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 of Laplace transforms provides an
alternative method to those discussed in Chapters 46 to
51 for solving linear differential equations.
61.2 Definition of a Laplace transform
The Laplace transform of the function f (t ) is defined
by the integral
∞
0 e −st f (t ) dt , where s is 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 of f (t ) and these include:
(i) L{ f (t )} or L{ f (t )}
(ii) L( f ) or L f
(iii) f (s) or f (s)
Also, the letter p is sometimes used instead of s as
the parameter. The notation adopted in this book will
be f (t ) for the original function and L{ f (t )} for its
Laplace transform.
Hence, from above:
L{ f (t)} =
∞
0
e
−st f (t)dt
(1)
61.3 Linearity property of the
Laplace transform
From equation (1),
L{k f (t )} =
∞
0
e
−st k f (t ) dt
= k
∞
0
e
−st f (t ) dt
i.e. L{k f (t )} =kL{ f (t )}
(2)
where k is any constant.
Similarly,
L{a f (t ) + bg(t )} =
∞
0
e
−st
(a f (t ) + bg(t )) dt
= a
∞
0
e
−st f (t ) dt
+ b
∞
0
e
−st g(t ) dt
i.e. L{a f (t ) + bg(t )} = aL{ f (t )} + bL{g(t )}, (3)
where a and b are any real constants.
The Laplace transform is termed a linear 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 equation (1) a number of elementary functions may be
transformed. For example: