电子科学与技术计算机知识点拉氏变换

THELAPLACETRANSFORM(拉氏变换)

PierreSimonLaplace(1791-1872)DefinitionofTheLaplaceTransformTheLaplacetransformisanintegraltransformationofafunctionf(t)fromthetimedomainintothecomplexfrequencydomain,givingF(S).Itisalsoknownasone-sidedLaplacetransform.f(t)：原函数；F(S)：f(t)在S域中的象函数。TheinverseLaplacetransform:PropertiesofTheLaplaceTransformLinearity（线性定理）Timedifferentiation（微分定理）Timeintegration（积分定理）Timeshift（时域位移定理）Initialandfinalvalues（初值定理与终值定理）Inorderforthefinal-valuetheoremtohold,allpolesofF(S)mustbelocatedinthelefthalfoftheSplane,theonlyexceptionisthecaseinwhichF(S)hasasimplepoleatS=0.Convolution（卷积定理）Frequencyshift（频域位移定理）Scaling（尺度变化定理）LaplacetransformsofsomefunctionsTheunit-stepfunction1(t)Theunit-impulsefunction(t)TherootsofP(S)=0arecalledthezerosofP(S),

whiletherootsofQ(S)=0arethepolesofQ(S).（零点和极点）TheInverseLaplaceTransformStepstofindtheinverseLaplacetransform:poseF(S)intosimpletermsusingpartialfractionexpansion.FindtheinverseofeachtembymatchingentriesinTable7-4.SimplePolesHowtofindtheexpansioncoefficientsKn?Methoda.（Theresiduemethod）Methodb.RepeatedPolesComplexPoles解：Or：APPLICATIONTOCIRCUITS(运算电路)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：图(b)：解：图(b)：例：如图电路，R1=30，R2=R3＝5，L1=0.1H，C=1000F，E=140V，开关闭合已久，求开关打开后的uk(t)和uC(t)。解：图(b)：例：如图电路，R=10，＝0.9，L=0.05H，iL(0)=0，e(t)=100sin(2000t+60)V，求电感中的过渡电流。解：图(b)：图(c)