Blank ExamExam SolutionsEE 350 EXAM IV 14 December 2009Last 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) Using the metho d of Laplace transforms, determine the impulse response h(t) of the circuit in Figure1, where f(t) is the input, y (t ) is the output, R1= R2= 1 kΩ, and C =1µF.Figure 1: Passive RC circuit with input voltage f(t) and output voltage y(t).22. (17 p oints) The output of the circuit in Figure 2 is the voltage y(t). Prior to the switch opening at time t =0,the capacitor voltage v(t) and the inductor current i(t), as defined in Figure 2, have reached steady-state values.Figure 2: Passive RLC circuit with output voltage y(t). The switch is opened at time t =0.(a) (4 points) Determine the values of v(0−) and i(0−).3(b) (8 points) After the switch is opened, determine the Laplace transform representation Y (s) of the outputresponse y( t). Express your answer in the standard formY ( s)=bmsm+ ···+ b1s + bosn+ an−1sn−1+ ···+ a1s + a0.4(c) (5 points) Given that R =2Ω,L =1H,C = 1 F, and Vo= 1 V, determine y(t) for t ≥ 0.5Problem 2: (25 points)1. (13 points) Figure 3 shows a closed-loop system containing a plant with the transfer functionGp(s)=Y ( s)F ( s)=10s +5and the proportional-plus-integral (PI) controller with proportional gain KPand integral gain KI. Determinethe transfer function of the closed-loop system in Figure 3 and express your answer in 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. (12 points) For the closed-loop system in Figure 3 it can be shown that the transfer function from the referenceinput r( t) to the closed-loop output error e(t)=r(t) − y(t)isE(s)R(s)=s (s +5+10KP)s2+(5+10KP) s +10KI(a) (6 points) Assuming that the controller gains KIand KPhave been chosen so that the closed-loop systemis BIBO stable, determine the steady-state erroress= limt→∞e(t)for the ramp input r(t)=tu(t).(b) (6 points) Choose the controller gains KPand KIso that the poles of the closed-loop transfer functionE(s)/R(s) have ζ =3/4 and ωn= 10.8Problem 3: (25 points)1. (13 points) A system has the transfer function representationH(s)=1000 s (s + 100)(s + 10) (s2+ 1010 s +10, 000).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 the straight-line segments and corner frequencies of the final magnitude and phaseplots.• Do not show the 3 dB corrections in the magnitude plot.9Figure 4: Semilog paper for Bode magnitude and phase plots.10Figure 5: Semilog paper for Bode magnitude and phase plots.112. (12 points) Figure 6 shows the straight-line approximation of the magnitude and phase plot of a transferfunction G(s). Determine the transfer function G(s), and place your answer in the standard formG(s)=Y ( s)R(s)=bmsm+ ···+ b1s + bosn+ an−1sn−1+ ···+ a1s + a0.Figure 6: Straight-line approximation of the magnitude and phase plot of G(s).1213Problem 4: (25 points)1. (6 points) Figure 7 shows the p ole-zero map of a linear time-invariant system with unity DC gain. Determinethe transfer function representation H(s) of the system and express your answer in the standard formH(s)=bmsm+ ···+ b1s + bosn+ an−1sn−1+ ···+ a1s + a0.Figure 7: Pole-zero map of H(s).142. (12 points) Another system with input f(t) and output y(t) has the transfer function representationY ( s)F ( s)=s +75s2+6s +25.(a) (4 points) Represent the system as an ordinary differential equation and express your result in the standardformdnydtn+ an−1dn−1ydtn−1+ ···+ aoy = bmdmfdtm+ bm−1dm−1fdtm−1+ ···+ bof.(b) (4 points) Determine whether or not the system is BIBO stable.(c) (4 points) Determine the steady-state value of the unit-step response of the system.153. (7 points) Using the MATLAB command>> [r, p, k] = residue(num,den)to determine the partial fraction expansion of H(s), an engineer determines that the corresponding impulseresponse function ish(t)=4δ(t)+2e−tu(t)+3te−tu(t)+e−3tu(t);Specify the numeric value of the vectors r, p, and
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