LAPLACE TRANSFORM AND ITS APPLICATION IN …

1 Pan11 LAPLACE LAPLACE TRANSFORM AND TRANSFORM AND ITS APPLICATION ITS APPLICATION IN circuit IN circuit Definition of the LAPLACE Definition of the LAPLACE Useful LAPLACE TRANSFORM Useful LAPLACE TRANSFORM circuit analysis in S circuit analysis in S The Transfer Function and the Convolution The Transfer Function and the Convolution The Transfer Function and the Steady state The Transfer Function and the Steady state Sinusoidal ResponseSinusoidal The Impulse Function in circuit The Impulse Function in circuit Definition of the LAPLACE Definition of the LAPLACE TransformPierre Simon LAPLACE (1749-1827) :A French astronomer and mathematician Firstpresented the LAPLACE TRANSFORM and its applications to differential equations in Definition of the LAPLACE Definition of the LAPLACE TransformDefinition:[]0()()() a complex variablestLftFsftedtsj ===+ The LAPLACE TRANSFORM is an integral transformation of a function f(t) from the time domain into the complex frequency domain, F(s).

2 Definition of the LAPLACE Definition of the LAPLACE TRANSFORM []1111()()()2 Look-up table ,an easier way for circuit APPLICATION ()()jstjLFsftFsedsjftFs+ == One-sided (unilateral) LAPLACE transformTwo-sided (bilateral) LAPLACE Definition of the LAPLACE Definition of the LAPLACE TransformSimilar to the APPLICATION of phasortransform to solve the steady state AC circuits , LAPLACE TRANSFORM can be used to TRANSFORM the time domain circuits into S domain circuits to simplify the solution of integral differential equations to the manipulation of a set of algebraic Pan88 Functionsf(t) , t>F(s)impulse1stepramptexponentialsine0 ()t ()ut21 Sate 1Sa+1 Ssint 22S + Useful LAPLACE TRANSFORM Useful LAPLACE TRANSFORM Pan99 Functionsf(t) , t>F(s)cosinedamped rampdamped sinedamped cosine0 cost 22SS +atte cosatet ()21Sa+()22Sa ++sinatet ()22 SaSa +++ Useful LAPLACE TRANSFORM Useful LAPLACE TRANSFORM Useful LAPLACE TRANSFORM Useful LAPLACE TRANSFORM Pairs[][][][]-000 ()()(0)() () ()()(),0 ()()1 ()(),0 ()() ()()limlimlimlimtasatsttsdLftsFsfdtFsLfd SLftautaeFsaLeftFsasLfatFaaaftsFsftsFs+ = = => =+ => = = Useful LAPLACE TRANSFORM Useful LAPLACE TRANSFORM PairsExampleUse the LAPLACE TRANSFORM to solve the differential equation.

3 []222682()(0)1'(0)2 Take LAPLACE transfrom2()(0)'(0)6()(0)8()dvdvvutdtdtv vsVssvvsVsvVss++=== + += Useful LAPLACE TRANSFORM Useful LAPLACE TRANSFORM Pairs[]222222242()(0)'(0)6()(0)8()42(68) ()4242()(2)(4)(68)1()(12)()4ttsVssvvsVsv VssssssVssssssVsssssssvteeut + += ++++=++++ ==++++=++ circuit analysis in S circuit analysis in S Domain(1) KCL , ()0 , for any node. Take LAPLACE TRANSFORM ()0 , for any node.(2) KVL , ()0 , for any loop. Take LAPLACE TRANSFORM ()0 , knknkmkmitIsvtVs==== for any circuit analysis in S circuit analysis in S Domain(3) circuit Component Modelsresistor ()() ()() ()()RRRRRRvtRitVsRIsIsGVs=== circuit analysis in S circuit analysis in S Domain0inductor ()1 ()(0)() ()()(0)()(0) ()LLtLLLLLLLL divtLdtitivdLVssLIsLiVsiIssLs ==+= =+ (0)LLi (0)Lis circuit analysis in S circuit analysis in S Domain0capacitor 1 ()(0)() ()()(0)(0)1 ()()CCtCCCCCcCCCdviCdtvtvidCIssCVsCvvVsI sssC ==+= =+ 1sC(0)Cvs 1sC(0)CCv circuit analysis in S circuit analysis in S DomainMi2i1L1L2v1v2+--+Coupling inductorsV1(s)=L1SI1(s)-L1i1(0)+MSI2(s)- Mi2(0)V2(s)= MSI1(s)-Mi1(0)+L2SI2(s)-L2i2(0)

4 DtdLdtdMdtdMdtdL22122111iiviiv+=+= circuit analysis in S circuit analysis in S DomainFor zero initial conditions()impedance ()()()1admittance ()()()()()()'VsZsISISYsVsZsVsZsIsohmslaw insdomain==== @ circuit analysis in S circuit analysis in S DomainThe elegance of using the LAPLACE TRANSFORM in circuit analysis lies in the automatic inclusion of the initial conditions in the transformation process, thus providing a complete (transient and steady state) circuit analysis in S circuit analysis in S DomainCircuit analysis in s domainnStep 1 : TRANSFORM the time domain circuit into s-domain 2 : Solve the s-domain Nodal analysis or mesh 3 : TRANSFORM the solution back into time circuit analysis in S circuit analysis in S DomainExampleFind vo(t) given vo(0)=5VS-domain equivalent circuit101s+ circuit analysis in S circuit analysis in S Domain2 Nodal analysis10()()() ()(1)(2)12()(1015)() VoooottoVsVsVssssVsssssvteeut +++=++ ==+++++ =+101s+ The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution IntegralGiven a linear circuit N in s domain as shown belowTransfer function H(s) is defined as)s(X)s(Y)s(H= The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution Integral ()() , ()() ; () ()() , ()() ; () ()() , ()() ; () ()() , ()().

