homework8_solutions.pdf - ECE 460 Homework 8 Instructor...

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ECE 460: Homework 8 Instructor: Stan Baek Due: Wednesday, March 30, at 2:10pm Note 1. Your solutions should be neat and legible; always show your work. Problem 1. ( 40 points) Sketch the asymptotes of the Bode plot magnitude and phase for each of the following transfer functions using the semi-log graph attached at the end of this homework assignment. Find the gain margin, the phase margin, the gain crossover frequency, and the phase crossover frequency of the systems. (a) G ( s ) = ( s + 6) ( s + 0 . 5)( s + 1)( s + 20) Solution: G ( s ) = 6( s/ 6 + 1) 0 . 5( s/ 0 . 5 + 1)( s + 1)20( s/ 20 + 1) = 3 5 ( s/ 6 + 1) ( s/ 0 . 5 + 1)( s + 1)( s/ 20 + 1) 20 log(3 / 5) = - 4 . 4 dB. There is no gain crossover frequency. Although the asymptotes of the phase plot reaches -180 , the actual plot never reaches -180 . Therefore, there is no phase crossover frequency. Since both gain and phase crossover frequencies do not exist, the gain margin and phase margin are both the infinity. Compared with the Bode plot generated by MATLAB shown in Figure 1a, the asymptotes are good approximations. (b) G ( s ) = ( s + 20)( s + 100) ( s + 1)( s 2 + 6 s + 25) Solution: G ( s ) = 20( s/ 20 + 1)100( s/ 100 + 1) ( s + 1)25(( s/ 5) 2 + 6 / 5( s/ 5) + 1) = 80( s/ 20 + 1)( s/ 100 + 1) ( s + 1)(( s/ 5) 2 + 6 / 5( s/ 5) + 1) 20 log(80) = 38 dB, ζ = 3 / 5 = 0 . 6. The gain crossover frequency is 12 rad/s and the phase crossover frequency is 10 rad/s. The gain margin is approximately -8 dB and the phase margin is 0 according to the asymptotes of the Bode plot. Compared with the Bode plot generated by MATLAB shown in Figure 1b, the phase plot is quite different. The main reason is that we have not incorporated the effect of ζ shown in Figure 10.17 on page 547 of our textbook. 1
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(c) G ( s ) = ( s + 4) s ( s + 2 s 2 + 10) Solution: G ( s ) = ( s + 4) s (2 s 2 + s + 10) = 4( s/ 4 + 1) 2 s ( s 2 + s/ 2 + 5) = 2( s/ 4 + 1) 5 s (( s/ 5) 2 + 1 / (2 5)( s/ 5) + 1) = 2 5 ( s/ 4 + 1) s (( s/ 5) 2 + 1 / (2 5)( s/ 5) + 1) 20 log(2 / 5) = - 8 dB, ζ = 1 / (4 5) = 0 . 11. The gain crossover frequency is 0.4 rad/s and the phase crossover frequency is 7 rad/s. The gain margin is approximately 36 dB and the phase margin is 55 according to the asymptotes of the Bode plot. Compared with the
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