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1、Ch3TransientAnalysisofCircuits -电路的暂态分析电路的暂态分析transientstate暂态暂态steadystate稳态稳态transientprocess暂态过程暂态过程lawofswitch换路定理换路定理first-ordercircuit一阶电路一阶电路three-factormethod三要素法三要素法timeconstant时间常数时间常数integralcircuit积分电路积分电路differentialcircuit微分电路微分电路Terms:2 2 GraspGrasp lawofswitchandcomputeinitialvalue;l
2、awofswitchandcomputeinitialvalue;3 3 ConceptsofConceptsof Zero-inputResponse,Zero-stateZero-inputResponse,Zero-stateResponse,CompleteResponseResponse,CompleteResponse; ;1 1 UnderstandtheconceptsoftransientstateUnderstandtheconceptsoftransientstate、steadystateandphysicalsignificanceoftimesteadystatea
3、ndphysicalsignificanceoftimeconstant;constant;Outline:4 4 Applicationofthree-factormethod.Applicationofthree-factormethod.Forthedirectcurrentcase, Lisshort-circuitedForthedirectcurrentcase, Cisopen-circuited+-LinductoruCiC+_capacitorRu+_resistorRu+_resistorRu+_resistorresistorresistorRu+_resistorBef
4、orethemovementofKi=0,uC=0i=0,uC=UsTransitionprocessofcircuitK+uCUsRCi t= 0LongafterKisclosedTransitionprocess：theprocessbeundergonebythecircuittransientingfromonesteadystatetoanothersteadystate.Switching：thecircuit(structureorparameter)changes.i+uCUsRCKisclosed2.the structureorparametersofthecircuit
5、arechangedDifferencesbetweensteady-stateanalysisandtransientanalysisSteadystateTransient1.Longafterswitching；1.Justafterswitching;2.iL、uCtime-varying;3.Circuitdescribedwithalgebraicequations;3.Circuitdescribedwithdifferentialequations;2.IL、UCkeepunchanged;Reasonsfortransitionprocess1.Thecircuitconta
6、insenergy-storageelementsL、C3.1 lawofSwitchandInitialValuesRCRC circuitcircuit: :Note：lawofswitchisjustusedtodeterminetheinitialvalue uC、 iLatthemomentofswitching。 Assumptions：t=0 switchinginstant t=0 beforeswitchinginstant t=0+ afterswitchinginstantRLRL circuitcircuit：LawofSwitch：capacitorvoltagean
7、dinductorcurrentcannotchangeabrubtlyattheinstantofswitching.Determinationofinitialvalue:OutlineofsolutionOutlineofsolution：InitialvalueInitialvalue：thevalueofthevalueof eacheach u u、i i at at t t=0=0+ + inthecircuitinthecircuit。1.FinduC(0-)oriL(0-)withtheaidofthecircuit(steadystate)beforeswitching.2
8、.DetermineuC(0+) oriL(0+)withlawofswitch.3.Drawequivalentcircuitdiagramatt=0+.(1)ifuC(0-)=0, , Replacecapacitorbyshortcircuit;iL(0-)=0, Replaceinductorbyopencircuit.3.Drawequivalentcircuitdiagramatt=0+. (2)ifuC(0-) 0, , Replacecapacitorbyvoltagesource;iL(0-) 0, Replaceinductorbycurrentsource.Getthev
