Lecture 31-33_v2 - Lectures 31-33 LaPlace Transform Partial...

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Spring 2011 ECE 2100, Cornell Univ. Prof. Molnar 1 Lectures 31-33 LaPlace Transform Partial fractions Poles, zeros
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Spring 2011 ECE 2100, Cornell Univ. Prof. Molnar 2 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
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Spring 2011 ECE 2100, Cornell Univ. Prof. Molnar 3 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 + + = + + - - + = + + - - - + - = - - - - = - + = + - = - - = - = - = 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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Spring 2011 ECE 2100, Cornell Univ. Prof. Molnar 4 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 + + + - + + - + + a s a s t at a s t at s s t s t ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 a s s a at a s at t a s at s t s t u t + - - + - + - exp 1 1 exp 1 exp 1 1 1 2 2 δ 1 2 3 4 5 6 7 8 9 10 But note there are many, many more. (see the internet)
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Spring 2011 ECE 2100, Cornell Univ. Prof. Molnar 5 Some operators: • LaPlace passes: – Addition: – multiplication by a scalar – Implies L() is linear: superposition works!! • LaPlace also transforms: – Time derivatives – Time Integrals – Time shifts (delays) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 s aF t af L s G s F t g t f L = + = + ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 as s F a t u a t f L s F s dx x f L f s sF dt t df L t - = - - = - = - exp 1 0 0
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Spring 2011 ECE 2100, Cornell Univ. Prof. Molnar 6 An Example 1 st find Diff. Eq. Convert to LaPlace: d/dt b s – Define V C0 as initial condition Solve for Vout Insert Vin = u(t) b 1/s Convert back to time domain, by using table of functional transforms (here #4 and #6) Vin(t) Vout ( 29 ( 29 ( 29 ( 29 RC t V RC t V t V RC a s s RC V RC s RC V s V s RC s V RC s RC V sRC s V RCV s V CV s CsV R s V s V dt dV C R t V t V in C out in C out in C in C out C out out in out out in - - + - = = + + + = + + + = + + = + - - = - - = exp 1 exp ) ( 1 1 1 1 ) ( 1 ) ( 1 1 1 ) ( ) ( ) ( ) ( ) ( 0 ) ( ) ( 0 0 0 0 0 0 R C
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Spring 2011 ECE 2100, Cornell Univ. Prof. Molnar 7 Impulse (delta) functions What is the derivative of u(t)? Slope is infinite, but only at t=0,
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Lecture 31-33_v2 - Lectures 31-33 LaPlace Transform Partial...

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