11.1 The Laplace Transform 11.2 Applications of Laplace Transform Chapter11 The Laplace Transform 拉普拉斯变换

1. Definition of the Laplace Transform The Laplace transform is an integral transformation of a function f(t) from the time domain into the complex frequency domain, giving F(s) The Laplace Transform 拉普拉斯变换 The inverse Laplace transform 拉普拉斯反变换 Laplace transform pair 拉普拉斯变换对 complex frequency 复频率

(1) (3) (2) Example 11.1 Determine the Laplace transform of each of the following functions: (1) u(t), (2), and (3) Solution:

2. Properties of the Laplace Transform (1) Linearity 线性性质 If (2) Time Shift 时移 (3) Frequency Shift 频移 (4) Time Differentiation 微分 (5) Time Integration 积分

3. The Inverse Laplace Transform 拉普拉斯反变换 The inverse Laplace transform The general form of (1) Decompose F(s)into simple terms using partial fraction expansion 部分分式展开. Steps to find the inverse Laplace transform: (2) Find the inverse of each term by matching entries in table 14.2

If when p i is Simple Poles the residue method: 留数法 (1) Simple Poles 单根 Partial fraction expansion 部分分式展开

Example 11.2 Find the inverse Laplace transform of By the residue method Solution:

p is repeated poles The inverse transform (2) Repeated Poles 重根

Example 11.3 Obtain g(t) if Solution:

The inverse transform

When The poles (3) Complex Poles 共轭复根 let

11.2 Applications of Laplace Transform (3) Take the inverse transform of the solution and thus obtain the solution in the time domain. Steps in applying the Laplace transform: (1) Transform the circuit from the time domain to the s domain. (2) Solve the circuit using any circuit analysis technique with which we are familiar.

(1) For a resistor In the time domain In the s domain 1. Circuits Element Models

(2) For an inductor In the time domain In the s domain

(3) For a capacitor In the time domain In the s domain

(4) For the impedance Under zero initial conditions:

Example 11.4 Find v o (t) in the circuit. Assume v o (0 - )=5V Solution: Transform the circuit to the s-domain Apply nodal analysis. At the top node Thus

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