9、alueatt=0+，thedirectionisidenticaltoassumeddirectionofcapacitorvoltageandinductorcurrent.4.Findthevalueofdesiredvariablesatt=0+within0+circuitdiagrambyknownmethods(Ohmslawetal.).Example1Example1SolutionSolution： (1)Solvefor(1)SolveforAccordingtoAccordingtotheconditions:theconditions:Bylawofswitch:By
10、lawofswitch:ForthecircuitinForthecircuitin FigFig(a). (a). letusletusdeterminetheinitialvaluesofeachdeterminetheinitialvaluesofeachvoltagesandcurrents.Supposethatvoltagesandcurrents.Supposethatthecircuitisinsteadystatebeforethecircuitisinsteadystatebeforecircuitswitching,andUcircuitswitching,andUC C
11、=0=0、I IL L=0.=0.S S(a(a) )C CU R R2 2R R1 1t t=0=0+-L LiC、uLchangedabruptly(2) Accordingto t=0+circuit，solveforinitialvaluesofothervoltagesandcurrents.S SC CU R R2 2R R1 1t=0t=0+-L L(a) (a) iL(0+ )U iC (0+ )uC (0+)uL(0+)_u2(0+)u1(0+)i1(0+ )R R2 2R1+_+-(b) (b) t= 0+ExampleExample2 2Determinetheiniti
12、alvaluesofcurrentsandvoltagesinDeterminetheinitialvaluesofcurrentsandvoltagesincircuitshowninFigcircuitshowninFig. . SupposethatthecircuitisinSupposethatthecircuitisinsteadystatebeforecircuitswitchingsteadystatebeforecircuitswitching. .4 4 2 2 +_R RR R2 2R R1 1U U8V8V+4 4 i i1 14 4 i iC C_u uC C_u u
13、L Li iL LR R3 3L LC Ct= 0 equivalentcircuit2 2 +_R RR R2 2R R1 1U U8V8Vt t=0=0+4 4 i i1 14 4 i iC C_u uC C_u uL Li iL LR R3 34 4 Solution：(1) Accordingto t= 0-circuit, andfind uC(0)、iL(0) ContinuedContinuedAccordingtolawofswitch：4 4 2 2 +_R RR R2 2R R1 1U U8V8V+4 4 i i1 14 4 i iC C_u uC C_u uL Li iL
14、 LR R3 3L LC Ct= 0 equivalentcircuitContinued：(2) Accordingtot=0+circuit,find iC(0+)、uL(0+),u uc c (0(0+ +) )Substitutei iL L (0(0+ +) )C C2 2 +_R RR R2 2R R1 1U U8V8Vt t=0=0+4 4 i i1 14 4 i iC C_u uC C_u uL Li iL LR R3 34 4 L Lt= 0+ circuit4V1A4 4 2 2 +_R RR R2 2R R1 1U U8V8V+4 4 i iC C_i iL LR R3
15、3i it= 0+ circuit4V1A4 4 2 2 +_R RR R2 2R R1 1U U8V8V+4 4 i ic c_i iL LR R3 3i i Weget2 2 +_R RR R2 2R R1 1U U8V8Vt t=0=0+4 4 i i1 14 4 i iC C_u uC C_u uL Li iL LR R3 34 4 ResultsResults：Electricquantity2 2 +_R RR R2 2R R1 1U U8V8Vt t=0=0+4 4 i i1 14 4 i iC C_u uC C_u uL Li iL LR R3 34 4 Attheinstan
16、tofswitchingAttheinstantofswitching，keepunchanged.keepunchanged.canchangeabruptly.Summary1. 1. 换路瞬间，换路瞬间，换路瞬间，换路瞬间，u uC C、 i iL L 不能跃变不能跃变不能跃变不能跃变, , 但其它电量均可以跃但其它电量均可以跃但其它电量均可以跃但其它电量均可以跃 变。变。变。变。 3. 3. 换路前换路前换路前换路前, , 若若若若uC(0(0-) ) 0 0, , 换路瞬间换路瞬间换路瞬间换路瞬间 ( (t t=0=0+ +等效电路中等效电路中等效电路中等效电路中), ), 电容元件
