examI_f99 - EXAMINATION I EE 350 Continuous Time Linear...

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Unformatted text preview: EXAMINATION I EE 350 Continuous Time Linear Systems October 4, 1999 Name (Print): ID. number : Section number : Do not turn this page unit you are told to do so Test form A Instructions 1. You have 2 hours (120 min. total) to complete this exam. 2. This is a closed book exam. You are allowed one sheet of 8.5” x 11” of paper for notes as mentioned in the syllabus. 3. Solve each part of the problem in the space following the question. Place your ﬁnal answer in the box when provided. If you need more Space, continue your solution on the reverse side labeling the page with the question number; for example, “Question 4.2) Continued”. NO credit will be given to solution that does not meet this requirement. 4. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a grade of ZERO will be assigned. Problem 1: (25 Points) Let the voltage source f (:5) denote the input of the system in Figure 1 and the voltage y(t) across the capacitor denote the output. The switch in the circuit has been at Position A for a long time and it is switched to Position B at time t=0. Let R1 = 6 169, R2 = 2 1:9, and C = 10oF. f(t) Figure 1: The circuit for problem 1. Question 1.1 (10 Points) Find an ordinary differential equation(ODE) relating the input f (t) and the output y(t) for t 2 0. (continued) Question 1.2 (3 Points) Determine the initial condition y(0+) required to solve the ODE. Question 1.3 (5 Points) Solve the following ODE for the zero-input response yz_,-(t). dny1 + 2.5y(t) = f6), y(0) = 1-5V Question 1.4 (7 Points) Solve the following ODE for the zero-state response yz_s(t). ‘12—? + 2.5y(t) = (3 + e“)U(t)- Problem 2: (25 Points) Question 2.1 (10 Points) Consider the circuit shown in Figure 2. Find the input-output model for the circuit using 2'(t) as the output. Express the relationship in terms of the parameters R,L, and C. Figure 2: The circuit for problem 2. Question 2.2 (8 Points) Let f (t) = (0.003u (—t)+0.005u (15)) amps. Find the initial conditions 2' (0+) and #9, , when R = 4 M), L = 1 mH and C = 2.5 “F. t=0+ Figure 3: The circuit for problem 2 (same as Figure 2). Question 2.3 (7 Points) Consider a. second order circuit whose model is 1 36 (t)=(p2+io+i)y(t). RC L0 The zero state response of the circuit to f (t) = u (t) V is shown in Figure 4. If L = 60mH and can = 630%, ﬁnd (, R, and C. Output Vottage in Volts 1.8 1.6 1.4 .1. N .3 F3 on P a) 0.4 0.2 0 0.005 0.01 0.015 0.02 0.025 0.03 Time in Seconds Figure 4: The zero state step response for problem 2.3. 0.035 0.04 Problem 3: (25 Points) Question 3.1 (10 Points) A zero-state response of a certain system is described by If“ f (T)U(T)d'r, t2 0; yz—a(t) = 5%9 +tf(t— 2), t < 0. where f (t) is the system input and u(t) is the unit step function. Is the system zero-state linear or nonlinear? Justify your answer. No partiaI credit will be given for a yes-or-no answer. 10 Question 3.2 (6 Points) Determine if the system whose input f (t) and output y(t) are related via the RC circuit shown below in Figure 5 is instantaneous (memoryless) or dynamic? Why? (Note that R = 2 Q and C = 4 F) R f(t) Figure 5: RC circuit for problem 3.2 11 Question 3.3 (5 Points) Is the system described by the equation W - 3) = (15+ 1)“H)+ f0?) causal or noncausal? Question 3.4 (4 Points) Give an example of a time-varying, dynamic system. 12 Problem 4: (25 Points) Question 4.1 (10 Points) Consider the function f (2?) shown in Figure 6. Sketch y (t) = f (—t — 2) in Figure 7. fﬁ) Figure 6: The function f (t) for problem 4.1. W) Figure 7: A blank graph for sketching y (t) for problem 4.1. 13 Question 4.2 (10 Points) Determine if the signal, f (t) = 38112“, is a power signal, an energy signal, or neither. If it is a power or energy signal, determine the measure of the signal. 14 _ 3‘12“ Question 4.3 (5 Points) Find the even and odd component of f (t) - 3e . 15 ...
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