Gain Margin Calculator Formula
Understand the math behind the gain margin calculator. Each variable explained with a worked example.
Formulas Used
Gain Margin (absolute)
gm_abs = 1 / gain_at_180Gain Margin
gm_db = -20 * log10(gain_at_180)Maximum Gain Increase Before Instability
max_gain_increase = (1 / gain_at_180 - 1) * 100Variables
| Variable | Description | Default |
|---|---|---|
gain_at_180 | Open-Loop Gain at Phase = -180° (|G|) | 0.5 |
How It Works
Gain Margin
Gain margin measures how much the open-loop gain can increase before the system becomes unstable. It is evaluated at the phase crossover frequency (where the open-loop phase equals -180°).
Formula
GM = 1 / G(j*omega_pc) = -20*log10( G(j*omega_pc)
where omega_pc is the phase crossover frequency. A positive gain margin (in dB) indicates stability. The larger the gain margin, the more robust the system is to gain variations.
Worked Example
The open-loop gain magnitude at -180° phase is 0.5.
- 01GM = 1 / 0.5 = 2.0 (absolute)
- 02GM = -20 × log10(0.5) = -20 × (-0.301) = 6.02 dB
- 03The gain can be increased by 100% before instability
Frequently Asked Questions
What is a good gain margin?
A gain margin of at least 6 dB (factor of 2) is commonly recommended. For safety-critical systems, 8-12 dB is preferred. Very high gain margins (>20 dB) may indicate an overly conservative (slow) design.
How do gain margin and phase margin relate?
Both are stability margins but measured differently. A system can have good phase margin but poor gain margin, or vice versa. Both should be adequate for robust stability. Typically, GM > 6 dB and PM > 30° are used together.
What if the system has no phase crossover?
If the open-loop phase never reaches -180° (as in a first-order system), the gain margin is infinite—the system is stable for any gain. This is a desirable property but uncommon in complex systems.
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