## Answers

(a) C/R = 28 + 5/S^2 + 2S + 4 C/R Vertical-bar _S = jw = 2jw + 5/(4 - w^2) + 2 jw Vertical-bar C/R Vertical-bar = squareroot 25 + 4 w^2/squareroot (4 - w^2)^2 + 4 w^2 angle C/R = tan^-1 (2w/5) - tan^-1 (2w/4 - w^2) (b) Let C/R = (28 + 5)/(s^2 + 2s + 4) = (1 + s/(5/2))/4(1 + s/2 + s^2/4) Break points are at w = 5/2 = 2.5 rad/sec and w^2_n = 4 = > w_n = 2 rad/sec (c) Vertical-bar C/R Vertical-bar _in dB = 20 log Vertical-bar C/R Vertical-bar = 1.93 for w = 0 1.955 for w = 0.1 2.37 for w = 0.5 3.23 for w = 0.9 4.5 for w = 1.5 4.48 for w = 1.8 4.08 for w = 1.2 3.24 for w = 2.2 2.64 for w = 2.4 2.2 for w = 2.5 0 for w = 3 -1.99 for w = 3.5 -6.36 for w = 5

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