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Laplace Transform (1)

Laplace Transform (1). Hany Ferdinando Dept. of Electrical Eng. Petra Christian University. Overview. Introduction Laplace Transform Convergence of Laplace Transform Properties of Laplace Transform Using table Inverse of Laplace Transform. Introduction.

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Laplace Transform (1)

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

  2. Overview • Introduction • Laplace Transform • Convergence of Laplace Transform • Properties of Laplace Transform • Using table • Inverse of Laplace Transform Laplace Transform (1) - Hany Ferdinando

  3. Introduction • It was discovered by Pierre-Simon Laplace, French Mathematician (1749-1827) Laplace Transform (1) - Hany Ferdinando

  4. Introduction • It transforms signal/system from time-domain to s-domain for continuous-time LTI system • It is analogous to Z Transform in discrete-time LTI system • It is similar to Fourier Transform, but ‘jw’ is substituted by s Laplace Transform (1) - Hany Ferdinando

  5. Introduction • Laplace Transform is continuous sum of exponential function of the form est, where s = s + jw is complex frequency • Therefore, Fourier can be viewed as a special case in which s = jw Laplace Transform (1) - Hany Ferdinando

  6. Laplace Transform Laplace Transform (1) - Hany Ferdinando

  7. Laplace Transform • For h(t) = e-at, find H(s) • What is your assumption in finishing the integration? • If you do not have that assumption, then what you can do? • Is it important to have that assumption? Laplace Transform (1) - Hany Ferdinando

  8. Convergence… The two-sided Laplace Transform exists if is finite Therefore, is finite Laplace Transform (1) - Hany Ferdinando

  9. Convergence… Suppose there exists a real positive numberRso that for some realaandbwe know that f(t) <Reat for t > 0, and f(t) < Rebt for t > 0 Laplace Transform (1) - Hany Ferdinando

  10. Convergence… Laplace Transform (1) - Hany Ferdinando

  11. Convergence… • How did you make your assumption in order to solve the equation? • Can you solve it without that assumption? The negative portion converges fors < bwhile the positive one converges fors > a Laplace Transform (1) - Hany Ferdinando

  12. Region of Convergence (RoC) Laplace Transform (1) - Hany Ferdinando

  13. Region of Convergence (RoC) Laplace Transform (1) - Hany Ferdinando

  14. Region of Convergence (RoC) Laplace Transform (1) - Hany Ferdinando

  15. Region of Convergence (RoC) Laplace Transform (1) - Hany Ferdinando

  16. Properties • Linearity • Scaling • Time shift • Frequency shift Laplace Transform (1) - Hany Ferdinando

  17. Properties • Time convolution • Frequency convolution • Time differentiation Laplace Transform (1) - Hany Ferdinando

  18. Properties • Time integration • Frequency differentation Laplace Transform (1) - Hany Ferdinando

  19. Properties • One-sided time differentiation • One-sided time integration Laplace Transform (1) - Hany Ferdinando

  20. Using Standard Table • Use table from books both for transform and for its inverse • No RoC is needed • Find the general form of the equation • Properties of Laplace transform are helpful • You use that table also to find the inverse Laplace Transform (1) - Hany Ferdinando

  21. Exercise Laplace Transform (1) - Hany Ferdinando

  22. Next… The Laplace Transform is already discussed. It transforms continuous-time LTI system from time-domain to s-domain. There are two types, one-sided (unilateral) and two-sided Next, we will study the application of Laplace Transform in Electrical Engineering. Read the Electric Circuit handout to prepare yourself! • Signals and Linear Systems by Alan V. Oppenheim, chapter 9, p 603-616 • Signals and System by Robert A. Gabel, chapter 6, p 373-394 Laplace Transform (1) - Hany Ferdinando

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