Library / Control engineering / CTL-15
Control engineering
Lead compensator design from required phase
Lead ratio and time constant that give a required phase lead at a chosen centre frequency, usually the gain crossover frequency.
- Ref
- CTL-15
- Rev
- v1
- Field
- Control engineering
- Result
- α
This inverts the lead-compensator formulas. For a required phase lead φmax the ratio is α = (1 − sin φmax)/(1 + sin φmax), and placing the maximum at the centre frequency ωmax (usually the gain crossover) gives τ = 1/(ωmax√α). The zero is then at 1/τ and the pole at 1/(ατ).
Lead ratio and time constant that give a required phase lead at a chosen centre frequency, usually the gain crossover frequency. The required lead is φmax and the centre frequency is ωmax.
Calculator
- Lead time constant, τ = 1/(ωmax√α)0.1732 s
- Zero frequency, 1/τ5.774 rad/s
- Pole frequency, 1/(ατ)17.32 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.
- To gain 30° at 10 rad/s the zero goes at 5.774 rad/s and the pole at 17.32 rad/s, symmetrically placed about the centre on a log scale.
- The high-frequency gain rises by 1/α = 3. Asking for 50° instead needs α = 0.132, a gain increase of 7.55: extra phase gets progressively more expensive in noise.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| φmax Required phase lead (below 90°) | 10 to 60 ° | 0.0718 | 0.7041 | 0.6323 | +10% | −11.6 % |
| ωmax Centre frequency | 1 to 100 rad/s | 0.3333 | 0.3333 | 0 | +10% | 0 % |
Sweep of Required phase lead (below 90°)
| Required phase lead (below 90°) (°) | Lead ratio |
|---|---|
| 10 | 0.7041 |
| 16.25 | 0.5627 |
| 22.5 | 0.4465 |
| 28.75 | 0.3504 |
| 35 | 0.271 |
| 41.25 | 0.2053 |
| 47.5 | 0.1512 |
| 53.75 | 0.1071 |
| 60 | 0.0718 |
Sweep one variable
Variables and how each one contributes
Lead ratio
What the equation gives you. Everything else feeds it.
Required phase lead (below 90°)
The phase lead the compensator must supply at the centre frequency, in degrees, below 90°. More lead needs a smaller α. Raising it by 10 (to 40) moves the result by −34.8 %, from 0.3333 to 0.2174.
Centre frequency
The frequency at which the phase lead is wanted, in rad/s, usually the gain crossover. No effect on α, which depends on the phase only. It sets the time constant: τ = 0.1732 s at the defaults.
Worked example with the values above
- Sine of the required lead: sin 30° = 0.5.
- Ratio: α = (1 − 0.5)/(1 + 0.5) = 0.3333.
- Time constant: τ = 1/(ωmax√α) = 1/(10 × 0.5774) = 0.1732 s.
Assumptions and limits
- A single lead stage with the maximum phase placed at the chosen frequency.
- The compensator gain is set separately so that the new crossover is at or near the centre frequency.
Common mistakes
- Asking a single stage for a very large lead. The noise gain 1/α becomes impractical.
- Entering the centre frequency in Hz, not rad/s. Multiply by 2π.
Statistics and uncertainty
The ratio α depends on the sine of the angle, so it is insensitive to small errors in φmax when the angle is small and more sensitive when it is large. A common design margin is to ask for several degrees more than needed (typically 5° to 10°) to allow for the shift in the crossover caused by the lead's own gain, rather than rely on a tight calculation.
Where it is used
- Designing a lead network to raise a loop's phase margin to a target.
- Cross-checking the pole and zero of a lead compensator supplied by another tool.
Maths you need
- Lead compensator: zero at 1/τ, pole at 1/(ατ)
- Gain crossover frequency: where the open-loop gain is 1