Stability margins – interpretation of negative gain margin In frequency-response analysis, a negative gain margin expressed in decibels indicates what about the closed-loop system?

Introduction / Context:
Gain and phase margins quantify how far a feedback system is from instability. They are read from Bode or Nyquist plots and provide robust stability information in the presence of modeling uncertainty or gain variations.

Given Data / Assumptions:

• Gain margin (GM) is defined at the phase crossover frequency (phase = −180 degrees).
• GM in decibels is 20log10(GM_linear).
• Negative gain margin means GM_linear < 1.

Concept / Approach:
If the loop transfer function magnitude at the −180° phase point already exceeds unity, then any additional positive gain is not required to reach instability; in fact, the system would need gain reduction to achieve marginal stability. A negative gain margin therefore implies the current loop is beyond the stability boundary, i.e., unstable (or would be unstable under standard loop closure assumptions).

Step-by-Step Solution:

Verification / Alternative check:
Nyquist criterion: encirclement of −1 point with |L| > 1 at −180° phase predicts closed-loop instability, consistent with negative gain margin.

Why Other Options Are Wrong:

• Stable/Critically damped: Would require positive gain margin (GM > 1, GM_dB > 0).
• None of these: Incorrect because negative gain margin has a clear stability implication.

Common Pitfalls:
Confusing definitions: phase margin is evaluated at gain crossover (|L| = 1), while gain margin is evaluated at phase crossover (−180°). Keep these distinct.

Final Answer:
Unstable