Calculation
T=−t/ln(1−y(t)/y∞)
Inverse evaluation turns a recorded step response into a model parameter for simulation and controller design.
Zacher/Reuter 2024, Abschnitt 3.2: PT1-Sprungantwort
PT1 time constant from a step test
Inverse evaluation turns a recorded step response into a model parameter for simulation and controller design.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Identified time constant T — Lag parameter of the PT1 equivalent model. Use it afterwards in the PT1 step-response calculator or a simulation.
- Measured output change y(t) — Measured change at time t relative to the pre-step output, not the absolute sensor reading.
- Final output change y∞ — New steady output minus pre-step output. Read it only after the process has settled sufficiently.
- Measurement time t — Time between input step and measurement y(t), using the same time basis desired for the time constant.
Example
After 5 s, 6.3212 of the final 10 units have been reached. Thus T=−5/ln(1−0.63212)≈5 s.
Assumptions and limits
PT1 behaviour without dead time, initial value correctly subtracted and 0<y(t)<y∞ for a positive step response. Noise and an incorrectly estimated final value strongly affect T.
Understand PT1 time constant from a step test
Determine the time constant of a PT1-like process from one measured point, the final value and elapsed time.
What does this quantity describe?
Inverse evaluation turns a recorded step response into a model parameter for simulation and controller design. This calculator represents a clearly bounded technical relationship between the displayed quantities. The definition helps put inputs into the same reference state before interpreting the result.
Formula and variables
T=−t/ln(1−y(t)/y∞)
| Symbol / input | Meaning |
|---|---|
| Identified time constant T | Lag parameter of the PT1 equivalent model. Use it afterwards in the PT1 step-response calculator or a simulation. |
| Measured output change y(t) | Measured change at time t relative to the pre-step output, not the absolute sensor reading. |
| Final output change y∞ | New steady output minus pre-step output. Read it only after the process has settled sufficiently. |
| Measurement time t | Time between input step and measurement y(t), using the same time basis desired for the time constant. |
Choose the inputs correctly
Identified time constant T: Lag parameter of the PT1 equivalent model. Use it afterwards in the PT1 step-response calculator or a simulation. Measured output change y(t): Measured change at time t relative to the pre-step output, not the absolute sensor reading. Final output change y∞: New steady output minus pre-step output. Read it only after the process has settled sufficiently. Measurement time t: Time between input step and measurement y(t), using the same time basis desired for the time constant.
How to use the calculator
Select the target quantity, enter the other known values with units, then check the result against the worked example and model limits.
Worked example
After 5 s, 6.3212 of the final 10 units have been reached. Thus T=−5/ln(1−0.63212)≈5 s.
Understand the result and units
Inverse evaluation turns a recorded step response into a model parameter for simulation and controller design. Read the result as a model value for the selected operating point and check units, sign, order of magnitude and application boundary conditions.
Use the displayed units and convert afterwards. Prefixes such as k-, m- and µ- are common sources of mistakes.
Typical applications
PT1 time constant from a step test: Such basic calculations support plausibility checks, early component selection and preparation of a complete verification.
Assumptions, limits and common mistakes
PT1 behaviour without dead time, initial value correctly subtracted and 0<y(t)<y∞ for a positive step response. Noise and an incorrectly estimated final value strongly affect T.
Common mistake: A formally correct result can still be unsuitable when load case, reference state or units do not match the application.
Frequently asked questions
How do I check the result?
Compare unit and order of magnitude with a second calculation and vary inputs one at a time.
Are assumptions automatically satisfied?
No. The calculator exposes a model; real boundary conditions require separate review.
Can I mix arbitrary units?
Only when the calculator converts within the same physical quantity type.