Laplace transform - Formula, Calculator, Inverse Laplace Transform and Applications

The Laplace transform is named after Pierre Simon De Laplace, a famous French mathematician (1749-1827) who formulated a transformation that can convert one signal into another using a set of laws or equations. The Laplace transformation is the most effective method for converting differential equations to algebraic equations. In electronics engineering, the Laplace transformation is very important to solve problems related to signal and system, digital signal processing, and control system. In studying the dynamic control system, the characteristics of the Laplace transform and the inverse Laplace transformation are both employed.

To understand Laplace transform, we must first learn about piecewise continuous function. A piecewise continuous function is a function that has a finite number of breaks, and this consistency remains till the function reaches infinity.

Let the function be f(t) (piecewise continuous), therefore the Laplace transform of the function f(t) will be L f(t) or F(s). The Laplace transform will change the differential equation into an easy-to-solve algebraic function. This transform converts any signal into the frequency domain 's', where the complexity of the problem reduces. Whenever you encounter any function written inside the capital letter L, instantly identify that it is the Laplace transform of a function. It can also be represented as the capital letter of the function related to frequency 's'. For instance, F (s), A (s), G (s), X (s), etc.

Laplace transform is formulated as the integration of the product of the function with e^-st. dt, with limits ranging from 0 to infinity.

Laplace transform of a piecewise continuous function can be given as f(t) = F (s) = ₀ʃ^∞ f(t). e^-st.dt.

Laplace transform is divided into two types, namely one-sided Laplace transformation and two-sided Laplace transformation. Let us learn the fundamentals of both:

One-sided Laplace transformation: The Laplace transformation with limits 0 to infinity is known as one-sided. This is also known as unilateral Laplace transformation.

Two-sided Laplace transformation: The Laplace transformation with limits ranging from -infinity to +infinity is considered as two-sided. This transformation is also known as bilateral Laplace transformation.

The Laplace transformation follows the Linearity Property, Frequency Shifting Property, Integration, Multiplication by Time, Complex Shift Property, Time Reversal Property, and the Time Scaling Property.

Inverse Laplace Transform

The inverse Laplace transform is represented by using the symbol L^-1; it is defined as the converse of Laplace transform. If a Laplace transform is given, say F(s), the inverse of this will be L^-1 F(s), this will give us the original function, i.e., f(p) (before Laplace transformation was applied to it)

Applications

Laplace transform is an essential topic from the perspective of engineering. Engineering courses dealing with electricity and signals are based on the concept of Laplace transform. Below are some applications of Laplace Transform:

In the telecommunications industry, it is utilized to deliver signals to both sides of the medium.
It is also utilized for a variety of technical jobs, including electrical circuit analysis, digital signal processing, system modelling, and more.
Laplace Transform is used to convert complex equations into simple polynomial equations.