Question

The gain margin of a system can be determined from the gain at

• A. $0\text{ dB}$
• B. $3\text{ dB}$
• C. Gain crossover frequency
• D. Phase crossover frequency

Hint:

- Gain Margin $\rightarrow$ measured at Phase Crossover Frequency ($\omega_{pc}$).
- Phase Margin $\rightarrow$ measured at Gain Crossover Frequency ($\omega_{gc}$).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question is from control systems engineering and relates to the frequency domain stability margins: Gain Margin (GM) and Phase Margin (PM).

Step 2: Key Formula or Approach:
- Phase Crossover Frequency ($\omega_{pc}$) is the frequency where the phase angle of the loop transfer function is $-180^\circ$:
\[ \angle G(j\omega_{pc})H(j\omega_{pc}) = -180^\circ \]
- Gain Margin is defined as the reciprocal of the gain magnitude at this frequency:
\[ GM = \frac{1}{|G(j\omega_{pc})H(j\omega_{pc})|} \]

Step 3: Detailed Explanation:

• To calculate how much the system gain can be increased before the system becomes unstable, we look at the frequency where the phase shift is exactly $-180^\circ$ (which is the phase crossover frequency).

• If the gain magnitude at this phase crossover frequency is less than 1 ($0\text{ dB}$), the system is stable.

• The gain margin is measured specifically at this phase crossover frequency.

• In contrast, the Phase Margin is measured at the gain crossover frequency (where the loop gain is $0\text{ dB}$).

Step 4: Final Answer:
The gain margin is determined from the gain at the phase crossover frequency, corresponding to Option (D).