solutions_test3_section001_spring08

solutions_test3_section001_spring08 - ECE 220 Test 3...

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ECE 220 Test 3 Solutions 14 April 2008 Problem 1. (20 points) A system is described by the differential equation given below: dv ( t ) dt + 4 v ( t ) = v s ( t ) (1) where v s ( t ) is the input. Let v s ( t ) = 10cos(5 t ) and v (0) = 0. Determine the solution in the Laplace domain; in other words, find V ( s ). Solution Take Laplace Transforms (LT) of both sides of Equation 1. We have L " dv ( t ) dt + 4 v ( t ) # = L [ v s ( t )] sV ( s ) - v (0) + 4 V ( s ) = 10 s s 2 + 5 2 (2) sV ( s ) - 0 + 4 V ( s ) = 10 s s 2 + 5 2 V ( s ) = 10 s ( s 2 + 5 2 )( s + 4) In deriving Equation 2 we used the following facts: 1. The linearity property of the LT 2. The differentiation property of the LT: L " dv ( t ) dt # = sV ( s ) - v (0) 3. The LT of the cos function is L [cos(5 t )] = s s 2 + 5 2 So, finally, V ( s ) = 10 s ( s 2 +5 2 )( s +4) 1
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Problem 2. (20 points) The solution (in the Laplace domain) is given by: V ( s ) = 1 ( s + 4)( s 2 + 4) (3) Determine the solution in the time domain, i.e., find v ( t ). Express v ( t ) as a real-valued signal. Solution The roots of ( s 2 + 4) are ± 2 j . Use Partial Fraction Expansion to rewrite Equation 3 as V ( s ) = K 1 s + 4 + K 2 s - 2 j + K * 2 s + 2 j (4) In Equation 4 we used the fact that the coefficients K 2 and K * 2 are complex conjugates of each other. Now K 1 = ( s + 4) V ( s ) | s = - 4 = ( s + 4) 1 ( s + 4)( s 2 + 4) | s = - 4 = 1 ( s 2 + 4) | s = - 4 = 1 (16 + 4) = 1 20 (5) K 2 = ( s - 2 j ) V ( s ) | s =2 j = ( s - 2 j ) 1 ( s + 4)( s 2 + 4) | s =2 j = 1 ( s + 4)( s + 2 j ) | s =2 j = 1 (2 j + 4)(2 j + 2 j ) = 1 4 j (4 + 2 j ) = 1 320 e (6) where θ = - tan - 1 ± 16 - 8 (7) Substitute Equations 5, 6 and 7 into Equation 4. We have V ( s ) = 1 20 1 s + 4 + 1 320 e 1 s - 2 j + 1 320 e - 1 s + 2 j and, in the time domain: v ( t ) = 1 20 e - 4 t u ( t ) + 1 320 e e 2 jt u ( t ) + 1 320 e - e - 2 jt u ( t ) 2
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= 1 20 e - 4 t u ( t ) + 1 320 [ e 2
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solutions_test3_section001_spring08 - ECE 220 Test 3...

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