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Laplace Transform (1)

Laplace Transform (1). Definition of Bilateral Laplace Transform. (b for bilateral or two-sided transform). Let s= σ +j ω. Consider the two sided Laplace transform as the Fourier transform of f(t) e - σ t . That is the Fourier transform of an exponentially windowed signal.

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Laplace Transform (1)

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  1. Laplace Transform (1)

  2. Definition of Bilateral Laplace Transform (b for bilateral or two-sided transform) Let s=σ+jω Consider the two sided Laplace transform as the Fourier transform of f(t)e-σt. That is the Fourier transform of an exponentially windowed signal. Note also that if you set the evaluate the Laplace transform F(s) at s= jω, you have the Fourier transform (F(ω))

  3. Unilateral Laplace Transform (Implemented in Mathematica)

  4. Difference Between the Unilateral Laplace Transform and Bilateral Laplace transform • Unilateral transform is used when we choose t=0 as the time during which significant event occurs, such as switching in an electrical circuit. • The bilateral Laplace transform are needed for negative time as well as for positive time.

  5. Laplace Transform Convergence • The Laplace transform does not converge to a finite value for all signals and all values of s • The values of s for which Laplace transform converges is called the Region Of Convergence (ROC) • Always include ROC in your solution! • Example: 0+ indicates greater than zero values Remember: e^jw is sinusoidal; Thus, only the real part is important!

  6. Example of Unilateral Laplace

  7. Bilateral Laplace

  8. Example – RCO may not always exist! Note that there is no common ROC  Laplace Transform can not be applied!

  9. Laplace Transform & Fourier Transform • Laplace transform is more general than Fourier Transform • Fourier Transform: F(ω). (t→ ω) • Laplace Transform: F(s=σ+jω) (t→ σ+jω, a complex plane)

  10. How is Laplace Transform Used (Building block of a negative feedback system) This system becomes unstable if βH(s) is -1. If you subsittuted s by jω, you can use Bode plot to evaluate the stability of the negative feedback system.

  11. Understand Stability of a system using Fourier Transform (Bode Plot) (unstable)

  12. Understand Stability of a System Using Laplace Transform Look at the roots of Y(s)/X(s)

  13. Laplace Transform • We use the following notations for Laplace Transform pairs – Refer to the table!

  14. Table 7.1

  15. Table 7.1 (Cont.)

  16. Laplace Transform Properties (1)

  17. Laplace Transform Properties (2)

  18. Model an Inductor in the S-Domain • To model an inductor in the S-domain, we need to determine the S-domain equivalent of derivative (next slide)

  19. Differentiation Property

  20. Model a Capacitor in the S-Domain If initial voltage is 0, V=I/sC 1/(sC) is what we call the impedance of a capacitor.

  21. Integration Property (1)

  22. Integration Property (2)

  23. Application • i=CdV/dt (assume initial voltage is 0) • Integrate i/C with respect to t, will get you I/(sC), which is the voltage in Laplace domain • V=Ldi/dt (assume initial condition is 0) • Integrate V/L with respect to t, get you V/(sL), which is current in Laplace domain.

  24. Next time

  25. Example – Unilateral Version • Find F(s): • Find F(s): • Find F(s): • Find F(s):

  26. Example

  27. Example

  28. Extra Slides

  29. Building the Case…

  30. Applications of Laplace Transform • Easier than solving differential equations • Used to describe system behavior • We assume LTI systems • Uses S-domain instead of frequency domain • Applications of Laplace Transforms/ • Circuit analysis • Easier than solving differential equations • Provides the general solution to any arbitrary wave (not just LRC) • Transient • Sinusoidal steady-state-response (Phasors) • Signal processing • Communications • Definitely useful for Interviews!

  31. Example of Bilateral Version Find F(s): ROC S-plane Re(s)<a a Find F(s): Remember These! Note that Laplace can also be found for periodic functions

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