GATE 2026 IN – Question 60
The system $G(s)=K/(s+1)^4$ has a gain margin of 20 dB. The value of $K$ is ________. (Rounded off to two decimal places.)
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Correct answer: 0.39 to 0.41
Explanation
For positive $K$, the phase is $-4\arctan\omega$. It reaches $-180^\circ$ at $\omega=1\ \mathrm{rad/s}$. At this frequency, $|G(j)|=K/|1+j|^4=K/4$.
A gain margin of 20 dB is a factor $10^{20/20}=10$. Hence $4/K=10$, giving **K = 0.40**.