5 ()oioiIfYsVsXsVsthenHsvoltagegainIfYsIsX sIsthenHscurrentgainIfYsVsXsIsthenHsimpe danceIfYsIsXsVsthenHsadmittance========= === The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution IntegralGiven the transfer funtionH(s) and input X(s) , then Y(s)=H(s)X(s)If the input is (t) , then X(s)=1 and Y(s)=H(s)Hence , the physical meaning of H(s) is in fact the LAPLACE TRANSFORM of the impulse response of the corresponding The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution Integralin time domainY(s)=H(s)X(s) , in s-domain()()()()() ythtxdhtxt = @Geometrical interpretation of finding the convolution integral value at t=tkis based on : The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution Integral(1)Approximating the input function by using a series of impulse functions.

6 (2)Shifting property of linear systems input x(t) outputy(t)x(t- ) output y(t- )(3)Superposition theorem for linear systems(4)Definition of integral : finding the The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution Integral(1)Input x( ) is approximated using impulsefunctions , x( )=0 , for <0 0111222333 , () , 0 , (0) , (0) , () , ()2 , () , ()3 , () , ()kkinstantxvalueareaxxxxxxxx ====VVVVVVV01 2 3 4 5 (0)x()x 01 ()x 2 011220()()()()()()kxfffxkk = + + + L@VVV The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution Integral(2) Use the linearity property11112222 ()(0)() (0)()()() ()()()() ()() inputoutputresponsexxhxxhxxhupto VVVVVVMMkt= The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution Integral(3) Use superposition theoremto find the total approximate response$0()() ()[]nkkkkytxkhtktninteger== = VVVV The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution Integral(4) Take the limit , d ,()()()kkytxhtd = Due to causality principle , h(t- )=0 for t tkand x( )=0 for <00()()ktkhtxd= $()()kkytyt The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution Integral ()() , ()() , (4),()()()(4)(4)()

7 TGivenxtuthteutfindyythtxdyhxd === = The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution IntegralStep1. 4 2024 4 2024 ()x ()h 2 2 4 4 The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution IntegralStep2. Shift to tk=4 4 2024 4 (4)h 4 2 The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution IntegralStep3. Find the product h(4- )x( ) The Transfer Function and the The Transfer Function and the Convolution IntegralConvolution IntegralStep4. Find the integral (area)444(4)44000444(4)==(1)(1)yedeedeee ee == = Step5. Check41111()()()1()(1) , 1(4)(1) taaYsHsXssassasyteaaye == =+++ = = = The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal ResponseFrom definition of transfer function()()()()()()YsHsXsYsHsXs= =Assume input X(t)=Acos(wt+Ф) and H(s) is given, then one can get the steady state solution without needing a separate The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal Response22*11*111:()coscossinsin(cossin) ()()()().

8 ()1()2()(cossin)ssjproofXtAtAtAsXssYsHsX sKKsjsjothertermsduetopolesundersteadyst ateKKYssjsjKHjAesjHsAssj = =+ ==+ + + =+ + == = + The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal Response()()|()|()|()|cos[()]..(())(())| ()|()jssssLetHjHjeandtakeinverseLaplacet ransformThenytAHjtiePXtAthenPytAHj = = ++ = = + Pan4040 Example : The transfer function H(s) of the circuit given below is the steady state solution of Vo(t) for the given Vg(t). The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal Response261000(5000)()60002510()120cos(5 00030)gsHsssVttV +=++ =+ The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal Response66.

9 50001000(50005000)(5000)25105000(6000)25 1024562()120cos(50003045)6202cos(500015) ssoSolutionLetsjjjEvaluateHjjThenVtttV = =+= ++ = = + = Pan4343nIn theory , the relationship between H(s) and H(jw) provides a link between the time domain and the frequency some cases , we can determine H(jw) experimentally and then construct H(s) from the The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal ResponseExample: Find the impulse response of the Example: Find the impulse response of the following +_ov+ The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal Response(a) Time domain solution(a) Time domain solution0() 0, (0)01() 0, (0) 1 oootodvRCvtdtAttvtAttvdtCRVRC +++====== ()t () The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal Response 0, ()0 Fortt +>=R1() , (0)ovtvRC+=+_C01 ()()ootRCodvRCvdtvteutRC += = The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal Response(b) s(b) s--domain solutiondomain solution.

10 ()()()oinzeroICVsHsVs=Find the transfer functionFind the transfer functionTransform into sTransform into s--domain circuitdomain circuit ()inVsR+_() The Transfer Function and the The Transfer Function and the Steady State Sinusoidal ResponseSteady State Sinusoidal Response-11()1 ()1()1 ()[()]1 ()ointRCVsCsHsVsRCsRCshtHseutRC ===++ == L()inVsR+_()oVs1 CsSame The Impulse Function in circuit The Impulse Function in circuit AnalysisAnalysisExample 1: Impulse voltage source excitationExample 1: Impulse voltage source excitation()OVt ()it(0)0i = The Impulse Function in circuit The Impulse Function in circuit AnalysisAnalysis(a) Time domain solution(a) Time domain solution0 0, (0)01 (0)() ()tOOAttiiVxdxLVAL +== == ()OVt (0)i+The impulse voltage source has stored energy,The impulse voltage source has stored energy,, in the inductor as an initial current, in the inductor as an initial currentin an infinitesimal an infinitesimal (0)()iL+ The Impulse Function in circuit The Impulse Function in circuit AnalysisAnalysis 0, ()0 Fortt +>=()it/0 , (0) ()()