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Module 2 · Lesson 3 · 13 min video · 30 min practice

PT1 step responses: the meaning of K and T

Connect gain and time constant to the full curve, then make the curve move yourself.

  • Read steady-state gain and time constant from a step response.
  • Predict how K and T reshape the response.
  • Verify the 63.2% time-constant rule analytically and numerically.
Unofficial TUM ACE alignment: ACE §2: PT1-Glied und Sprungantwort

Visual lesson storyboard ready · video recording next

8–15 minute visual explanation. The interactive experiment below is usable now.

Prediction

If K doubles and T halves, what changes in a unit-step response?

Sketch your prediction: mark the final value, the value at one time constant, and the initial slope.

One curve, two independent meanings

For the first-order model Ty˙+y=KuT\dot y+y=Ku, initially at rest, a unit step produces

y(t)=K(1−e−t/T),t≥0.y(t)=K\left(1-e^{-t/T}\right),\qquad t\ge 0.

The gain KK sets the final value. The time constant TT stretches the horizontal axis. At t=Tt=T, every PT1 response has completed the same fraction of its journey:

y(T)K=1−e−1≈0.632.\frac{y(T)}{K}=1-e^{-1}\approx 0.632.

Concept check

A unit-step response settles at 4.0. What output marks one time constant?

Reactive experiment: reshape the response

This plot is already present in the built HTML. When Python activates, both sliders update it without rerunning any cells manually.

Worked example

A heater has K=3 K/kWK=3\,\mathrm{K}/\mathrm{kW} and T=40 sT=40\,\mathrm{s}. A 2 kW2\,\mathrm{kW} step eventually raises the temperature by 6 K6\,\mathrm{K}. After 40 s40\,\mathrm{s}, the rise is 0.632⋅6=3.79 K0.632\cdot 6=3.79\,\mathrm{K}.

Derivation

Solve the homogeneous and forced parts

The homogeneous equation T ẏ + y = 0 gives yₕ = C exp(-t/T). A constant unit input has the particular solution yₚ = K. Applying y(0)=0 gives C=-K, so y=K(1-exp(-t/T)).

Unofficial TUM ACE exam prep

Use the tangent and 63% constructions carefully

State which construction you use and measure changes relative to the pre-step baseline. A delayed first-order response needs a separate delay parameter; forcing it into T biases the model.

Python exercise: implement the analytical response

Complete the function. All execution stays in your browser. Syntax and runtime errors appear below the editor and cannot break the surrounding lesson.

Summary

  • KK sets the response scale; TT sets the time scale.
  • At t=Tt=T, a PT1 response reaches 63.2% of its total change.
  • The analytical formula, numerical plot, and physical interpretation must agree.

Retrieve it later

Derive the 63.2% rule and explain what would happen to the curve if K doubles while T halves.

Answer from memory tomorrow, then return to check your reasoning.