电路基础chapter7

1Chapter7First-OrderCircuits7.0introduction7.1TheSource-FreeRCCircuit7.2TheSource-FreeRLCircuit7.3Unit-StepFunction7.4TheStep-ResponseofaRCCircuit7.5TheStep-ResponseofaRLCircuit27.0introductionInthischapter,wewillfocusoncircuitsthatconsistonlyofsources,resistors,andeither(butnotboth)inductorsorcapacitors.WecallthemasRC(resistor-capacitor)circuitsandRL(resistor-inductor).Networksofthisformfinduseinelectronicamplifiers,automaticcontrolsystems,operationalamplifiers,communicationsequipment,andmanyotherapplications.3theinstantwhenthecircuitchanges换路时刻t0t=0+t=0-theinstantjustpriortocircuitchanges换路前瞬间Initialstate初始状态OnlyiL(0-)oruC(0-)theinstantimmediatelyfollowingtocircuitchanges换路后瞬间Initialvalue初始值orInitialcondition初始条件y(0+)ydenoteanycurrentorvoltage47.1TheSource-FreeRCCircuitAfirst-ordercircuitischaracterizedbyafirst-orderdifferentialequation.Anycircuitwithasingleenergystorageelement（1个储能元件，R或L）anarbitrarynumberofsources（任意个电源）anarbitrarynumberofresistors（任意个电阻）isacircuitoforder1.ApplyKirchhoff’slawstopurelyresistivecircuitresultsinalgebraicequations.ApplythelawstoRCandRLcircuitsproducesdifferentialequations.5OhmslawCapacitorlaw0dtdvCRv0CRiiByKCL1-dvdtvRCRearrangethetermsasIntegratingbothsidesgives1ln-lnvtARC/()tRCvtAeV6Thekeytoworkingwithasource-freeRCcircuitisfinding:1.Theinitialvoltagev(0)=V0acrossthecapacitor.2.Thetimeconstant=RC./0)(teVtvCRwhere7•Thenaturalresponseofacircuitreferstothebehavior(intermsofvoltagesandcurrents)ofthecircuititself,withnoexternalsourcesofexcitation.Thetimeconstantofacircuitisthetimerequiredfortheresponsetodecaybyafactorof1/eor36.8%ofitsinitialvalue.vdecaysfasterforsmallandslowerforlarge.CRTimeconstantDecaysmoreslowlyDecaysfaster8Example1Refertothecircuitbelow,determinevC,vx,andiofort≥0.AssumethatvC(0)=30V.Pleaserefertolectureortextbookformoredetailelaboration.Answer:vC=30e–0.25tV;vx=10e–0.25t;io=–2.5e–0.25tA9Example2Theswitchincircuitbelowisopenedatt=0,findv(t)fort≥0.•Pleaserefertolectureortextbookformoredetailelaboration.Answer:V(t)=8e–2tV1011PracticeproblemIftheswitchinFig.belowopensatt=0,findv(t)fort0andWc(0)127.2TheSource-FreeRLCircuitAfirst-orderRLcircuitconsistsofainductorL(oritsequivalent)andaresistor(oritsequivalent)0RLvvByKVL0iRdtdiLInductorslawOhmslawdtLRidiLtReIti/0)(13•Thetimeconstantofacircuitisthetimerequiredfortheresponsetodecaybyafactorof1/eor36.8%ofitsinitialvalue.•i(t)decaysfasterforsmallandslowerforlarge.•ThegeneralformisverysimilartoaRCsource-freecircuit./0)(teItiRLAgeneralformrepresentingaRLwhere14/0)(teItiRLARLsource-freecircuitwhere/0)(teVtvRCARCsource-freecircuitwhereComparisonbetweenaRLandRCcircuit15Thekeytoworkingwithasource-freeRLcircuitisfinding:1.Theinitialvoltagei(0)=I0throughtheinductor.2.Thetimeconstant=L/R./0)(teItiRLwhere16Example3Findiandvxinthecircuit.Assumethati(0)=5A.•Pleaserefertolectureortextbookformoredetailelaboration.•Answer:i(t)=5e–53tA17Example4Forthecircuit,findi(t)fort0.•Pleaserefertolectureortextbookformoredetailelaboration.Answer:i(t)=2e–2tA18Assumingthati(0)=10A,calculatei(t)andix(t)inthecircuitintheFig.below197.3Unit-StepFunction•Theunitstepfunctionu(t)is0fornegativevaluesoftand1forpositivevaluesoft.0,10,0)(tttuooottttttu,1,0)(ooottttttu,1,0)(20Voltagesource.Representanabruptchangefor:Currentsource21•Initialcondition:v(0-)=v(0+)=V0ApplyingKCL,or•Whereu(t)istheunit-stepfunction7.4TheStep-ResponseofaRCCircuit•Thestepresponseofacircuitisitsbehaviorwhentheexcitationisthestepfunction,whichmaybeavoltageoracurrentsource.()0svVutdvCdtR)(tuRCVvdtdvs22•Integratingbothsidesandconsideringtheinitialconditions,thesolutionoftheequationis:0)(0)(/00teVVVtVtvtssFinalvalueatt-∞Initialvalueatt=0Source-freeResponseCompleteResponse=Naturalresponse+ForcedResponse(storedenergy)(independentsource)=V0e–t/τ+Vs(1–e–t/τ)23ThreestepstofindoutthestepresponseofanRCcircuit:1.Theinitialcapacitorvoltagev(0).2.Thefinalcapacitorvoltagev()—DCvoltageacrossC.3.Thetimeconstant./)]()0([)()(tevvvtvNote:Theabovemethodisashort-cutmethod.YoumayalsodeterminethesolutionbysettingupthecircuitformuladirectlyusingKCL,KVL,ohmslaw,capacitorandinductorVIlaws.24Example5Findv(t)fort0inthecircuitinbelow.Assumetheswitchhasbeenopenforalongtimeandisclosedatt=0.Calculatev(t)att=0.5.Answer:andv(0.5)=0.5182V515)(2tetv257.5TheStep-ResponseofaRLCircuit•Thestepresponseofacircuitisitsbehaviorwhentheexcitationisthestepfunction,whichmaybeavoltageoracurrentsource.Initialcurrenti(0-)=i(0+)=IoFinalinductorcurrenti(∞)=Vs/RTimeconstant=L/R)()()(tueRVIRVtitsos26ThreestepstofindoutthestepresponseofanRLcircuit:1.Theinitialinductorcurrenti(0)att=0+.2.Thefinalinductorcurrenti().3.Thetimeconstant.Note:Theabovemethodisashort-cutmethod.YoumayalsodeterminethesolutionbysettingupthecircuitformuladirectlyusingKCL,KVL,ohmslaw,capacitorandinductorVIlaws./)]()0([)()(teiiiti27Example6Theswitchinthecircuitshownbelowhasbeenclosedforalongtime.Itopensatt=0.Findi(t)fort0.Answer:teti102)(