Critically Damped System

Topics: Feedback Control Systems

A critically damped system corresponds to $\zeta=1$ in standard form. It is the point of transition between the underdamped case that experiences oscillations and the overdamped case.

Step Response

$$y(t) = 1 - (\omega_n\,t+1) \; e^{-\omega_n \, t}$$

It is clear from the step response that the critically damped system does not have oscillations nor overshoot.

Poles

The transfer function of a critically damped system with an undamped natural frequency $\omega_n$ is

$$T(s) = \bigg[ \frac{\omega_n}{s + \omega_n} \bigg]^2$$

So, the system has 2 real poles at $s=-\omega_n$.