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Control engineering
Gain margin in decibels
How much the loop gain can rise, expressed in decibels, before the loop becomes unstable.
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- CTL-17
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- Result
- GMdB [dB]
|L(jωpc)| is the open-loop gain at the phase-crossover frequency, where the open-loop phase is −180°. The closed loop is stable when this gain is below 1. The gain margin GM = 1/|L(jωpc)| is the factor by which the loop gain could be raised before the loop becomes unstable, and GMdB = 20 log₁₀ GM expresses it in decibels. A negative result means |L| exceeds 1 at the phase crossover.
How much the loop gain can rise, expressed in decibels, before the loop becomes unstable. It is measured where the open-loop phase crosses -180°; a negative value means the loop is already unstable on this test.
Calculator
- Gain margin as a ratio, 1/|L|2
- Gain increase tolerated before instability100 %
- Margin above a 6 dB working target0.0206 dB
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At |L| = 0.5 the loop gain can be multiplied by 2 (6.02 dB) before the loop reaches the stability limit.
- Each halving of |L| adds 6.02 dB. At |L| = 0.25 the margin is 12 dB. A margin of at least about 6 dB is a common working target, but it does not replace the phase margin.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (dB) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Lpc Open-loop magnitude |L(jω) | at the phase crossover | 0.1 to 1.5 | −3.522 | 20 | 23.52 | +10% | −0.828 dB |
Sweep of Open-loop magnitude |L(jω) | at the phase crossover
| Open-loop magnitude |L(jω) | at the phase crossover | Gain margin (dB) |
|---|---|
| 0.1 | 20 |
| 0.275 | 11.21 |
| 0.45 | 6.936 |
| 0.625 | 4.082 |
| 0.8 | 1.938 |
| 0.975 | 0.2199 |
| 1.15 | −1.214 |
| 1.325 | −2.444 |
| 1.5 | −3.522 |
Sweep one variable
Variables and how each one contributes
Gain margin
What the equation gives you. Everything else feeds it.
Open-loop magnitude |L(jω) | at the phase crossover
The magnitude of the open-loop transfer function at the frequency where its phase is −180°, as a plain ratio (not in dB). A larger open-loop gain at the phase crossover leaves less margin. Raising it by 0.1 (to 0.6) moves the result by −26.3 %, from 6.021 to 4.437 dB.
Worked example with the values above
- Gain margin as a ratio: 1/|L| = 1 / 0.5 = 2.
- Common logarithm: log₁₀(2) = 0.301.
- GMdB = 20 × 0.301 = 6.021 dB.
Assumptions and limits
- A loop with a single phase crossover, stable when the gain is reduced, so that the gain margin has its usual meaning.
- The magnitude is taken at the frequency where the phase is −180° (modulo 360°); other crossings need separate assessment.
Common mistakes
- Assuming a large gain margin means the loop is robust. A loop can have a big gain margin yet be close to instability at another frequency, so the phase margin and the sensitivity peak must be checked too.
- Using 10 log₁₀ for a gain: amplitude ratios use 20 log₁₀, power ratios use 10 log₁₀.
Statistics and uncertainty
Decibels turn products into sums, so a gain uncertainty of ±2 dB is a factor of about 1.26 either way. A 10 % error in the measured |L| is only 0.83 dB (20 log₁₀ 1.1), so a gain margin read from a measured frequency response is reliable to within a decibel or so if the frequency response is, but the phase crossover frequency itself is sensitive to delay and noise.
Where it is used
- Specifying and reporting the robustness of a loop to gain uncertainty.
- Converting a measured open-loop gain at the phase crossover into a margin for a design review.
Maths you need
- Decibels: 20 log₁₀ of an amplitude ratio
- Phase crossover: the frequency where the open-loop phase is −180°