Laplace transform circuit analysis, Slides of Fourier Transform and Series

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Prof. Yogesh Kumar

Lecture: 13

Laplace Transform-I

Dr. Yogesh Kumar Choukiker School of Electronics Science Engineering Microwave and Photonics Division VIT University, Vellore, India

Prof. Yogesh Kumar 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.

Laplace Transform Definition

Prof. Yogesh Kumar

Key Points

 What is the definition of the Laplace transform?  What are the Laplace transforms of unit step, impulse, exponential, and sinusoidal functions?  What are the Laplace transforms of the derivative, integral, shift, and scaling of a function?  How to perform partial fraction expansion for a rational function F ( s ) and perform the inverse Laplace transform?

Prof. Yogesh Kumar Transforming a real function f ( t ) of real variable t to a complex function F ( s ) of complex variable s :  The integral will converge ( 1 ) over a portion of the s-plane (e.g. Re( s ) > 0 ), and ( 2 ) for most of the functions except for those of little interest (e.g. 𝑒 𝑡 ).  F ( s ) is determined by f ( t ) only for t > 0. Thus we use it to predict the response after initial conditions have been established.

What is Laplace Transform?

Prof. Yogesh Kumar

Unit step function u ( t )

Prof. Yogesh Kumar Representation of time shift and reversal

Prof. Yogesh Kumar

Impulse function 𝜹 ( t )

𝛿( t ) is the derivative of u ( t )

Prof. Yogesh Kumar

Impulse function 𝜹 ( t )

Laplace transform of 𝛿 ' ( t )

Prof. Yogesh Kumar

Laplace Transform of Specific Functions

E.g. Single-sided exponential function

Prof. Yogesh Kumar

Laplace Transform of Specific Functions

E.g. Sinusoidal function

Prof. Yogesh Kumar

List of Laplace transform pairs