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1、PHYS370–AdvancedElectromagnetismPart6:TransmissionLinesTransmissionLinesTransmissionlinesareused(asarewaveguides)toguideelectromagneticwavesfromoneplacetoanother.Acoaxialcable(used,forexample,toconnectaradioortelevisiontoanaerial)isanexampleofatransmissionline.Transmissi
2、onlinesmaybelessbulkyandlessexpensivethanwaveguides;buttheygenerallyhavehigherlosses,soaremoreappropriateforcarryinglow-powersignalsovershortdistances.AdvancedElectromagnetism1Part6:TransmissionLinesTransmissionLinesInthispartofthecourse,weshallconsider:•asimpleLCmodelof
3、ageneraltransmissionline;•thespeedofpropagationofawaveinatransmissionline;•thecharacteristicimpedanceofatransmissionline;•impedancematchingattheterminationofatransmissionline;•practicaltransmissionlines(parallelwires;coaxialcable).AdvancedElectromagnetism2Part6:Transmiss
4、ionLinesLCModelofaTransmissionLineConsideraninfinitelylong,parallelwirewithzeroresistance.Ingeneral,thewirewillhavesomeinductanceperunitlength,L,whichmeansthatwhenanalternatingcurrentIflowsinthewire,therewillbeapotentialdifferencebetweendifferentpointsalongthewire.IfVisthepo
5、tentialatsomepointalongthewirewithrespecttoearth,thenthepotentialdifferencebetweentwopointsalongthewireisgivenby:∂V∂I∆V=δx=−Lδx(1)∂x∂tAdvancedElectromagnetism3Part6:TransmissionLinesLCModelofaTransmissionLineIngeneral,aswellastheinductance,therewillalsobesomecapacitancepe
6、runitlength,C,betweenthewireandearth.Thismeansthatthecurrentinthewirecanvarywithposition:∂I∂Vδx=−Cδx(2)∂x∂tAdvancedElectromagnetism4Part6:TransmissionLinesLCModelofaTransmissionLineLetustakeEquations(1)and(2)above:∂V∂I=−L(3)∂x∂t∂I∂V=−C(4)∂x∂tDifferentiate(3)withrespecttot
7、:∂2V∂2I=−L(5)∂x∂t∂t2and(4)withrespecttox:∂2I∂2V=−C(6)∂x2∂t∂xAdvancedElectromagnetism5Part6:TransmissionLinesCurrentandVoltageWavesHence:∂2I∂2I=LC(7)∂x2∂t2Similarly(bydifferentiating(3)withrespecttoxand(4)withrespecttot),wefind:∂2V∂2V=LC(8)∂x2∂t2Equations(7)and(8)arewaveequ
8、ationsforthecurrentinthewire,andthevoltagebetweenthewireandearth.Thewavestravelwithspeedv,givenby:1v=√(
