Gain and Phase Margin
Stability margins measure how much extra gain or phase lag a stable loop can tolerate before it begins to oscillate.
How Close to the Edge
A loop can be stable yet fragile, sitting just barely on the safe side of instability. Stability margins quantify the buffer. They answer how much the plant can change, in gain or in phase lag, before the closed loop goes unstable.
Gain margin
The gain margin is the factor by which the loop gain can be increased before the system reaches the brink of instability. It is measured at the phase-crossover frequency, where the open-loop phase equals minus 180 degrees. On a Bode plot it is how far below 0 dB the magnitude sits at that frequency. A larger gain margin means more tolerance to gain error.
Phase margin
The phase margin is the additional phase lag that can be added before instability. It is measured at the gain-crossover frequency, where the open-loop magnitude equals 0 dB. On a Bode plot it is how far above minus 180 degrees the phase sits at that frequency. Extra phase lag comes from unmodeled dynamics and time delays, so phase margin protects against those.
Typical targets
- Phase margin of 30 to 60 degrees is a common design range; near 60 degrees gives a well-damped response.
- Gain margin of roughly 6 dB (a factor of two) or more is typical.
- Very small margins produce ringing, sensitivity to parameter drift, and poor disturbance rejection.
- Very large margins usually mean the loop is sluggish, sacrificing bandwidth.
Relationship to damping
For a second-order system, phase margin correlates directly with damping ratio and thus with overshoot: roughly, phase margin in degrees divided by 100 approximates the damping ratio for margins up to about 60 degrees. This rule of thumb links a frequency-domain margin to a time-domain overshoot.
Margins alone can mislead: a loop can have healthy gain and phase margins yet still pass dangerously close to the minus-one point along a diagonal. The vector or disk margin, which measures the actual shortest distance to minus one, closes that gap and is preferred in robust design.