s-parametersS参数

*IntroductiontoFrequencyDomainAnalysis(3Classes)ManythankstoSteveHall,IntelfortheuseofhisslidesReferenceReading:PosarCh4.5http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdfSlidecontentfromStephenHallInstructor:RichardMellitz*OutlineMotivation:WhyUseFrequencyDomainAnalysis2-PortNetworkAnalysisTheoryImpedanceandAdmittanceMatrixScatteringMatrixTransmission(ABCD)MatrixMason’sRuleCascadingS-MatricesandVoltageTransferFunctionDifferential(4-port)ScatteringMatrix*Motivation:WhyFrequencyDomainAnalysis?TimeDomainsignalsonT-lineslinesarehardtoanalyzeManyproperties,whichcandominateperformance,arefrequencydependent,anddifficulttodirectlyobserveinthetimedomainSkineffect,Dielectriclosses,dispersion,resonanceFrequencyDomainAnalysisallowsdiscretecharacterizationofalinearnetworkateachfrequencyCharacterizationatasinglefrequencyismucheasierFrequencyAnalysisisbeneficialforThreereasonsEaseandaccuracyofmeasurementathighfrequenciesSimplifiedmathematicsAllowsseparationofelectricalphenomena(loss,resonance…etc)*KeyConceptsHerearethekeyconceptsthatyoushouldretainfromthisclassTheinputimpedance&theinputreflectioncoefficientofatransmissionlineisdependenton:TerminationandcharacteristicimpedanceDelayFrequencyS-ParametersareusedtoextractelectricalparametersTransmissionlineparameters(R,L,C,G,TDandZo)canbeextractedfromSparametersVias,connectors,sockets-parameterscanbeusedtocreateequivalentcircuits=ThebehaviorofS-parameterscanbeusedtogainintuitionofsignalintegrityproblems*Review–ImportantConceptsTheimpedancelookingintoaterminatedtransmissionlinechangeswithfrequencyandlinelengthTheinputreflectioncoefficientlookingintoaterminatedtransmissionlinealsochangeswithfrequencyandlinelengthIftheinputreflectionofatransmissionlineisknown,thenthelinelengthcanbedeterminedbyobservingtheperiodicityofthereflectionThepeakoftheinputreflectioncanbeusedtodeterminelineandloadimpedancevalues*TwoPortNetworkTheoryNetworktheoryisbasedonthepropertythatalinearsystemcanbecompletelycharacterizedbyparametersmeasuredONLYattheinput&outputportswithoutregardtothecontentofthesystemNetworkscanhaveanynumberofports,however,considerationofa2-portnetworkissufficienttoexplainthetheoryA2-portnetworkhas1inputand1outputport.Theportscanbecharacterizedwithmanyparameters,eachparameterhasaspecificadvantageEachParametersetisrelatedto4variables2independentvariablesforexcitation2dependentvariablesforresponse*NetworkcharacterizedwithPortImpedanceMeasuringtheportimpedanceisnetworkisthemostsimplisticandintuitivemethodofcharacterizinganetworkPort1Port2Case1:InjectcurrentI1intoport1andmeasuretheopencircuitvoltageatport2andcalculatetheresultantimpedancefromport1toport2Case2:InjectcurrentI1intoport1andmeasurethevoltageatport1andcalculatetheresultantinputimpedance2-portNetworkI1I2+-V1V2+-2-portNetworkI1I2+-V1V2+-*ImpedanceMatrixAsetoflinearequationscanbewrittentodescribethenetworkintermsofitsportimpedancesWhere:Iftheimpedancematrixisknown,theresponseofthesystemcanbepredictedforanyinputOpenCircuitVoltagemeasuredatPortiCurrentInjectedatPortjZiitheimpedancelookingintoportiZijtheimpedancebetweenportiandjOr*ImpedanceMatrix:Example#2Calculatetheimpedancematrixforthefollowingcircuit:Port1Port2R1R2R3*ImpedanceMatrix:Example#2Step1:CalculatetheinputimpedanceR1R2R3I1V1+-Step2:CalculatetheimpedanceacrossthenetworkR1R2R3I1V2+-*ImpedanceMatrix:Example#2Step3:CalculatetheImpedancematrixAssume:R1=R2=30ohmsR3=150ohms*MeasuringtheimpedancematrixQuestion:Whatobstaclesareexpectedwhenmeasuringtheimpedancematrixofthefollowingtransmissionlinestructureassumingthatthemicro-probeshavethefollowingparasitics?Lprobe=0.1nHCprobe=0.3pFAssumeF=5GHz*MeasuringtheimpedancematrixT-linePort2Answer:OpencircuitvoltagesareveryhardtomeasureathighfrequenciesbecausetheygenerallydonotexistforsmalldimensionsOpencircuitcapacitance=impedanceathighfrequenciesProbeandviaimpedancenotinsignificant0.1nH106ohms106ohmsZo=50WithoutProbeCapacitanceZo=50WithProbeCapacitance@5GHzZ21=50ohmsZ21=63ohmsPort1Port2Port1Port2*Advantages/DisadvantagesofImpedanceMatrixAdvantages:TheimpedancematrixisveryintuitiveRelatesallportstoanimpedanceEasytocalculateDisadvantages:RequiresopencircuitvoltagemeasurementsDifficulttomeasureOpencircuitreflectionscausemeasurementnoiseOpencircuitcapacitancenottrivialathighfrequenciesNote:TheAdmittanceMatrixisverysimilar,however,itischaracterizedwithshortcircuitcurrentsinsteadofopencircuitvoltages*ScatteringMatrix(S-parameters)Measuringthe“power”ateachportacrossawellcharacterizedimpedancecircumventstheproblemsmeasuringhighfrequency“opens”&“shorts”Thescatteringmatrix,or(S-parameters),characterizesthenetworkbyobservingtransmitted&reflectedpowerwaves2-portNetwork2-portNetworkPort1Port2airepresentsthesquarerootofthepowerwaveinjectedintoportia1a2b2b1bjrepresentsthepowerwavecomingoutofportjRR*ScatteringMatrixAsetoflinearequationscanbewrittentodescribethenetworkintermsofinjectedandtransmittedpowerwavesWhere:Sii=theratioofthereflectedpowertotheinjectedpoweratportiSij=theratioofthepowermeasuredatportjtothepowerinjectedatporti*MakingsenseofS-Parameters–ReturnLossWhenthereisnoreflectionfromtheload,orthelinelengthiszero,S11=ReflectioncoefficientS11ismeasureofthepowerreturnedtothesource,andiscalledthe“ReturnLoss”R=ZoZ=-lZ=0ZoR=50*MakingsenseofS-Parameters–ReturnLossWhenthereisareflectionfromtheload,S11willbecomposedofmultiplereflectionsduetothestandingwavesRLZ=-lZ=0ZoIfthenetworkisdrivenwitha50ohmsource,thenS11iscalculatedusingtheinputimpedanceinsteadofZo50ohmsS11ofatransmissionlinewillexhibitperiodiceffectsduetothestandingwaves*Example#3–InterpretingthereturnlossBasedontheS11plotshownbelow,calculateboththeimpedanceanddielectricconstant00.050.10.150.20.250.30.350.40.451.01.52.02.53..03.54.04.55.0Frequency,GHzS11,MagnitudeR=50L=5inchesZoR=50*Example–InterpretingthereturnlossStep1:Calculatethetimedelayofthet-lineusingthepeaksStep2:CalculateErusingthevelocity00.050.10.150.20.250.30.350.40.451.01.52.02.53.03.54.04.55.0Frequency,GHzS11,Magnitude1.76GHz2.94GHzPeak=0.384*Example–InterpretingthereturnlossStep3:CalculatetheinputimpedancetothetransmissionlinebasedonthepeakS11at1.76GHzNote:Thephaseofthereflectionshouldbeeither+1or-1at1.76GHzbecauseitisalignedwiththeincidentStep4:Calculatethecharacteristicimpedancebasedontheinputimpedanceforx=-5inchesEr=1.0andZo=75ohms*MakingsenseofS-Parameters–InsertionLossWhenpowerisinjectedintoPort1withsourceimpedanceZ0andmeasuredatPort2withmeasurementloadimpedanceZ0,thepowerratioreducestoavoltageratio2-portNetwork2-portNetworkV1a1a2=0b2b1V2ZoZoS21ismeasureofthepowertransmittedfromport1toport2,andiscalledthe“InsertionLoss”*LossfreenetworksForalossfreenetwork,thetotalpowerexitingtheNportsmustequalthetotalincidentpowerIfthereisnolossinthenetwork