5054.54320 18 16 14 12 10z/z08 6 4 2 0Freq=40GHzs/h=2.0s/h=5.0s/h=10.0TLT: s/h=2.0TLT: s/h=5.0TLT: s/h=10.0Figure33. Total crosstalkandtransmission-linetheory(TLT)currentsversusz/l0forvariouslineseparations: s/h=2, 5, 10;er=2.2,w1=w2=h=1.0 mm, frequency=40GHz.LEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2287increases, the TLT model loses accuracy as the CS currentbecomes stronger. It is clear by comparing Figs. 29 and 33that as the line separationincreases, the frequencyatwhich the CS current becomes appreciable decreases.Hence,simpletransmission-linetheorylosesaccuracyatlowerfrequencieswhentheline separationislarger.5. CROSSTALKFIELDSInthissection,theeldsradiatedbythecurrentsonthestripconductorofaprinted-circuittransmissionlineareexamined, toseehowcrosstalkeldsmaybeproduced,especially at high frequency, due to the radiation from thecurrents that are induced on the strip conductor excited byagapvoltagesource[39,40].Thestructurethatisexaminedisthecoveredmicro-stripstructureofFig. 1b.Thecurrentsonthisstructurethatareproducedbyagapvoltagesourcehavealreadybeen examined in some detail in Section 3.2, where it wasshownthat the total current consists of abound-mode(BM) transmission-line modetogether with a continuous-spectrum (CS) current. The CS current consists of a phys-ical leaky-mode(LM)current, whensuchamodeexists,togetherwithaTM0residual-wave(RW)current.Hence,thereareamaximumofthreedifferenttypesofcurrentsontheline,theBM,LM,andRWcurrents.The BMcurrent is normally regarded as a nonradiatingtypeofcurrent,sincetheeldsareboundtotheguidingstructure.Indeed,forsuchamodepropagatingonanin-niteline,therewouldbenoradiationandno powerloss(assuminglossless conductors anddielectric). However,the1-Vgapvoltagesourceatz =0isadiscontinuity,andradiationingeneral occursfromdiscontinuities. Inpar-ticular,thecurrentinEq.(2)hasaslopediscontinuity(adiscontinuityinthederivative) at z =0. Becauseof thisdiscontinuity, the BMcurrent will produce aradiationeld, with radiation occurring into the TM0 parallel-platesubstrate mode. As shown in the results below, the level ofthisradiationincreaseswith frequency.TheRWcurrentincreaseswithfrequency, asdemon-strated in Section 3.2. This current is a rather complicatedfunctionof distancezfromthesource, althoughasymp-totically it propagates witha wavenumber of kTM0, asshowninEq.(4).Forthisreason,theradiationfromthiscurrentissomewhatendredirected, concentratedmoretoward thedirectionof thestripaxis,aswillbeshowninthe results below. However, the radiation is somewhat dif-fuse,sincethecurrentisactuallycomposedofacontinu-ousspectrumof wavenumbers.When a physical LM current exists on the line, the ra-diation from this current is typically directive, in the formof a beam, since the LM current has a single wavenumberbthatislessthankTM0.TheLMcurrentmaythusbere-sponsible for significant crosstalk elds in the direction ofthisbeam. Thiswill bedemonstratedintheresultspre-sentedbelow.5.1. CalculationofCrosstalkFieldsTo calculate the crosstalk eld, it is rst assumed that onlyradiationintotheTM0parallel-platemodeissignificant.Alloftheotherhigher-orderparallel-platemodesareas-sumedtobesufcientlyfarbelowcutoffthattheseeldsdecay very rapidly away from the strip current, and hencearenegligibleatanysignificantdistanceawayfromthestrip(inpractice, more thanahalf-wavelengthor so).Hence, the crosstalk eld is assumed to be the eld of theTM0parallel-platemodethatisproducedbythecurrenton the strip conductor. To calculate this eld, the eld of aunitamplitudeinnitesimalelectricdipole inthez direc-tionontopofthesubstrateatx =0, z =0, y =hisconsi-dered. TheTM0eldfromthedipolehasthefollowingformincylindrical(r, f,y)coordinates:Ey(r; f; y) =Af (y)H(2)1(kTM0r) cos f (53)where, for0oyoh,weobtainf (y) =cos(ky1y) (54)withky1=(k21k2TM0)1=2(55)wherek1isthewavenumberinthesubstrate. Thecon-stant A can be determined by spectral-domain techniques,butthevalueisnotimportanthere.The eld from the strip current is found by integratingoverthestripcurrent, usingtheabovementioneddipoleeld as a Greens function. Assuming the line width to besmall relativetoawavelength, theintegral hasthefol-lowingformEy(x; y; z) =Af (y)_ooI(z/)H(2)1(kTM0r/) cos f/dz/(56)wherer/=x2(z z/)2_(57)tan(f/) =xz z/(58)Integratingacrossthesubstrateheight,thevoltagedropbetweenthestripand thegroundplaneatanypointisVh(x; z) = Ah sinc(ky1h)_ooI(z/)H(2)1(kTM0r/) cos f/dz/(59)The integral can be accelerated by using a technique suchastheShankstransform[41].5.2. ResultsAll thecasespresentedinthissectionareforacoveredmicrostripwither=2.2andw=h, withacover heighthc=h. Thedispersionplotforthisstructureisshownin2288 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITSFig. 34. This gure shows three distinct regions of behavior.In the rst region, h/l0o0.021, there is no leaky mode; onlya pair of nonphysical improper real modes exist, in additiontothe boundmode. Inthe region0.021oh/l0o0.052, aleakymodeexists, butitisnonphysical (sincethewave-numberoftheleakymodeislargerthanthatoftheTM0parallel-platemode). Intheregionh/l040.052theleakymodeisphysical.Resultswillbeshownforeachregiontoillustrate how the eld surrounding the strip behaves.Therstcaseisthatforh/l0=0.005.Forthislow-fre-quencycase, theoperatingpoint iswell insidethenon-physical spectral-gap region in the dispersion plot ofFig.34.Asaresultofthelowfrequency,thecontinuous-spectrumcurrentisverysmall,asshowninFig.35.Theoscillations inthetotal current duetotheinterferencebetweentheBMandCScurrents areapparent,althoughthey are small in amplitude, since the CS current is quitesmallcomparedwiththeBMcurrent.Figure36showsaplotof theeldsurroundingthestripforthiscase. Theeldisfairlydiffuseawayfromthestrip,withnoticeableoscillations close to the strip as the observation distance zincreases. This oscillationis duetoaninterferencebe-tween two components that make up the overall eld pro-duced by the BM current. One component is the eld dueto a bound mode propagating on an innite line. This eldpropagates in the z direction with the wavenumber of thebound mode, and decays exponentially away from the line.The secondcomponent is aradiationtype of eldthatpropagatesradiallywithawavenumberkTM0anddecaysas1= r_, andarisesbecauseoftheslopediscontinuityoftheBMcurrentatthe origin.Figure37showsthecurrentcomponentsatahigherfrequency of h/l0=0.04. At this frequency the leaky modeexists, but isnonphysical. Figure37showsthat at thishigherfrequencytheCScurrenthasbecomesignificant,because of the strong RW current that now exists. Figure 38showsthattheeldsurroundingthestriphasbeguntobecomemore directivein shape, althoughthe eld is stillsomewhatdiffuse.Figure 39 shows the current components at a still high-er frequencyof h/l0=0.08. At this frequencytheleakymodeexistsandisphysical. BecauseofthephysicalLMcurrent, theCScurrent isnowquitestrong. Figure40Normalized phase constant1.351.41.31.251.21.151.10.