Calculation
y(t) = K · Δu · (1 − e^(−t/T))
The calculator shows how a lagging system approaches its new final value exponentially after an input step and how much has been reached at the selected time.
Zacher/Reuter 2024, Abschnitt 3.2: P-Strecke 1. Ordnung
PT1 element: calculate step response
The calculator shows how a lagging system approaches its new final value exponentially after an input step and how much has been reached at the selected time.
MINTSI
02
Result
Select a target and calculate.
Technical principle schematic
Understand the inputs
- Output change y(t) — Instantaneous output change at time t. If required, combine this dimensionless model value with the physical reference unit of your model, such as bar, °C or rpm.
- Static gain K — Ratio of the final output change to input-step height. Obtain K from two steady operating points as Δy∞/Δu or from an identified plant model.
- Input step Δu — Change in the input signal at t=0, i.e. new minus old input. Use scaled or normalised quantities when input and output have different physical units.
- Time constant T — Lag measure of the first-order element. After one time constant, 63.2% of the total output change has been reached. T usually comes from a step test, datasheet or model identification.
- Time since the step t — Elapsed time between the input step and the output point being evaluated. t=0 represents the instant immediately after the step.
Example
K=2, Δu=3 and T=5 s give y=6·(1−e⁻¹)≈3.793 at t=5 s. This is 63.2% of the final value 6.
Assumptions and limits
Ideal linear time-invariant first-order lag initially at rest before an instantaneous input step; initial value, dead time, saturation, disturbances and additional time constants are excluded.
Understand PT1 element: calculate step response
This calculator makes the time behaviour of a system with one effective lag visible. Typical approximations include a temperature sensor, a well-mixed vessel or a low-pass measuring element.
What does this quantity describe?
A first-order lag cannot respond instantaneously. After an input step Δu, output change y(t) approaches final value K·Δu with time constant T. The curve reaches that value only asymptotically in theory.
Formula and variables
y(t) = K · Δu · (1 − e^(−t/T))
G(s) = K/(1 + T·s)y(t)/(K·Δu) = 1 − e^(−t/T)
| Symbol / input | Meaning |
|---|---|
| Output change y(t) | Instantaneous output change at time t. If required, combine this dimensionless model value with the physical reference unit of your model, such as bar, °C or rpm. |
| Static gain K | Ratio of the final output change to input-step height. Obtain K from two steady operating points as Δy∞/Δu or from an identified plant model. |
| Input step Δu | Change in the input signal at t=0, i.e. new minus old input. Use scaled or normalised quantities when input and output have different physical units. |
| Time constant T | Lag measure of the first-order element. After one time constant, 63.2% of the total output change has been reached. T usually comes from a step test, datasheet or model identification. |
| Time since the step t | Elapsed time between the input step and the output point being evaluated. t=0 represents the instant immediately after the step. |
Choose the inputs correctly
Obtain K from the final change between two steady operating points. Δu is new input minus old input. From a step test, read T as the time required to reach 63.2% of the total output change. t is the instant at which the output is needed.
How to use the calculator
First bring the plant to a steady operating point, change the input as quickly as practical and record the output. Enter the resulting model parameters K and T. The calculated y value is a change; add the pre-step output when an absolute output is required.
Worked example
With K=2, Δu=3, T=5 s and t=5 s, y≈3.793. The final value is 6; exactly one time constant reaches 63.2% of it.
Understand the result and units
y(t) answers what portion of a setpoint or manipulated-variable change has arrived at the output at a selected time. It supports estimating waiting times and checking whether a measured curve behaves like a first-order lag.
T and t must have the same time dimension; available time units are converted internally. K, Δu and y are normalised here. In a dimensional model, K carries output units per input unit.
Typical applications
Plant modelling before controller design, plausibility checking of a step test and estimating the response duration of simple thermal, hydraulic or electrical systems.
Assumptions, limits and common mistakes
The model applies only to a linear time-invariant plant with one dominant time constant and no dead time. Saturation, hysteresis, multiple storage elements, disturbances and a non-zero initial output are excluded.
Common mistake: y(t) is an output change, not automatically the absolute measured value. Also do not read T as time to 100%: t=T reaches 63.2%, 3T about 95%, and 5T about 99.3%.
Frequently asked questions
What is “PT1 step response” used for?
Plant modelling before controller design, plausibility checking of a step test and estimating the response duration of simple thermal, hydraulic or electrical systems.
Where do the input values come from?
Obtain K from the final change between two steady operating points. Δu is new input minus old input. From a step test, read T as the time required to reach 63.2% of the total output change. t is the instant at which the output is needed.
What does the result not cover?
The model applies only to a linear time-invariant plant with one dominant time constant and no dead time. Saturation, hysteresis, multiple storage elements, disturbances and a non-zero initial output are excluded.