network analysis and synthesis

1. Chapter 2
Network Transform Representation and
Analysis
In analysis of electrical networks containing linear elements, one is usually required to solve integro-
differential equations. Solution of such differential equations can sometimes be difficult or time
consuming. Transform methods simplify this process by converting the differential equation into
an algebraic one. One such method is the Laplace transform. The Laplace transform transforms
the time-domain (the independent variable is time t) differential equations describing a given
network into frequency-domain algebraic equation in which the independent variable is s which
represent the complex frequency given as
s = σ + jω
where σ describes the growth and decay of the amplitude of a given signal and ω is the angular
frequency of the signal.
2.1 Review of Laplace Transform
The Laplace transform of a time-domain function (signal) f(t) is defined as:
L [f(t)] = F(s) =
Z ∞
0
f(t)e−st
dt (2.1)
where s is the generalized complex frequency variable. In the above form of the Laplace transform
the lower limit of integration is made 0 to account for the possibility that f(t) may be an impulse
functon or one of its higher derivatives.
Existence Condition
The Laplace transform of a given function exists only if
Z ∞
0
|f(t)|e−σt
dt < ∞ (2.2)
The Laplace transform differs from the Fourier transform in the following ways:
1. Because of the added convergence factor e−σt
a broader class of signals have Laplace trans-
form.
2. Laplace transform can directly take into account initial conditions at t = 0
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2. 2.2 Inverse Laplace Transform
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