,thetotalpowerleavingthenetworkmustbeaccountedforinthepowerreflectedfromtheincidentportandthepowertransmittedthroughnetworkSinces-parametersarethesquarerootofpowerratios,thefollowingistrueforloss-freenetworksIftheaboverelationshipdoesnotequal1,thenthereislossinthenetwork,andthedifferenceisproportionaltothepowerdissipatedbythenetwork*InsertionlossexampleQuestion:Whatpercentageofthetotalpowerisdissipatedbythetransmissionline?EstimatethemagnitudeofZo(boundit)*InsertionlossexampleWhatpercentageofthetotalpowerisdissipatedbythetransmissionline?WhatistheapproximateZo?Howmuchamplitudedegradationwillthist-linecontributetoa8GT/ssignal?Ifthetransmissionlineisplacedina28ohmsystem(suchasRambus),willtheamplitudedegradationestimatedaboveremainconstant?Estimatealphafor8GT/ssignal*InsertionlossexampleAnswer:Sincethereareminimalreflectionsonthisline,alphacanbeestimateddirectlyfromtheinsertionlossS21~0.75at4GHz(8GT/s)Whenthereflectionsareminimal,alphacanbeestimatedIfthereflectionsareNOTsmall,alphamustbeextractedwithABCDparameters(whicharereviewedlater)Thelossparameteris“1/A”forABCDparametersABCDwillbediscussedlater.IfS11<~0.2(-14dB),thentheaboveapproximationisvalid*ImportantconceptsdemonstratedTheimpedancecanbedeterminedbythemagnitudeofS11Theelectricaldelaycanbedeterminedbythephase,orperiodicityofS11ThemagnitudeofthesignaldegradationcanbedeterminedbyobservingS21Thetotalpowerdissipatedbythenetworkcanbedeterminedbyaddingthesquareoftheinsertionandreturnlosses*Anoteabouttheterm“Loss”TruelossescomefromphysicalenergylossesOhmic(I.e.,skineffect)Fielddampeningeffects(LossTangent)Radiation(EMI)InsertionandReturnlossesincludeeffectssuchasimpedancediscontinuitiesandresonanceeffects,whicharenottruelossesLossfreenetworkscanstillexhibitsignificantinsertionandreturnlossesduetoimpedancediscontinuities*Advantages/DisadvantagesofS-parametersAdvantages:EaseofmeasurementMucheasiertomeasurepowerathighfrequenciesthanopen/shortcurrentandvoltageS-parameterscanbeusedtoextractthetransmissionlineparametersnparametersandnUnknownsDisadvantages:Mostdigitalcircuitoperateusingvoltagethresholds.Thissuggestthatanalysisshouldultimatelyberelatedtothetimedomain.Manysiliconloadsarenon-linearwhichmakethejobofconvertings-parametersbackintotimedomainnon-trivial.Conversionbetweentimeandfrequencydomainintroduceserrors*CascadingSparameterWhileitispossibletocascades-parameters,itgetsmessy.Graphicallywejustflipeveryothermatrix.Mathematicallythereisabetterway…ABCDparametersWewillanalyzedthislaterwithsignalflowgraphsa11b11a21b21a12b12a22b22a13b13a13b133cascadedsparameterblocks*ABCDParametersThetransmissionmatrixdescribesthenetworkintermsofbothvoltageandcurrentwaves2-portNetwork2-portNetworkV1I1I2V2Thecoefficientscanbedefinedusingsuperposition*Transmission(ABCD)MatrixSincetheABCDmatrixrepresentstheportsintermsofcurrentsandvoltages,itiswellsuitedforcascadingelementsV1I1I2V2ThematricescanbecascadedbymultiplicationI3V3Thisisthebestwaytocascadeelementsinthefrequencydomain.Itisaccurate,intuitiveandsimplistic.