17、可用一理想电压源替代电容元件可用一理想电压源替代电容元件可用一理想电压源替代电容元件可用一理想电压源替代, , 其电压为其电压为其电压为其电压为uc(0(0+); ); 换路前换路前换路前换路前, , 若若若若iL(0(0-) ) 0 0 , , 在在在在t t=0=0+等效电路中等效电路中等效电路中等效电路中, , 电感元电感元电感元电感元件件件件 可用一理想电流源替代可用一理想电流源替代可用一理想电流源替代可用一理想电流源替代，其电流为，其电流为，其电流为，其电流为iL(0(0+) )。2. 2. 换路前换路前换路前换路前, , 若储能元件没有储能若储能元件没有储能若储能元件没有储能若储能
18、元件没有储能, , 换路瞬间换路瞬间换路瞬间换路瞬间( (t t=0=0+ +的等的等的等的等 效电路中效电路中效电路中效电路中) )，可视电容元件短路，电感元件开路。，可视电容元件短路，电感元件开路。，可视电容元件短路，电感元件开路。，可视电容元件短路，电感元件开路。3.2 Responseof RCCircuitSolvingmethodofone-ordertransientcircuitSolvingmethodofone-ordertransientcircuit1.Classicmethod:accordingtoexcitation(source)，findtheresponse
19、ofcircuitthroughsolvingdifferentialequation。2.Three-factormethodInitialvalueSteadyvalueTimeconstantFind（threefactors)containingonlyoneenergy-storageelement,containingonlyoneenergy-storageelement, describedbyone-orderdifferentialequation,describedbyone-orderdifferentialequation, iscallediscalledone-o
20、rderlinearcircuit.one-orderlinearcircuit.One-ordercircuitOne-ordercircuitsubstitutingGiven(1) ApplyingKVL1. Capacitorvoltage uC(t 0) Zero-inputresponse:excitationisswitchedoff, theresponseiscausedonlybyinitialenergy-storageofcapacitor.EssenceEssence：DischargingofRCcircuitDischargingofRCcircuit3 .2 .
21、1 Zero-inputResponseof RCCircuit+-SRU21+ +(2(2) ) SolvingSolvingequationequation：eigenequationhence： u uCCwoulddecaywoulddecay exponentiallyfrominitialvalue.exponentiallyfrominitialvalue.(3(3) ) CapacitorvoltageCapacitorvoltage u uC CInitialvalue：Weobtainresistorvoltagedischargingcurrentcapacitorvol
22、tagecapacitorvoltage2.2.CapacitorcurrentandresistorvoltageCapacitorcurrentandresistorvoltagetO3. 、 、 variation curvecurve4.4.TimeconstantTimeconstant(2)PhysicalsignificanceLetunitunit: : S S(1)dimensionwhenThevalueoftimeconstantreflectsthelengthoftimeThevalueoftimeconstantreflectsthelengthoftimespen
23、tontransientprocess.spentontransientprocess.：thelenghthoftimethatisspentbycapacitorvoltageondecayingto36.8%oftheoriginalvoltagewhen t t=5=5 ，transientprocessisnearlyovertransientprocessisnearlyover, , u uCCreachesreachestosteady-statevalue.tosteady-statevalue.(3)(3)TransienttimeTransienttimet0.368U
24、0.135U 0.050U 0.018U 0.007U 0.002UTheoretically,Theoretically,、circuitreachestosteadystate.circuitreachestosteadystate.Inengineeringperspective,atransientlasts3 -5 . 3.2.2 Zero-stateresponseof RCcircuitZero-stateresponse: Theresponsecausedbytheexcitationofenergy-storagecomponentwithoutinitialenergy.
