Lesson+12

# Lesson+12 - Yes I used MATLAB too 1 Lesson 12 Lesson 12...

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Lesson 12 1 Yes, I used MATLAB too!

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Inverse Laplace Transform Ch. 12.6-7 Lesson 12 2 Lesson 12
Lesson 12 3 V S R L C Find V C (s) t=0 V dt dV RC d V L R V dt dV C d V L R V V KCL C t C C C t C C 0 0 ) ( or 0 ) ( 1 : System at-rest. IC’s = 0

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Lesson 12 4 V S R L C Find V C (s) t=0 ) / ( ) / ( ) / ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( LC s RC s s V RC s V LV s V RCLs s RV s sLV s s V s RCsV s s V L R s V C C C C C C C 1 1 1 or or 2 2 V dt dV RC d V L R V C t C C 0 τ τ ) (
Lesson 12 5 V S R L C Find V C (s) t=0 1 1 ) / 1 ( ) / 1 ( ) / 1 ( ) ( 2 2 s s s LC s RC s s V RC s V C But what is v C (t)? If V=R=C=L=1.0

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Lesson 12 OK. I can do that Laplace thing but what does it mean? Laplace transform Lesson 12 6 dt e t x s X st c j c st ds e s X j t x 2 1 ) ( Inverse Laplace transform
We think that maybe Messr Laplace may have invented a most useful transform. It apparently can be used to solve and analyze ODEs and circuits. But to be practical, an inverse mapping process between the s-domain to the time-domain will need to be developed. Why, because the #%\$^@* solution lives in the time domain. Lesson 12 7

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Lesson 12 8 Formally, the Inverse Laplace Transform is given by the contour integral: Computing this integral is, in general, impossible in a practical context. Something better is required. ds e s X j t x st j j 0 0 2 1
We assume that Laplace transform pairs of primitive signals have been pre-computed and reduced to published tables, or determined using a tool such as MATLAB. We hope that through the intelligent understanding of the Laplace transform, a Laplace transformed X(s) can be returned to the time-domain (i.e., x(t)). But how? The standard mechanism to achieve this goal is called the Heaviside expansion (a.k.a., partial fraction). Lesson 12 9

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Heaviside Lesson 12 10 Oliver Heaviside (1850 –1925) was a self-taught English electrical engineer who adapted complex numbers to the study of electrical circuits , invented mathematical techniques to the solution of differential equations (later found to be equivalent to Laplace transforms ), plus other innovations.
Lesson 12 11 Assume the Laplace transform common signal types are pre-computed and tabled. Lesson 12

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Lesson 12 12 The trick, of course, is representing some arbitrary Laplace transform X(s) as a collection of terms having known Laplace transforms. This is facilitated using Heaviside’s method.
For a given X(s) of the form: If M N, X(s) is said to be proper . If M < N, X(s) is said to be strictly proper . If M > N, X(s) is said to be improper . N i i M i i N i i M i i p s z s s a s b s X 1 1 0 1 0 1 ) ( ) ( ) ( Lesson 12 13

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The terms z i are called zeros (N(s)=0 at s s=z i ).
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• Fall '07
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