Calculation
y(t)=0 for t<Td; y(t)=K·Δu for t≥Td
The output remains unchanged until dead time expires; the idealised step then appears without additional settling.
Zacher/Reuter 2024, Abschnitt 3.8: Totzeitstrecke
Dead-time element: delayed step response
The output remains unchanged until dead time expires; the idealised step then appears without additional settling.
MINTSI
02
Result
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Calculated step response
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Understand the inputs
- Output change y(t) — Output change visible at the selected time. It is zero before dead time and equals the amplified input-step height afterwards.
- Static gain K — Ratio of transmitted output change to input step after dead time has elapsed.
- Input step Δu — Change at the plant input at time zero, i.e. new minus old value.
- Dead time Td — Time between a detectable input change and the start of output response. Obtain it from step-test timestamps or transport distance and velocity.
- Time since the step t — Observation time from the input change. At t=Td the step has just taken effect in the ideal model.
Example
K=2, Δu=2 and Td=3 s give y=0 at t=2 s, but y=4 at t=5 s.
Assumptions and limits
Ideal pure dead time with unchanged signal shape; real plant dynamics, dispersion, sampling, saturation and a non-zero initial value are excluded.
Understand Dead-time element: delayed step response
Determine when and at what level an input step appears at the output after a pure transport or propagation delay.
What does this quantity describe?
The output remains unchanged until dead time expires; the idealised step then appears without additional settling. 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(t)=0 for t<Td; y(t)=K·Δu for t≥Td
| Symbol / input | Meaning |
|---|---|
| Output change y(t) | Output change visible at the selected time. It is zero before dead time and equals the amplified input-step height afterwards. |
| Static gain K | Ratio of transmitted output change to input step after dead time has elapsed. |
| Input step Δu | Change at the plant input at time zero, i.e. new minus old value. |
| Dead time Td | Time between a detectable input change and the start of output response. Obtain it from step-test timestamps or transport distance and velocity. |
| Time since the step t | Observation time from the input change. At t=Td the step has just taken effect in the ideal model. |
Choose the inputs correctly
Output change y(t): Output change visible at the selected time. It is zero before dead time and equals the amplified input-step height afterwards. Static gain K: Ratio of transmitted output change to input step after dead time has elapsed. Input step Δu: Change at the plant input at time zero, i.e. new minus old value. Dead time Td: Time between a detectable input change and the start of output response. Obtain it from step-test timestamps or transport distance and velocity. Time since the step t: Observation time from the input change. At t=Td the step has just taken effect in the ideal model.
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=2, Δu=2 and Td=3 s give y=0 at t=2 s, but y=4 at t=5 s.
Understand the result and units
The output remains unchanged until dead time expires; the idealised step then appears without additional settling. 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
Dead-time element: delayed step response: Such basic calculations support plausibility checks, early component selection and preparation of a complete verification.
Assumptions, limits and common mistakes
Ideal pure dead time with unchanged signal shape; real plant dynamics, dispersion, sampling, saturation and a non-zero initial value 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.