Worked problem

First-Order Sensor Response: Time Constant and Settling

Calculate the indicated temperature after a step change for a sensor with a 12-second time constant, and find the time to reach 95% of the final change.

Controls & Instrumentation · Last reviewed

Given

A temperature sensor behaves as a first-order system with a time constant τ = 12 s. It sits at 20 °C and is suddenly placed in a process stream at 80 °C.

Find

The indicated temperature after 10 s, the time to reach 95% of the final change, and the values at one, three and four time constants.

Solution

T(t) = T(final) − (T(final) − T(start)) × e^(−t ÷ τ)
  1. The step is 80 − 20 = 60 °C.
  2. At t = 10 s: T = 80 − 60 × e^(−10 ÷ 12) = 80 − 60 × 0.4346 = 53.9 °C.
  3. 95% of the change means 60 × 0.95 = 57 °C of rise, so T = 77 °C. Solve e^(−t ÷ τ) = 0.05: t = −τ × ln(0.05) = 12 × 2.996 = 35.9 s.
  4. At t = τ = 12 s: the sensor has completed 63.2% of the change, 57.9 °C.

Answer

53.9 °C at 10 s | 95% of the change (77 °C) after 35.9 s, about 3τ
TimeFraction of the changeIndicated temperature
1τ = 12 s63.2%57.9 °C
2τ = 24 s86.5%71.9 °C
3τ = 36 s95.0%77.0 °C
4τ = 48 s98.2%78.9 °C

Why it matters

  • A sensor never reaches the final value in finite time. The usual practice is to quote the time to reach 95% (3τ) or 98% (4τ) of the change.
  • A control loop sees the lagging reading, not the true temperature. A slow sensor in a fast loop adds phase lag and can make the loop unstable.
  • The time constant is not a property of the sensor alone. A thermowell, the fluid velocity and the mounting all increase it.

Common mistakes

  • Treating the time constant as the time to reach the final value. It is the time to reach 63.2%.
  • Using the time constant measured in water for a sensor in still air, where it can be many times longer.
  • Taking a reading in the middle of a transient as the true process value.
  • Forgetting that alarm and trip delays add to the sensor delay.

Related tools and labs

Calculators, labs and simulators are part of the All-Access Pass.

More worked problems

Questions

Is every sensor first-order?

It is a good model for many temperature sensors. Sensors with a thermowell or a protective sheath often behave as higher-order systems, with a slower start.

How can I speed up the response?

Use a smaller, thinner sensor, better thermal contact, a faster fluid velocity, or a fast-response design without a thick thermowell.

How is the time constant measured?

Apply a step change to the sensor and record the time to reach 63.2% of the change.

Educational material. Examples use simplified, hypothetical inputs and are not a design, a construction document or professional engineering advice. Confirm requirements with the codes and standards adopted where the work is built and with a licensed engineer.

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