gainmarge

Gain margin, sometimes encountered in some texts as gainmarge, is a measure of robustness in a feedback control system. The standard term is gain margin. It quantifies how much the loop gain can be increased before the closed-loop system becomes unstable. For a continuous-time system with loop transfer function L(s), the gain margin is defined at the phase crossover frequency—where the phase of L(jω) is -180 degrees (mod 360). The gain margin gm is the reciprocal of the magnitude at that frequency: gm = 1/|L(jω_pc)|, where ω_pc satisfies ∠L(jω_pc) = -180°. In decibels, GM_dB = 20 log10(gm) = -20 log10(|L(jω_pc)|). If the phase never crosses -180° for positive frequencies, the gain margin is infinite, indicating no gain increase would destabilize the system within the ideal model.

Gain margin can be determined from Bode or Nyquist plots as well. On a Bode plot, gm is read at the phase crossover frequency from the inverse of the magnitude there. On a Nyquist plot, gm corresponds to the distance from the critical point -1 on the real axis to the plot at the phase-crossing condition. For discrete-time systems, the analogous calculation uses the unit circle and the point -1 accordingly.

Usage and interpretation: gain margin reflects how much the loop gain can be amplified before instability occurs, with larger margins indicating greater robustness to gain variations. It is commonly reported together with phase margin, which measures robustness to phase variations at the gain crossover. Both concepts assume a linear time-invariant model and can be affected by model uncertainty and measurement noise.