Question

If the phase margin of a system is zero degrees, then the system is

• A. stable.
• B. unstable.
• C. marginally stable.
• D. conditionally stable.

Hint:

For marginal stability, the open-loop frequency response curve passes exactly through the critical point $(-1, j0)$ on a Nyquist plot.
This corresponds to a Gain Margin of $0\text{ dB}$ and a Phase Margin of $0^\circ$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question concerns frequency domain stability criteria for linear time-invariant control systems, specifically relating the phase margin value to the stability state.

Step 2: Key Formula or Approach:
Phase margin ($PM$) is defined as:
\[ PM = 180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc}) \]
where $\omega_{gc}$ is the gain crossover frequency where magnitude $|G(j\omega_{gc})H(j\omega_{gc})| = 1$ ($0\text{ dB}$).

Step 3: Detailed Explanation:

• If the phase margin is positive ($PM > 0^\circ$), the system is stable.

• If the phase margin is negative ($PM < 0^\circ$), the system is unstable.

• If the phase margin is exactly $0^\circ$:
\[ 180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc}) = 0^\circ \implies \angle G(j\omega_{gc})H(j\omega_{gc}) = -180^\circ \]

• This means at the gain crossover frequency, the phase angle is exactly $-180^\circ$.

• Thus, the gain crossover frequency ($\omega_{gc}$) is identical to the phase crossover frequency ($\omega_{pc}$).

• Under this condition, the system sustained oscillations at frequency $\omega_{gc}$, which is the definition of a marginally stable system.

Step 4: Final Answer:
The system is marginally stable, which corresponds to Option (C).