20101ee102_1_hw2W

20101ee102_1_hw2W - the sinusoidal input u H t L = 5 Sin 20...

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EE 102 Winter HW ± 2 Do not use decimals - leave the answer in fractions, as necessary ! All the problems are on the system with IPOP : dv dt + k v H t L = u H t L , t > 0 1. Given k = 1 v H 0 L = 2 Input u H t L = 3 Cos H 2 Π t + Π ± 4 L t > 0 H i L Calculate the total response v H t L , t > 0 H ii L Use your answer in H i L to Find the steady state response, amplitude and phase. Ans H 1 L H i L v H t L = 2 e - t + ² 0 t e - H t - s L 3 Cos H 2 Π s + Π ± 4 L ² s Worked out in the Lecture Notes : k = 1; f = 1 ; Φ = Π 4 v H t L = 2 e - t + A H t L Cos 2 Π t + Π 4 - Ψ H t L Tan Ψ H t L = Sin2 Π t + 2 Π e - t - 2 Π Cos2 Π t Cos2 Π t - e - t + 2 Π Sin2 Π t Amplitude A H t L = 3 1 + 4 Π 2 - JI Sin2 Π t + 2 Π e - t - 2 Π Cos2 Π t M 2 + H Cos2 Π ft - e - t + 2 Π Sin2 Π t N 2 Simplify further H ii L Again worked out in the notes Printed by Mathematica for Students
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v H t L = A Cos 2 Π t + Π 4 - Ψ A = 3 1 + 4 Π 2 Tan Ψ = 2 Π 2. Continuing problem 1, find the steady state response H steady state only ! L for the input u H t L = 0 0 < t < 2 = Cos 2 2 Π t, t > 2 Ans Cos 2 2 Π t = 1 2 + Cos 4 Π t 2 v H t L = ± 0 ¥ e 1 2 + Cos 4 Π H t - Τ L 2 ±Τ = 1 2 + 1 2 1 I 1 + 16 Π 2 M Cos H 4 Π t - Ψ L Tan Ψ = 4 Π 3. Calculate the value of k, given the steady state amplitude response to
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Unformatted text preview: the sinusoidal input u H t L = 5 Sin 20 t 2 hw2W.nb Printed by Mathematica for Students A = 5 k 2 + 400 2 k 2 + 400 2 = 25 A 2 k = + 25 A 2-400 2 Plus sign because of stability ! 4. True or False ? Indicate your reasoning ! ' The more stable the system, the smaller is the steady state response to a the Unit Step Input b sinusoid Input c Unit Impulse Input Ans a True; response = 1 k ; larger the k more stable the system b True Amplitude = 1 k 2 + 4 2 f 2 c Response steady state = 0. However timeconstant gets smaller-so we can still say True hw2W.nb 3 Printed by Mathematica for Students...
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This note was uploaded on 09/03/2010 for the course EE ee102 taught by Professor Levan during the Spring '09 term at UCLA.

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20101ee102_1_hw2W - the sinusoidal input u H t L = 5 Sin 20...

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