hwsln-421-s06-12 - ECE 421 Spring 2006 Solutions to HW...

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ECE 421, Spring 2006 Solutions to HW Assignment #12 Problem #B-8-26 The forward transfer function is G ( s ) = as + 1 s 2 (1) The two poles at s = 0 provide 180 phase shift at all frequencies. The requirement on phase margin of PM = 45 means that at the gain crossover frequency ω x the magnitude | G ( x ) | = 1 and the phase 6 G ( x ) = 135 . Therefore, the term as + 1 must provide +45 phase shift when s = x . tan (45 ) = 1 = x 1 a = 1 ω x (2) | G ( x ) | = 1 = q ( x ) 2 + 1 ω 2 x = 2 ω 2 x ω x = 2 1 / 4 = 1 . 1892 a = 0 . 8409 (3) Problem #B-8-27 Bode plots are shown in Fig. 1. The closed-loop system is stable. Note the beginning and ending slopes of the magnitude curve and the beginning and ending values of the phase curve. Relate that information to the transfer function. Problem #B-8-28 Bode plots are shown in Fig. 2. The closed-loop system is stable. Note the beginning and ending slopes of the magnitude curve and the beginning and ending values of the phase curve. Relate that information to the transfer function.
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