GATE 2020 CH – Question 63
A negative unity-feedback loop has proportional controller G꜀ = K꜀, final control element $G_f=10/(0.1s+1)$ and process $G_p=1/(0.1s+1)^2$. For gain margin 1.6, find K꜀, to one decimal place.
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Correct answer: 0.49 to 0.51
Explanation
The loop transfer is $L(s)=10K_c/(0.1s+1)^3$. Phase crossover occurs when $-3\tan^{-1}(0.1\omega)=-180^\circ$, so $0.1\omega=\sqrt3$. At that frequency each first-order denominator has magnitude 2, making $|L|=10K_c/8=1.25K_c$. Gain margin is its reciprocal, $1/(1.25K_c)=1.6$. Hence K꜀ = 0.5.
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