Laplace Transform (2)

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

2. 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. 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. 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. a < Re(s) < b a b Inverse (two-sided) • 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. Inverse (two-sided) • For the positive part, use the unilateral approach • For negative part, use the following chart… f(-t) F(-s) F(s) f(t) 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. Stability • What is stability? • Is it important? Why? A B Laplace Transform (2) - Hany Ferdinando

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