GATE EE 2024 Set 1 — Question 59
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Control Systems → Stability Analysis → Gain & Phase Margin
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Question
Consider the stable closed-loop system shown in the figure. The magnitude and phase values of the frequency response of are given in the table. The value of the gain for a phase margin is _______ (rounded off to 2 decimal places).
| 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 |
Correct answer
1.11 to 1.13
Solution
The open-loop transfer function is .
The phase of the open-loop system is .
For a phase margin of , the phase at the gain crossover frequency must be:
.
Substituting the expression for phase:
.
From the given table, at rad/sec. Thus, rad/sec.
At the gain crossover frequency, the magnitude of the open-loop transfer function must be 1:
.
From the table, dB. Converting to linear scale:
.
Now, solve for :
.
Rounding to two decimal places, .
The phase of the open-loop system is .
For a phase margin of , the phase at the gain crossover frequency must be:
.
Substituting the expression for phase:
.
From the given table, at rad/sec. Thus, rad/sec.
At the gain crossover frequency, the magnitude of the open-loop transfer function must be 1:
.
From the table, dB. Converting to linear scale:
.
Now, solve for :
.
Rounding to two decimal places, .
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