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Solution of ECE 316 Test #7 S03 3/5/03 1. What numerical value of K in this system will make it marginally stable? X(s) 1 s 1 s Y(s) K 10 −K ± K 2 − 40 1 The transfer function is H( s) = 2 . The poles are at s = . For a system to 2 s + Ks + 10 be marginally stable the poles must lie on the ω axis which means the real parts must be zero. That happens when K = 0 . 200 s is realized in the parallel s + 9 s + 18 configuration below. What are the numerical values of K1 , K 2 and K 3 ? 2. A system whose transfer function is H( s) = 2 1 s K1 3 X(s) 1 s K3 Y(s) K2 . The transfer function can be expanded in partial fractions as H( s) = 200 s 200 400 =− + . s+3 s+6 (s + 3)(s + 6) Therefore K1 = −200 , K 2 = 6 and K 3 = 400 . 3. A normalized lowpass Butterworth filter has a pole at s = e the transfer function of this filter? j 3π 5 . How many poles are there in For a Butterworth lowpass filter the poles lie at angles which are integer multiples of the number of poles. Since this pole lies at an angle of 3π , n must be 5. 5 π where n is n