This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 33+34 Inverse LaPlace: partial fractions General idea: • Take differential equation • Convert derivatives + integrals to sdomain • Convert inputs to sdomain (look up f(t) b F(s) • Solve for outputs in sdomain • 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 ?...
View
Full
Document
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.
 Spring '05
 KELLEY/SEYLER

Click to edit the document details