*RelatingtheABCDMatrixtoCommonCircuitsZPort1Port2Port1YPort2Z1Port1Port2Z2Z3Y1Port1Port2Y2Y3Port1Port2Assignment6:Convertthesetos-parameters*ConvertingtoandfromtheS-MatrixTheS-parameterscanbemeasuredwithaVNA,andconvertedbackandforthintoABCDtheMatrixAllowsconversionintoamoreintuitivematrixAllowsconversiontoABCDforcascadingABCDmatrixcanbedirectlyrelatedtoseveralusefulcircuittopologies*ABCDMatrix–Example#1Createamodelofaviafromthemeasureds-parameters*ABCDMatrix–Example#1ThemodelcanbeextractedaseitheraPioraTnetworkL1CVIAL2TheinductancevalueswillincludetheLofthetraceandtheviabarrel(itisassumedthatthetestsetupminimizesthetracelength,andsubsequentlythetracecapacitanceisminimalThecapacitancerepresentstheviapads*ABCDMatrix–Example#1Assumethefollowings-matrixmeasuredat5GHz*ABCDMatrix–Example#1Assumethefollowings-matrixmeasuredat5GHzConverttoABCDparameters*ABCDMatrix–Example#1Assumethefollowings-matrixmeasuredat5GHzConverttoABCDparametersRelatingtheABCDparameterstotheTcircuittopology,thecapacitanceandinductanceisextractedfromC&AZ1Port1Port2Z2Z3*ABCDMatrix–Example#2Calculatetheresultings-parametermatrixifthetwocircuitsshownbelowarecascaded2-portNetworkNetworkX5050Port1Port22-portNetworkNetworkY5050Port1Port22-portNetworkNetworkX50Port12-portNetworkNetworkY50Port2*ABCDMatrix–Example#2Step1:ConverteachmeasuredS-MatrixtoABCDParametersusingtheconversionspresentedearlierStep2:MultiplytheconvertedT-matricesStep3:ConverttheresultingMatrixbackintoS-parametersusingtheeconversionspresentedearlier*Advantages/DisadvantagesofABCDMatrixAdvantages:TheABCDmatrixisveryintuitiveDescribesallportswithvoltagesandcurrentsAllowseasycascadingofnetworksEasyconversiontoandfromS-parametersEasytorelatetocommoncircuittopologiesDisadvantages:DifficulttodirectlymeasureMustconvertfrommeasuredscatteringmatrix*Signalflowgraphs–Startwith2portfirstThewavefunctions(a,b)usedtodefines-parametersforatwo-portnetworkareshownbelow.Theincidentwavesisa1,a2onport1andport2respectively.Thereflectedwavesb1andb2areonport1andport2.Wewillusea’sandb’sinthes-parameterfollowslides*SignalFlowGraphsofSParameters“Inasignalflowgraph,eachportisrepresentedbytwonodes.Nodeanrepresentsthewavecomingintothedevicefromanotherdeviceatportn,andnodebnrepresentsthewaveleavingthedeviceatportn.Thecomplexscatteringcoefficientsarethenrepresentedasmultipliers(gains)onbranchesconnectingthenodeswithinthenetworkandinadjacentnetworks.”*ExampleMeasurementequipmentstrivestobematchi.e.reflectioncoefficientis0See:http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf*Mason’sRule~Non-TouchingLoopRuleTisthetransferfunction(oftencalledgain)TkisthetransferfunctionofthekthforwardpathL(mk)istheproductofnontouchingloopgainsonpathktakenmkattime.L(mk)|(k)istheproductofnontouchingloopgainsonpathktakenmkatatimebutnottouchingpathk.mk=1meansallindividualloops*VoltageTransferfunctionWhatisreallyofmostrelevancetotimedomainanalysisisthevoltagetransferfunction.Itincludestheeffectofnon-perfectloads.Wewillshowhowthevoltagetransferfunctionsfora2portnetworkisgivenbythefollowingequation.NoticeitisnotS21*ForwardWavePath*ReflectedWavePatha1b1b2a2VsGSGLs21s12s11s22*Combineb2anda2*ConvertWavetoVoltage-Multiplybysqrt(Z0)*VoltagetransferfunctionusingABCDLet’sseeifwecangetthisresultsanotherway*Cascade[ABCD]todeterminesystem[ABCD]*ExtractthevoltagetransferfunctionSameaswithflowgraphanalysis*CascadingS-ParameterAspromisedwewillnowlookathowtocascades-parametersandsolvewithMason’sruleTheproblemwewilluseiswhatwaspresentedearlierTheassertionisthatthelossofcascadechannelcanbedeterminejustbyaddingupthelossesindB.Wewillshowhowwecangaininsightaboutthisassertionfromtheequationandgraphicformofasolution.