chapter15TheLaplaceTransform.ppt
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1、Chapter 15 The Laplace Transform,15-1 Introduction,15-2 Definition of the Laplace Transform,15-4 The Inverse Laplace Transform,15-5 Application to Circuits,15-6 Transfer Functions,15-7 The Convolution Integral,15-10 Summary and Review,15-3 Properties of the Laplace Transform,15-1 Introduction,1. Ter
2、ms,2. Introduction of this chapter,convolution integral,卷积,asterisk,星号 *,1. terms,commutative,可交换的,distributive,可分配的,associative,结合的,We use the Laplace transformation to transform the circuit from the time domain to the complex frequency (s) domain, obtain the solution, and apply the inverse Laplace
3、 transform to the result to transform it back to the time domain.,2. Introduction of this chapter,15-2 Definition of the Laplace Transform,Given a function f(t), its Laplace transform, denoted by F(s) or Lf(t), is given by,Where s is a complex variable given by,The Laplace transform is an integral t
4、ransformation of a function f(t) from the time domain into the complex frequency domain, giving F(s).,Eq. (1),1. Definition,A companion to the direct Laplace transform in Eq. (1) is the inverse Laplace transform given by,2.Laplace transform of the unit step function (t) and that of the difference or
5、der derivative of (t).,L : s-1, s0, s1, s2, ,Laplace transform regular pattern of the unit step function:,15.5 Application to circuits,Steps in applying the Laplace transform:,1. Transform the circuit from the time domain to the s domain.,2. Solve the circuit using nodal analysis, mesh analysis, sou
6、rce transformation, superposition, or any circuit analysis technique with which we are familiar.,3. Take the inverse transform of the solution and thus obtain the solution in the time domain.,Time domain Phasor domain s domain,uR=RiR,iR,R,uR,R,IR(s),UR(s),uR=RiR,iR=GuR,UR(s)=RIR(s),IR(s)=GUR(s),1.R,
7、VCR relationship of elements in the s domain,IL(s),sL,UL(s),LiL(0-),IL(s),2. L,iC,C,uC,3. C,IC(s),sC=Y,UC(s),IC(s),UC(s),KCL, KVL in the s domain:,The series connection of RLC circuit:,u(t),R,IS(s),U(s),(b),Example: The parallel circuit of RC in Fig. (a).,find the response of u(t).,Solution: we tran
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