25、EssenceEssence： chargingprocessofRCcircuitchargingprocessofRCcircuitGivenGiven：u uC C(0(0- -)=0)=0uC(0 -) = 0sRU+_C+_iuCCaculateCaculate：capacitorvoltagecapacitorvoltage u uC C(t)andcurrent(t)andcurrenti(t)i(t)CompletesolutionCompletesolution = =paticularsolutionpaticularsolution + +complementarysol
26、utionofhomogeneousequations1.Capacitorvoltage uC(1)ApplyingKVLuC(0 -) = 0sRU+_C+_iuc(2)ParticularsolutionKuC= = Assume:Weget(Forcedresponse、steady-stateresponse)强制分量、稳态分量强制分量、稳态分量(3)ComplementarysolutionHomogeneousequation,Complementarysolution:Completesolution:(4)CompletesolutionFrominitialvalueuC(
27、0+)=0,So A=-Ufreeresponse，transientresponseTransientresponseSteady-stateresponseSteady-statevoltage-U+UOnlyexitsinTransientprocess 63.2%U-36.8%Uto3. 3. 、 variationcurvevariationcurvetwhen t= describesthelengthoftimespentonthevaluedescribesthelengthoftimespentonthevalueofof u uC C risingfrominitialva
28、luetorisingfrominitialvalueto 6 63.23.2% ofsteady-stateofsteady-statevaluevalue。2. 2. CurrentCurrent i iC C4. 4. TimeconstantTimeconstant UU0.632Uthelargererthe ,thelongerthetransientdurationConclusionConclusion：0.9980.998U Ut t00 00.6320.632U U 0.8650.865U U 0.9500.950U U 0.9820.982U U 0.9930.993U
29、UtO3 .2 .3 CompleteResponse of RCCircuitCompleteresponse：energy-storagecomponentshaveinitialenergyattheinstantofswitching,andthecurcuitcontainsexcitationafterswitchingoperation.AccordingtoSuperposition:AccordingtoSuperposition:CompleteresponseCompleteresponse =Zero-inputresponse+Zero-stateresponseZe
30、ro-inputresponse+Zero-stateresponseuC(0 -) = U0sRU+_C+_iuCSteady-stateresponseZero-inputresponseZero-stateresponseTransientresponseConclusion2Conclusion2： CompleteresponseCompleteresponse =Steady-stateresponse+Transientresponse Conclusion1Conclusion1： CompleteresponseCompleteresponse=Zero-inputrespo
31、nse+Zero-stateresponse=Zero-inputresponse+Zero-stateresponseSteady-statevalueInitialvalueSolutionofFirst-orderCircuits(Classicmethod):1.Determinetheinitialvalueoftheenergystorageelement;2.Writethedifferentialequationforthecircuitfort0;3.Determinethetimeconstantofthecircuitfort0;4.Writethecompletesol
32、utionasthesumofthenaturalandforcedresponse;5.Applytheinitialvaluetothecompletesolution,todeterminetheconstanK;FinalvalueInitialvalue3.3 Three-factorMethodforOne-orderCircuitAccordingtoresultsofClassicmethodAccordingtoresultsofClassicmethodCompleteresponseCompleteresponseuC(0 -) = UosRU+_C+_iuc：repre
33、sentsvoltagerepresentsvoltage、currentfunctionsofcurrentfunctionsofone-ordercircuit.one-ordercircuit.wherewhere, ,initialvalueinitialvalue-（threefactorsthreefactors） steadystatevalue-timeconstanttimeconstant -WhencircuitWhencircuit d drivenbyDC,thesolutionofone-orderrivenbyDC,thesolutionofone-orderli
34、nearcircuitdifferentialequationisgenerallyexpressedlinearcircuitdifferentialequationisgenerallyexpressed：Allone-ordercircuitcanbesolvedbythree-factormethod.Allone-ordercircuitcanbesolvedbythree-factormethod.Keypointsofthree-factormethodEndEndpointpointStartStartpointpoint(1)Findinitialvalue,steady-s
35、tatevalue,andtimeconstant；(3)Drawvoltage、currentvariationcurve.(2)Substitutethesevaluesintogeneralexpression;tf(t)O FindcurrentsandvoltagesafterswitchingFindcurrentsandvoltagesafterswitching, , wherewhere capacitorcapacitorCbehavesasopencircuitbehavesasopencircuit, , inductor Lbehavesasshortcircuit.