*CreatingthesignalflowgraphWemapoutputatoinputbandvisaversa.NextwedefinealltheloopsLoop“A”and“B”donottoucheachother*UseMason’sruleThereisonlyoneforwardpatha11tob23.Thereare2nontouchinglooksMason’sRule*EvaluatethenatureofthetransferfunctionIfresponseisrelativelyflatandreflectionisrelativelylowResponsethroughachanneliss211*s212*213…Assumptionisthattheseare~0*JitteranddBBudgetingChanges21intoaphasorInsertionlossindb=i.e.Forabudgetjustaddupthedb’sandjitter=*DifferentialS-Parametersarederivedfroma4-portmeasurementTraditional4-portmeasurementsaretakenbydrivingeachport,andrecordingtheresponseatallotherportswhileterminatedin50ohmsAlthough,itisperfectlyadequatetodescribeadifferentialpairwith4-portsingleendeds-parameters,itismoreusefultoconverttoamulti-modeportDifferentialS-Parameters4-porta1a2b1b2S11S22S21S12S33S44S43S34S31S42S41S32S13S24S23S14b1b2b3b4*DifferentialS-ParametersMatrixassumesdifferentialandcommonmodestimulusItisusefultospecifythedifferentialS-parametersintermsofdifferentialandcommonmoderesponsesDifferentialstimulus,differentialresponseCommonmodestimulus,CommonmoderesponseDifferentialstimulus,commonmoderesponse(akaACCMNoise)Commonmodestimulus,differentialresponseThiscanbedoneeitherbydrivingthenetworkwithdifferentialandcommonmodestimulus,orbyconvertingthetraditional4-ports-matrixDS11DS22DS21DS12CS11CS22CS21CS12CDS11CDS22CDS21CDS12DCS11DCS22DCS21DCS12bdm1bdm2bcm1bcm2*ExplanationoftheMulti-ModePortDifferentialMatrix:DifferentialStimulus,differentialresponsei.e.,DS21=differentialsignal[(D+)-(D-)]insertedatport1anddiffsignalmeasuredatport2CommonmodeMatrix:Commonmodestimulus,commonmodeResponse.i.e.,CS21=Com.modesignal[(D+)+(D-)]insertedatport1andCom.modesignalmeasuredatport2CommonmodeconversionMatrix:DifferentialStimulus,Commonmoderesponse.i.e.,DCS21=differentialsignal[(D+)-(D-)]insertedatport1andcommonmodesignal[(D+)+(D-)]measuredatport2differentialmodeconversionMatrix:CommonmodeStimulus,differentialmoderesponse.i.e.,DCS21=commonmodesignal[(D+)+(D-)]insertedatport1anddifferentialmodesignal[(D+)-(D-)]measuredatport2DS11DS22DS21DS12CS11CS22CS21CS12CDS11CDS22CDS21CDS12DCS11DCS22DCS21DCS12bdm1bdm2bcm1bcm2adm1adm2acm1acm2=*DifferentialS-ParametersConvertingtheS-parametersintothemulti-moderequiresjustalittlealgebraExampleCalculation,DifferentialReturnLossThestimulusisequal,butopposite,therefore:2-portNetwork4-portNetwork2134AssumeasymmetricalnetworkandsubstituteOtherconversionsthatareusefulforadifferentialbusareshownDifferentialInsertionLoss:DifferentialtoCommonModeConversion(ACCM):Similartechniquescanbeusedforallmulti-modeParameters*Nextclasswewilldevelopmoredifferentialconcepts*backupreview*Advantages/DisadvantagesofMulti-ModeMatrixoverTraditional4-portAdvantages:Describes4-portnetworkintermsof4twoportmatricesDifferentialCommonmodeDifferentialtocommonmodeCommonmodetodifferentialEasiertorelatetosystemspecificationsACCMnoise,differentialimpedanceDisadvantages:Mustconvertfrommeasured4-portscatteringmatrix*HighFrequencyElectromagneticWavesInordertounderstandthefrequencydomainanalysis,itisnecessarytoexplorehowhighfrequencysinusoidsignalsbehaveontransmissionlinesTheequationsthatgovernsignalspropagatingonatransmissionlinecanbederivedfromAmperesandFaradayslawsassumimngauniformplanewaveThefieldsareconstrainedsothatthereisnovariationintheXandYaxisandthepropagationisintheZdirectionZXYDirectionofpropagationThisassumptionholdstruefortransmissionlinesaslongasthewavelengthofthesignalismuchgreaterthanthetracewidthFortypicalPCBsat10GHzwith5miltraces(W=0.005”)*HighFrequencyElectromagneticWavesForsinusoidaltimevaryinguniformplanewaves,AmperesandFaradayslawsreduceto:AmperesLaw:AmagneticFieldwillbeinducedbyanelectriccurrentoratimevaryingelectricfieldFaradaysLaw:AnelectricfieldwillbegeneratedbyatimevaryingmagneticfluxNotethattheelectric(Ex)fieldandthemagnetic(By)areorthogonal*HighFrequencyElectromagneticWavesIfAmperesandFaradayslawsaredifferentiatedwithrespecttozandtheequationsarewrittenintermsoftheEfield,thetransmissionlinewaveequationisderivedThisdifferentialequationiseasilysolvableforEx:*HighFrequencyElectromagneticWavesTheequationdescribesthesinusoidalEfieldforaplanewaveinfreespacePortionofwavetravelingInthe+zdirectionPortionofwavetravelingInthe-zdirectionNotethepositiveexponentisbecausethewaveistravelingintheoppositedirection=permittivityinFarads/meter(8.85pF/mforfreespace)(determinesthespeedoflightinamaterial)=permeabilityinHenries/meter(1.256uH/mforfreespaceandnon-magneticmaterials)Sinceinductanceisproportionalto&capacitanceisproportionalto,thenisanalogoustoinatransmissionline,whichisthepropagationdelay*HighFrequencyVoltageandCurrentWavesThesameequationappliestovoltageandcurrentwavesonatransmissionlineRLz=-lz=0Ifasinusoidisinjectedontoatransmissionline,theresultingvoltageisafunctionoftimeanddistancefromtheload(z).ItisthesumoftheincidentandreflectedvaluesVoltagewavetravelingtowardstheloadVoltagewavereflectingofftheLoadandtravelingtowardsthesourceIncidentsinusoidReflectedsinusoidNote:isaddedtospecificallyrepresentthetimevaryingSinusoid,whichwasimpliedinthepreviousderivation*HighFrequencyVoltageandCurrentWaves=AttenuationConstant(attenuationofthesignalduetotransmissionlinelosses)=PhaseConstant(relatedtothepropagationdelayacrossthetransmissionline)=Complexpropagationconstant–includesallthetransmissionlineparameters(R,LCandG)(Forthelossfreecase)(lossycase)(Forgoodconductors)(Forgoodconductorsandgooddielectrics)Theparametersinthisequationcompletelydescribethevoltageonatypicaltransmissionline*HighFrequencyVoltageandCurrentWavesSubsequently:Thevoltagewaveequationcanbeputintomoreintuitivetermsbyapplyingthefollowingidentity:TheamplitudeisdegradedbyThewaveformisdependentonthedrivingfunction()&thedelayoftheline*Interaction:transmissionlineandaload(Assumealinelengthofl(z=-l))Thisisthereflectioncoefficientlookingintoat-lineoflengthlZlZ=-lZ=0ThereflectioncoefficientisnowafunctionoftheZodiscontinuitiesANDlinelengthInfluencedbyconstructive&destructivecombinationsoftheforward&reversewaveformsZo*Thisistheinputimpedancelookingintoat-lineoflengthlRLZ=-lZ=0Interaction:transmissionlineandaloadIfthereflectioncoefficientisafunctionoflinelength,thentheinputimpedancemustalsobeafunctionoflengthNote:isdependentonandZin*Line&loadinteractionsInchapter2,youlearnedhowtocalculatewaveformsinamulti-reflectivesystemusinglatticediagramsPeriodoftransmissionline“ringing”proportionaltothelinedelayRemember,thelinedelayisproportionaltothephaseconstantInfrequencydomainanalysis,thesameprinciplesapply,however,itismoreusefultocalculatethefrequencywhenthereflectioncoefficientiseithermaximumorminimumThiswillbecomemoreevidentastheclassprogressesTodemonstrate,letsassumealossfreetransmissionline*Line&loadinteractionsThefrequencywherethevaluesofthereal&imagin