2Definition of Bilateral Laplace Transform (b for bilateral or two-sided transform)Let s=σ+jωConsider 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))
3Unilateral Laplace Transform (Implemented in Mathematica)
4Difference 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.
5Laplace Transform Convergence The Laplace transform does not converge to a finite value for all signals and all values of sThe 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 valuesRemember: e^jw is sinusoidal; Thus, only the real part is important!
8Example – RCO may not always exist! Note that there is no common ROC Laplace Transform can not be applied!
9Laplace Transform & Fourier Transform Laplace transform is more general than Fourier TransformFourier Transform: F(ω). (t→ ω)Laplace Transform: F(s=σ+jω) (t→ σ+jω, a complex plane)
10How is Laplace Transform Used (Building block of anegative feedback system)This system becomes unstable if βH(s) is -1. If you subsittuteds by jω, you can use Bode plot to evaluate the stability ofthe negative feedback system.
11Understand Stability of a system using Fourier Transform (Bode Plot) (unstable)
12Understand Stability of a System Using Laplace Transform Look at the roots of Y(s)/X(s)
13Laplace TransformWe use the following notations for Laplace Transform pairs – Refer to the table!
23Application 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 domainV=Ldi/dt (assume initial condition is 0)Integrate V/L with respect to t, get you V/(sL), which is current in Laplace domain.
30Applications of Laplace Transform Easier than solving differential equationsUsed to describe system behaviorWe assume LTI systemsUses S-domain instead of frequency domainApplications of Laplace Transforms/Circuit analysisProvides the general solution to any arbitrary wave (not just LRC)TransientSinusoidal steady-state-response (Phasors)Signal processingCommunicationsDefinitely useful for Interviews!
31Example of Bilateral Version Find F(s):ROCS-planeRe(s)<aaFind F(s):RememberThese!Note that Laplace can also be found for periodic functions