Electronic Circuits with MATLAB, PSpice, and Smith Chart

Appendix ALaplace Transform

A.1 Definition of Laplace Transform

The (unilateral or one‐sided) Laplace transform is defined for a function x(t) of a real variable t (often meaning the time) as

(A.1)

where s is a complex variable, the lower limit, t₋, of integration interval is the instant just before t = 0, and x(t) is often assumed to be causal in the sense that it is zero for all t < 0.

A.2 Inverse Laplace Transform

Suppose an s‐function X(s) is given in the form of a rational function, i.e. the ratio of an Mth‐degree polynomial Q(s) to an Nth‐degree polynomial P(s) in s and it is expanded into the partial fraction form as

(A.2)

(A.3)

where

(A.4a)

(A.4b)

Then the inverse Laplace transform of X(s) can be obtained as

(A.5)

Example A.1 Inverse Laplace Transform

Let us find the inverse Laplace transform of the following s‐functions.

1. (1)

   We can use Formula (A.4a) to find the ...