Convolution Integral in Network Analysis

Convolution Integral in Network Analysis:

The Convolution Integral in Network Analysis in Laplace transform states that

Convolution Integral in Network Analysis

where f(t) * f(t) = Convolution of f(t) and f(t)

From the definition of system function,

Convolution Integral in Network Analysis

Taking inverse Laplace,

Convolution Integral in Network Analysis

From the convolution theorem,

Convolution Integral in Network Analysis

where

Convolution Integral in Network Analysis

Thus with the help of convolution of e(t) and h(t), the response of the network can be obtained directly.

Solving Differential Equations:

Consider the differential equation of order two, for simplicity of understanding as,

where K1, K0 are the constants.

We are interested to obtain the solution x(t) in time domain which satisfies the differential equation.

Take Laplace transform of both sides,

Let the initial conditions be zero for simplicity.

AssumeConvolution Integral in Network Analysis

In this equation Y(s) is Laplace transform of y(t) which is known and called Forcing Function.

Hence taking Laplace inverse of X(s), using suitable method, the required solution can be obtained.

As most of the equations describing various networks are the differential equations, this technique is very effective in the network analysis. It takes into account, any initial conditions of the network as well.

Updated: October 15, 2019 — 9:43 pm