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Unformatted text preview: 1 1 Summary of Lecture #4 8/30/2010 Inv. Laplace trans. by partial fraction expansion of rational functions Double poles Triple and higher order poles Complex poles Examples Basic properties of Laplace transform Physical meaning and Examples Use Laplace transform to solve differential equations , an example Use Laplace transform to solve circuit problems, examples 2 Integral is from - to + The initial value, f(0- ) , is accounted for in L (f(t)). But f(t) for t < 0 is excluded in, and has no effect on, L (f(t)). Inv. Lap. trans. always gives for t < 0 , i.e., f(t) = L- 1 (F(s)) = 0 for t < 0 . 10 basic properties of Lap. trans. are given in Table 12.2 (p. 584). dt e ) t ( f ) s ( F )) t ( f ( st -- L ) t ( f ds e ) s ( F 2 j 1 )) s ( F ( st 1- = + L 3 p. 564 Simple pole Double pole Higher order pole Complex pole Imaginary pole x j x j x j x x j x . (Top half of Table 13.1) x j x Table 12.1 4 2 at f (t) F(s) K (t) K 1 r(t) tu(t) K Ku(t) or K s 1 e u(t s s ) a- = + 2 2 at 2 2 2 2 at 2 2 f (t) F(s) sin t u(t) s e sin t u(t) (s s cos t u(t) s s a e cos t u(t) (s a) a)-- + + + + + + + Selected part of Table 12.1 5 p. 584 Table 12.2 6 Rational functions (RF) Set b n = 1 Proper RF, m n Strictly proper RF, m < n Partial fraction expansion of RF Constant Simple poles Repeated poles (double, triple poles etc.) Complex poles 1 2 n 2 n 1 n 1 n n 1 2 m 2 m 1 m 1 m m m b s b s b s b s a s a s a s a s a ) s ( F + + + + + + + + + + =-------- 2 7 Strictly proper rational functions, m < n and if roots are distinct, p 1 p 2 p 3 p 4 p 1 , p 2 , etc. are roots of the denominator polynomial. How do we determine A 1 , A 2 . etc.?...
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- Spring '06