Blank ExamSolutionsEE 350 EXAM IV 19 December 2007Last Name (Print):First Name (Print):ID number (Last 4 digits):Section:DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SOProblem Weight Score1 252 253 254 25Total 100Test Form AINSTRUCTIONS1. You have 2 hours to complete this exam.2. This is a closed book exam. You may use one 8.5” × 11” note sheet.3. Calculators are not allowed.4. Solve each part of the problem in the space following the question. If you need more space, continue your solutionon the reverse side labeling the page with the question number; for example, Problem 1.2 Continued . NOcredit will be given to solutions that do not meet this requirement.5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not b e accepted and agrade of ZERO will be assigned.6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be preciseand clear; your complete English sentences should convey what you are doing. To receive credit, you mustshow your work.1Problem 1: (25 Points)1. (10 p oints) Determine the transfer function for the circuit in Figure 1 and express your answer in the standardformY (s)F (s)=bmsm+ bm−1sm−1+ ···+ b1s + b0sn+ an−1sn−1+ ···+ a1s + a0.Figure 1: Passive circuit with input voltage f(t) and output voltage y(t).22. (8 points) The input current f(t) and the stored energy determine the total output current y(t) for the networkshown in Figure 2. Given that C =0.5F, R = 1Ω, L = 5H, vc(0−) = 2V, and y (0−) = 2A, determine just thezero-input response yzi(t) of the circuit for t ≥ 0.Figure 2: Passive circuit with input current f(t) and output current y(t).33. (7 points) An engineer seeks your help in determining the impulse response h(t) of a LTI system, and providesthe MATLAB data in Figure 3.>> [r,p,k] = residue(num,den)r=82p=-2-1k=4Figure 3: Expanding H(s) using MatLab.• (3 points) What is the impulse response function h(t)?• (4 points) Specify the transfer function, and express your answer in the standard formH(s)=bmsm+ bm−1sm−1+ ···+ b1s + b0sn+ an−1sn−1+ ···+ a1s + a0.4Problem 2: (25 points)1. (10 p oints) Figure 4 shows a feedback control system with reference input r(t) and output y(t). Determine theclosed-loop transfer function and express your answer in the standard formY (s)R(s)=bmsm+ bm−1sm−1+ ···+ b1s + b0sn+ an−1sn−1+ ···+ a1s + a0.Figure 4: Feedback control system.562. (15 points) Figure 5 shows another closed-loop system where the parameters α and K are real-valued constants.Figure 5: Feedback control system with real-valued constant parameters K and α.(a) (7 p oints) Given that the parameter α is chosen so that the closed-loop system is BIBO stable, specifythe value of K so that the DC gain of the closed-loop system is four.(b) (8 points) Set K = 1 and choose the value of α so that the unit-step response of the closed-loop systemis under-damped with ζ =0.6.7Problem 3: (25 points)1. (13 points) An engineer determines the frequency response function of a LTI system by applying the inputf(t) = cos(ωot)and recording the sinusoidal steady-state responsey(t)=A cos(ωot + θ)for a range of frequencies ωo. Table 1 shows the experimental data. Determine the frequency response functionof the system and express your answer in the standard form, for example,H(ω)=K(ω /ω1+1)ω2n(ω /ωa+1)(ω )2+2ζωn(ω )+ω2n.Carefully read note 6 on page 1 of the exam. In order to receive partial credit, clearly describe your approach.ωo[rad/sec] A [V] θ [deg]0.1 20 -1800.5 20 -1901.0 20 -2005.0 10 -27010 4 -30650 0.2 -350100 0.05 -354500 0.002 -3591000 0.0005 -360Table 1: Frequency dependence of the sinusoidal steady-state resp onse.82. (12 points) Consider the system whose transfer function representation isH(s)=100 (s − 1)s +10.The zero of the transfer function H(s) is called an unstable zero because it resides within the right-half ofthe s-plane.(a) (1 point) What is the magnitude and sign of the DC gain?(b) (1 point) What is the magnitude and sign of the high frequency gain?(c) (3 points) Provide a hand sketch of the magnitude and phase reponse of the zero termω/1 − 1.Carefully label the slope of the magnitude and phase plots and indicate the location of the corner frequency.(d) (8 points) Sketch the straight-line approximation of the magnitude and phase response of H(s) in Figure6 (page 10) and Figure 7 (page 11), respectively. Label all relevant features of the graphs.910 10 10 10 10 10 10Magnitude construction plot.Magnitude [dB]10 10 10 10 10 10 10Magnitude plot.Magnitude [dB]Frequency [rad/sec]Figure 6: Graphs for constructing the Bode magnitude response.1010 10 10 10 10 10 10Phase construction plot.Phase [Degrees]10 10 10 10 10 10 10Phase plot.Phase [Degrees]Frequency [rad/sec]Figure 7: Graphs for constructing the Bode phase response.11Problem 4: (25 points)1. (13 points) A LTI system has the transfer function representationH(s)=13s +45s2+10s +9.(a) (3 points) Specify the system pole(s) and zero(s).(b) (1 point) Is the system BIBO stable?(c) (2 points) What is the DC gain of the system?(d) (3 points) Is the zero-state unit-step response under-damped, over-damped, or critically damped?(e) (4 points) Find the impulse response representation of the system.122. (12 points) A LTI system with input f(t) and output y(t) has the ODE representationd2ydt2+5dydt+6y(t)=f(t).(a) (3 points) Find the transfer function representation of the system and express your answer in the standardformH(s)=bmsm+ bm−1sm−1+ ···+ b1s + b0sn+ an−1sn−1+ ···+ a1s + a0.(b) (5 points) Given the initial states y (0−) = 0 and ˙y(0−) = 1, find the Laplace transform Y (s) of theunit-step response, and express your result in the standard formY ( s)=bmsm+ bm−1sm−1+ ···+ b1s + b0sn+ an−1sn−1+ ···+ a1s + a0.(c) (4 points) Using Y (s), determine the steady-state value of the unit-step
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