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Lecture 31+32

# Lecture 31+32 - LaPlace Transform The basic transform...

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Lectures 31+32 LaPlace Transform

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The basic transform LaPlace transform converts functions from time domain to “s” (s is a sort of generalized frequency) Formally an integral: Also an inverse form But rarely use it Big benefit of LaPlace transform: it works on both functions and operations ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 t f ds st s F j s F L s F dt st t f t f L jT jT T = = = - = - - exp lim 2 1 exp 1 0 π
Some functions: Step Function: Decaying exponential: Decaying sinusoid: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 0 0 0 0 0 1 1 2 1 2 exp exp exp exp 2 exp exp sin exp 1 exp exp exp exp 1 exp ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ + + = + + - - + = + + - - - + - = - - - - = - + = + - = - - = - = - = a s j s a j s a j dt j j s a t j s a t dt st at j t j t j t at L a s dt a s t dt st at at L s dt st t u t u L

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Table of functional transforms: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 cos exp sin exp cos sin ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ + + + - + + - + + a s a s t at a s t at s s t s t ( 29 ( 29 ( 29 ( 29 ( 29 (
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