Stability margins from Bode plots: In frequency-response analysis, the gain margin is numerically equal to which of the following quantities (using amplitude ratio at the phase-crossover frequency)?
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AAmplitude ratio
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BReciprocal of amplitude ratio
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CGain in a proportional (P) controller
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DGain in a proportional–integral (PI) controller
Answer
Correct Answer: Reciprocal of amplitude ratio
Explanation
Introduction / Context:Gain margin quantifies how much the loop gain can increase before a closed-loop system becomes unstable. It is read from Bode plots at the phase-crossover frequency, where the phase equals −180 degrees. Understanding the numerical relationship ensures correct computation and interpretation.
Given Data / Assumptions:
- Open-loop transfer function L(jω) = G(jω)H(jω).
- Phase-crossover frequency ωpc satisfies ∠L(jωpc) = −180°.
- Amplitude ratio (AR) denotes |L(jω)| at the specified frequency.
Concept / Approach:By definition, gain margin (GM) = 1/|L(jωpc)| when |L| is expressed as amplitude ratio. In dB, GM(dB) = −20 log10|L(jωpc)|. If |L| is less than one at phase crossover, GM > 1 (or positive in dB), indicating a stable reserve.
Step-by-Step Solution:
Locate ωpc where phase = −180°.Read the magnitude |L| (AR) at ωpc.Compute GM = 1/AR, or in dB: GM(dB) = −20 log10(AR).Verification / Alternative check:Example: if AR = 0.2 at ωpc, then GM = 5 or +14 dB; if AR = 1, GM = 1 (0 dB), indicating marginal stability.
Why Other Options Are Wrong:
Amplitude ratio: this is inverted to obtain GM; not equal to GM.Gains in P or PI controllers are design parameters, not the definition of gain margin.Common Pitfalls:Confusing crossover frequencies: gain margin uses phase crossover; phase margin uses gain crossover (|L| = 1).
Final Answer:Reciprocal of amplitude ratio