Critical step response of a galvanometer

For $\ddot{\theta }+2{\omega }_0\dot{\theta }+{\omega }_0^2\theta =F$ switched on from rest, $\theta (t)=(F/{\omega }_0^2)[1-(1+{\omega }_{0}t)e^{-{\omega }_{0}t}]$. critical damping is the fastest nonoscillatory step response at fixed natural frequency within the ordinary second-order model.

With $e^{i\omega t}$ phasors, critical damping gives $\Theta (\omega )=F_0/({\omega }_0+i\omega {)}^2$, amplitude $F_0/({\omega }_0^2+{\omega }^2)$ and phase $-2\arctan (\omega /{\omega }_0)$. The response strongly suppresses alternating currents well above the mechanical frequency.