电路原理课件讲义英文版-Chapter-7

Ioftensaythatyoucanmeasurewhatyouarespeakingabout,andexpressitinnumbers,youknowsomethingaboutit;butwhenyoucannotexpressitinnumbers,yourknowledgeisofameagerandunsatisfactorykind;itmaybethebeginningofknowledge,butyouhavescarcely,inyourthoughts,advancedtothestageofascience,whateverthemattermaybe..---LordKelvinChapter7First-orderCircuits7.1Introduction7.2TheSource-freeRCCircuit7.3TheSource-freeRLCircuit7.4SingularityFunctions7.5StepResponseofanRCCircuit7.6StepResponseofanRLCircuit7.1IntroductionRCcircuits:acircuitcomprisingaresistorandcapacitorRLcircuits:acircuitcomprisingaresistorandaninductorAfirst-ordercircuit:ischaracterizedbyafirst-orderdifferentialequation(一阶微分方程).Asecond-ordercircuits:ischaracterizedbyasecond-orderdifferentialequation(二阶微分方程).Itconsistsofresistorsandtheequivalentoftwoenergystorageelements.RecallCCdviCdt1tCCvidtC001()tCCCtvidtvtCCapacitor:Inductor:LLdivLdt1tLLivdtL001()tLLLtivdtitLThecircuitisinsteadystate(稳定状态)beforeswitchKisclosed.i=0,vL=0vL=0,i=Vs/RThecircuitwillreachnewsteadystateafterKhasbeenclosedforalong.FormersteadystateTransienttermNewsteadystatet1VS/Rit0?VLSVTransientprocess(过渡过程)K+–vLVsRLi(t=0)+–vLVsRLi(t→)RLcircuit1.AnalysisoftransientprocessThecircuitisinsteadystatebeforeswitchKisclosed.i=0,vC=0i=0,vC=VsK+–vCVsRCi(t=0)ThecircuitwillreachnewsteadystateafterswitchKhasbeenclosedforalongtime.+–vCVsRCi(t→)FormersteadystateTransienttermNewsteadystatet1VSvct0?iSVRTransientProcessRCcircuitWhydoestransientprocessexit?1.Thecircuitcomprisesenergystorageelementsofcapacitorsorinductors.2.Energystoredwillchangewhenthecircuitchanges,anditneedstimetostoreorreleaseenergytwpTheconstructionorstateofacircuitischangedCircuitswitching()ccSdvRCvvtdt()cSRivvtByKVL,thereiscdviCdtTakingcurrentascircuitvariable:1()SRiidtvtC+–vCvs(t)RCi(t0)()SdvtdiiRdtCdt2.DerivationofdifferentialEquationsRCcircuitFirst-ordercircuit+–vLvsRLi(t0)()LSRivvt()SdiRiLvtdtFirst-ordercircuitLdivLdt()LLSRvdtvvtL()SLLdvtdvRvLdtdtRLcircuitTakingvoltageascircuitvariable:+–vLvS(t)RLi(t0)CvC＋－＋－22()cccSdvdvLCRCvvtdtdt()LcSRivvvtcdviCdtLdivLdt1()SdiRiLidtvtdtC22()1SdvtdidiRLidtdtCdtSecond-ordercircuitTakingcurrentascircuitvariable:First-ordercircuit0)(01ttexadtdxa0)(01222ttexadtdxadtxdaSecond-ordercircuitHigh-ordercircuit0)(01111ttexadtdxadtxdadtxdannnnnn(1)Conceptsoft=0＋andt=0－Acircuitisswitchedatt=