Sensor Step Response

Sensors cannot instantaneously respond to changes in the values that they measure. There is some delay, which may be very important to characterize.

First-order Response

For some sensors, the speed of sensor response to an input is quantified by a time constant $\tau$, similar to RL circuits. This is expressed as $10\%\to 90\%$ of the rise time, such that:

$$t_{10\%\to 90\%}=2.2\tau$$

If needed, a first-order factor may be included in the sensor’s transfer function to model this low-pass behaviour:

$$\frac{1}{(s\tau +1)}$$

Second-Order Response

Some sensors may be modeled as a second-order system if the step response exhibits overshoot or “ringing”, which occurs when a system is underdamped. Usually, adding damping (resistance, friction, rotational damper) will eliminate oscillations. Ringing can lead to instability if feedback is present. A second-order factor with appropriate natural frequency.

A second-order factor with appropriate natural frequency and damping factor, $\omega$and $\zeta$ , can be included in the transfer function.