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Published byCaitlin Whinery Modified about 1 year ago

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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: Remember: e^jw is sinusoidal; Thus, only the real part is important! 0+ indicates greater than zero values

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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):

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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 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!

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Example of Bilateral Version Find F(s): Re(s)

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