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0 0.01h/z0Bound modeTM0 modeLeaky modeImproper real modesFigure 34. Dispersion plot showing the normalized phaseconstants (b/k0)versusnormalizedfrequency (h/l0)foracoveredmicrostrip (hc=h,w=h, er=2.2)Current magnitude (mA)765432100 1 2 3z/z04 5 8 7 6TotalBound modeContinuous spectrumFigure 35. Strip current for a covered microstrip at a normalizedfrequency ofh/l0=0.005 (hc=h,w=h, er=2.2).x/z0z/z02.521.510.500 5 10 15 203.5344.550.033+0.03 to 0.033 0.027 to 0.03 0.024 to 0.0270.021 to 0.024 0.018 to 0.0210.015 to 0.018 0.012 to 0.015 0.009 to 0.012 0.006 to 0.009 0.003 to 0.006 0 to 0.003Figure36. Normalizedsubstratevoltageforthecoveredmicro-stripof Fig. 35. (Thisgureisavailableinfull colorat http://www.mrw.interscience.wiley.com/erfme.)LEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2289showsthattheeldsurroundingthestriphasbeguntoassumeabeamlikeshape, dueto theradiationof theLMcurrent. The overall level of the eld is also higher. Clear-ly, significant crosstalkcouldoccur withcircuit compo-nents that are placed within the beamlike region of strongleakageelds. Theangleof leakageof theleakymode,measuredfromthestripaxis,isgivenapproximately(forsmallattenuationconstants)asf0=cos1bkTM0_ _(60)Current magnitude (mA)765432100 1 2 3z/z04 5 8 7 6TotalBound modeContinuous spectrumFigure 37. Strip current for a covered microstrip at a normalizedfrequency ofh/l0=0.04 (hc=h,w=h, er=2.2).x/z0z/z02.521.510.500 5 10 15 203.5344.550.033+0.03 to 0.033 0.027 to 0.03 0.024 to 0.0270.021 to 0.024 0.018 to 0.0210.015 to 0.018 0.012 to 0.015 0.009 to 0.012 0.006 to 0.009 0.003 to 0.006 0 to 0.003Figure38. Normalizedsubstratevoltageforthecoveredmicro-stripof Fig. 37. (Thisgureisavailableinfull colorat http://www.mrw.interscience.wiley.com/erfme.)Current magnitude (mA)765432100 1 2 3z/z04 5 8 7 6TotalBound modeLeaky modeResidual waveFigure 39. Strip current for a covered microstrip at a normalizedfrequency ofh/l0=0.08 (hc=h,w=h, er=2.2).x/z0z/z02.521.510.500 5 10 15 203.5344.550.044+0.04 to 0.044 0.036 to 0.04 0.032 to 0.0360.028 to 0.032 0.024 to 0.0280.02 to 0.024 0.016 to 0.02 0.012 to 0.016 0.008 to 0.012 0.004 to 0.008 0 to 0.004Figure40. Normalizedsubstratevoltageforthecoveredmicro-stripof Fig. 39. (Thisgureisavailableinfull colorat http://www.mrw.interscience.wiley.com/erfme.)2290 LEAKY MODES ANDHIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATED CIRCUITSwherebis thephaseconstant of theleakymode. Thisequationactuallypredictstheleakageangleforaleakymodeexistingbyitself onaninniteline, withasmallattenuationconstant. Thepredictedleakageanglefromthis equation is B13.51, whereas the actual beam angle inFig. 40 is B181. The discrepancy is caused by several fac-tors: (1) the attenuation constant of the leaky mode is notzero, as is assumed in Eq. (60); and (2) the total eld con-sists of the BM and RW elds in addition to the LM eld.Finally, thedirectionofmaximumradiationfortheeldradiatedbytheLMcurrentofEq.(3)isnotthesameasthedirectionof leakageforaleakymodeonaninniteline,sincetheeldradiatedbytheLMcurrentofEq.(3)consists of a source discontinuity radiation term due to theslopediscontinuityofthe currentatthe origin.6. CONCLUSIONSThisarticlehasexaminedthenatureofthecurrentandelds excited by gap voltage source on an innite printed-circuit transmissionline, in orderto ascertain the natureofthecurrentandtoshowwhattypesofspuriouseffectsmay be produced at high frequency. Three particularstructures have been examined: microstrip, coveredmicrostrip, andstriplinewithanairgapabovethestripconductor.A semianalytical method for determining the strip cur-renthasbeenused, basedonaFouriertransformofthecurrent in the longitudinal (z) direction. The strip currentis thus expressed as an inverse Fourier transform integralinthecomplexlongitudinalwavenumber(kz)plane.Thismethodprovidesanaccuratecalculationofthestripcur-rent, aswell asphysical insight intothenatureof thecurrent on the strip. In particular, deforming the originalpath of integration in the kz plane allows for a convenientdecomposition of the total strip current into a sum of con-stituentparts.It was shown that the total current consists of a bound-mode(BM)currenttogetherwithacontinuous-spectrum(CS) current. TheBMcurrent correspondstotheusualtransmission-line mode, whose elds are bound to thestripregion, decayingawayfromthestripinthetrans-verse(x) direction. Theamplitudeof theBMcurrent isaccuratelycalculatedatanyfrequencybytakingtheres-idue in the kz plane at the BM pole. The amplitude of theBMcurrentobtainedthiswayisaccurateevenforhighfrequencies, where the definition of the characteristic im-pedance becomes ambiguous. The CS current is, generallyspeaking, a high-frequency radiation type of currentwhoseexistencecannotbepredictedbysimpletransmis-sion-linetheory.TheCScurrentconsistsof twotypesof current. Therst is a set of any physical leaky-mode currents. A phys-ical leakymodeisdenedhereasonethat satisesapathconsistencycondition, meaningthatthenatureoftheleakage(asdeterminedbythepathofintegrationinthetransversewavenumber(kx) planeusedtocalculatethe wavenumber of the leaky mode) is consistent with thephaseconstant of theleakymode. For openstructuressuch as microstrip, there are two types of leaky modes thatmayinprinciplebephysical. OnetypehasleakageintoonlytheTM0surface-wavemodeof the groundedsub-strate(assumingherethatonlytheTM0modeisabovecutoff). This type of leaky mode is physical when the phaseconstant of the leaky mode is less than the wavenumber oftheTM0surface-wavemode,andgreaterthanthewave-number of free space. The secondtype of leaky mode,which can exist on open (uncovered) structures, has leak-ageintoboththeTM0surface-wavemodeandalsointospace. This type of leaky mode is physical when the phaseconstant of the leaky mode is less than the wavenumber k0of freespace.Forthecasespresentedhere,onlythersttype of physical leaky mode has been found to exist for theparameters used in the results. For closed structures(meaningthatagroundplaneexistsbothaboveandbe-lowthestructure) suchascoveredmicrostrip, onlythersttypeofleakymodecanexist.ThesecondtypeofcurrentthatmakesuptheCScur-rent is theresidua1-wave (RW) current. This currentphysicallyrepresentsthecurrentthatisleftoverwhenthephysical leaky-modecurrentsareremovedfromtheCScurrent. Foropenstructuressuchasmicrostrip, theRW current consists of a sum of two different RW currents:a TM0 RW current and ak0 RW current. Mathematically,these currents arise from the corresponding branchpointsinthecomplexkzplane,atthelocationsoftheTM0sur-face-wavewavenumberandthek0wavenumber, respec-tively. Although there is no closed-form expressionavailableforthesecurrents, anasymptotic analysisre-veals howthey behave for large distances z fromthesource. TheTM0RWcurrentpropagateswiththewave-numberoftheTM0surface-wavemode,anddecayswithdistance along the strip as z3/2. The k0 RW current prop-agateswiththewavenumberk0of freespace,anddecayswith distance along the strip as z2. For a closed structuresuchas covered microstrip, only the TM0RWcurrentexists.Hence, foranopenstructuresuchasmicrostrip, thestripcurrentproducedbythegapvoltagesourceconsistsofasumof vepossiblecurrentwaves:1. Thebound-mode(transmission-linemode)current2. Aphysical leakymode(if any) thatleaksintotheTM0substratemode3. Aphysicalleakymode(ifany)thatleaksintoboththeTM0substratemodeand intospace4. A TM0residual-wave current, which asymptoticallypropagateswiththewavenumberoftheTM0modeanddecaysasz3/25. Ak0residual-wave current, whichasymptoticallypropagateswiththewavenumberof freespaceanddecaysasz2For a closed structure