SolutionsBlank ExamEE 350 EXAM I 24 September 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 2 hours 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. (10 points) Consider the signal f(t) shown in Figure 1.Figure 1: The signal f(t).(a) (5 points) Write an expression for the signal f(t).(b) (5 points) Determine whether f(t) is an energy signal, a power signal, or neither. If f(t) is either anenergy or power signal, calculate the corresponding metric Efor Pf, respectively.22. (15 points) A system with input f(t) has the zero-state resp onsey(t)=Z2t−2−∞cos(t − τ)f(τ )dτ.(a) (4 points) Is the system zero-state linear or nonlinear? To receive partial credit you must justify youranswer.(b) (4 p oints) Is the system time-invariant or time-varying? To receive partial credit you must justify youranswer.(c) (4 points) Is the system causal or noncausal? To receive partial credit you must justify your answer.(d) (3 points) Is the system dynamic or instantaneous? To receive partial credit you must justify your answer.3Problem 2: (25 points)1. (14 points) Figure 2 shows a ferromagnetic sphere with mass M suspended by the attractive force of anelectromagnet. The current F (t)=Fo+ f(t) is the sum of a constant current Fothat is much larger thanan adjustable current f(t). The displacement Y (t)=Yo+ y(t) between the sphere and the electromagnet isrepresented as the sum of a constant distance Yoand a variable distance y (t ) that is much smaller than Yo. Therelationship between the input F (t) and the output Y (t) is represented by a nonlinear differential equation.However, at an operating point defined by the fixed current Foand position Yo, small changes y(t) in theposition Y (t) can be related to small changes f(t) in the current F (t) by a LTI small-signal model whose ODErepresentation isd2ydt2− 4 y(t)=f(t).Figure 2: Attractive magnetic suspension system.(a) (4 p oints) Is the small signal mo del asymptotically stable, marginally stable, or unstable? Justify youranswer.4(b) (4 points) An engineer recommends forming a new system with output y(t) and input r (t ) by feeding backthe output so thatf(t)=−KPy(t)+r(t ),where the proportional gain KPis a real-valued constant that can be chosen by the designer. Find anODE representation of the resulting system and place your result in the formQ(D)y(t)=P (D)r(t)by specifying the polynomials Q(D) and P (D). Is it possible to choose a value of proportional gain KPfor which the new system is asymptotically stable? Justify your answer.5(c) (6 p oints) After further studying the problem, the engineer recommends forming another system bychoosingf(t)=−KPy(t) − KDdydt+ r(t),where the proportional gain KPand the derivative gain KDare real-valued constants that can be chosenby the designer. Find an ODE representation of the resulting system and place your result in the formQ(D)y(t)=P (D)r(t)by specifying the polynomials Q(D) and P (D). Choose the controller gains KPand KDso that the newsecond-order system is critically damped with a natural frequency of 4 rad/sec.62. (11 points) The following line in an m-file generates the zero-state response, y(t), of a circuit driven by someinput.y = lsim([10], [1,0.5], exp(-x/2), x );• (4 points) The response y generated by MATLAB can be expressed as the sum of a natural and forcedresponse. What is the form of the forced response? You may leave your answer in terms of undeterminedcoefficents, but you must specify the numeric value of each exponent.• (4 points) What is the rise-time and settling time of the zero-state unit-step response of the circuit?• (3 points) What is the DC gain of the circuit?7Problem 3: (25 points)1. (8 points) Find the ODE representation of the circuit shown in Figure 3, where f(t) is the input voltage andy(t) is the output voltage. Assume that the operational amplifier is ideal and place your answer in the standardformdnydtn+ an−1dn−1ydtn−1+ ···+ aoy = bmdmfdtm+ bm−1dm−1fdtm−1+ ···+ bof.If you introduce a no de voltage or mesh current in your analysis, clearly indicate this variable in Figure 3 andlabel its polarity or reference direction.Figure 3: Active filter circuit with input f(t) and output y(t).82. (8 points) The circuit in Figure 4 is driven by two independent voltage sources that have constant strengths ofV1and V2. Immediately prior to the switch closing at time t = 0, the currents and voltages within the circuithave reached steady-state values. The output of the circuit is the voltage y (t ) across the capacitor.Figure 4: Passive RC circuit with output voltage y(t).(a) (4 points) Derive an expression for the output voltage y(0+) in terms of the circuit parameters. Stateany assumptions made, and show sufficient steps in the derivation to allow the grader to understand yoursolution path.(b) (4 points) Derive an expression for the output voltage y(∞) in terms of the circuit parameters. Stateany assumptions made, and show sufficient steps in the derivation to allow the grader to understand yoursolution path.93. (9 points) A first-order system with input f(t) and output y(t) is represented by the ODEdydt+ a0y(t)=f(t)where a0is a constant, real-valued parameter. For a certain input f(t) and some initial condition y(0), it isknown that for t ≥ 0 the zero-input response isyzi(t)=4e−5t,while the total response isy(t)=5e−5t− e−4t.(a) (4 points) Determine the numeric value of the parameter a0.(b) (5 points) Determine the input f(t) for t ≥ 0.10Problem 4: (25 points)1. (10 points) Find the ODE representation of the circuit shown in Figure 5, where f(t) is the input current
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