电路基础chapter8

1Chapter8SinusoidalSteady-StateAnalysis8.1TheSinusoidalSource8.2TheSinusoidalResponse8.3Phasor8.4Kirchhoff’sLawsintheFrequencyDomain8.5ThePassiveCircuitElementsintheFrequencyDomain8.6NodalandMeshAnalysis2Chapter8SinusoidalSteady-StateAnalysis8.7Superposition,SourceTransformations,andThevenin’stheorem8.8PhasorDiagrams8.9ACCircuitPowerAnalysis3Chapter8SinusoidalSteady-StateAnalysis(正弦稳态分析)ACAnalysisDCAnalysis:voltageandcurrentareconstantwithrespecttotime.ACAnalysis:voltageandcurrentvarywithtime.ACcanbesinusoidal,squarewaves,orarbitraryperiodicwaveforms.SinusoidalisparticularlyimportantCommonlyused,e.g.,powersystems,communications,etc.Simpleperiodicfunction(e.g.,derivativeandanti-derivativeofasinusoidalisalsoasinusoidal)Anyperiodicfunctioncanberepresentedasthesumofsinusoidalfunction=FourierSeries4Example:1kΏ1μFsin1000tGivenuc(0)=0,solveforuc(t)fort0)451000sin(2121)(001.0/otctetu)()(tutusstrSinusoidalSteady-StateAnalysis001.0/21)(ttretuapproachzeroasttransientresponse)451000sin(21)(osstturemainsaststeadystateresponse56Chapterobjectives1.Understandphasor（相量）conceptsandbeabletoperformaphasortransformandaninversephasortransform.2.Beabletotransformacircuitwithasinusoidalsourceintothefrequencydomain（频域）usingphasorconcepts.3.Knowhowtousethefollowingcircuitanalysistechniquestosolveacircuitinthefrequencydomain.Kirchhoff’slaws;Voltageandcircuitequivalent;ThéveninandNortonequivalents;Node-voltagemethodandmesh-currentmethod.4.CircuitAnalysis7Phaseangleutimeshift8.1TheSinusoidalSourceSinusoidalSources:voltageorcurrentsourcesthatvarysinusoidallywithtime.Thesinusoidalvoltagemaybewrittenasu(t)=Umcos(t+u)amplitudeangularfrequencyπradian=180oUm-Umcos(t1+u)=1t1+u=0t1=-u/ifu0,functionshiftstotheleft.phaseangleuifu0,functionshiftstotheright.Unit:radian0u(t)t8T=period=timeintervalofacycleT.Angularfrequencyinradians/sisrelatedtothefrequencyfinhertzvia=2f=2/TUm0u(t)t-Um2Acosineorsinefunctionrepeatsitselfevery2.Frequency(numberofcyclespersecond)f=1/T=/2Rootmeansquareofaperiodicfunctionedinpowercalculations2)(102mTUdttuTU98.2TheSinusoidalResponsetransientcomponentsteady-statecomponentAssumeus(t)=Umcos(t+u),applyKVL)451000sin(2121)(001.0/otteti)451000sin(2121)(001.0/otctetuRLus(t)ulLdi/dt+Ri=Umcos(t+u)Solvetheequation,wehave10ResponseCharacteristicsThesteadystateresponsesignalisasinusoidalfunction.Thefrequency()oftheresponsesignalisidenticaltothefrequencyofthesourcesignal(forcingfunction).Onlytrueforlinearcircuits,i.e.,whenthecircuitparametersR,L,andCareconstant.Amplitudeandphaseangleoftheresponsesignalareusuallydifferentfromamplitudeandphaseangleofthesource.So,thesteadystateanalysistaskisreducedtofindingtheamplitudeandphaseangleoftheresponse(thewaveformandfrequencyoftheresponsearealreadyknown).118.3Phasor（相量）Conceptually:Actualinputus(t)(timedomain)Actualresponseu(t)(timedomain)“Complex”inputsUsU(complexnumberdomainorfrequencydomain)“Complex”inputU(complexnumberdomainorfrequencydomain)Expressincomplex(phasor)formTransferbacktoD.E.(hard)Algebra(simple)128.3.1RepresentationofComplexNumbersComplexnumber=realnumber+imaginarynumberz=x+jy1j(1)RectangularformReIm(j)34Z=3+4j(2)PolarFormzzrjyxzReIm(j)34ojz13.53543where22yxzrθ=tan-1(y/x)13(3)Exponentialformjrez148.3.2PhasorNotationSupposeasinusoidalvoltagesource:u(t)=Umcos(t+u))}({Re}){(Re}){(Re}Re{)cos()()(tUeUeeUeUtUtusmtjsmtjjsmtjsmsmscapturesthetwounknownsoftheresponsesignal(amplitudeandphaseangle)phasorstandardpolarformstandardpolarform15}){(Re)cos()(tjsmsmseUtUtusmUrsmUsmUtus168.3.3DefinitionofPhasorPhasorforsinusoidalfunction}Re{)cos()(tjjsmsmseeUtUtu)}cos({jsmsmseUtUPUsmsUUphasortransform(transfersthesinusoidalfunctionfromthetimedomaintothecomplexnumberorfrequencydomain).ororsincossmsmsjUUUInversephasortransform(transfersthesinusoidalfunctionfromthefrequency/complexdomaintothetimedomain)(multiplybyejtandextracttherealpart).)cos(}{1tUeUPsmjsm178.3.4UsefulnessofPhasorNotationtoSolveDEsjmsmmseUUtUtu)cos()()cos(tURidtdiLm}Re{}Re{)cos()()(jmtjmmeIeItItiKVL:Weknowthatthesteadystatesolutionforiisintheformoftheforcingfunction(inthiscase,cos),andsamefrequency(w)):substituting(2)in(1)}Re{}Re{}][Re{tjsmtjmtjmeUeIReIdtdL)451000sin(2121)(001.0/otctetuRLus(t)ul18}Re{}]{Re[}][Re{buttjmtjmtjmeIjeIdtdeIdtd}Re{}Re{}][Re{tjsmtjmtjmeUeIReIdtdL0}]Re{[tjsmmmeUIRILj0}Re{}Re{}Re{tjsmtjmtjmeUeIReILjjustacomplexnumber,apointonthecomplexplaneSo,theequationsaysthattherealpartof{}hastothezeroatalltimes.0smmmUIRILj19)(2222)()(jsmjjsmsmmeLRUeLReULjRUI)cos()()(}{)(221tLRUtiIPtismmsameasthesteadystatesolutionderivedbeforesolvingDE208.4Kirchhoff’sLawsintheFrequencyDomain（频域中的基尔霍夫定律）0)(1tiknk8.4.1KCL0}Re{1tjkmnkeI01kmnkItimedomainfrequencydomain01knkI21Example1.)1.53cos(25)(1tti)9.36cos(210)(2tti)()()(21tititi)(1ti)(2ti)(tiFindSoluti