In this post I will discuss about the Laplace Transform Definition and Inverse Laplace Transform. In the last post I have discussed about the transfer function and bode plot in matlab and there I have mentioned that the Transfer Function in MATLAB and bode plot in matlab
After reading this post you will learn about the laplace transformation,and different examples of laplace transformation. So sit back, keep reading and enjoy learning.
A signal or a system can be represented in either the time domain or frequency domain. The representation of the signal or the system in time domain is a function of time and the representation of the signal or system in frequency domain is a function of frequency. The representation of the system in terms of the frequency variable is commonly called the transfer function. The transfer function of the system compliances the knowledge about the behavior of the system in the frequency domain. Thus the transfer function gives the knowledge about the magnitude and phase relationship between the input and output of the system for a particular range of frequency thus the transfer function of the system can be thought of as the frequency response of the system.
The concept of transform is very important in Mathematics with the help of which one can alter the representation of data without affecting the specific nature of the data. In engineering circuit analysis and design there are various transform tools available some important and relevant ones are mentioned below:
- Laplace Transform.
- Fourier Transform.
introduction to Laplace Transform:
The discussion here is oriented on Laplace Transform and Fourier transform and Z- transform are out of the scope of this discussion.
With the help of this transform technique one can alter the representation of data for example the concept of Laplace transform is used to alter the representation of system or signal from time domain to frequency domain. Thus the Laplace transform is a tool for transforming the time domain representation of the system or signal to the frequency domain representation. It must be noted here that the Laplace transform of the system represents the frequency response of the system and in the similar way the Laplace Transform of the signal in time domain represents the spectrum of the signal. The representation of the Laplace transform is as shown in the following figure:
The expression on the right hand side shows the Laplace transform of the time domain representation of the system. The symbol ‘s’ in this representation is used to denote the complex frequency variable.
Laplace Transform Definition:
The Laplace transform is defined by the following equation;
The Integral transform shown in the above equation converts the time domain representation of the system into the frequency domain representation of the system.
s-Domain Circuit Analysis:
The concept of Laplace transform is also used in the circuit analysis. The Laplace transform of the time domain representation of the system gives the different perspective of the system. Anyhow the use of the concept of Laplace transform of the system is depicted in the following figure:
The circuit analysis can be performed in two ways:
- Classical Approach for circuit analysis.
- Laplace method for circuit analysis.
In the classical approach for circuit analysis the mathematical operations are performed on the time domain representation of the system. As can be seen in the above image that the input and output relationship of the system can be represented in the time domain using the differential equation. In order to solve these differential equations one has to involve in critical calculations before they get the required information. The Laplace transform method for circuit analysis and design bypasses these hectic calculations. The Laplace transform of the differential equation converts the representation of the system in the frequency domain and also converts the differential equations into the simple algebraic equations with variable ‘s’ (complex frequency variable) which can be solved using relatively simple algebraic manipulations. Once the algebraic manipulations have been carried out on the Laplace Transform of the system and have got the final response of the system in s domain the inverse Laplace transform is carried out to obtain the final response of the system in the time domain. Thus in this way Laplace transform bypasses the hectic calculations involved in the time domain calculations.
Inverse Laplace Transform:
The Laplace transform of the system or signal can be converted back to the time domain representation of the system with the help of the Inverse Laplace Transform. The Inverse Laplace transform is represented as follows:
The definition of the Inverse Laplace transform is shown in the following figure:
Laplace Transform Pairs:
The Laplace transform of the commonly known functions are listed in the following table:
Laplace Transform Properties:
The Laplace transform follow commonly known properties. The following list shows some of the properties that the Laplace Transform posses:
That is all for now I hope this article would be helpful for you. In the next post I will come up with more interesting educational topics. Till then stay connected, keep reading and enjoy learning.