lecture 4

# lecture 4 - 1 1 Summary of Lecture#4 • Inv Laplace trans...

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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- = π ≡ ∫ Γ + L 3 p. 564 Simple pole Double pole Higher order pole Complex pole Imaginary pole x σ j ϖ x σ j ϖ x σ j ϖ x x σ j ϖ x …. (Top half of Table 13.1) x σ j ϖ 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- δ = + 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)-- ϖ ϖ + ϖ ϖ ϖ + + ϖ ϖ + ϖ + ϖ + + ϖ 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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lecture 4 - 1 1 Summary of Lecture#4 • Inv Laplace trans...

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