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Control engineering
Frequency response of a first-order lag
Gain and phase of a first-order system K/(τs + 1) for a sinusoidal input of frequency ω.
- Ref
- CTL-18
- Rev
- v1
- Field
- Control engineering
- Result
- |G|
For a sinusoidal input at angular frequency ω, a stable first-order system K/(τs + 1) gives a steady output of the same frequency with amplitude multiplied by |G| = K/√(1 + (ωτ)²) and shifted in phase by −arctan(ωτ). The corner frequency 1/τ is where the gain has fallen to 70.7 % of K and the phase lag is 45°. Well above it the gain falls at 20 dB per decade.
Gain and phase of a first-order system K/(τs + 1) for a sinusoidal input of frequency ω. The gain falls to 70.7 % of K at the corner frequency 1/τ, where the phase lag is 45°.
Calculator
- Gain in decibels, 20 log₁₀|G|3.01 dB
- Phase, −arctan(ωτ)−45 °
- Corner frequency, 1/τ2 rad/s
- Frequency ratio ωτ1
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 default frequency 2 rad/s equals the corner 1/τ = 2 rad/s, so the gain is 0.707 K = 1.414 and the phase lag is 45°. Ten times higher (20 rad/s) the gain is 0.199, about a tenth of K.
- Below about a tenth of the corner frequency the system passes signals almost unchanged. The system's bandwidth is therefore about 1/τ rad/s, which is also why it responds to a step with a time constant of τ.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| K DC gain | 0.5 to 5 | 0.3536 | 3.536 | 3.182 | +10% | +10 % |
| ω Frequency | 0.1 to 20 rad/s | 0.199 | 1.998 | 1.798 | +10% | −4.87 % |
| τ Time constant | 0.1 to 2 s | 0.4851 | 1.961 | 1.476 | +10% | −4.87 % |
Sweep of Frequency
| Frequency (rad/s) | Gain magnitude |
|---|---|
| 0.1 | 1.998 |
| 2.587 | 1.223 |
| 5.075 | 0.7333 |
| 7.562 | 0.5113 |
| 10.05 | 0.3904 |
| 12.54 | 0.3151 |
| 15.02 | 0.2639 |
| 17.51 | 0.2269 |
| 20 | 0.199 |
Sweep one variable
Variables and how each one contributes
Gain magnitude
What the equation gives you. Everything else feeds it.
DC gain
The DC gain, the gain at zero frequency. Directly proportional: the gain curve scales with K at every frequency. Raising it by 10 % (to 2.2) moves the result by +10 %, from 1.414 to 1.556.
Time constant
The time constant in s. The corner frequency is 1/τ. A larger τ lowers the corner frequency, so the gain at a given frequency is lower. Raising it by 10 % (to 0.55) moves the result by −4.87 %, from 1.414 to 1.345.
Frequency
The angular frequency in rad/s (2π times the frequency in Hz). Gain falls as frequency rises, slowly below the corner and at 20 dB per decade above it. Raising it by 10 % (to 2.2) moves the result by −4.87 %, from 1.414 to 1.345.
Worked example with the values above
- Frequency ratio: ωτ = 2 × 0.5 = 1.
- Denominator: √(1 + 1) = 1.414.
- |G| = K / 1.414 = 2 / 1.414 = 1.414, with a phase of −45°.
Assumptions and limits
- A stable linear first-order system in sinusoidal steady state, after any transient has died out.
- Constant K and τ over the frequencies of interest.
Common mistakes
- Using Hz for ω. The formula needs rad/s, so multiply a frequency in hertz by 2π.
- Expecting the phase lag to exceed 90°. A single first-order lag approaches but never reaches 90°.
Statistics and uncertainty
Reading τ from the −3 dB point is sensitive to the gain measurement near the corner: a gain read 5 % too low gives ωτ = 1.1 instead of 1, an error of about 10 % in τ. Measurements should be made over a range of frequencies and a line fitted, not read at one point.
Where it is used
- Reading the bandwidth and lag of a sensor, filter or actuator from a Bode plot.
- Checking how much of a vibration or noise signal a first-order filter removes.
Maths you need
- Complex numbers: magnitude and angle of 1 / (1 + jωτ)
- Decibels: 20 log₁₀ of an amplitude ratio