19电阻矩阵+地损耗文档格式.docx

19电阻矩阵+地损耗文档格式.docx

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andarethecomplexcurrentamplitudevectorandcomplexvoltageamplitudevector,respectively,

andarethecurrentandvoltageatthei-thconductor,respectively,

isthepropagationconstantofthetransmissionline,

istheattenuationconstant,andisthephaseconstant.

●Substitutionofthesetwoexpressionsintothetelegraphequationsgives

or

●Thatwillleadto

●Ifthetransmissionlineislossless,i.e.

then

givingthefollowingeigenvalueequations:

whereandrepresent,respectively,thecomplexcurrentamplitude

vectorandthecomplexvoltageamplitudevectorforthelosslessline,andthesetwovectorsshouldberealsincethetransmissionlinedoesnothaveanylosses.

denotesthephasevelocityoftheguidedwavealongtheline.

●Sincethetransmissionlineiscomposedofconductors,theseeigenvalueequationswillproduceeigenvectorsforcurrent,,andeigenvectorsforvoltage,:

●Sinceandarerealsymmetrical,thematricesandshouldbemutuallytransposeandcomplexconjugate.Infact,

wherethesuperscriptTsymbolizesthetransposeand*theconjugate.

●Itfollowsthattheeigenvectorsofandtheeigenvectorsofshouldbemutuallyorthogonal,namelybi-orthogonal,

wheretheisKroneckerDelta，

●Italsofollowsthattheeigenvaluesofandtheeigenvaluesofshouldbemutuallycomplexconjugate.Sincethephasevelocityisreal,thesetwomatrices,and,possessthesameeigenvalues,

●Therefore,thistransmissionlinesystemcomposedofconductorsandonegroundplanepossesspropagatingmodes.

●Inalosslesslinesystem,alleigenvectors,eitherand,shouldbereal.

●Foralosslesssystem,

and

theearliermentionedformulas

arereducedto

●Ifdissipationisinvolvedinatransmissionline,thenthecomplexpowerpropagatedalongthelongitudinaldirection,i.e.axis,is

andtheaveragepoweris

●Thepowerlossperunitlengthofthelinesystemis

whereanddenote,respectively,theconductinglossperunitlength

andthedielectriclossperunitlength,

●Itwillbeprovenbelowthattheattenuationconstantinadissipatedtransmissionlineisgivenby

whereandindicatetheattenuationconstantforconductorsandthe

attenuationconstantfordielectrics,respectively.

[Proof]

Since

then

hence

●Tocalculatetheresistancematrix,theattenuationconstantforconductorsisconsideronly,

●Thenumeratorinthisquotientisthedissipatedconductingpowerperunitlength,,which,accordingtotheperturbationtheory,isapproximatelyequalto

isthenumberofconductors,

isthecontourofcrosssectionofthei-thconductor,

isthesurfaceresistanceofconductors,

isthecurrentdensityofthej-thconductor,whichisapproximately

equalto

，

isthefreechargedensityonthej-thconductor.

●Ifeachofconductorsisdrivenbytheelementsofthei-theigenvector,,here,,representsthevoltagebetweenthej-thconductorandtheground,thenthefreechargedensityonthej-thconductor,,canbedeterminedinawaygivenearlier,andthesurfacecurrentdensityflowingonthej-thconductor,,canbefoundinanabove-mentionedformula,

●Theaveragepowertransmittedalongthetransmissionline,,appearedinthequotientexpressionofisapproximatelyequalto

●Thereforetheattenuationconstantforconductors,,becomes

●Sincethereexistcurrentandvoltagemodesinthetransmissionline,and,thenthereareattenuationconstants,,givenby

●Tocalculatetheresistancematrixalone,atransmissionlineisassumedtohaveconductinglossonly,viz.,thetelegraphequationforvoltage

isreducedto

●Foralowlosstransmissionline,,theaboveequationbecomes

andtheequalityofimaginarypartsofbothsidesgivesriseto

namely

●Itisinferredfromthat

●Ifthedielectriclossisnotconsideredforthetimebeing,,thentheattenuationconstantfordielectricsbecomeszero,

andtheattenuationconstantissimplifiedas

●Asaresult,theequationistransferredas

●Asmentionedabove,therearemodesfor,,whichgiverisetoequations,

aremodalcurrentsgivenby

aremodalvoltagesgivenby

aremodalattenuationconstantsgivenby

aremodalcurrentdensitiesgivenby

aremodalchargedensitiesdeterminedbyputtingeachelementofmodalvoltage,,totheeachconductor.

●Eachofthefollowingm