SolutionsBlank ExamEE 350 EXAM I 23 September 2010Last 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 100Te st Form AINSTRUCTIONS1. You have 2 hours to complete thi s 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 fol lowing 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 solutio ns 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 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 po ints) Consider the signalf(t) = e−|2t|[u(t + 2) − u(t − 2)] .(a) (4 poi nts) Carefully sketch f(t) using appropriate axes labels.(b) (6 points) Determine w hether f(t) is an energy signal, a power signal, o r neither. If f(t) is either anenergy or a power signal, calculate the corresponding metric Efor Pf, respectively.22. (15 po ints) A system with input f(t) has the zero-state responsey(t) = e−tf (2| t | − 3) .(a) (4 points) Is the system zero-state l inear or nonlinear? To receive partial credit you must justify youranswer.3(b) (4 points) Is the system, with zero-state responsey(t) = e−tf (2| t | − 3) ,time-invariant or time-varying? To receive partial credit you must justify your answer.4(c) (4 po ints) Is the system, with zero-state responsey(t) = e−tf (2| t | − 3) ,causal or noncausal? To receive partial credit you must justify your answer.(d) (3 points) Is the system, with zero-state responsey(t) = e−tf (2| t | − 3) ,dynamic or instantaneous? To receive parti al credit you must justify your answer.5Problem 2: (25 points)1. (10 points) An engineer measures the zero-state unit-step response y(t) of a strictly proper first-order LTIsystem. The engineer notes that the steady-state value of the respo nse is -100 , and thaty ( 2 ln(9) ) = −10y ( 7 ln(9) ) = −90y ( 5 ln(100) ) = −99(a) (2 poi nts) Determine the DC gain of the system.(b) (2 points) W hat i s the numeric value of the system rise-time?(c) (2 po ints) What is the numeric value of the system settling-time?(d) (2 points) Specify the root of the characteristic equation of the ODE that describes the input-outputbehavior of the system.(e) (2 po ints) Represent the system asQ(D)y(t) = P (D)f(t)by specifying the polynomials P (D) and Q(D).62. (5 points) Consider another LTI system, with input f(t) and output y(t), that is represented by the ODEd3ydt3−d2ydt2− 6dydt= f(t).(a) (3 po ints) Sketch the roots of the characteristic equation i n the λ-plane. To receive credit you must labelthe axes appropriately.(b) (2 points) Specify whether the system is asymptotically stable, marginally stable, or unstable. To receivecredit, justify your answer i n one or two short sentences.73. (10 po ints) Consider another LTI system, with input f(t) and output y(t), that is represented by the ODEd2ydt2+ a1dydt+ a0y(t) = 5f(t).It is known that the zero-state unit- step response of this system yields an exponentially decaying sinusoid. Thefrequency of the observed oscillation is 4 rad/sec and the time constant of the expo nential decay is 1/3 sec.(a) (4 poi nts) What is the natural frequency of the system?(b) (4 points) W hat i s the dimensionless damping ratio of the system?(c) (2 po ints) Specify the DC gain of the system.8Problem 3: (25 points)1. (12 points) The circuit in Figure1 is driven by two independent voltage sources that have constant strengthsV1and V2. At time t = 0, switch 1 closes and switch 2 opens. You may assume that the voltag es and currentsin the circuit reached steady-state values imm ediately prior to time t = 0. When answering parts (a) through(c) below, state any assumptions made, and show sufficient steps in the derivation to allow the grader tounderstand your solution path. Please underline or box your final answer.Figure 1: At time t = 0, switch 1 closes while switch 2 opens.(a) (4 poi nts) Determine y(0−), VA(0−), and iL(0−) in terms of the circuit parameters.(b) (4 points) Determine y(0+), VA(0+), and iL(0+) in terms of the circuit parameters.(c) (4 po ints) Determine y(∞), VA(∞), and iL(∞) in terms of the circuit parameters.92. (13 po ints) A LTI system with i nput f(t) and output y(t) is represented by the ODEdydt+ 3 y(t) = 9 f(t).The system is driven by the inputf(t) = (3 + t) u(t)and the initial value of the output is y(0) = 6 .(a) (6 poi nts) Determine the zero-input response yzi(t) for t ≥ 0.(b) (6 points) Determine the zero-state response yzs(t) for t ≥ 0.(c) (1 po int) Determine the total response y(t) for t ≥ 0.10Problem 4: (25 points)1. (15 poi nts) The circuit in Figure 2 is driven by a dependent current source with strength g v(t), where g is aconstant parameter that has units of conductance. At time t = 0 there is no energy stored in the capacitor,and the current through inductor is non-zero.Figure 2: RLC circuit driven by a dependent current source.(a) (9 poi nts) Determine a second-order ODE in the form¨v + a1˙v + a0v = 0that describes the time evolution of the voltage v(t) for t ≥ 0. Specify the parameters a1and a0in termsof R, L, C, and g.11(b) (6 points) Using your result in part (a), specify the range of the pa rameter g, in terms of the resistanceR, for which the system is• (2 points) asymptotically stable,• (2 points) margi nally stable, and• (2 points) unstable.122. (10 points) Consider a second-order system, with input f(t) and output y(t), that has the ODE representationQ(D) = D2+ 8D + 16P (D) = 32.(a) (5 poi nts) Determine the zero-state unit-step respo nse of the system.(b) (5 points) Write an m -file that numerically computes and plots the zero-state unit-step response of thesystem. Use a time vector of three hundred points over the range 0 ≤ t ≤ 3, and add appropriate
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