Blank ExamSolutionsEE 350 EXAM IV 15 December 2010Last Name (Print):First Name (Print):ID number (Last 4 digits):Section:DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO D O SOProblem Weight Score1 252 253 254 25Total 100Te st Form AINSTRUCTIONS1. You have one hour and fifty minutes 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 be accepted and agrade of ZERO will be assigned.6. The quality of your analysis and eva luation 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. (9 points) Using the m ethod of Laplace transforms, determine the transfer function H(s) of the active filter inFigure 1, where f(t) is the input and y(t) is the output. Express your answer in the standard fo rmH(s) =bmsm+ · · · + b1s + bosn+ an−1sn−1+ · · · + a1s + a0.Figure 1: Active RC filter with input voltage f(t) and output voltage y(t).22. (9 points) The circuit in Figure 2 has input f(t) and output y(t). Using Laplace transform anal ysis, determinethe zero-input response of the system given that R = 1/3 Ω, L = 1/4 H, C = 1/2 F, i(0−) = −3 A, andy(0−) = 2 V.Figure 2: Passive RLC circuit w ith output voltage y(t).343. (7 points) In order to determi ne the partial fraction expansion of the transfer functionH(s) =b1s + bos2+ a1s + a0,an engineer used the MATLAB command>> [r, p, k] = resi due([b1, b0], [1, a1, a0])and obtainedr =2.00001.0000p =-3.0000-2.0000k =[](a) (3 points) Write down the partial fraction expansion of H(s).(b) (4 points) Specify the numeric values of the parameters b1, b0, a1, and a0.5Problem 2: (25 points)1. (12 points) Determine the closed-loop transfer function of the feedback control system in Fig ure 3, and specifyyour final answer using the standard formY (s)R(s)=bmsm+ · · · + b1s + bosn+ an−1sn−1+ · · · + a1s + a0.Figure 3: Feedback control system with reference input r(t) and controlled output y(t).672. (13 points) Another feedback control system, different from the one considered in part 1, has the closed-looptransfer function representationY (s)R(s)=K1s2+ (K2− K1)s + K1,where R(s) is the command input, Y (s) is the controlled output, and K1and K2represent controll er gains.(a) (6 points) Choose the controller gains so that the zero-state unit-step response of the closed-loop systemis underdamped with a natural frequency of 10 rad/sec and a dimensionless damping ratio of 1/2.(b) (7 points) Suppose that K1= −8, K2= −6, and define the closed-loop system error as e(t) = r (t) − y(t).For a ramp-input r(t) = tu(t), what value does e(t) approach as time increases?8Problem 3: (25 points)1. (15 points) A system has the transfer function representationH(s) =1014(s + 100)(s + 104) (s + 106)2.Construct the Bode magnitude and phase plots using the semilog graphs provided in Figure 4 (a duplicate copyappears in Figure 5).In order to receive credit:• In both your magnitude and phase plots, indicate each term separately using dashed lines.• Indicate the slope of each straight-line segment and the corner frequencies of the final magnitude andphase plots.• Do not show the 3 dB corrections in the magnitude plot.9Figure 4: Semi log paper for Bode magnitude and phase plots.10Figure 5: Semi log paper for Bode magnitude and phase plots.112. (10 points) Figure 6 shows the straight-line approximation of the ma gnitude and phase plots of a transferfunction H(s). The transfer function H(s) was generated in MATLAB using the script shown below, wherethe parameters a, b, and c are real-valued constantsH1 = tf([1,a], [1,b])H2 = tf([10], [1,c])H = seri es(H1, H2);100101102103104105106−100−60−2020Magnitude [dB]100101102103104105106−180−135−90−45045Phase [Deg]frequency [rad/sec]Figure 6: Straight-line approxima tion of the magnitude and phase plot of H(s).12(a) (3 points) From the MATLAB script, specify the transfer function in terms of the parameters a, b, and c.Express your answer in standard formH(s) =bmsm+ · · · + b1s + bosn+ an−1sn−1+ · · · + a1s + a0.(b) (2 points) Using Figure 6, specify the numeric value of the DC gain of the system represented by H(s).(c) (5 points) Determine H(s), and sp ecify the numeric values of the param eters a, b, and c.13Problem 4: (25 points)1. (13 points) A LTI system has the impulse response function representationh(t) = δ(t) + e−tu(t) − 2e−2tu(t).(a) (4 points) Determine the transfer function representation of the system and express your answer in thestandard formH(s) =bmsm+ · · · + b1s + bosn+ an−1sn−1+ · · · + a1s + a0.(b) (4 points) Sketch the p ole-zero m ap of the system transfer function. To receive credit, you must label theaxes and clearly specify the location of each pole and zero.14(c) (2 points) Specify the DC gain and high frequency gain of the system.(d) (3 points) If the system input and output are denoted by f(t) and y(t) respectively, specify the ODErepresentation of the LTI system.152. (12 points) Consider another LTI system, that is di fferent from the one in part 1. The zero-state response ofthe system to the unit-step input f(t) = u(t) isy(t) =2 − 2e−t− 2te−tu(t).For another input,¯f(t), the observed zero-state response is¯y(t) =2 − 3e−t+ e−3tu(t).Determine the
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