Library / Control engineering / CTL-16
Control engineering
Delay margin from phase margin
Extra pure time delay a single-crossover loop can tolerate before its phase margin is used up.
- Ref
- CTL-16
- Rev
- v1
- Field
- Control engineering
- Result
- Tmax [s]
A pure delay T adds a phase lag of ωT without changing the gain. A loop with a single gain crossover at ωgc becomes unstable once the added lag equals its phase margin, so the extra delay it can tolerate is Tmax = PM/ωgc, with PM in radians. This is the margin for additional delay, beyond any delay already in the model.
Extra pure time delay a single-crossover loop can tolerate before its phase margin is used up. A delay of T adds a phase lag of ωT, so the margin in radians divided by the crossover frequency gives the limit.
Calculator
- Phase margin in radians0.7854 rad
- Crossover period, 2π/ωgc0.6283 s
- Delay margin as a fraction of the crossover period12.5 %
- Phase lost by a 10 ms delay at ωgc5.73 °
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.
- A phase margin of 45° at 10 rad/s tolerates 78.5 ms of extra delay, about 12.5 % of the crossover period.
- Doubling the crossover frequency to 20 rad/s halves the delay margin to 39.3 ms. Faster loops are much less tolerant of sensor, computation and communication delay.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ωgc Gain crossover frequency | 1 to 100 rad/s | 0.007854 | 0.7854 | 0.7775 | +10% | −9.09 % |
| PM Phase margin | 20 to 80 ° | 0.03491 | 0.1396 | 0.1047 | +10% | +10 % |
Sweep of Gain crossover frequency
| Gain crossover frequency (rad/s) | Delay margin (s) |
|---|---|
| 1 | 0.7854 |
| 13.38 | 0.05872 |
| 25.75 | 0.0305 |
| 38.13 | 0.0206 |
| 50.5 | 0.01555 |
| 62.88 | 0.01249 |
| 75.25 | 0.01044 |
| 87.63 | 0.008963 |
| 100 | 0.007854 |
Sweep one variable
Variables and how each one contributes
Delay margin
What the equation gives you. Everything else feeds it.
Phase margin
The phase margin of the loop in degrees: 180° plus the open-loop phase at the gain crossover. Directly proportional: more phase margin means more tolerable delay. Raising it by 10 % (to 49.5) moves the result by +10 %, from 0.07854 to 0.08639 s.
Gain crossover frequency
The gain crossover frequency in rad/s, where the open-loop gain is 1. Inversely proportional: a faster loop has less tolerance for delay. Raising it by 10 % (to 11) moves the result by −9.09 %, from 0.07854 to 0.0714 s.
Worked example with the values above
- Convert the margin to radians: PM = 45° × π/180 = 0.7854 rad.
- Divide by the crossover frequency: Tmax = 0.7854 / 10.
- Tmax = 0.07854 s, or 78.5 ms.
Assumptions and limits
- A conventional loop with a single gain crossover, where the phase falls with frequency.
- A pure time delay, which changes phase but not gain.
Common mistakes
- Using the phase margin in degrees. The formula needs radians.
- Treating it as the total allowed delay. It is the extra delay on top of what the model already contains.
Statistics and uncertainty
The delay margin carries the errors of its inputs. A phase margin read from a plot to within ±5° gives a delay margin uncertain by ±11 % at the defaults. Delays are often poorly known, so a design is normally given a delay margin several times larger than the delays it expects, not a margin just above them.
Where it is used
- Judging how much sensor, computation or network delay a control loop can absorb.
- Setting a limit on the sample interval and latency of a digital controller.
Maths you need
- Phase margin: 180° plus the open-loop phase at crossover
- Pure delay: e^(−sT) has gain 1 and phase −ωT