Question:
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 below. The value of the gain \( K_I \; (>0) \) required for a 50° phase margin is __________ (rounded off to two decimal places).
Frequency response data of \( G(s) \):
\(\omega\) (rad/s) Magnitude (dB) Phase (degrees) 0.5 -7 -40 1.0 -10 -80 2.0 -18 -130 10.0 -40 -200
System diagram:
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 below. The value of the gain \( K_I \; (>0) \) required for a 50° phase margin is __________ (rounded off to two decimal places).
Frequency response data of \( G(s) \):
| \(\omega\) (rad/s) | Magnitude (dB) | Phase (degrees) |
|---|---|---|
| 0.5 | -7 | -40 |
| 1.0 | -10 | -80 |
| 2.0 | -18 | -130 |
| 10.0 | -40 | -200 |
System diagram:
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For phase margin calculations, adjust the gain such that the phase lag matches the required value at the crossover frequency.
Updated On: Feb 3, 2026
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Verified By Collegedunia
Correct Answer: 1.11 - 1.13
Solution and Explanation
Step 1: Determine the gain crossover frequency. From the phase response table, identify the frequency at which the phase reaches \( -180^\circ + 50^\circ = -130^\circ \).
Step 2: Calculate the gain \( K_I \). Using the magnitude response at this frequency, calculate \( K_I \) to achieve the required phase margin: \[ K_I = 1.11 \, \text{to} \, 1.13. \]
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