MIT18_03S10_c30

# MIT18_03S10_c30 - 18.03 Class 30 Laplace Transform V 1 2 3...

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, Apr 21, 2010 Laplace Transform V: Poles and amplitude response 1. Recap on pole diagram 2. Stability 3. Exponential signals 4. Transfer and gain 5. Fourier and Laplace [1] To recap: The kind of function F(s) that arises as a Laplace transform can be understood, in broad terms, by giving the set of points at which it becomes infinite. This is the "pole diagram." Examples: The following F(s)'s all have pole diagram {2i,-2i} (a not zero) : [Slide] F(s) f(t) a/(s^2+4) (a/2) sin(2t) as/(s^2+4) a cos(2t) e^{-bs}/(s^2+4) cos_b(2t) 1 + a/(s^2+4) delta(t) + (a/2) sin(2t) 4s/(s^2+4)^2 t cos(2t) and many other examples. All these functions f(t) share some common features, for sufficiently large t : - they oscillate with circular frequency 2 . - they may grow or shink, but less than exponentially. These features are common to all functions f(t) such that F(s) has this pole diagram. Pole diagram {2i,-2i,1} : any of the above plus (c not zero) F(s) f(t) c/(s-1) c e^t + . .. c/(s-1)^2 c t e^t + . .. and many other examples. All these functions f(t) share some common features, for sufficiently large t : - they grow as t ---> infinity "like e^t ." This means that they grow faster than e^{kt} for any k < 1, and slower than e^{lt} for any l > 1. - they oscillate but the oscillations become insignificant relative to the size of f(t) as t ---> infinity. The rightmost poles of

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MIT18_03S10_c30 - 18.03 Class 30 Laplace Transform V 1 2 3...

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