Blank ExamSolutionsEE 350 EXAM IV 17 December 2008Last 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 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 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. (8 points) Determine the zero-state unit-step response y(t) of the circuit in Figure 1 where R =3Ω,C =0.5F,and L =1H.Figure 1: Passive RLC circuit with input voltage f (t) and output voltage y(t).22. (9 points) Determine the zero-input response y(t) for the circuit in Figure 2, where R =5Ω,C =16F, andL = 1 H, given that the initial conditions are i(0−)=−2 A and y (0−)=0V.Figure 2: Passive RLC circuit with initial conditions i(0−) and y(0−).33. (8 points) An engineer utilizes MATLAB to analyze the transfer function of a particular system representedby the numerator and denominator polynomials num and den, respectively, and obtains:>> [r, p, k] = residue(num,den)r=3-2p=-4-3k=1>>Determine the impulse resp onse function h(t ) and the transfer function representation H(s) of the system.Express the transfer function in the standard formH(s)=bmsm+ ···+ b1s + bosn+ an−1sn−1+ ···+ a1s + a0.4Problem 2: (25 points)1. (10 points) Determine the transfer function of the closed-loop system in Figure 3 and express your answer inthe standard formH(s)=Y ( 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).52. (15 points) A closed-loop system, different from the one considered in part 1, is represented by the transferfunctionY (s)R(s)=50s2+ βs +β24+25,where β is a real-valued constant.(a) (5 points) For what range of β is the closed-loop system BIBO stable?(b) (5 points) Determine the value of β for which the steady-state unit-step response of the closed-loop systemis one, that is, yss= limt→∞y(t)=1.(c) (5 points) Find the value of β for which the zero-state unit-step response of the closed-loop system is asinusoid that decays exponentially as e−3t.6Problem 3: (25 points)1. (13 points) A FET amplifier has the transfer function representationH(s)=−200, 000 s(s + 20)(s +20, 000).Construct the Bo de magnitude and phase plots using the semilog graphs provided in Figures 4 (a duplicatecopy appears 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 the straight-line segments and corner frequencies of the final magnitude and phaseplots.• Do not show the 3 dB corrections in the magnitude plot.7Figure 4: Semilog paper for Bode magnitude and phase plots.8Figure 5: Semilog paper for Bode magnitude and phase plots.92. (13 points) For a particular linear time-invariant system, Figure 6 shows the exact magnitude and phaseresponse generated by the MATLAB scriptomega = logspace(a,b,1000);num = c*[1,1];den = [d,1];bode(num,den,omega)grid on10−210−11001011021031040306090Phase (deg)Bode DiagramFrequency (rad/sec)202530354045505560Magnitude (dB)Figure 6: Exact magnitude and phase plot generated by MATLAB.10(a) (4 points) What are the values of the parameters a and b?(b) (4 p oints) Using the infromation available from Figure 6, specify the DC gain and high frequency gain.Express your answers in both |H|dBand |H|.(c) (4 points) Using your results in part (b), determine the values of the parameters c and d.11Problem 4: (25 points)1. (12 points) A linear time-invariant system is represented by the transfer functionH(s)=2s +10s2+2s +5.(a) (3 points) Sketch the pole-zero map of the transfer function and label the value of all poles and zeros.(b) (2 points) Is the zero-state unit-step response of the system underdamped, critically damped, or over-damped?(c) (2 points) Determine the DC gain and high frequency gain of the system.(d) (5 points) Find h(t), the impulse response representation of the system.122. (13 points) Another linear time-invariant system with input f(t) and output y(t) has the ODE representationd2ydt2+ adydt+ by( t)=cf(t)where a, b, and c are constant parameters.(a) (4 points) Find the transfer function representation of the system and place your answer in the standardformH(s)=bmsm+ ···+ b1s + bosn+ an−1sn−1+ ···+ a1s + a0.(b) (9 points) Given that the initial conditions y(0−) = 0 and ˙y(0−)=a result in the unit-step responseY ( s)=5s +2s3+5s2+10s,determine the numeric values of the constants a, b, and c.132. (13 points) For a particular linear time-invariant system, Figure 6 shows the exact magnitude and phaseresponse generated by the MATLAB scriptomega = logspace(a,b,1000);num = c*[1,1];den = [d,1];bode(num,den,omega)grid on10−210−11001011021031040306090Phase (deg)Bode DiagramFrequency (rad/sec)202530354045505560Magnitude (dB)Figure 6: Exact magnitude and phase plot generated by
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