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MathematicalAnalysisofWaveguides

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Inthissection,wewilltakeamathematicallookatwaveguidesandderivesomeoftheirkeyproperties.Wewillagainbeconcernedwithmetal(perfectlyconducting)waveguideswitharectangularcrosssectionasshowninFigure1.

Figure1.GeometryforWaveguideAnalysis.

Ifyouweren'taware,electromagneticsisgovernedbyMaxwell'sEquations,andMaxwell'sEquationsarenoteasytosolve.Hence,everymathtricksomeonecanthinkofwillbeusedinordertomaketheanalysistractable.We'llstartwithdiscussingtheelectricvectorpotential,F.Inasourcefreeregion(i.e.,anareathroughwhichwavespropagatethatisawayfromsources),weknowthat:

Intheabove,DistheElectricFluxDensity.Ifavectorquantityisdivergenceless(asintheabove),thenitcanbeexpressedasthecurlofanotherquantity.ThismeansthatwecanwritethesolutionforDandthecorrespondingelectricfieldEas:

Intheabove,epsilonisthepermittivityofthemediumthroughwhichthewavepropagates.Wearepurelyintheworldofmathematicsnow.ThequantityFisnotphysical,andisoflittlepracticalvalue.Itissimplyanaidinperformingourmathematicalmanipulations.

Itturnsoutthatwaves(orelectromagneticenergy)cannotpropagateinawaveguidewhenbothHzandEzareequaltozero.Hence,whatfieldconfigurationsthatareallowedwillbeclassifiedaseitherTM(TransverseMagnetic,inwhichHz=0)andTE(TransverseElectric,inwhichEz=0).ThereasonthatwavescannotbeTEM(TransverseElectromagnetic,Hz=Ez=0)willbeshowntowardstheendofthisderivation.

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Toperformouranalysis,we'llassumethatEz=0(i.e.,wearelookingataTEmodeorfieldconfiguration).Inthiscase,workingthroughMaxwell'sequations,itcanbeshownthattheEandHfieldscanbedeterminedfromthefollowingequations:

Therefore,ifwecanfindFz(thezcomponentofthevectorF),thenwecanfindtheEandHfields.Intheaboveequation,kisthewavenumber.

WorkingthroughthemathofMaxwell'sEquations,itcanbeshownthatinasourcefreeregion,thevectorpotentialFmustsatisfythevectorwaveequation:

[1]

Tobreakthisequationdown,wewilllookonlyatthezcomponentoftheaboveequation(thatis,Fz).Wewillalsoassumethatwearelookingatasinglefrequency,sothatthetimedependenceisassumedtobeoftheformgivenby(wearenowusingphasorstoanalyzetheequation):

Thentheequation[1]canbesimplifiedasfollows:

[2]

Tosolvethisequation,wewillusethetechniqueofseparationofvariables.HereweassumethatthefunctionFz(x,y,z)canbewrittenastheproductofthreefunctions,eachofasinglevariable.Thatis,weassumethat:

[3]

(Youmightask,howdoweknowthattheseparationofvariablesassumptionaboveisvalid?

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Wedon'twejustassumeitscorrect,andifitsolvesthedifferentialequationwhenwearedonedoingtheanalysisthentheassumptionisvalid).NowwepluginourassumptionforFz(equation[3])intoequation[2],andweendupwith:

[4]

Intheaboveequation,theprimerepresentsthederivativewithrespecttothevariableintheequation(forinstance,Z'representsthederivativeoftheZfunctionwithrespecttoz).Wewillbreakupthevariablek^2intocomponents(again,justtomakeourmatheasier):

[5]

Usingequation[5]tobreakdownequation[4],wecanwrite:

[6]

Thereasonthattheequationsin[6]arevalidisbecausetheyareonlyfunctionsofindependentvariableshence,eachequationmustholdfor[5]tobetrueeverywhereinthewaveguide.Solvingtheaboveequationsusingordinarydifferentialequationstheory,weget:

[7]

TheformofthesolutionintheaboveequationisdifferentforZ(z).Thereasonisthatbothforms(thatforXandY,andthatforZ),arebothequallyvalidsolutionsforthedifferentialequationsinequation[6].However,thecomplexexponentialtypicallyrepresentstravellingwaves,andthe[real]sinusoidsrepresentstandingwaves.Hence,wechoosetheformsgivenin[7]forthesolutions.Nomathrulesareviolatedhereagain,wearejustchoosingformsthatwillmakeouranalysiseasier.