00-:themomentjustbeforeswitching0+:themomentjustafterswitching3.Initialcondition)(lim)0(00tfftt)(lim)0(00tffttInitialcondition:valuesofv,iandtheirderivativeatt=0＋0－0＋0tf(t))0()0(ff)0()0(ff1()()dtCvtiC0011()d()dtiiCC01(0)()dtCviCWhent=0+001(0)(0)()dCCvviCi()isfinite,soivcC+-q(0+)=q(0－)vC(0+)=vC(0－)(2)InitialconditionofRCcircuit00()d0iInaddition,1()()dtLitvL0011()d())dtvvLL001(0)(0)()LLiivdLL(0＋)=L(0－)iL(0＋)=iL(0－)ivL+-LWhent=0+01(0)()tLivdLLLi(3)InitialconditionofRLcircuitv()isfinite,so00()d0vL(0+)=L(0－)iL(0+)=iL(0－)qc(0+)=qc(0－)vC(0+)=vC(0－)Conclusion(2)由换路定律uC(0+)=uC(0－)=8V+-10ViiC+8V-10k0+等效电路mA2.010810)0(Ci(1)由0－电路求uC(0－)或iL(0－)+-10V+uC-10k40kuC(0－)=8V(3)由0+等效电路求iC(0+)iC(0－-)=0iC(0+)例1求iC(0+)+-10ViiC+uC-k10k40k电容开路电容用电压源替代4.Determinationofinitialcondition0)0(0)0(LLuuiL(0+)=iL(0－)=2AVuL842)0(例2t=0时闭合开关k,求uL(0+)iL+uL-L10VK14+uL-10V140+电路2A先求AiL24110)0(由换路定律:电感用电流源替代)0(Li10V14解电感短路求初始值的步骤:1.由换路前电路（一般为稳定状态）求uC(0－)和iL(0－)；2.由换路定律得uC(0+)和iL(0+)。3.画0+等效电路。4.由0+电路求所需各变量的0+值。b.电容（电感）用电压源（电流源）替代。a.换路后的电路（取0+时刻值，方向与原假定的电容电压、电感电流方向相同）。iL(0+)=iL(0－)=ISuC(0+)=uC(0－)=RISuL(0+)=-RIS求iC(0+),uL(0+)0)0(RRIIiSsC例3K(t=0)+–uLiLC+–uCLRISiC解0+电路uL+–iCRISRIS+–0－电路RIS由0－电路得：由0＋电路得：VuuCC24122)0()0(AiiLL124/48)0()0(例3iL+uL-LK2+-48V32C求K闭合瞬间各支路电流和电感电压解由0－电路得：12A24V+-48V32+-iiC+-uL由0+电路得：AiC83/)2448()0(Ai20812)0(VuL2412248)0(iL2+-48V32+－uC例4求K闭合瞬间流过它的电流值。iL+200V-LK100+uC100100C－解（1）确定0－值AiiLL1200200)0()0(VuuCC100)0()0(（2）给出0＋等效电路Aik21100100100200)0(1A+200V-100+100V100100－ki+uL－iCViuLL100100)0()0(AuiCC1100/)0()0(7.2Thesource-freeRCcircuitiK(t=0)+–vRC+–vCR0Initialvoltage:(0)vV201Initialenergystored:(0)2CwCV,,CCRRdvvvvRiiCdtCCdvvRCdt1CCdvdtvRCIntegretingbothsides,weget:1CInvtInARCistheintegrationconstant.Thus,InACvtInARC()tRCCvtAe0(0)vAV0()tRCCvtVeNaturalResponse(固有相应)Thenaturalresponseofacircuitreferstothebehavior(intermsofvoltagesandcurrents)ofthecircuititself,withnoexternalsourcesofexcitation(外部激励源).TimeConstant(时间常数)Thetimeconstantofacircuitisthetimerequiredfortheresponsetodecay(衰减)byafactorof1/eor36.8percentofitsinitialvalue.At,t1000()0.368tRCCvtVeVeVOrRC0Then()tCvtVetv0V0tVe000.368VDecayRatet0()tvteV0.3678820.1353430.0497940.0183250.00674tv0V0tVe000.368V=RCIftheinitialvoltageV0isdetermined,then：t0tveV012345210.512(withdetermined)2(withdetermined)CvCRWvRCiRenergyreleasetimeCurrentiR(t)Powerdissipatedintheresistor00()()