36、(1)CaculateDeterminationofThreeFactorsExample：uC+-t=0C10V5k5k 1 FS5k +-t=03 6 6 6mAS1H1H1) Find at t=0- inthe circuit;2)Findbythelawofswitch.3) Findthevalueofdesiredvariablesatt=0+within0+circuitdiagrambyknownmethods(Ohmslawetal.).InequivalentcircuitInequivalentcircuit at at t t =(=(0 0+ +): ):Cbeha
37、vesasshortcircuit.Thevalueis(1)IfCbehavesasvoltagesource，Thevalueis I0 ; , Lbehavesasopencircuit.(2)If,Lbehavesascurrentsource，Note:(2)Caculate 1)1)Forsimpleone-ordercircuitsForsimpleone-ordercircuits，R0=R; ;2)2) Forcomplexone-ordercircuitsForcomplexone-ordercircuits， R R0 0 equalsequalsequivalenteq
38、uivalent resistancebetweenterminalsofenergy-storageresistancebetweenterminalsofenergy-storageelementbyzeroingthesourcesandenergy-storageelementbyzeroingthesourcesandenergy-storageelement.element.(3)(3)CaculateCaculate Forone-orderRCcircuitForone-orderRCcircuitForone-orderRLcircuitForone-orderRLcircu
39、itNote:R0ThecaculationofRThecaculationofR0 0 issimilartothatofTheveninissimilartothatofTheveninresistence,bylookingbackintothetwoterminalsofresistence,bylookingbackintothetwoterminalsofenergy-storageelement.energy-storageelement.R1U+-t=0CR2R3SR1R2R33)Alltheresponseswithinacircuitbearthesametimeconst
40、ant。Solution：Applythree-factormethodApplythree-factormethodExample1Example1：CloseSatt=0，thecircuithasreachedsteadystatebeforecloseS.Find、and .(1)Initialvalue(1)Initialvaluet=0- circuit9mA+-6k RS9mA6k 2 F3k t=0+-CR teuuuuCCCC- - - -+ + = =+ +)()0()(2)(2)SteadystateSteadystatevaluevalue(3) (3) Timecon
41、stantTimeconstant t circuit9mA+-6k R3k t=0- circuit9mA+-6k RThreeThreefactorsfactorsu uC C variationcurveshownasFigure1variationcurveshownasFigure1. .18V54VFigure1.uFigure1.uC C curvecurvetOApplythree-factormethodfor directly54V18V2k t t=0=0+ +-S9mA6k 2 F3k t=0+-CR3k 6k +-54 V9mAt=0+circuitExample2：
42、Accordingtocircuitat t=0-ThecircuithasreachedsteadystatebeforeSclosed.ThecircuithasreachedsteadystatebeforeSclosed.SclosedwhenSclosedwhen t t=0=0，FinFind d uC C、iC C、i1 1 anand d i2 2 . .Find+-St=06V1 2 3 +-t=0- circuit1 2 + +- -6V3 +-Solution：Applythree-factorApplythree-factormethodmethodFindFind +
43、-St=06V1 2 3 +-2 3 +-( 、 in parallel)+-St=06V1 2 3 +-3.4 DifferentialCircuitandIntegralCircuit微分电路和积分电路微分电路和积分电路3.4.1 Differentialcircuit ThesetwocircuitsareRCcircuitsdrivenbysquareThesetwocircuitsareRCcircuitsdrivenbysquarepulsewithdifferenttimeconstants.pulsewithdifferenttimeconstants.1. 1. circui
44、tcircuitConditions:Conditions:(2) (2) Output voltage is the voltage across Output voltage is the voltage across R RTtU0tpCR+_+_+_2.2.analysisanalysisAccording to KVLAccording to KVLFromtheformulaabove outputvoltageisapproximatelyinpropotionto thedifferentialofinputvoltagetothetime.3.3.waveformwavefo
45、rmtt1UtpOtOCR+_+_+_3.4.2 integralcircuitconditonsconditons(2) (2) OutputvoltageisthevoltageacrosscapacitorOutputvoltageisthevoltageacrosscapacitor C C。1. 1. circuitcircuitoutputvoltageistheintegrationofinputvoltage。2.analysisTtU0tpCR+_+_+_3.waveform3.waveformt2Utt1tt2t1Utt2t1UThiskindofwaveformcanbeusedinoscillograph.Application:Application:u1
