電子科學(xué)與技術(shù)計(jì)算機(jī)知識(shí)點(diǎn)拉氏變換

THELAPLACETRANSFORM(拉氏變換)

PierreSimonLaplace(1791-1872)DefinitionofTheLaplaceTransformTheLaplacetransformisanintegraltransformationofafunctionf(t)fromthetimedomainintothecomplexfrequencydomain,givingF(S).Itisalsoknownasone-sidedLaplacetransform.f(t)：原函數(shù)；F(S)：f(t)在S域中的象函數(shù)。TheinverseLaplacetransform:PropertiesofTheLaplaceTransformLinearity（線性定理）Timedifferentiation（微分定理）Timeintegration（積分定理）Timeshift（時(shí)域位移定理）Initialandfinalvalues（初值定理與終值定理）Inorderforthefinal-valuetheoremtohold,allpolesofF(S)mustbelocatedinthelefthalfoftheSplane,theonlyexceptionisthecaseinwhichF(S)hasasimplepoleatS=0.Convolution（卷積定理）Frequencyshift（頻域位移定理）Scaling（尺度變化定理）LaplacetransformsofsomefunctionsTheunit-stepfunction1(t)Theunit-impulsefunction(t)TherootsofP(S)=0arecalledthezerosofP(S),

whiletherootsofQ(S)=0arethepolesofQ(S).（零點(diǎn)和極點(diǎn)）TheInverseLaplaceTransformStepstofindtheinverseLaplacetransform:poseF(S)intosimpletermsusingpartialfractionexpansion.FindtheinverseofeachtembymatchingentriesinTable7-4.SimplePolesHowtofindtheexpansioncoefficientsKn?Methoda.（Theresiduemethod）Methodb.RepeatedPolesComplexPoles解：Or：APPLICATIONTOCIRCUITS(運(yùn)算電路)TransformacircuitinthetimedomaintotheSdomainResistorThevoltage-currentrelationshipforaresistor:

L[U]=L[Ri]，L[U]=U(S)，L[i]=I(S)U(S)=RI(S)，orI(S)=GU(S)CapasitorIC(S)=SCUC(S)CuC(0)InductorUL(S)=SLIL(S)LiL(0)Theinitialconditionismodeledasavoltageorcurrentsource.CoupledinductorKCL、KVLintheSdomainTheimpedanceintheSdomainTheadmittanceintheSdomainZ(S)I(S)=Useg(S)，Y(S)U(S)=Iseg(S)StepsinapplyingtheLaplacetransform:TransformthecircuitfromthetimedomaintotheSdomain.Solvethecircuitusingnodalanalysis,meshanalysis,sourcetransformation,superposition,oranycircuitanalysistechniquewithwhichwearefamiliar.Taketheinversetransformofthesolutionandthusobtainthesolutioninthetimedomain.解：圖(b)：解：圖(b)：例：如圖電路，R1=30，R2=R3＝5，L1=0.1H，C=1000F，E=140V，開(kāi)關(guān)閉合已久，求開(kāi)關(guān)打開(kāi)后的uk(t)和uC(t)。解：圖(b)：例：如圖電路，R=10，＝0.9，L=0.05H，iL(0)=0，e(t)=100sin(2000t+60)V，求電感中的過(guò)渡電流。解：圖(b)：圖(c)