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Laplace Transform (1)

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**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))

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**Unilateral Laplace Transform**

(Implemented in Mathematica)

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**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.

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**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!

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**Example of Unilateral Laplace**

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Bilateral Laplace

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**Example – RCO may not always exist!**

Note that there is no common ROC Laplace Transform can not be applied!

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**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)

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**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.

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**Understand Stability of a system using Fourier Transform (Bode Plot)**

(unstable)

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**Understand Stability of a System Using Laplace Transform**

Look at the roots of Y(s)/X(s)

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

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Table 7.1

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Table 7.1 (Cont.)

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**Laplace Transform Properties (1)**

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**Laplace Transform Properties (2)**

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**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)

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**Differentiation Property**

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**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.

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**Integration Property (1)**

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**Integration Property (2)**

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**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.

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Next time

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**Example – Unilateral Version**

Find F(s): Find F(s):

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Example

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Example

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Extra Slides

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Building the Case…

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**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!

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**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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9.0 Laplace Transform 9.1 General Principles of Laplace Transform linear time-invariant Laplace Transform Eigenfunction Property y(t) = H(s)e st h(t)h(t)

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