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Laplace Transform (1).

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Presentation on theme: "Laplace Transform (1)."— Presentation transcript:

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)

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):

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 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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