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### 332-3Solutions

Course: ECE 332, Spring 2008
School: Wisconsin
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Word Count: 357

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332 ECE Homework #3 1) Design a second-order transfer function Hd (s) to meet all of the following specifications. Choose n as small as possible. a) ess0 = 0 b) Mr 1.3 c) Mp 1.3 d) ess1 .8 s e) b 1.5 rad/s f) r .7 rad/s g) Tr 2.3 s h) Tp 3.5 s Find the poles of the system. 2) Let HR (s) = Hd (s) where Hd was found in 1). a) Find the poles and zeros of HR for R = 5. What is the damping factor for the...

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332 ECE Homework #3 1) Design a second-order transfer function Hd (s) to meet all of the following specifications. Choose n as small as possible. a) ess0 = 0 b) Mr 1.3 c) Mp 1.3 d) ess1 .8 s e) b 1.5 rad/s f) r .7 rad/s g) Tr 2.3 s h) Tp 3.5 s Find the poles of the system. 2) Let HR (s) = Hd (s) where Hd was found in 1). a) Find the poles and zeros of HR for R = 5. What is the damping factor for the high-frequency poles? b) Use MATLAB to find the smallest value of R so that i) 20 log |HR (j)| < -40 dB and 20 log |Hd (j)| < -40 dB for all such that |20 log |HR (j)| - 20 log |Hd (j)|| > 1 dB ii) |yd (t) - yR (t)| .01, where yd and yR are the step responses corresponding to Hd and HR . To do this, enter "xi = ; wn = n ; R = 5; hw32", where and n are the values of damping ratio and natural frequency chosen in part 1). MATLAB will display the frequency and step responses for Hd and H5 . Increase R until i) and ii) are Print met. the final plots and identify which curves correspond to each transfer function. c) Find the poles and zeros of HR for the value of R selected in b). 3) Consider each of the following all-pass transfer functions Ha . s-1 s+1 2 ii) s2 -2s+1 s + 2s+1 (s-1)(s2 - 2s+1) iii) (s+1) s2 +2s+1 ( ) s2 R(s + R) , + .05Rs + R2 i) For each case, let H = Hd Ha ,where Hd was obtained in problem 1). a) Find the poles and zeros of H. b) Using MATLAB, draw Bode plots for Hd and H. Also plot yd (t) and (-1)n y(t), where yd and y are the step responses corresponding to Hd and H, and n is the order of Ha . For case i), this is done by entering "xi = ; wn = n ; hw33i". Compare the responses of the minimum-phase and the non-minimum-phase systems. Print the graphs and identify which correspond to Hd and H. Repeat a) and b) for cases ii) and iii) using the commands "xi = ; wn = n ; hw33ii" and "xi = ; wn = n ; hw33iii".
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