Lecture 33+34_rev3

Lecture 33+34_rev3 - Inverse LaPlace partial fractions...

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Lecture 33+34 Inverse LaPlace: partial fractions
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General idea: Take differential equation Convert derivatives + integrals to s-domain Convert inputs to s-domain (look up f(t) barb2right F(s) Solve for outputs in s-domain Convert back to time: G(s) barb2right g(t) But what if G(s) isn’t in your table?
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An example Find diff. eq. using KCL LaPlace transform it Solve for Vc(s) Convert into a set of ratios of polynomials And then…? We don’t necessarily have the inverse LaPlace for this formula in our table. ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 0 2 2 0 2 2 2 2 0 2 0 2 0 0 2 0 0 0 0 0 0 2 2 2 1 1 1 1 0 0 0 o L o C o o in C L C in C L C in C L in C C C C L t in C C C L t in C C C s s C I s s sV s s s V s V LC RC s s C I LC RC s s sV LC RC s s LC s V s V LI sLCV s V R sL LC s s V s I sL s V sL s V R s V CV s sCV I dx L x V x V R t V dt t dV C L I dx L x V x V R t V dt t dV C ϖ α ϖ α ϖ α ϖ + + + + + + + + = + + + + + + + + = + + = + + + - + + + = + - + + = + - + + = V in (t) L C R
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Partial Fraction Decompostion 1 st Thing: superposition works, so can solve for each piece separately we’ll start with the last term Factor the denominator to find its roots Now “guess” that this can be written as a sum of constants over the roots Now just need to solve for the constants in the numerator Can do this directly, which is easier here, and obvious, but doesn’t scale well for more complicated problems Can use a more general technique ( 29 ( 29 ( 29 ( 29 ( 29( 29 ( 29( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29( 29 a b C I K K K s K s K C I a K b K s b s a s a K s b K s b K s a K s b K s a K s b s a C I b a s b s a C I s V s s C I s V s s C I s s sV s s s V s V L L L o o L I C o L I C o L o C o o in C L L - = - = = + = + + + + + + = + + + + + + = + + + - = + + = + + = + + = + + + + + + + + = 0 1 2 1 2 1 0 2 1 2 1 2 1 2 1 ? 0 2 2 2 2 0 2 2 0 2 2 0 2 2 0 2 2 2 , 0 , , 2 2 2 2 0 0 ϖ α α ϖ α α ϖ α ϖ α ϖ α ϖ α ϖ L C R V in (t) I L0
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Solving for “weights” of partial fractions, in general In general, for each root, have one constant, K
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