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Question

# Consider a unity feedback system with forward path transfer function G(s) = K(s+3)(s+5)(sâˆ’2)(sâˆ’4) The range of K to ensure stability is

A
K>34
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B
K<1 (or) K>34
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C
k < - 1
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D
1< K<34
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Solution

## The correct option is B K<−1 (or) K>34Closed loop transfer function is, T(s)=G(s)1+G(s)=K(s+3)(s+5)(K+1)s2+(8K−6)s+(15K+8) For stability all coefficients should be positive or all negative Case-I: All coefficient are positive, K+1>0⇒K>−1 8 K−6>0⇒K>0.75 15 K+8>0⇒K>−815 ∴ K > 0.75 Case-II : All coefficients are negative: K+1<0⇒K<−1 8K−6<0⇒K<0.75 15 K+8<0⇒K<−815 ∴ K < - 1 So, for stability either K > 0.75 or K < - 1

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