2
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ "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."... not that i disagree with Matt's judgment (i don't), i just don't think that the statement is strictly or objectively true. you can have a low-pass filter that is over-damped or critically-damped (so it has only real poles) and have a meaningful cut-off frequency. it's frequency selective to an extent, but is sorta sloppy about it. $\endgroup$ – robert bristow-johnson Dec 10 '14 at 19:48
  • $\begingroup$ @robertbristow-johnson: Yes, but the filters with a relatively quick transition from passband to stopband all have complex conjugate poles. So I meant all practically useful filters for which the frequency-selective property is of importance. $\endgroup$ – Matt L. Dec 10 '14 at 21:39
  • 1
    $\begingroup$ 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. $\endgroup$ – robert bristow-johnson Dec 10 '14 at 22:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.