GATE 2024 EE – Question 59

Control Systems · Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci · 2 marks · Numerical answer

Consider the stable closed-loop system shown in the figure. The magnitude and phase values of the frequency response of $G(s)$ are given in the table. The value of the gain $K_I$ ($>0$) for a 50° phase margin is _______ (rounded off to 2 decimal places).

ω in rad/sec	Magnitude in dB	Phase in degrees
0.5	−7	−40
1.0	−10	−80
2.0	−18	−130
10.0	−40	−200

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Correct answer: 1.11 to 1.13

Explanation

The loop is $L(s)=\dfrac{K_I}{s}G(s)$. A phase margin of 50° needs $\angle L=-130^\circ$ at the gain crossover. The integrator contributes $-90^\circ$, so $\angle G=-40^\circ$, which occurs at $\omega=0.5$ rad/s where $|G|=-7$ dB. Unity loop gain there requires $20\log\dfrac{K_I}{0.5}-7=0$, so $\dfrac{K_I}{0.5}=10^{7/20}=2.239$ and $K_I=1.12$.