blanksolutionsEE 350 Exam # 1 25 September 2014Last 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 100INSTRUCTIONS1. You have 2 hours to complete thi s exam.2. This is a closed book exam. You may use one 8.5”× 1 1” 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 exampl e, Pro blem 1.2 Continued. NOcredit will be given to solutions tha t 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.7. Any student caught cheating on an exam will receive a grade o f zero for the exam. Additional sanctions,including assigning an XF grade, will be pursued following university guidelines.1Problem 1: (25 Points)1. (6 points) Consider the si gnalf(t) =1 − e−(t + T )/T[u(t + T ) − u(t)] +1 − e−1e−t/T[u(t) − u(t − T )] .(a) (4 points) Sketch f(t) in Figure 1. Label the values of f(0) and f(T ) on the y axes in terms of the numbere (do not provide a numeric value).Figure 1: Sketch of f(t).(b) (2 points) State whetherf(t) is a causal or noncausal signal. Justify your answer using a short sentence.22. (5 points) Consider the si gnalg(t) = 4e−(t + 3T )/Tu(t + 3T ).Determine whether g(t) is an energy signal, a power signal, or neither. If g(t) is either an energy or a powersignal, determine the corresponding metric Egor Pg, respectively.33. (14 points) A system with input f(t) and output y(t) has the zero-state responsey(t) =Zt−∞f2(τ)e−(t − τ )dτ.(a) (7 points) Is the system zero-state linear or nonlinear? To receive partial credit justify your answer.(b) (7 points) Is the system time invariant or time-varying? To receive partial credit justify your answer.4Problem 2: (25 points)1. (8 points) A linear time-invariant system with input f(t) and output y(t) is represented by the ODE¨y − 2 ˙y + 1 0y(t) = f(t)(a) (6 points) Determine the roots of the cha racteristic equation and sketch these ro ots in the λ-plane providedin Figure 2. To obtain full credit, appropriately label the real and imaginary axes to specify the locationof the characteristics roots.Figure 2: Sketch o f the characteristic roots in the λ-plane.(b) (2 points) Based on the location of the characteristic roots, specify if the system is asymptotically stable,marginally stable, or unstable. Justify your answer using a short sentence.52. (8 points) Figure 3 show the roots of the characteristic equation for a certain fourth-order LTI system.Figure 3: Location of the characteristic roots in the λ-plane.(a) (3 points) State the form of the homog eneous solution yh(t).(b) (2 points) Which characteristic root do minates the transient response characteristics of the homogeneoussolution? Justify you answer using a short sentence.(c) (3 points) Suppose that you are determining the zero-state response of the system for the inputf(t) = 1 + e−2t, t ≥ 0Specify the form of the particular solution yp(t) for t ≥ 0.63. (9 points) A LTI system w ith input f(t) and output y(t) is represented by the ODEd5ydt5+ 7d4ydt4+ 18d3ydt3+ 23d2ydt2+ 17dydt+ 6y(t) = 8d2fdt2− 3f(t)Compl ete the m-file in Figure 4 so that it accomplishes the following tasks.(a) (3 points) D etermines and display the roots of the characteristic equation.(b) (6 points) Determines and plots the zero-state unit-step response for the inputf(t) = e−2tu(t)over the time range 0 ≤ t ≤ 1 00 using a time vector of one thousand points. Appropriately label the xand y axes of the figure.% EE 350 Fall 2014% Exam 1% Problem 2 Part 3%clc; close all; clear%Q = ; % specify Q(D)P = ; % specify P(D)%% Determine and display root s of the characteri stic equationdisplay(’roots of Q ar e’)disp( )%% Generate the tim e vectort = ;%% Generate the inp utf = ;%% Determine the zero-st ate re sponsey = ;%% Plot the respons e y(t)% plot y versus t% label the x-axis% label the y-axisFigure 4: MTALB m-file.7Problem 3: (25 points)1. (12 points) The circuit in Figure 5 is driven by an independent voltage source with a constant strength Vs.Prior to the switch closing at time t = 0, the current and voltages within the circuit have reached steady-stateval ues.Figure 5: The switch i n the passive RL circuit closes at time 0.(a) (8 points) Determine expressions for the current i(0+) and voltage y(0+) in terms of the circuit parameters.(b) (4 points) Determine expressions for the current i(∞) and voltage y(∞) in terms of the circuit parameters.82. (13 points) A LTI system with input f(t) and output y(t) has the ODE representationdydt+ 2 y(t) = 4 f(t).The system is driven by the inputf(t) = 1 + e−2t, t ≥ 0,and the initial value y(0) = 2.(a) (3 points) D etermine the the zero-input response yzi(t) for t ≥ 0.(b) (9 points) Determine the the zero-state response yzs(t) for t ≥ 0.9Problem 4: (25 points)1. (12 points) The circuit in Figure 6 has input voltage f(t) and output voltage y(t).Figure 6: Passive RLC filter circuit with input f(t) and output y(t).(a) (2 points) Determine the DC gain and high frequency gai n of the network. Justify your answers usingshort sentences.(b) (10 points) Determine a second-order ODE representation of the form¨y + a1˙y + a0y = boffor the circuit. Specify the parameters b0, a1and a0in terms of R, L, and C. Obtain the ODE represen-tation by applyi ng nodal anal ysis at nodes A and B, which are indicated in Figure 6.10112. (13 points) Consider another second-order system with input f(t), output y(t), and ODE representation¨y + 4 ˙y + 8 y = 24f(t).(a) (2 points) D etermine the natural frequency and dimensionless damping ratio for the system.(b) (1 po int) Based on your result in part 1, will the zero-state respo nse be overdamped, critically damped,or underdamped?(c) (10 points) Given that that y(0) = 1, ˙y(0) = 8, and f(t) = u(t), determine the total response y(t). Toreceive full credit, your solution must not contain any complex-valued terms.1213EE 350 Exam # 1 25 September 2014Last Name (Print):First Name (Print):ID number
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