Question

A Gain Margin of 0 dB or a Phase Margin of 0 degrees indicates the system is:

• A. Stable
• B. Unstable
• C. Marginally stable
• D. Highly damped

Hint:

For a stable system: \( \text{GM} > 0\text{ dB} \) and \( \text{PM} > 0^\circ \).
For an unstable system: \( \text{GM} < 0\text{ dB} \) or \( \text{PM} < 0^\circ \).
For a marginally stable system: \( \text{GM} = 0\text{ dB} \) and \( \text{PM} = 0^\circ \).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks to identify the stability state of a closed-loop system when its Gain Margin (GM) is exactly 0 dB or its Phase Margin (PM) is exactly 0 degrees.
Gain and Phase margins are frequency-domain indices that indicate how close a stable feedback system is to becoming unstable.

Step 2: Key Formula or Approach:
Let the loop transfer function be \( G(j\omega)H(j\omega) \).
- The Phase Crossover Frequency (\( \omega_{pc} \)) is the frequency at which the phase angle is \( -180^\circ \).
- The Gain Crossover Frequency (\( \omega_{gc} \)) is the frequency at which the loop gain magnitude is 1 (or 0 dB).
At the stability boundary:
\[ \text{Gain Margin (GM)} = 0\text{ dB} \quad (\text{or Magnitude } = 1) \] \[ \text{Phase Margin (PM)} = 0^\circ \quad (\text{or Phase } = -180^\circ) \]

Step 3: Detailed Explanation:
Let us analyze the physical and analytical meaning of these margins:

• Stable System Condition:
- A system is stable if the open-loop gain is less than 0 dB at the phase crossover frequency, and the phase lag is less than \( 180^\circ \) at the gain crossover frequency. This corresponds to a positive Gain Margin (in dB) and a positive Phase Margin.

• Unstable System Condition:
- If the GM is negative (in dB) or PM is negative, the system is unstable.

• Boundary Condition (GM = 0 dB, PM = 0 degrees):
- Under this condition, the gain crossover frequency is identical to the phase crossover frequency (\( \omega_{gc} = \omega_{pc} \)).
- The polar plot of the loop transfer function passes exactly through the critical point \( (-1, j0) \).
- The characteristic equation has roots located directly on the imaginary (\( j\omega \)) axis of the s-plane.
- As a result, the system experiences sustained, non-decaying oscillations, which corresponds to a marginally stable state.

Step 4: Final Answer:
A Gain Margin of 0 dB or a Phase Margin of 0 degrees indicates that the system is marginally stable.