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20101ee102_1_hw2W

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

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EE102Winter HW 2 Donotusedecimals - leavetheanswerinfractions, asnecessary ! AlltheproblemsareonthesystemwithIPOP: dv dt + kv H t L = u H t L , t > 0 1.Given k = 1 v H 0 L = 2 Input u H t L = 3Cos H 2 Π t + Π 4 L t > 0 H i L Calculate thetotalresponse v H t L , t > 0 H ii L Useyouranswerin H i L to Find thesteadystateresponse, amplitudeandphase. Ans H 1 L H i L v H t L = 2e - t + 0 t e - H t - s L 3Cos H 2 Π s + Π 4 L s WorkedoutintheLectureNotes: k = 1; f = 1; Φ = Π 4 v H t L = 2e - 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 Againworkedoutinthenotes Printed by Mathematica for Students

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v H t L = ACos 2 Π t + Π 4 - Ψ A = 3 1 + 4 Π 2 Tan Ψ = 2 Π 2.Continuing problem1, findthesteadystateresponse H steadystateonly ! L fortheinput u H t L = 0 0 < t < 2 = Cos 2 2 Π t, t > 2 Ans Cos 2 2 Π t = 1 2 + Cos4 Π t 2 v H t L = 0 ¥ e 1 2 + Cos4 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. Calculatethevalueofk, giventhesteadystateamplituderesponseto
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Unformatted text preview: the sinusoidal input u H t L = 5 Sin 20 Π t 2 hw2W.nb Printed by Mathematica for Students A = 5 k 2 + 400 Π 2 k 2 + 400 Π 2 = 25 A 2 k = + 25 A 2-400 Π 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 Π 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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