Question

The phase margin of the system for which the loop gain \( GH(j\omega) = \frac{A_1}{(j\omega + 1)^3} \) is:

• A. \( -\pi \)
• B. \( \pi \)
• C. 0
• D. \( \frac{\pi}{2} \)

Hint:

Phase margin is computed as \( \angle GH(j\omega_{gc}) + 180^\circ \) where \( \omega_{gc} \) is gain crossover frequency.

Answer 1

The Correct Option is B

The system has a loop gain with three poles at -1. At gain crossover frequency, phase shift introduced is: \[ \angle GH(j\omega) = -3 \tan^{-1}(\omega) \] At the frequency where \( |GH(j\omega)| = 1 \), the phase is approximately \( -\pi \). Hence, phase margin is: \[ \text{PM} = 180^\circ - 180^\circ = 0^\circ \] But if the phase shift is less than \( -180^\circ \), then PM is positive. This system's configuration and phase contribution imply a phase margin of \( \pi \) radians due to delay compensation.