Blank ExamSolutionsEE 350 EXAM IV 7 May 2004Last Name:First Name:ID number (Last 4 digits):Section:DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SOProblemWeight Score125225325425Total100Test Form AInstructions1. You have two hours to complete this exam.2. This is a closed-bo ok exam. You are allowed one 8.5" by 11" note sheet.3. Calculators arenotallowed.4. Solve each part of the problem in the space following the question. If you need more space,continue your solution on the reverse side lab eling the page with the question numb er, forexample, \Problem 2.1) Continued." No credit will b e given to a solution that does not meetthis requirement.5. Do not remove any pages from this exam. Lo ose papers will not b e accepted and a grade ofzero will be assigned.6. The quality of your analysis and evaluation is as imp ortant as your answers. Your reasoningmust be precise and clear; your complete English sentences should convey what you are doing.To receive credit, you must show your work.1Problem 1: (25 Points)1. (13 points) The switch in Figure 1 has b een closed suciently long so that the voltages and currents inthe circuit have reached steady-statevalues at timet= 00. At timet= 0 the switch is op ened. UsingLaplace transform analysis techniques, determine the output voltagey(t) fort0 given thatVs= 6 V,R= 1 ,C= 2 F, andL= 2 H. Express your answer in terms of sine and cosine functions that donot include a phase angle; your answers mustnotcontain complex exp onential terms.No creditwillbe given for solutions using the classical approach covered on Exam I.Vs−−t = 0R+C L+y(t)Figure 1: The switch in the RLC circuit is opened at timet= 0.232. (12 p oints) Figure 2 shows the circuit diagram of a bandpass lter with input voltagef(t) and outputcurrenty(t). Assuming an ideal op erational amplier, determine the transfer function of the lter andplace your answer in the standard formH(s) =Y(s)F(s)=bmsm+bm01sm01+1 1 1+b1s+b0sn+an01sn01+1 1 1+a1s+a0:+−f(t)+−C1C2R2R1R3y(t)Figure 2: Active bandpass lter with input voltagef(t) and output currenty(t).45Problem 2: (25 p oints)1. (7 p oints) The feedback control system in Figure 3 has a command inputF(s), an outputY(s), anda proportional controller with gainK. Determine the closed-lo op transfer function and express youranswer in the formH(s) =Y(s)F(s)=bmsm+bm01sm01+1 1 1+b1s+b0sn+an01sn01+1 1 1+a1s+a0:Σ+_Σ+_Ks0.1F(s)s + 11Y(s)Figure 3: Feedback control system with command inputF(s) and outputY(s).672. (11 p oints) Another closed-loop system, dierent from the one considered in part 1, has the closed-lo optransfer functionY(s)F(s)=300s2+ s+2+ 75;whereis a real-valued parameter.(6 points) For what value ofwill the steady-state resp onse of the closed-loop system to aunit-step input be equal to three ?(5 points) For the value ofthat you obtained, is the zero-state unit-step resp onse underdamped,critically-damped, or overdamped ?83. (7 points) In order to determine the partial fraction expansion of the transfer functionH(s) =Y(s)F(s)=2s+ 1s3+ 5s2+ 8s+ 4;the MatLabCommand residue is used:[R, P, K] = residue(B, A);R = 1.00003.0000-1.0000P = -2.0000-2.0000-1.0000K = [ ](2 p oints) Sp ecify the numeric value of the row vectors B and A.(2 p oints) rite down the partial fraction expansion ofH(s).(3 p oints) Findh(t), the inverse Laplace transform ofH(s).9Problem 3: (25 p oints)1. (15 points) A system has the transfer functionH(s) =105s(s+ 10) (s+ 1000)2:Sketch the Bo de magnitude and phase plots in Figure 4.In order to receive credit:Use dashed lines to show the Bo de plot of each term of the transfer function and a solid line toshow the comp osite Bo de plot.Indicate the slope of the straight-line segments and corner frequencies of the nal magnitude andphase plots.Do not show the 3 dB corrections in the magnitude plot.10Figure 4: Semilog pap er for constructing the ode magnitude and phase plots.112. (10 p oints) The Bode magnitude and phase response of a second-order LTI system with no zeros(m= 0) is shown in Figure 5. Find the steady-state responsey(t) of the system for the inputf(t) = 3 + cos(102t045) + 106cos(106t+ 45):1210−1100101102103104105−100−80−60−40−20020Magnitude (dB)Bode Diagram10−110010110210310410504590135180Frequency (rad\sec)Phase (deg)Figure 5: ode magnitude (in d ) and phase (in degrees) plots for the second-order LTI system.13Problem 4: (25 p oints)1. (12 points) A LTI system with 0< <1 has the transfer functionH(s) =!2ns2+ 2!ns+!2n:(6 p oints) Draw a p ole-zero diagram of the transfer functionH(s), and clearly specify the realand imaginary parts of the p ole lo cations in terms of and!n.(6 p oints) hat is the distance between the origin of the s-plane (s= 0) and each pole ? Expressyour answer in terms of and/or!n.142. (13 points) A LTI system with inputf(t) and outputy(t) is represented by the DEy+ 20 _y+ 200y(t) = 400f(t):(4 points) Find the transfer functionH(s) =Y(s)=F(s) of the system and express your result inthe formH(s) =Y(s)F(s)=bmsm+bm01sm01+1 1 1+b1s+b0sn+an01sn01+1 1 1+a1s+a0:(5 p oints) Sp ecify the impulse resp onse functionh(t) of the system.(4 points) Supp ose we need to exp erimentally determine the DC gain of the system. A unit-stepinput is applied to the system, and we waits= 5seconds to measure the steady-state responsein order to calculate the DC gain. The parameteris the largest time constant associated withthe natural response of the system. hat is the numerical value
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