Question

5. Consider the feedback system as follows G2(s) where Gi(s) K( where G1(s) K(1and G2(s) and G2 (s) - S+3 s2+2s+2 K(s+3) (s+5) (a) Show that the transfer function of the above system is +(2+8K)s+15K (b) What value of K the system stability can be maintained?

Answer 1

G_1 (s) = K [1 + 5/s] = K (s + 5/s) G_2 (s) = (s + 3)/s^2 + 2s + 2 Here H (s) = Feedback K = 1 a) Ans Closed loop Transfer Function Forward path = G_1 (s) G_2 (s) Feedback path = H (s) = L Loop gain = -G_1(s) G_2 (s) C (s)/R(s) = G_1 (s) G_2 (s)/1 [-G_1 (s) G_2 (s) H (s)] = G_1 (s) G_2 (s)/1 + G_1 (s) G_2 (s) [Therefore H (s) = L] = K (s + 5/s) (s + 3)/s^2 + 2s + 2/1 + k (s + 5)/s (s + 3)/s^2 + 2s + 2 = K (s + 5)/3 (s + 3)/s^2 + 2s + 2 /s(s^2 + 2s + 2) + k(s + 5) (s + 3) /s(s^2 + 2s + 2) = K (s + 5) (s + 3)/ s^3 + 2s^2 + 2s + k [s^2 + 3s + 5s + 15] \r\n = K (s + 3) (s + 5)/ s^3 + 2s^2 + 2s + ks^2 + 8ks + 15k
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