Page 119 - Circuit Analysis II with MATLAB Applications

Chapter 4

The Laplace Transformation

T his chapter begins with an introduction to the Laplace transformation, definitions, and proper-

ties of the Laplace transformation. The initial value and final value theorems are also discussed
and proved. It concludes with the derivation of the Laplace transform of common functions
of time, and the Laplace transforms of common waveforms.

4.1 Definition of the Laplace Transformation

The two-sided or bilateral Laplace Transform pair is defined as

f
L ft = Fs = ³ ft e – st dt (4.1)
`
^
– f
– 1 1 V + jZ st
`
L ^ Fs = ft = -------- ³ Fs e ds (4.2)
2Sj V – jZ

,
where L ft ^ ` denotes the Laplace transform of the time function f t L – 1 ^ Fs ` denotes the
s
Inverse Laplace transform, and is a complex variable whose real part is , and imaginary part ,
Z
V
that is, s = V + jZ .
In most problems, we are concerned with values of time greater than some reference time, say
t
t = t = 0 , and since the initial conditions are generally known, the two-sided Laplace transform
0
pair of (4.1) and (4.2) simplifies to the unilateral or one-sided Laplace transform defined as

f f
L ft = F s = ³ ft e – st dt = ³ f t e – st dt (4.3)
^
`
t 0 0

– 1 1 V + jZ st
`
L ^ Fs = f t = -------- ³ Fs e ds (4.4)
2Sj V – jZ
The Laplace Transform of (4.3) has meaning only if the integral converges (reaches a limit), that is, if

f – st
³ ft e dt f (4.5)
0
To determine the conditions that will ensure us that the integral of (4.3) converges, we rewrite (4.5)

Circuit Analysis II with MATLAB Applications 4-1
Orchard Publications