antenna tutorials - mathematical analysis of waveguides
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4/20/2015 AntennaTutorialsMathematicalAnalysisofWaveguides
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MathematicalAnalysisofWaveguides
Back:IntrotoWaveguides Waveguides(TableofContents) Antennas(Home)
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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4/20/2015 AntennaTutorialsMathematicalAnalysisofWaveguides
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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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4/20/2015 AntennaTutorialsMathematicalAnalysisofWaveguides
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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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4/20/2015 AntennaTutorialsMathematicalAnalysisofWaveguides
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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.
Next:MoreAnalysisofWaveguideFields
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