What is Laplace Transform? Formula, Properties, Conditions and Applications

Mathematics plays a decisive role to understand the behavior and working of electrical and electronic systems. Polynomials, Algebra, Probability, Integrations, and Differentiations etc…forms a significant part of the tools used to solve the systems. With the increasing complexity of systems, very sophisticated methods are required. Differential equations are prominently used for defining control systems. These equations are simple to solve. But complexity arises while solving higher order differential equations. To solve such complex higher order differential equations, the mathematical method that proved to be effective is Laplace Transform. As this transform is widely employed, it is useful to know what they really meant for and how do they work.

What is a Laplace Transform?

In mathematics, transforms are applied for transforming a variable from one form to another to make the equation easy to handle. Laplace transforms pretty much does the same thing. They transform higher order differential equation into a polynomial form which is far easy than solving differential equation directly.

But there is various transform like Fourier transform, z transforms what makes Laplace transform special? The major advantage of Laplace transform is that, they are defined for both stable and unstable systems whereas Fourier transforms are defined only for stable systems.

 Laplace Transform Formula

A Laplace transform of function f (t) in a time domain, where t is the real number greater than or equal to zero, is given as F(s), where there s is the complex number in frequency domain .i.e. s = σ+jω
The above equation is considered as unilateral Laplace transform equation. When the limits are extended to the entire real axis then the Bilateral Laplace transform can be defined as
In practical circuits like RC and RL circuits usually, initial conditions are used so, one-sided Laplace transforms are applied for analysis purpose.
As s= σ + jω , when σ = 0 Laplace transforms behaves as Fourier transform.

Conditions For Applicability of Laplace Transform

Laplace transforms are called integral transforms so there are necessary conditions for convergence of these transforms.
i.e. f must be locally integrable for the interval [0, ∞) and depending on whether σ is positive or negative, e^(-σt) may be decaying or growing. For bilateral Laplace transforms rather than a single value the integral converges over a certain range of values known as Region of Convergence.

Properties of Laplace Transform:

Linearity

Time Shifting

Shift in S-domain

Time-reversal

Differentiation in S-domain

Convolution in Time

Initial Value Theorem

Initial value theorem is applied when in Laplace transform the degree of the numerator is less than the degree of the denominatorFinal Value Theorem:

If all the poles of sF(s) lie in the left half of the S-plane final value theorem is applied.

Inverse Laplace Transform

Due to convergence characteristic Laplace transform also have an inverse transform. Laplace transforms exhibit one-to-one mapping from one function space to another. The formula for Inverse Laplace transform is;

How to Calculate Laplace Transform?

Laplace transform makes the equations simpler to handle. When a higher order differential equation is given, Laplace transform is applied to it which converts the equation into an algebraic equation, thus making it easier to handle. Then we calculate the roots by simplification of this algebraic equation. Now inverse Laplace transform of simpler expression is found which solves the given higher order differential equation.

Applications of Laplace Transform

• Analysis of electrical and electronic circuits.
• Breaking down complex differential equations into simpler polynomial forms.
• Laplace transform gives information about steady as well as transient states.
• In machine learning, the Laplace transform is used for making predictions and making analysis in data mining.
• Laplace transform simplifies calculations in system modeling.

Application of Laplace Transform In Signal Processing

Laplace transforms are frequently opted for signal processing. Along with the Fourier transform, the Laplace transform is used to study signals in the frequency domain. When there are small frequencies in the signal in the frequency domain then one can expect the signal to be smooth in the time domain. Filtering of a signal is usually done in the frequency domain for which Laplace acts as an important tool for converting a signal from time domain to frequency domain.

Application of Laplace Transform In Control Systems

Control systems are usually designed to control the behavior of other devices. Example of control systems can range from a simple home heating controller to an industrial control system regulates the behavior of machinery.

Generally, control engineers use differential equations to describe the behavior of various closed loop functional blocks. Laplace transform is used here for solving these equations without the loss of crucial variable information.

Characterization of Linear Time-Invariant Systems Using Laplace Transform

For a casual system ROC associated with the system, the function is the right half plane. A system is anti-casual if its impulse response h(t) =0 for t > 0.

If ROC of the system functions H(s) includes the jω axis then the L.T.I. the system is called a stable system. If a casual system with rational system functions H(s) have negative real parts for all of its poles then the system is stable.

Thus Laplace transform is a crucial tool in analyzing circuits. We can say as a stethoscope is to doctor Laplace transforms are to control engineer. What do you consider Laplace transforms as? In what way were they helpful to you?