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Chapter 4 The Laplace Transformation
as
f – Vt – jZt
³ ft e e dt f (4.6)
0
The term e – jZt in the integral of (4.6) has magnitude of unity, i.e., e – jZt = 1 , and thus the condition
for convergence becomes
f – Vt
³ ft e dt f (4.7)
0
*
Fortunately, in most engineering applications the functions f t are of exponential order . Then, we
can express (4.7) as,
f – Vt f V t – Vt
0
³ ft e dt ³ ke e dt (4.8)
0 0
and we see that the integral on the right side of the inequality sign in (4.8), converges if V V ! 0 .
Therefore, we conclude that if f t is of exponential order, L ft ^ ` exists if
Re s^` = V ! V 0 (4.9)
where Re s^` denotes the real part of the complex variable . s
Evaluation of the integral of (4.4) involves contour integration in the complex plane, and thus, it will
not be attempted in this chapter. We will see, in the next chapter, that many Laplace transforms can
be inverted with the use of a few standard pairs, and therefore, there is no need to use (4.4) to obtain
the Inverse Laplace transform.
In our subsequent discussion, we will denote transformation from the time domain to the complex
frequency domain, and vice versa, as
f t Fs (4.10)
4.2 Properties of the Laplace Transform
1. Linearity Property
The linearity property states that if
f t f t } f t
2
n
1
have Laplace transforms
* A function f t is said to be of exponential order if ft ke V 0 t for all t t 0 .
4-2 Circuit Analysis II with MATLAB Applications
Orchard Publications
