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
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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 ) (
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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
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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
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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.
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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.
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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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