such as covered microstrip, onlycurrents1,2, and4exist.Formicrostrip, onlythedesiredBMcurrentexistsatlowfrequency. However, asthefrequencyincreases, theCScurrent becomes increasinglyimportant. For mode-rate frequencies, theCSincreases inamplitude as thefrequencyincreasesbecausetheRWcurrentincreasesinLEAKY MODES AND HIGH-FREQUENCY EFFECTS INMICROWAVE INTEGRATEDCIRCUITS 2291amplitudewithfrequency. Forhighfrequenciesaleakymode(havingleakageintotheTM0surface-wavemode)becomesphysical, andtheCScurrentbecomesverysig-nificant above this frequency. This frequency occursroughlywhenthesubstrateisatenthof awavelengththickelectricallyinthedielectric. Asthefrequencyin-creases, a significant oscillation is observed in a plot of thestripcurrentversusthedistancezfromthesource.ThisoscillationisduetoaninterferencebetweenthedesiredBMcurrentand theCScurrent.For covered microstrip, the oscillations in the strip cur-rent alsoincreasewithfrequency, sincetheCScurrentincreaseswithfrequency. However, theeffect of thetopcover also significantlyincreases the magnitude of thespuriousoscillation. Thisisduetotwoeffects: (1)asthetopcoverisloweredtowardthemicrostripline, theam-plitudeof theRWcurrent increases; and(2) evenmoreimportantly,asthetopcoverislowered,thefrequencyatwhich a physical leaky mode exists is lowered. In fact, forasufcientlysmall coverheight, aphysical leakymodeexistsatallfrequencies. Therefore, evenatlowfrequen-cies, asignificantoscillationinthestripcurrentmaybeobserved when the cover height is sufciently small, due tothe leaky mode. For a certainvalue of the cover height,acompletedestructiveinterferencewilloccurbetweentheleaky mode and the bound mode at some distance from thesource; thatis, thetwocurrentswill haveanequal am-plitudeanda1801 phasedifferenceat somedistancezfrom the source. At this point on the line the total currentwillessentiallyhaveacompletenull.Homogeneous stripline supports only the desired TEMmodeof propagation. However, duringmanufacture, anairgapmaybeinadvertentlyintroducedabovethestripconductor. It has been known for many years that such anairgapmayleadtopoorperformance, althoughtherea-sonswere, perhaps, not completelyclear. It wasshownhere that the presence of the airgap causes the TEM strip-line mode to become a leaky mode. For a small airgap theleaky mode still resembles the usual TEM stripline mode,but the mode nowhas acomplex wavenumber due toleakageloss(leakageintotheTM0parallel-platemodeofthebackgroundairgapstructure). Thisresultsinpowerlossasthemodepropagates, whichisinadditiontoanyattenuation due to conductor or dielectric loss. In additionto the quasi-TEM leaky mode, there is also a bound modethatexistsonthestructurewithasmallairgap,buttheboundmodehas aeld congurationthat resemblesthatofaparallel-platemode,andisthusonlyweaklyexcitedby the gap source (or by a conventional stripline connector,in practice). The bound mode does not attenuate with dis-tance from the source, however, and thus a significant in-terference may occur between the quasi-TEM leaky modeandtheboundmodeat somedistancefromthesource.This results in a spurious dip in the observed transmissionresponse (S12) of the transmission line at a particular dis-tancefromthesource(orataparticularfrequency,ifthedistanceisxed).Theanalysisandresultswereextendedtothecaseoftwocoupledmicrostriplines,whereagapvoltagesourcewasplacedononeline(thesourceline)andthecurrentthat was produced on the second line (the victim line) wasexamined. It was concluded that at lowfrequency, aneven/odd-modetransmission-lineanalysisissufcienttodeterminethecrosstalkcurrentonthevictimline.Inter-estingly, for larger