1. CIRCUITS and SYSTEMS – part II
Prof. dr hab. Stanisław Osowski
Electrical Engineering (B.Sc.)
Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

2. Laplace transformation
Lecture 10
Laplace transformation

3. Laplace transformation
Basic notations
original time function: f(t)
Laplace transform: F(s)
complex frequency:
simple Laplace transformation
inverse Laplace transformation

4. Properties of Laplace transformation
Linearity
Derivative of time function
Integral
Convolution
where

5. Table of chosen Laplace transforms

6. Determination of inverse Laplace transform
Transform is given in rational form of s
Degree of numerator smaller than degree of denominator (m<n). It it is not true you should divide the higest power of numerator by the highest power of denominator.
The poles si of transfer function are known.

7. Residue method
Assume the Laplace transform function F(s) in rational form. The poles and zeros of it are known. Then
Residues of the function are calculated on the basis of the poles si of the function

8. Residue calculation
At lth multiplicity of si
At single si

9. Example Determine the inverse Laplace transform of the function
There are 2 poles:
Hence
And finally

10. Application of table of transforms
In this approach we should present the transform in the forms that are present in the table of basic trasforms. The time function is just read from the table.
Example
Transform function
Transformation to the basic form of transform in the table
The original time function read from the table