# For a standard second order transfer function, what is the equivalent time domain significance of $\zeta>0.707$?

In frequency domain $\zeta>0.707$ implies no resonant peak in frequency response. But I am unable to correlate what is the exact time domain effect of this? If $\zeta=0.707$ acts like an 'important damping value' in the frequency domain, shouldn't its importance be reflected in time domain as well. But in time domain $\zeta=1$ is where system behavior changes.

You are absolutely right that the damping value $\zeta=1/\sqrt{2}$ is important for the frequency domain behavior of a second-order system. However, as you've also noted, there is no one-to-one correspondence in the time domain. Generally we distinguish over-damped ($\zeta>1$), under-damped ($\zeta<1$), and critically damped ($\zeta=1$) systems, and these categories have distinct time-domain behaviors. Over-damped and critically damped systems only have real poles, whereas under-damped systems have complex poles, which results in oscillations in the impulse and step responses.
Note that only under-damped systems are useful for implementing frequency-selective filters, because the pole angles are related to the cut-off frequency, i.e. you need complex conjugate poles. So for frequency-selective filters you only consider under-damped systems, and then you can find different categories of under-damped systems according to their behavior in the frequency domain. A second-order Butterworth low pass filter has $\zeta=1/\sqrt{2}$, i.e. it is maximally flat at $\omega=0$, a Bessel low pass is even smoother and has larger damping ($\zeta=\sqrt{3}/2$), and a Chebyshev low pass filter is less damped than both of these filters. All of these filters satisfy $\zeta<1$, i.e. they are under-damped.
• i'm just dotting the t's and crossing the i's. yes, filter selectivity increases with $Q$ and and any $Q>\frac{1}{2}$ will correspond to complex conjugate poles. but you can have a sloppy LPF with two real poles and there is a (sloppy) cut-off frequency related to them. – robert bristow-johnson Dec 10 '14 at 22:42