Topic: Control Systems
Frequency response of a system is given in the following table:
Frequency | Gain | Phase
1 | 8 | -50
10 | 1 | -155
100 | 0.25 | -180
1000 | 0.1 | -235
The Phase margin and Gain margin respectively are
- A. 4, 25
- B. 25, 4
- C. 25, 0.25
- D. 0.25, 25
Answer: B. 25, 4
Explanation: Phase Margin (PM) is evaluated at the gain crossover frequency where the magnitude is 1. From the table, at $\omega = 10$ rad/s, Gain = 1 and Phase = $-155^{\circ}$. $PM = 180^{\circ} + \angle G(j\omega) = 180^{\circ} + (-155^{\circ}) = 25^{\circ}$.
Gain Margin (GM) is evaluated at the phase crossover frequency where the phase is $-180^{\circ}$. At $\omega = 100$ rad/s, Phase = $-180^{\circ}$ and Gain = $0.25$. GM in absolute magnitude is $\frac{1}{Gain} = \frac{1}{0.25} = 4$.
Thus, Phase Margin = $25^{\circ}$ and Gain Margin = 4.