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)?

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:

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:

Common Pitfalls:Confusing crossover frequencies: gain margin uses phase crossover; phase margin uses gain crossover (|L| = 1).

Final Answer:Reciprocal of amplitude ratio