Fornow,wecansetc5=0,becausewewanttoanalyzewavespropagatinginthe+zdirection.Theanalysisisidenticalforwavespropagatinginthezdirection,sothisisfairlyarbitrary.ThesolutionforFzcanbewrittenas:

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[8]

Ifyourememberanythingaboutdifferentialequations,youknowthereneedstobesomeboundaryconditionsappliedinordertodeterminetheconstants.Recallingourphysics,weknowthatthetangentialElectricfieldsatanyperfectconductormustbezero(why?because

,soiftheconductivityapproachesinfinity(perfectconductor),thenifthetangentialEfieldisnotzerothentheinducedcurrentwouldbeinfinite).

Thetangentialfieldsmustbezero,soExmustbezerowheny=0andwheny=b(seeFigure1above),nomatterwhatthevalueforyandzare.Inaddition,Eymustbezerowhenx=0andwhenx=a(independentofxandz).WewillcalculateEx:

Exisgivenbytheaboveequation.Theboundaryconditiongivenby

Ex(x,y=0,z)=0[9]

impliesthatc4mustbeequaltozero.Thisistheonlywaythatboundaryconditiongivenin[9]willbetrueforallxandzpositions.Ifyoudon'tbelievethis,trytoshowthatitisincorrect.Youwillquicklydeterminethatc4mustbezerofortheboundaryconditionin[9]tobesatisfiedeverywhereitisrequired.

Next,thesecondboundarycondition,

Ex(x,y=b,z)=0[10]

impliessomethingveryunique.Theonlywayfortheconditionin[10]tobetrueforallvaluesofxandzwhenevery=b,wemusthave:

Ifthisistobetrueeverywhere,c3couldbezero.However,ifc3iszero(andwehavealreadydeterminedthatc4iszero),thenallofthefieldswouldendupbeingzero,becausethefunctionY(y)in[7]wouldbezeroeverywhere.Hence,c3cannotbezeroifwearelookingforanonzerosolution.Hence,theonlyalternativeisiftheaboveequationimpliesthat:

Thislastequationisfundamentaltounderstandingwaveguides.Itstatesthattheonlysolutions

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forY(y)functionmustendupbeingsinusoids,thatanintegernumberofmultiplesofahalfwavelength.Thesearetheonlytypeoffunctionsthatsatisfythedifferentialequationin[6]andtherequiredboundaryconditions.Thisisanextremelyimportantconcept.

Ifweinvokeourothertwoboundaryconditions:

Ey(x=0,y,z)=0

Ey(x=a,y,z)=0

Then(usingidenticalreasoningtothatabove),wecandeterminethatc2=0andthat:

Thisstatementimpliesthattheonlyfunctionsofxthatsatisfythedifferentialequationandtherequiredboundaryconditionsmustbeanintegermultipleofhalfsinusoidswithinthewaveguide.

Combiningtheseresults,wecanwritethesolutionforFzas:

Intheabove,wehavecombinedtheremainingnonzeroconstantsc1,c3,andc6intoasingleconstant,A,forsimplicity.Wehavefoundthatonlycertaindistributions(orfieldconfigurations)willsatisfytherequireddifferentialequationsandtheboundaryconditions.Eachofthesefieldconfigurationswillbeknownasamode.BecausewederivedtheresultsabovefortheTEcase,themodeswillbeknownasTEmn,wheremindicatesthenumberofhalfcyclevariationswithinthewaveguideforX(x),andnindicatesthenumberofhalfcyclevariationswithinthewaveguideforY(y).

Inthenextsection,we'llexplicitlywriteoutthefieldscorrespondingtothesemodes,discusswhichmodesareallowable,andlookintotheTM(transversemagnetic)case.

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