# lt3 - Boise State University Department of Electrical and...

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Department of Electrical and Computer Engineering ECE225 – Circuit Analysis and Design The Laplace Transform III Reading Assignment : Review Sec. 15.3 Lecture Objectives : 1. To derive and apply the initial-value and ﬁnal-value theorems. 2. To run a MATLAB session with the functions pzmap and residue . Initial-Value Theorem F ( s ) = L{ f ( t ) } = Z 0 f ( t ) e - st dt f (0+) = lim t 0+ f ( t ) = lim s →∞ sF ( s ) Proof : Assume that f ( t ) = f (0) + f 0 (0) 1! t + f 00 (0) 2! t 2 + f 000 (0) 3! t 3 + ... F ( s ) = f (0) s + f 0 (0) 1! 1 s 2 + f 00 (0) 2! 2! s 3 + f 000 (0) 3! 3! s 4 + ... lim s →∞ sF ( s ) = lim s →∞ f (0) + f 0 (0) s + f 00 (0) s 2 + f 000 (0) s 3 + ... = f (0) Example : F ( s ) = - 2 s 3 + 7 s 2 + 2 s + 9 3 s 4 + 3 s 3 + 2 s 2 + 6 f (0) = lim s →∞ sF ( s ) = lim s →∞ - 2 s 4 + 7 s 3 + 2 s 2 + 9 s 3 s 4 + 3 s 3 + 2 s 2 + 6 = lim s →∞ - 2 s 4 3 s 4 = - 2 3 or f (0) = lim s →∞ sF ( s ) = lim s →∞ - 2 + 7 /s + 2

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lt3 - Boise State University Department of Electrical and...

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