Lecture 33+34_rev3

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

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Unformatted text preview: Lecture 33+34 Inverse LaPlace: partial fractions General idea: • Take differential equation • Convert derivatives + integrals to s-domain • Convert inputs to s-domain (look up f(t) b F(s) • Solve for outputs in s-domain • Convert back to time: G(s) b g(t) • But what if G(s) isn’t in your table? 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 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 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 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- =- = = + = + ∴ + + + + + = + + + + + + = + + +- = + + = + + = + + = + + + + + + + + = 1 2 1 2 1 2 1 2 1 2 1 2 1 ?...
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This note was uploaded on 11/21/2011 for the course ECE 2100 taught by Professor Kelley/seyler during the Spring '05 term at Cornell.

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

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