Calculation
y=K·Δu·[1−e^(−D·t/T₀)·(cos(ωd·t)+D/√(1−D²)·sin(ωd·t))], ωd=√(1−D²)/T₀
The second-order model describes systems with two coupled storage elements whose output may overshoot the final value depending on damping.
Zacher/Reuter 2024, Abschnitt 3.5: schwingfähige P-Strecke 2. Ordnung
PT2 element: damped step response
The second-order model describes systems with two coupled storage elements whose output may overshoot the final value depending on damping.
MINTSI
02
Result
Select a target and calculate.
Technical principle schematic
Understand the inputs
- Output change y(t) — Instantaneous output change relative to the pre-step state. Add the former output value to obtain an absolute reading.
- Static gain K — Final output change divided by input step. Determine K from steady readings before and after the test.
- Input step Δu — Step height, new input minus old input. Together with K it sets the final output value.
- Damping ratio D — Dimensionless measure of oscillation decay. Obtain D from a step test or model identification; small values mean strong overshoot. This calculator applies for 0<D<1.
- Natural time T₀ — Reciprocal of undamped natural angular frequency: T₀=1/ω₀. Obtain it from model parameters or measured oscillation period.
- Time since the step t — Observation time measured from the input step.
Example
K=1, Δu=1, D=0.5 and T₀=1 s give y≈1.0746 at t=5 s: the output is still 7.46% above its final value.
Assumptions and limits
Linear time-invariant second-order element initially at rest before an ideal step and underdamped case 0<D<1; dead time, zeros, saturation and additional poles are excluded.
Understand PT2 element: damped step response
Calculate the time response of an underdamped second-order element from gain, damping, natural time, step height and time.
What does this quantity describe?
The second-order model describes systems with two coupled storage elements whose output may overshoot the final value depending on damping. 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^(−D·t/T₀)·(cos(ωd·t)+D/√(1−D²)·sin(ωd·t))], ωd=√(1−D²)/T₀
| Symbol / input | Meaning |
|---|---|
| Output change y(t) | Instantaneous output change relative to the pre-step state. Add the former output value to obtain an absolute reading. |
| Static gain K | Final output change divided by input step. Determine K from steady readings before and after the test. |
| Input step Δu | Step height, new input minus old input. Together with K it sets the final output value. |
| Damping ratio D | Dimensionless measure of oscillation decay. Obtain D from a step test or model identification; small values mean strong overshoot. This calculator applies for 0<D<1. |
| Natural time T₀ | Reciprocal of undamped natural angular frequency: T₀=1/ω₀. Obtain it from model parameters or measured oscillation period. |
| Time since the step t | Observation time measured from the input step. |
Choose the inputs correctly
Output change y(t): Instantaneous output change relative to the pre-step state. Add the former output value to obtain an absolute reading. Static gain K: Final output change divided by input step. Determine K from steady readings before and after the test. Input step Δu: Step height, new input minus old input. Together with K it sets the final output value. Damping ratio D: Dimensionless measure of oscillation decay. Obtain D from a step test or model identification; small values mean strong overshoot. This calculator applies for 0<D<1. Natural time T₀: Reciprocal of undamped natural angular frequency: T₀=1/ω₀. Obtain it from model parameters or measured oscillation period. Time since the step t: Observation time measured from the input step.
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, D=0.5 and T₀=1 s give y≈1.0746 at t=5 s: the output is still 7.46% above its final value.
Understand the result and units
The second-order model describes systems with two coupled storage elements whose output may overshoot the final value depending on damping. 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
PT2 element: damped step response: Such basic calculations support plausibility checks, early component selection and preparation of a complete verification.
Assumptions, limits and common mistakes
Linear time-invariant second-order element initially at rest before an ideal step and underdamped case 0<D<1; dead time, zeros, saturation and additional poles 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.