Introduction
The purpose of Laplace transformation is to solve different differential equations.
There are a number of methods to solve homogeneous and non-homogeneous
equations but Laplace and Fourier transform are used widely. Laplace transform comes
in to use when we have to solve the equations that cannot be solved by any of the
previous methods invented.
It can be used to solve various algebraic problems. Laplace transform is more expedient
when it comes to non-homogeneous equations. It is one of the easiest methods to
solve complicated non-homogeneous equations.
Definition
The Laplace transform converts integral and differential equations into algebraic
equations. Although it is a different and beneficial alternative of variations of
parameters and undetermined coefficients, the transform is most advantageous for
input terms that piecewise, periodic or pulsive.
Why use a transform method?
Some problems are difficult to solve directly.
With a transform method, the hope is that the transformed problem is easy
to solve. That is certainly the case for the simple example above. One must
also take into account the difficulty of transforming the original problem
and inverse transforming the solution to the transformed problem.
Applications of Laplace Transforms
Circuit Equations
There are two (related) approaches:
Derive the circuit (differential) equations in the time domain, then transform these ODEs to
the s-domain;
Transform the circuit to the s-domain, then derive the circuit equations in the s-domain (using
the concept of "impedance").
We will use the first approach. We will derive the system equations(s) in the t-plane, then
transform the equations to the s-plane. We will usually then transform back to the t-plane.
In simple words, Laplace transform converts time domain signal into
frequency domain, which is depicted more clearly in below figure: