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Control engineering
Gain of a filtered derivative term
Gain of a filtered derivative term at a given frequency.
- Ref
- CTL-10
- Rev
- v1
- Field
- Control engineering
- Result
- |CD|
The filtered derivative CD(s) = Kds/(1 + τfs) behaves like a true derivative below the corner frequency 1/τf and flattens out at a gain of Kd/τf above it. Without the filter the gain would keep rising with frequency and amplify high-frequency sensor noise without limit.
Gain of a filtered derivative term at a given frequency. The filter caps the gain at high frequency, which limits amplification of sensor noise. The term is Kds/(1 + τfs) and its gain tops out at Kd/τf.
Calculator
- High-frequency limit of the gain, Kd/τf20
- Filter corner frequency, 1/τf100 rad/s
- Gain without the filter, Kdω10
- Phase of the filtered derivative, 90° − arctan(ωτf)63.43 °
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.
- The corner is at 100 rad/s. At 50 rad/s the filtered gain is 8.944, 89.4 % of the unfiltered value; at 500 rad/s it is 19.61, against 100 with no filter.
- Because the gain is capped at Kd/τf = 20, a noise component of amplitude 0.01 at 500 rad/s gives an output of about 0.2 instead of 1 without the filter.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Kd Derivative gain | 0.05 to 2 s | 2.236 | 89.44 | 87.21 | +10% | +10 % |
| ω Frequency | 1 to 500 rad/s | 0.2 | 19.61 | 19.41 | +10% | +7.76 % |
| τf Derivative filter time constant | 0.001 to 0.1 s | 1.961 | 9.988 | 8.026 | +10% | −2.04 % |
Sweep of Frequency
| Frequency (rad/s) | Derivative term gain |
|---|---|
| 1 | 0.2 |
| 63.38 | 10.71 |
| 125.8 | 15.65 |
| 188.1 | 17.66 |
| 250.5 | 18.57 |
| 312.9 | 19.05 |
| 375.3 | 19.33 |
| 437.6 | 19.5 |
| 500 | 19.61 |
Sweep one variable
Variables and how each one contributes
Derivative term gain
What the equation gives you. Everything else feeds it.
Derivative gain
The derivative gain in s (for dimensionless signals). Directly proportional: the whole gain curve scales with Kd. Raising it by 10 % (to 0.22) moves the result by +10 %, from 8.944 to 9.839.
Derivative filter time constant
The filter time constant in s. The corner frequency is 1/τf, and a good starting value is a tenth to a fifth of Td. A larger τf cuts the gain at this frequency, because the filter starts to act at a lower corner frequency. Raising it by 10 % (to 0.011) moves the result by −2.04 %, from 8.944 to 8.762.
Frequency
The frequency at which the gain is evaluated, in rad/s. Gain rises with frequency but flattens above the filter corner, so the increase is less than proportional. Raising it by 10 % (to 55) moves the result by +7.76 %, from 8.944 to 9.638.
Worked example with the values above
- Product ωτf = 50 × 0.01 = 0.5, so the denominator is √(1 + 0.25) = 1.118.
- Unfiltered gain: Kd ω = 0.2 × 50 = 10.
- |CD| = 10 / 1.118 = 8.944.
Assumptions and limits
- A first-order filter on the derivative term, with the derivative acting on a dimensionless or consistently scaled signal.
- Sinusoidal steady state (a frequency-response gain, not a transient).
Common mistakes
- Choosing τf so large that the derivative stops working at the frequencies where it is needed. Useful action ends near 1/τf.
- Differentiating the reference as well as the measurement, which makes the output kick at every set-point change. Differentiating the measurement avoids it.
Statistics and uncertainty
Derivative action amplifies noise, and the amplification is set by the high-frequency gain Kd/τf, not by the gain at the frequency of interest. For white measurement noise the output noise power grows roughly with the filter bandwidth 1/τf, so halving τf doubles the noise gain's reach into the spectrum while improving the phase lead only slightly.
Where it is used
- Setting the derivative filter so that sensor noise is not amplified without limit.
- Checking how much phase lead a derivative term really delivers at the crossover frequency.
Maths you need
- Frequency response: gain and phase of a transfer function at s = jω
- First-order filter: corner at 1/τ