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 the time domain as well. But in time domain Zeta= 1 is where system behavior changes.
The quantity you call "zeta" is defined as Zeta=1/2Qp with Qp being the pole quality factor. A 2nd order function with Qp=0.7071 (Zeta=1/1.4142=0.7071) will exhibit no peaking at the pole frequency (Butterworth design) - however, there will be a small overshoot in the step response (time domain). This overshoot will disappear only if Qp1). Does this answer your question?
If your zeta is between 0 and 1, then your time response will result in an overshoot.
The quantity you call "zeta" is defined as Zeta=1/2Qp with Qp being the pole quality factor. A 2nd order function with Qp=0.7071 (Zeta=1/1.4142=0.7071) will exhibit no peaking at the pole frequency (Butterworth design) - however, there will be a small overshoot in the step response (time domain). This overshoot will disappear only if Qp1). Does this answer your question?
I think I misunderstood the question as I thought that zeta was the damping ratio.
I think, there was no misunderstanding on your side. Indeed, "zeta" is the damping ratio - and therefore: Step response with overshoot for zeta
Then my opinion is that if we will be talking about the time domain, a zeta value of 0.707 is not special at all. Similarly with other values of zeta between 0 to 1, there will still be an overshoot.
Yes - I agree to this, as far as the time domain is concerned.
However, in the frequency domain the value of zeta=0.7071 is something special because it is equivalent to a pole quality factor of 0.7071 (Butterworth response) with a maximum flat magnitude response.
I think you have compare the analytical solutions for the 2nd- order system in the frequency and time domains. In the frequency domain , you have a term sqrt( 1-2z**2) , z = damping ratio.
For z =z0=0.707 , you have the resonance. In the time domain you have instead the term sqrt (z**2-1) So , the modes are real ( no oscillatory solution) for z greater than or equal to one ( overdasmpled or critically damped solution). For values below one the modes are complex and the solution is oscillatory and decaying except for z=0 for which the solution is ( non-deacaying) pure , then purely oscillatory ( harrmonic oscillation).
You can check for reading related simple ideas:
http://en.wikipedia.org/wiki/Damping_ratio
https://www.physicsforums.com/threads/resonance-frequency-in-second-order-system.626088/
It is a very astute and fine observation. The following wiki gives a very good explanation.
http://en.wikibooks.org/wiki/Signals_and_Systems/Second_Order_Transfer_Function
For a quick reference the extract from above is quote"The amplitude of the poles gives the corner frequency of the filter. The corner frequency is defined as the abscissa of the point where the horizontal and the -40 [dB/decade] lines meet in the log-log magnitude response plot. Note that this is not necessarily the -3 [dB] attenuation frequency of the filter.The ratio between the real part of the poles and the corner frequency is proportional to the damping, or inversely proportional to the quality factor of the system.The closer the poles are to the imaginary axis, the more a resonance will appear at a frequency smaller but close to the corner frequency of the system." unquote
Please see also http://en.wikipedia.org/wiki/Cutoff_frequency
it is clear that corner frequency will not manifest in the time domain simulation in an obvious manner as you have observed.
Cheers
"The corner frequency is defined as the abscissa of the point where the horizontal and the -40 [dB/decade] lines meet in the log-log magnitude response plot. Note that this is not necessarily the -3 [dB] attenuation frequency of the filter".
I am not too happy with this "definition". This is because this "definition" neglects the fact that the width of the passband, normally, is defined by the user. In particular, for 2nd order Chebyshev responses (and all other possible responses, except Butterworth) this "definition" does not apply.
Zeta represents damping ratio.And other parameters are natural frequency (wn) and steady state gain(K).
So keeeping (wn) and (K) constant,if you find the poles of second order transfer function, you would see that:
1. for zeta = 0,poles lie only on imaginary axis +j and -j --> that means in time domain you would have pure oscillations.
2. for zeta >=1, there are only real poles --> that means in time domain we have exponential decay
3. for 0
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