GainMargin

Gain margin, sometimes written as gain margin, is a measure of robustness for a feedback control system. It indicates how much the open-loop gain can be multiplied before the closed-loop system becomes unstable.

For a linear time-invariant system with negative unity feedback and open-loop transfer function L(s), the gain margin GM is defined using the frequency response. Let ω_pc be a phase crossover frequency where the open-loop phase is −180 degrees (modulo 360). Then GM is the reciprocal of the magnitude of the loop at that frequency: GM = 1 / |L(jω_pc)|. In decibels, GM is expressed as GM_dB = −20 log10 |L(jω_pc)|. If the phase never reaches −180 degrees, the gain margin is infinite, indicating the system can tolerate any amount of gain increase without becoming unstable (within the model).

In practice, GM is determined from a Bode or Nyquist plot. On a Bode plot, locate the frequency where the phase crosses −180 degrees and read the corresponding magnitude; its reciprocal gives GM. If multiple phase crossovers exist, the most conservative margin is taken as the smallest positive GM across those frequencies. The Nyquist plot provides a similar assessment by examining how close the plot comes to encircling the critical point −1.

Relation to other margins, such as phase margin, is that gain margin measures tolerance to gain changes, while phase margin measures tolerance to phase changes at the gain crossover. GM is a fundamental concept in assessing stability robustness of feedback systems.