Jan 27

# Jan 27 - EE 102 WINTER 10 Laplace Transform Techniques...

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Unformatted text preview: EE 102 WINTER 10 Laplace Transform Techniques Wednesday 27 and Monday Feb1. Feb 3 is Mid Term Review Feb 8 is Midterm CHAPTER 3 IN BOOK RECALL W' s s W s W We got this formula by Integration b Parts What do we mean by the derivative ? We mean t W' Σ Σ W t W Let W . be the unit step. Then W' t We do not include delta functions in our definition of a function It is not the delta function On the other hand we have defined the delta function by t Δ Σ Σ 1 for t Just so you remember both If a function is an integral it must be continuous ... Printed by Mathematica for Students d 2 v t dt 2 a 1 d v t dt a v t b u t b 1 d u t dt Here we do NOT allow delta functions in the derivatives on both sides Plus ALL Coefficients area REAL. They may of course be zero in principle in the limiting sense. The spirit of the secondorder equation is that a is NOT zero You can see this if you consider a physical example. Define e st v t t V s Laplace Transform Both Sides e st d dt v t dt v s e st v t dt e st v t dt s V s v Hence e st d 2 dt 2 v t dt s s V s v v s 2 V s s v v 2 Jan 27.nb Printed by Mathematica for Students a 1 e st v t dt a 1 s V s v Hence LT of LHS e st d 2 dt 2 v t a 1 d dt v t a v t dt s 2 V s a 1 s V s a V s s v v a 1 v s 2 V s a 1 s V s a V s v s a 1 v RHS Transform b 1 sU s u b U s Hence s 2 V s a 1 s V s a V s b 1 s U s b U s b 1 u s v v a 1 v V s b 1 s b s 2 a 1 s a U s v s v a 1 v b 1 u s 2 a 1 s a IPOP TRANSFER FUNCTION s b 1 s b s 2 a 1 s a SYSTEM TRANSFER FUNCTION H s 1 s 2 a 1 s a Jan 27.nb 3 Printed by Mathematica for Students V s s U s H s v s v a 1 v b 1 u First Term : ZERO State Input Response Second Term : Zero Input Transient Response : Note the dependence on the input ZERO INPUT RESPONSE Inverse Transform of H s v s v a 1 v b 1 u H s v a 1 v b 1 u s H s v First Consider H s Partial Fraction Expansion Technique Inverse Transform of 1 k s IS e kt Inverse Transform of : 1 s 2 b s c Note that we now use b for a 1 and c for a as a concession to HighSchoolath in root finding...
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Jan 27 - EE 102 WINTER 10 Laplace Transform Techniques...

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