Laplace Transform
Laplace transformation is a technique for solving differential equations. Here differential equation of time domain form is first transformed to algebraic equation of frequency domain form. After solving the algebraic equation in frequency domain, the result then is finally transformed to time domain form to achieve the ultimate solution of the differential equation. In other words it can be said that the Laplace transformation is nothing but a shortcut method of solving differential equation.
In this article, we will be discussing Laplace transforms and how they are used to solve differential equations. They also provide a method to form a transfer function for an input-output system, but this shall not be discussed here. They provide the basic building blocks for control engineering, using block diagrams, etc.
Many kinds of transformations already exist but Laplace transforms and Fourier transforms are the most well known. The Laplace transforms is usually used to simplify a differential equation into a simple and solvable algebra problem. Even when the algebra becomes a little complex, it is still easier to solve than solving a differential equation.
There is always a table that is available to the engineer that contains information on the Laplace transforms. An example of Laplace transform table has been made below. We will come to know about the Laplace transform of various common functions from the following table .
When learning the Laplace transform, it’s important to understand not just the tables – but the formula too.
To understand the Laplace transform formula: First Let f(t) be the function of t, time for all t ≥ 0
Then the Laplace transform of f(t), F(s) can be defined as
Provided that the integral exists. Where the Laplace Operator, s = σ + jω; will be real or complex j = √(-1)
Laplace transforms can only be used to solve complex differential equations and like all great methods, it does have a disadvantage, which may not seem so big. That is, you can only use this method to solve differential equations WITH known constants. If you do have an equation without the known constants, then this method is useless and you will have to find another method.