Calculation
y=K·Δu·[1−e^(−t/T)·Σ(k=0…n−1)(t/T)^k/k!]
As order increases, the response starts more slowly and increasingly resembles a delay although the model contains no pure dead time.
Zacher/Reuter 2024, Abschnitt 3.4: P-Strecken höherer Ordnung
PTn element: equal-lag step response
As order increases, the response starts more slowly and increasingly resembles a delay although the model contains no pure dead time.
MINTSI
02
Result
Select a target and calculate.
Calculated step response
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Understand the inputs
- Output change y(t) — Output change reached at the selected time. Add the pre-step output to obtain the absolute process value.
- Overall gain K — Steady ratio of total output change to input step for the complete cascade, not the gain of each individual element.
- Input step Δu — New minus old input value applied to the first element in the cascade.
- Time constant per element T — Time constant of each identical first-order stage. Obtain it from the chosen equivalent model or identification of the individual stages.
- Order n — Number of identical lag elements in series. Only integers from 1 to 20 are allowed; n=1 equals the first-order model.
- Time since the step t — Time measured from the input step at which the cascade response is required.
Example
K=1, Δu=1, T=2 s, n=3 and t=5 s give y≈0.4562. At that time, 45.62% of final value 1 has been reached.
Assumptions and limits
Cascade of n linear time-invariant first-order elements with identical time constants, initially at rest and driven by an ideal step; unequal time constants, dead time and zeros are excluded.
Understand PTn element: equal-lag step response
Calculate and plot the step response of a cascade of n identical first-order lags at a selected time.
What does this quantity describe?
As order increases, the response starts more slowly and increasingly resembles a delay although the model contains no pure dead time. 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
y=K·Δu·[1−e^(−t/T)·Σ(k=0…n−1)(t/T)^k/k!]
| Symbol / input | Meaning |
|---|---|
| Output change y(t) | Output change reached at the selected time. Add the pre-step output to obtain the absolute process value. |
| Overall gain K | Steady ratio of total output change to input step for the complete cascade, not the gain of each individual element. |
| Input step Δu | New minus old input value applied to the first element in the cascade. |
| Time constant per element T | Time constant of each identical first-order stage. Obtain it from the chosen equivalent model or identification of the individual stages. |
| Order n | Number of identical lag elements in series. Only integers from 1 to 20 are allowed; n=1 equals the first-order model. |
| Time since the step t | Time measured from the input step at which the cascade response is required. |
Choose the inputs correctly
Output change y(t): Output change reached at the selected time. Add the pre-step output to obtain the absolute process value. Overall gain K: Steady ratio of total output change to input step for the complete cascade, not the gain of each individual element. Input step Δu: New minus old input value applied to the first element in the cascade. Time constant per element T: Time constant of each identical first-order stage. Obtain it from the chosen equivalent model or identification of the individual stages. Order n: Number of identical lag elements in series. Only integers from 1 to 20 are allowed; n=1 equals the first-order model. Time since the step t: Time measured from the input step at which the cascade response is required.
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
K=1, Δu=1, T=2 s, n=3 and t=5 s give y≈0.4562. At that time, 45.62% of final value 1 has been reached.
Understand the result and units
As order increases, the response starts more slowly and increasingly resembles a delay although the model contains no pure dead time. 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
PTn element: equal-lag step response: Such basic calculations support plausibility checks, early component selection and preparation of a complete verification.
Assumptions, limits and common mistakes
Cascade of n linear time-invariant first-order elements with identical time constants, initially at rest and driven by an ideal step; unequal time constants, dead time and zeros are excluded.
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.