SolutionsBlank ExamEE 350 EXAM I 22 September 2011Last 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. (12 po ints) Consider the signal f(t) shown in Figure 1.Figure 1: Signal f(t)(a) (2 poi nts) Is the signal f(t) a causal or noncausal signal? To receive credit you must justify your answerin a single sentence.(b) (5 points) Determine an expression for the signal f(t).(c) (5 points) D etermine whether f(t) is an energy signal, a power signal, or neither. If f(t) is either anenergy or a power signal, calculate the corresponding metric Efor Pf, respectively.22. (7 points) A linear time-invariant system generates the zero-state responses shown in Figure 2, where thezero-state responses y1(t) and y2(t) correspond to the inputs f1(t) and f2(t), respectively.Figure 2: Zero-state responses for a LTI system.(a) (2 points) Is the system causal or noncausal? To receive credit you must justify your answer in a shortsentence.(b) (5 points) The input f3(t) in Figure 3(A) is applied to the LTI system. Sketch the resulting zero-stateresponse y3(t) in Figure 3(B).Figure 3: For the input f3(t) in (A), the LTI system generates the zero-state response y3(t) in (B).33. (6 points) Consider another system whose zero-state response y(t) to the input f(t) isy(t) −12y(t − 2 ) = f(t).Is this system zero-state linear or no nlinear? To receive partial credit you must justify your answer.4Problem 2: (25 points)1. (13 po ints) A linear time-invariant system with input f(t) and output y (t) is represented by the ODE¨y + 8 ˙y + 20y(t) = 40f(t) + 20˙f(a) (5 poi nts) Determine the roots of the characteristic equation.(b) (6 points) Determine the value of the dim ensionless damping ratio and the natural frequency.(c) (2 points) Is the system asymptotically stable, marginally stable, or unstable? To receive credit you mustjustify your answer in a short sentence.52. (12 po ints) An engineer is studying the characteristics of a linear time-invariant system represented asQ(D)y(t) = P (D)f(t),where f(t) and y(t) represent the input and output, respectively. The engineer represents the polynomialsP (D) and Q(D) as row vectors P and Q in MATLAB. Execution of the m-filedisp(’roots(Q): ’), roots(Q)disp(’length(P): ’), length(P)disp(’P(1)/Q(2):’), P(1)/Q (2)results in the outputroots(Q):ans =-2length(P):ans =1P(1)/Q(2):ans =5Suppose that the system is driven by the input f(t) = 2u(t) and that y(0) = 0.(a) (4 poi nts) Determine the steady-state value of the response y(t).(b) (4 points) At what time will the respo nse y(t) reach and stay within 1% of the steady-state value of y(t)?(c) (4 po ints) Determine the ODE representation of the system and place your answer in the standard formdnydtn+ an−1dn−1ydtn−1+ · · · + aoy = bmdmfdtm+ bm−1dm−1fdtm−1+ · · · + bof,by providing numeric values for the coefficients aiand bi.6Problem 3: (25 points)1. (13 po ints) T he circuit in Figure 4 takes an input voltage f(t) and generates an output current y(t).Figure 4: RC circuit with i nput voltage f(t) and output current y(t).(a) (8 poi nts) Determine the ODE representation of the system and place your answer in the standard formdnydtn+ an−1dn−1ydtn−1+ · · · + aoy = bmdmfdtm+ bm−1dm−1fdtm−1+ · · · + bof,by expressing the coefficients aiand biin terms of the parameters R and C.7(b) (5 points) Suppose that f(t) = u(t), R = 1kΩ, and C = 1µF . Determine the initial value, y(0+), andfinal value, y(∞), of the zero-state output current.82. (12 po ints) T he response of a system represented by the ODEdydt+ a y(t) = b f(t).to the inputf(t) = e−3tu(t)isy(t) = 3e−2t+ 6e−3tfor t ≥ 0.(a) (8 poi nts) Determine the numeric value of the parameters a and b.(b) (4 points) In terms of the parameters a and/or b (do not use your numeric answer from part a), specifythe rise-time of the zero-state unit-step response of the system.9Problem 4: (25 points)1. (13 po ints) T he circuit in Figure 5 has input voltage f(t) and output voltage y(t).Figure 5: RC filter circuit with input f(t) and output y(t).(a) (2 poi nts) What is the DC g ain of the circuit? In order to receive credit you must justify your answer ina short sentence.(b) (11 points) Determine a second-order ODE representation of the form¨y + a1˙y + a0y =¨f + boffor the circuit. Specify the parameters b0, a1and a0in terms of R, and C. Obtain the ODE representationby using nodal analysis at the three nodes, a, b, and c, shown in Figure 5.10112. (12 points) Consider a second-order system, with input f(t) and output y(t), that has the ODE representation¨y + 2 ˙y + y = f(t).(a) (2 points) Determine if the zero-state unit-step response would be overdamped, critically damped, orunderdamped. Justify your answer by showing appropriate calculations, but do n ot attempt to calculatethe zero-state unit-step response.(b) (5 points) Suppose that y(0) = 1 and ˙y(0) = 0. Determine the zero-input response, yzi(t) of the system.12(c) (5 po ints) Suppose that f(t) = 6e−2tu(t). Determine the zero- state response, yzs(t) of the
View Full Document
Unlocking...