Chapter 61

Introduction to Laplace

transforms

61.1 Introduction

The solution of most electrical circuit problems can
be reduced ultimately to the solution of differential
equations. The use ofLaplace transformsprovides an
alternative method to those discussed in Chapters 46 to
51 for solving linear differential equations.

61.2 DefinitionofaLaplacetransform

The Laplace transform of the functionf(t)is defined
by the integral

∫∞

0 e

−stf(t)dt,wheresis a parameter

assumed to be a real number.

Common notations used for the Laplace
transform

There are various commonly used notations for the
Laplace transform off(t)and these include:

(i) L{f(t)}orL{f(t)}

(ii) L(f)orLf

(iii) f(s)orf(s)

Also, the letterpis sometimes used instead ofsas
the parameter. The notation adopted in this book will
bef(t)for the original function andL{f(t)}for its
Laplace transform.

Hence, from above:

L{f(t)}=

∫∞

0

e−stf(t)dt (1)

61.3 Linearity property of the

Laplace transform

From equation (1),

L{kf(t)}=

∫∞

0

e−stkf(t)dt

=k

∫∞

0

e−stf(t)dt

i.e. L{kf(t)}=kL{f(t)} (2)

wherekis any constant.

Similarly,

L{af(t)+bg(t)}=

∫∞

0

e−st(af(t)+bg(t))dt

=a

∫∞

0

e−stf(t)dt

+b

∫∞

0

e−stg(t)dt

i.e. L{af(t)+bg(t)}=aL{f(t)}+bL{g(t)}, (3)

whereaandbare any real constants.

The Laplace transform is termed alinear operator

because of the properties shown in equations (2) and (3).

61.4 Laplace transforms of

elementary functions

Using the definition of the Laplace transform in equa-

tion (1) a number of elementary functions may be

transformed. For example: