Library / Control engineering / CTL-23
Control engineering
Tustin frequency warping
The analogue frequency that the Tustin (bilinear) transform maps to a given digital frequency.
- Ref
- CTL-23
- Rev
- v1
- Field
- Control engineering
- Result
- ωa [rad/s]
The Tustin (bilinear) transform s ≈ (2/Ts)(z − 1)/(z + 1) maps the left half-plane into the unit circle, so it keeps stability. It does not keep frequencies: a digital frequency ωd corresponds to the analogue frequency ωa = (2/Ts)tan(ωdTs/2). Features of an analogue design are therefore shifted, more at high frequency, unless the design frequency is pre-warped. The digital frequency must be below the Nyquist limit π/Ts.
The analogue frequency that the Tustin (bilinear) transform maps to a given digital frequency. The mapping is not linear, so a digital filter designed this way has its features shifted unless the frequency is pre-warped.
Calculator
- Warping error, (ωa − ωd)/ωd14.02 %
- Digital frequency in rad per sample, ωdTs1.2 rad
- Nyquist frequency, π/Ts157.1 rad/s
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 make a digital filter act at 60 rad/s with a 0.02 s sample interval, the analogue prototype must be designed at 68.41 rad/s, 14 % higher. This is pre-warping.
- The error shrinks fast at low frequency. At a tenth of the Nyquist frequency the warping is under 1 % (about 0.83 %), while at the default, 38.2 % of Nyquist, it is 14 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ωd Digital angular frequency | 1 to 100 rad/s | 1 | 155.7 | 154.7 | +10% | +13.4 % |
| Ts Sample interval | 0.005 to 0.03 s | 60.45 | 84.01 | 23.56 | +10% | +3.13 % |
Sweep of Digital angular frequency
| Digital angular frequency (rad/s) | Analogue frequency (rad/s) |
|---|---|
| 1 | 1 |
| 13.38 | 13.46 |
| 25.75 | 26.33 |
| 38.13 | 40.09 |
| 50.5 | 55.28 |
| 62.88 | 72.72 |
| 75.25 | 93.63 |
| 87.63 | 120 |
| 100 | 155.7 |
Sweep one variable
Variables and how each one contributes
Analogue frequency
What the equation gives you. Everything else feeds it.
Digital angular frequency
The digital angular frequency in rad/s that the filter should act at. Rising frequency pushes ωa up faster than in proportion, because the tangent steepens towards Nyquist. Raising it by 10 % (to 66) moves the result by +13.4 %, from 68.41 to 77.61 rad/s.
Sample interval
The sample interval in s. A longer interval increases the warping, since ωdTs grows towards π. Raising it by 10 % (to 0.022) moves the result by +3.13 %, from 68.41 to 70.55 rad/s.
Worked example with the values above
- Half the angle per sample: ωdTs/2 = 60 × 0.02 / 2 = 0.6 rad.
- Tangent: tan(0.6) = 0.6841.
- ωa = (2/0.02) × 0.6841 = 68.41 rad/s.
Assumptions and limits
- The Tustin (bilinear) transform of an LTI filter or controller, at a frequency below the Nyquist limit.
- A single critical frequency is pre-warped. Other features of the response are shifted according to the same mapping.
Common mistakes
- Forgetting to pre-warp, so a notch or crossover lands at the wrong frequency.
- Using a frequency at or above the Nyquist frequency π/Ts, which has no analogue equivalent.
Statistics and uncertainty
For small ωdTs the relative warping is about (ωdTs)²/12, here 12 % against the exact 14 %. It depends on the square of the sample interval, so halving Ts cuts the warping by about four.
Where it is used
- Pre-warping a notch or lead compensator so that it acts at the intended frequency after discretisation.
- Estimating how much a sample interval distorts a frequency response.
Maths you need
- Tangent function: tan x ≈ x for small x
- Nyquist frequency: half the sample rate