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2 Problem 1: (25 Points) 1. (9 points) Using the method of Laplace transforms, determine the transfer function H(s) of the active filter in Figure 1, where f(t) is the input and y(t) is the output. Express your answer in the standard form H(s) = b m s m + + b 1 s + b o s n + a n 1 s n a 1 s + a 0. Figure 1: Active RC filter with input voltage f(t) and output voltage y(t). 2

3 2. (9 points) The circuit in Figure 2 has input f(t) and output y(t). Using Laplace transform analysis, determine the zero-input response of the system given that R = 1/3 Ω, L = 1/4 H, C = 1/2 F, i(0 ) = 3 A, and y(0 ) = 2 V. Figure 2: Passive RLC circuit with output voltage y(t). 3

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5 3. (7 points) In order to determine the partial fraction expansion of the transfer function an engineer used the MATLAB command H(s) = b 1 s + b o s 2 + a 1 s + a 0, >> [r, p, k] = residue([b1, b0], [1, a1, a0]) and obtained r = p = k = [] (a) (3 points) Write down the partial fraction expansion of H(s). (b) (4 points) Specify the numeric values of the parameters b 1, b 0, a 1, and a 0. 5

6 Problem 2: (25 points) 1. (12 points) Determine the closed-loop transfer function of the feedback control system in Figure 3, and specify your final answer using the standard form Y (s) R(s) = b m s m + + b 1 s + b o s n + a n 1 s n a 1 s + a 0. Figure 3: Feedback control system with reference input r(t) and controlled output y(t). 6

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8 2. (13 points) Another feedback control system, different from the one considered in part 1, has the closed-loop transfer function representation Y (s) R(s) = K 1 s 2 + (K 2 K 1 )s + K 1, where R(s) is the command input, Y (s) is the controlled output, and K 1 and K 2 represent controller gains. (a) (6 points) Choose the controller gains so that the zero-state unit-step response of the closed-loop system is underdamped with a natural frequency of 10 rad/sec and a dimensionless damping ratio of 1/2. (b) (7 points) Suppose that K 1 = 8, K 2 = 6, and define the closed-loop system error as e(t) = r(t) y(t). For a ramp-input r(t) = tu(t), what value does e(t) approach as time increases? 8

9 Problem 3: (25 points) 1. (15 points) A system has the transfer function representation H(s) = (s + 100) (s ) (s ) 2. Construct the Bode magnitude and phase plots using the semilog graphs provided in Figure 4 (a duplicate copy appears in Figure 5). In order to receive credit: In both your magnitude and phase plots, indicate each term separately using dashed lines. Indicate the slope of each straight-line segment and the corner frequencies of the final magnitude and phase plots. Do not show the 3 db corrections in the magnitude plot. 9

10 Figure 4: Semilog paper for Bode magnitude and phase plots. 10

11 Figure 5: Semilog paper for Bode magnitude and phase plots. 11

12 2. (10 points) Figure 6 shows the straight-line approximation of the magnitude and phase plots of a transfer function H(s). The transfer function H(s) was generated in MATLAB using the script shown below, where the parameters a, b, and c are real-valued constants H1 = tf([1,a], [1,b]) H2 = tf([10], [1,c]) H = series(h1, H2); 20 Magnitude [db] Phase [Deg] frequency [rad/sec] Figure 6: Straight-line approximation of the magnitude and phase plot of H(s). 12

13 (a) (3 points) From the MATLAB script, specify the transfer function in terms of the parameters a, b, and c. Express your answer in standard form H(s) = b m s m + + b 1 s + b o s n + a n 1 s n a 1 s + a 0. (b) (2 points) Using Figure 6, specify the numeric value of the DC gain of the system represented by H(s). (c) (5 points) Determine H(s), and specify the numeric values of the parameters a, b, and c. 13

14 Problem 4: (25 points) 1. (13 points) A LTI system has the impulse response function representation h(t) = δ(t) + e t u(t) 2e 2t u(t). (a) (4 points) Determine the transfer function representation of the system and express your answer in the standard form b m s m + + b 1 s + b o H(s) = s n + a n 1 s n a 1 s + a 0 (b) (4 points) Sketch the pole-zero map of the system transfer function. To receive credit, you must label the axes and clearly specify the location of each pole and zero. 14

15 (c) (2 points) Specify the DC gain and high frequency gain of the system. (d) (3 points) If the system input and output are denoted by f(t) and y(t) respectively, specify the ODE representation of the LTI system. 15

16 2. (12 points) Consider another LTI system, that is different from the one in part 1. The zero-state response of the system to the unit-step input f(t) = u(t) is y(t) = ( 2 2e t 2te t) u(t). For another input, f(t), the observed zero-state response is Determine the input f(t). ȳ(t) = ( 2 3e t + e 3t) u(t). 16

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### Problem Weight Total 100

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