FOPDT Step Test Calculator (Gain, Time Constant, Dead Time)
By Santaji GadepatilinUpdated October 29, 2026
An open-loop step test — change the controller output in manual and watch the process variable — is the standard way to model a process before tuning. This FOPDT step test calculator turns the step size and the times to reach 28.3% and 63.2% of the response into process gain, time constant and dead time — see step-test and auto-tune methods in DCS PID controllers.
How to use the FOPDT step test calculator
With the controller in manual, make a step in the output and record the PV until it settles. Enter the output step, the total PV change (both in % of span), and the times when the PV reached 28.3% and 63.2% of its final change. You get the process gain, time constant, dead time and the θ/τ ratio. Use the result in the lambda tuning calculator.
Two-point (Smith) method
K = ΔPV ÷ ΔOP · τ = 1.5 (t63.2 − t28.3) · θ = t63.2 − τ
The points come from the first-order response 1 − e−t/τ, which reaches 28.3% at τ/3 and 63.2% at τ after the dead time.
Example
A 10% output step raises a temperature PV by 20% of span; the PV reaches 28.3% of its change at 25 s and 63.2% at 40 s:
- K = 20 ÷ 10 = 2
- τ = 1.5 × (40 − 25) = 22.5 s
- θ = 40 − 22.5 = 17.5 s
- θ/τ = 0.778
First-order response reference
| Time after dead time | Response (% of final) |
|---|---|
| τ/3 | 28.3% |
| τ/2 | 39.3% |
| τ | 63.2% |
| 2τ | 86.5% |
| 3τ | 95.0% |
| 4τ | 98.2% |
| 5τ | 99.3% |
Step test tips
- Make the step large enough to see clearly above noise (typically 5–10% output), but small enough to stay safe.
- Wait for the process to be steady before the step, and repeat in the opposite direction to check for non-linearity.
- Valve deadband and stiction add apparent dead time — check valve response separately.
Frequently asked questions
How do I measure t28 and t63?
Measure both times from the moment the output was stepped. Find the final change in PV, then read the times when the PV has moved 28.3% and 63.2% of that change.
What is the two-point method formula?
Smith's method gives τ = 1.5 × (t63 − t28) and θ = t63 − τ. The process gain is the PV change divided by the output change, both in % of span.
What does the θ/τ ratio tell me?
Below about 0.2 the loop is easy to control; above 1 it is dead-time dominant and needs slow, careful tuning or advanced control.
What if the dead time comes out negative?
The response is not first order (for example, it overshoots or is integrating). Check the data, use a larger step, or fit the model by eye from the tangent at the steepest point.