Library / Control engineering / CTL-14
Control engineering
Maximum phase lead of a lead compensator
Peak phase advance of a lead compensator (1 + τs)/(1 + ατs) with 0 < α < 1, and the frequency where it occurs.
- Ref
- CTL-14
- Rev
- v1
- Field
- Control engineering
- Result
- φmax [°]
The lead compensator (1 + τs)/(1 + ατs) with 0 < α < 1 has a zero at 1/τ and a pole at 1/(ατ), a factor 1/α higher. Between them the phase is positive, with a peak of φmax = arcsin((1 − α)/(1 + α)) at ωmax = 1/(τ√α), the geometric mean of the two corner frequencies. The price of the phase is a higher gain at high frequency, by a factor 1/α compared with DC.
Peak phase advance of a lead compensator (1 + τs)/(1 + ατs) with 0 < α < 1, and the frequency where it occurs. It adds phase margin, at the cost of more high-frequency gain.
Calculator
- Frequency of maximum lead, 1/(τ√α)17.32 rad/s
- Zero frequency, 1/τ10 rad/s
- Pole frequency, 1/(ατ)30 rad/s
- High-frequency to DC gain ratio, 1/α3
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 lead with α = 1/3 gives 30° of phase and raises high-frequency gain 3 times (about 9.5 dB), which is also the factor by which sensor noise at high frequency is amplified relative to DC.
- More lead costs more noise gain: α = 0.1 gives 54.9° but 10 times the high-frequency gain, and α = 0.05 gives 64.8° for 20 times. Beyond about 60° it is usually better to use two stages.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| α Lead ratio (0 < α < 1) | 0.05 to 0.9 | 3.017 | 64.79 | 61.77 | +10% | −2.39 ° |
| τ Lead time constant | 0.01 to 1 s | 30 | 30 | 0 | +10% | 0 ° |
Sweep of Lead ratio (0 < α < 1)
| Lead ratio (0 < α < 1) | Maximum phase lead (°) |
|---|---|
| 0.05 | 64.79 |
| 0.1563 | 46.86 |
| 0.2625 | 35.74 |
| 0.3687 | 27.46 |
| 0.475 | 20.85 |
| 0.5813 | 15.36 |
| 0.6875 | 10.67 |
| 0.7938 | 6.603 |
| 0.9 | 3.017 |
Sweep one variable
Variables and how each one contributes
Maximum phase lead
What the equation gives you. Everything else feeds it.
Lead ratio (0 < α < 1)
The ratio of the compensator's pole and zero time constants, between 0 and 1. A smaller α gives more phase lead. A larger α gives less phase lead. Phase lead falls as α rises. Raising it by 0.1 (to 0.4333) moves the result by −22.4 %, from 30 to 23.29 °.
Lead time constant
The lead time constant in s: the zero is at 1/τ. No effect on the maximum phase lead, which depends on α only. It moves the frequency where the lead acts: ωmax = 17.32 rad/s at the defaults.
Worked example with the values above
- Ratio: (1 − α)/(1 + α) = (1 − 0.3333)/(1 + 0.3333) = 0.5.
- Phase lead: arcsin(0.5) = 30°.
- φmax = 30°, reached at ωmax = 1/(τ√α) = 17.32 rad/s.
Assumptions and limits
- A single lead stage (1 + τs)/(1 + ατs) with 0 < α < 1, and linear frequency response.
- The phase lead is the maximum only at ωmax; at other frequencies it is less.
Common mistakes
- Centring the lead away from the gain crossover frequency. The benefit is only at ωmax, which should sit at or near the crossover.
- Counting the full φmax as gained margin. The lead raises the gain near ωmax, which pushes the crossover up to where the loop has less phase, so designers add a few degrees of allowance.
Statistics and uncertainty
φmax depends on α alone. Near the default a 20 % error in α (0.267 to 0.4) changes the phase lead from 25.4° to 35.4°. Component tolerances in an analogue lead network therefore move the phase by a few degrees but not by tens of degrees.
Where it is used
- Raising the phase margin of a servo or attitude loop.
- Placing the zero and pole of a lead network from a required peak phase and frequency.
Maths you need
- Frequency response: phase of a transfer function at s = jω
- Geometric mean: the centre of two frequencies on a log scale