1. Hany Ferdinando Dept. of Electrical Eng. Petra Christian University
Laplace Transform (2)
Hany Ferdinando
Dept. of Electrical Eng.
Petra Christian University

2. Laplace Transform (2) - Hany Ferdinando
Overview
Unilateral Laplace Transform
Two-sided Laplace Transform
Application in electric circuit
Application in differential equation
Stability
Frequency response analysis
Laplace transform for periodic signal
Laplace Transform (2) - Hany Ferdinando

3. Unilateral Laplace Transform
This is applied for causal function only
The general form is from 0 to ∞ for the time variable
It is only positive part of the whole function
Laplace Transform (2) - Hany Ferdinando

4. Inverse (unilateral only)
Make the form of the function in s-domain as sum of rational function  use partial fraction expansion
From the table, find the formula with the highest similarity
Use the properties to help you to find the result
Laplace Transform (2) - Hany Ferdinando

5. Laplace Transform (2) - Hany Ferdinando
Inverse (two-sided)
This is applied for non causal function
The RoCs are needed
Make the form of the function in s-domain as sum of rational function  use partial fraction expansion
Laplace Transform (2) - Hany Ferdinando

6. Laplace Transform (2) - Hany Ferdinando
Inverse (two-sided)
The location of the poles of the F(s) with respect to the RoC determines whether a given singularity refers to a positive or negative region
Poles to the left of RoC give rise to a positive time portion of f(t)
Poles to the right of RoC give rise to a negative time portion of f(t)
Laplace Transform (2) - Hany Ferdinando

7. Laplace Transform (2) - Hany Ferdinando
Inverse (two-sided) a b
a < Re(s) < b
Pole ‘a’ lies to left of the RoC  it give rise the positive time portion of f(t)
Pole ‘b’ lies to right of the RoC  it give rise the negative time portion of f(t)
Laplace Transform (2) - Hany Ferdinando

8. Laplace Transform (2) - Hany Ferdinando
Inverse (two-sided)
For the positive part, use the unilateral approach
For negative part, use the following chart…
f(-t)
F(-s)
f(t)
F(s)
Laplace Transform (2) - Hany Ferdinando

9. Application in Electric Circuit
Transform all components to s-domain
R  R
L  sL
C  1/(sC)
Source  use table
Use DC analysis to write the standard equation (you can use node, mesh or superposition)
Laplace Transform (2) - Hany Ferdinando

10. Application in Electric Circuit
Solve the equation in ‘s’
Use inverse Laplace transform to get the result in time domain (do not forget to do this!!!)
Laplace Transform (2) - Hany Ferdinando

11. Application in Electric Circuit
Source: 20 cos (3t+1)
Calculate the current which flows in the circuit!
Laplace Transform (2) - Hany Ferdinando

12. Application in differential equation
Solving differential equation with ordinary way sometimes is difficult
We can use Laplace transform to simplify it
The differential equation is transformed to s-domain and then solve it
Do not forget to inverse the result…!!!
Laplace Transform (2) - Hany Ferdinando

13. Application in differential equation
Use the following property…
Apply that property to solve this…
Laplace Transform (2) - Hany Ferdinando

14. Laplace Transform (2) - Hany Ferdinando
Stability
What is stability?
Is it important? Why? A B
Laplace Transform (2) - Hany Ferdinando

15. Laplace Transform (2) - Hany Ferdinando
Stability
Simple poles of the form c/(s+a)
Complex conjugates poles of the form c/[(s+a)2+w2]
Complex conjugates poles of the form c/(s2+w2)
Laplace Transform (2) - Hany Ferdinando

16. Frequency Response Analysis
It is evaluated along the jw axis
Substitute ‘s’ with jw and solve it as you do in the Fourier analysis
Laplace Transform (2) - Hany Ferdinando

17. Laplace Transform (2) - Hany Ferdinando
Periodic Signal
If f(t) is periodic signal with period T, then the Laplace transform of f(t) is defined as
Laplace Transform (2) - Hany Ferdinando

18. Laplace Transform (2) - Hany Ferdinando
Periodic Signal
Find Laplace transform for f(t)
Then calculate the voltage across the inductor
T/2 T 1 -1
f(t)
Laplace Transform (2) - Hany Ferdinando