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?
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AStable
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BUnstable
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CCritically damped
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DNone of these
Answer
Correct Answer: Unstable
Explanation
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:
At phase crossover (−180°), measure |L(jω_pc)|.If |L| > 1 → GM_linear = 1/|L| < 1 → GM_dB = 20log10(GM_linear) < 0.Negative GM_dB indicates the loop gain must be reduced to reach marginal stability → the current system is unstable.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