08-16-2017, 10:34 PM
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.
An interesting analogy that may help in understanding Laplace is this. Imagine you come across an English poem which you do not understand. However, you have a Spanish friend who is excellent at making sense of these poems. So you translate this poem to Spanish and send it to him, he then in turn explains this poem in Spanish and sends it back to you. You understand the Spanish explanation and are then able to transfer the meaning of the poem back to English and thus understand the English poem.
Where are Laplace Transforms used in Real Life?
The Laplace Transform is derived from Lerch s Cancellation Law. In the Laplace Transform method, the function in the time domain is transformed to a Laplace function in the frequency domain. This Laplace function will be in the form of an algebraic equation and it can be solved easily. The solution can be again transformed back to the time domain by using an Inverse Laplace Transform.
This transform is most commonly used for control systems, as briefly mentioned above. The transforms are used to study and analyze systems such as ventilation, heating and air conditions, etc. These systems are used in every single modern day construction and building.
Laplace transforms are also important for process controls. It aids in variable analysis which when altered produce the required results. An example of this can be found in experiments to do with heat.
Apart from these two examples, Laplace transforms are used in a lot of engineering applications and is a very useful method. It is useful in both electronic and mechanical engineering.
The control action for a dynamic control system whether electrical, mechanical, thermal, hydraulic, etc. can be represented by a differential equation. The system differential equation is derived according to physical laws governing is a system. In order to facilitate the solution of a differential equation describing a control system, the equation is transformed into an algebraic form. This transformation is done with the help of the Laplace transformation technique, that is the time domain differential equation is converted into a frequency domain algebraic equation.