Lecture 13

# Lecture 13 - ECE 3101 Signals and Systems Reading...

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Lecture 13, 3/14/11, Dis ECE 3101, Copyright P. B. Luh 1 ECE 3101 Signals and Systems Reading Assignment: Sections 4.3-4.5 and 10.3.1 Problem Set 5: Due tomorrow Problem Set 6: 4.3-1, 4.3.6, 4.4-5, 4.4-10, 10.3-3, and 10.3-9. Due Thursday next week Exam. 2, Tuesday next week, end of Ch. 3; 6 cheat sheets Last Time: Inverse Laplace Transform F(s) = N(s)/D(s) ~ Rational function with real coefficients Table Lookup 0 0, δ (t) 1, δ (n) (t) s n u(t) 1/s, t 1/s 2 , … t n-1 /(n-1)! 1/s n e - α t 1/(s+ α ), t n-1 e - α t /(n-1)! 1/(s+ α ) n sin ϖ t ϖ /(s 2 + ϖ 2 ), cos ϖ t s/(s 2 + ϖ 2 )

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2 Inverse Laplace Transform by Partial Fraction Expansion Case 1. F(s) = N(s)/D(s) is not a strictly proper rational function. Long division. δ (t) and its derivatives Case 2. F(s) is strictly proper rational with distinct poles ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 n n 2 2 1 1 n 2 1 s k .. s k s k s .. s s ) s ( N ) s ( F λ - + + λ - + λ - = λ - λ - λ - = f(t) = (k 1 e λ 1 t + k 2 e λ 2 t + . . + k n e λ n t ) U(t) Case 3. F(s) is strictly proper with distinct complex poles Γ (s) F(s)(s - λ 1 )(s - λ 2 )/j β ; λ 1 = α + j β , Γ ( λ 1 ) Γ r + j Γ i f(t) = e α t ( Γ r cos β t - Γ i sin β t)U(t) + other terms Case 4. F(s) is strictly proper with multiple poles ( 29 ( 29 ( 29 ) s ( F s a .. s a s a ) s ( F 1 r 1 r 1 r 0 + λ - + + λ - + λ - = - - ( 29 ( 29 { } ) s ( F ) t ( U e a .. e ! 2 r t a e ! 1 r t a ) t ( f 1 t 1 r t 2 r 1 t 1 r 0 - λ - λ - λ - + + + - + - = L
Lecture 13, 3/14/11, Dis ECE 3101, Copyright P. B. Luh 3 Case 5. With Exponentials: f(t-t 0 )U(t-t 0 ) e -st 0 F(s) Applications of Laplace Transform: Find y(t), h(t), x(t) I/O Description y (N) + a 1 y (N-1) + . . + a N-1 y (1) + a N y = b 0 u (N) + b 1 u (N-1) + . . + b N-1 u (1) + b N u (s N + a 1 s N-1 + . . + a N-1 s + a N ) Y(s) = (b 0 s N + b 1 s N-1 + . . + b N-1 s + b N ) U(s) + f8e5 P(s) Q(s) P(s) Y ZI (s ) ) s ( U ) s ( Q ) s ( P ) s ( Q ) s ( P ) s ( Y + = Y ZS (s ) H(s) Y ZS (s) = H(s) U(s)

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Lecture 13, 3/14/11, Dis ECE 3101, Copyright P. B. Luh 4 State and Output Description 0 x ) 0 ( x with , Bu Ax x = + = - y = Cx + Du ) s ( BU ) s ( AX x ) s ( sX 0 + = - ) s ( BU x ) s ( AX ) s ( X sI 0 + = - ( 29 [ ] ) s ( BU x A sI ) s ( X 0 1 + - = - ) s ( DU ) s ( CX ) s ( Y + = ( 29 [ ] ) s ( DU ) s ( BU x A sI C 0 1 + + - = - ( 29 ( 29 [ ] ) s ( U D B A sI C x A sI C ) s ( Y 1 0 1 + - + - = - - Y ZI (s ) Y ZS (s ) H(s) H(s) = C(sI-A) -1 B + D
Lecture 13, 3/14/11, Dis ECE 3101, Copyright P. B. Luh 5 Today: Sections 4.3-4.5, and 10.3.1 The Transformed Network Method: A different look at circuits without having I/O description or State and Output description first Block Diagrams System Stability Next Thursday: Sections 2.6 and 4.8 Frequency Response to Sinusoidal Inputs

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Lecture 13 - ECE 3101 Signals and Systems Reading...

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