Laplace Transform (1)

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))

Unilateral Laplace Transform (Implemented in Mathematica)

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.

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!

Example of Unilateral Laplace

Bilateral Laplace

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

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)

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.

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

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

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

Table 7.1

Table 7.1 (Cont.)

Laplace Transform Properties (1)

Laplace Transform Properties (2)

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)

Differentiation Property

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.

Integration Property (1)

Integration Property (2)

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.

Next time

Example – Unilateral Version Find F(s): Find F(s):

Example

Example

Extra Slides

Building the Case…

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!

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