Odd Even Mode Analysis

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1、4/14/2006OddEvenModeAnalysis1/17Odd/EvenModeAnalysisConsiderthefollowingD1symmetrictwo-portdevice:200ΩI1100Ω100ΩI2++50ΩV2-Q:Yikes!Theplaneofreflectionsymmetryslicesthroughtworesistors.Whatcanwedoaboutthat?A:Resistorsareeasilysplitintotwoequalpieces:the200Ωresistorintotwo100Ωresistorsi

2、nseries,andthe50Ωresistorastwo100Ωresistorsinparallel.100Ω100ΩI1100Ω100ΩI2++V1100Ω100ΩV2--JimStilesTheUniv.ofKansasDept.ofEECS4/14/2006OddEvenModeAnalysis2/17Recallthatthesymmetryofthis2-portdeviceleadstosimplifiednetworkmatrices:⎡⎤SS1121⎡ZZ1121⎤⎡⎤YY1121S===⎢⎥Z⎢⎥⎢⎥YSSZZYY⎣⎦2111⎣211

3、1⎦⎣⎦2111Q:Yes,butcancircuitsymmetrylikewisesimplifytheprocedureofdeterminingtheseelements?Inotherwords,cansymmetrybeusedtosimplifycircuitanalysis?A:Youbet!First,considerthecasewhereweattachsourcestocircuitinawaythatpreservesthecircuitsymmetry:100Ω100ΩI1100Ω100ΩI2++++Vs-V1100Ω100ΩV2-Vs

4、--Or,JimStilesTheUniv.ofKansasDept.ofEECS4/14/2006OddEvenModeAnalysis3/17100Ω100ΩI1100Ω100ΩI2++IsV100Ω100ΩV2Is1--Or,100Ω100ΩI1100Ω100ΩI2Z0Z0++++Vs-V1100Ω100ΩV2-Vs--Butremember!Inorderforsymmetrytobepreserved,thesourcevaluesonbothsides(i.e,Is,Vs,Z0)mustbeidentical!Now,considerthevoltag

5、esandcurrentswithinthiscircuitunderthissymmetricconfiguration:JimStilesTheUniv.ofKansasDept.ofEECS4/14/2006OddEvenModeAnalysis4/17I1aI2a+V1a--V2a+I1III2bI2I1b1d2d+V1b--V2b+++++++Vs-V1V1cV2cV2-Vs----I1cI2cSincethiscircuitpossessesbilateral(reflection)symmetry(12→→,21),symmetriccurrents

6、andvoltagesmustbeequal:II=12V=V12II=12aaV=V12aaII=12bbV=V12bbII=12ccV=V12ccII=12ddQ:Wait!Thiscan’tpossiblybecorrect!LookatcurrentsI1aandI2a,aswellascurrentsI1dandI2d.FromKCL,thismustbetrue:II=−II=−12aa12ddYetyousaythatthismustbetrue:II==II12aa12ddJimStilesTheUniv.ofKansasDept.ofEECS4/

7、14/2006OddEvenModeAnalysis5/17Thereisanobviouscontradictionhere!Thereisnowaythatbothsetsofequationscansimultaneouslybecorrect,isthere?A:Actuallythereis!Thereisonesolutionthatwillsatisfybothsetsofequations:II==00II==12aa12ddThecurrentsarezero!Ifyouthinkaboutit,thismakesperfectsense!The

8、resul