About the definition of resonance frequency

[DaveE]

If this helps, it's a plot of a quadratic LPF pole locations as Q is varied. At Q=0,the poles are real at 0 and ∞. As Q is increased they move together and meet when Q=0.5 on the real axis at -ω_o. Then they split into a conjugate pair each with a magnitude of ω_o. At Q→∞ they get to the imaginary axis.

 

[LvW]

#30 is just words.

Yes - in many cases, definitions consists just of words.

#19 is the same as my definition, Q>0.5

No - I don`t think so.
In the definition according to #19 the circuit at resonance is pure resistive (no phase shift between input voltage and input current resp. output and input voltages).
But even lowpass functions with Bessel (Q=0.577) or Butterworth characteristics (Q=0.7071) exhibit a 90deg phase shift at the pole frequency wp.
I think the concept of pole frequency is a very important and helpful tool for analyzing/comparing the variuous different filter responses as well as defining the corresponding parameter Qp (visual interpretation of Qp in the s-plane).
But I think there is no good reason to link the defintion of pole frequencies with the definition of resonance. (I have not found a single reference for such a concept.)
In this conext, I like to mention that an universally accepted and logical definition is helpful and necessary to enable a technical/scientific communication without misunderstandings.

 

[DaveE]

But even lowpass functions with Bessel (Q=0.577) or Butterworth characteristics (Q=0.7071) exhibit a 90deg phase shift at the pole frequency wp.

But these are both just a simple quadratic pole. Of course they have resistive input impedance at ω_o. For higher order filters I may informally say the circuit is resonant, but I would really mean it contains one or more resonant quadratics in the transfer function. In that case there many be additional phase shift from other poles/zeros which makes the definition difficult.

But I think there is no good reason to link the defintion of pole frequencies with the definition of resonance. (I have not found a single reference for such a concept.)

Yes, I agree. But I also don't think anyone has done that here. Maybe I'm misunderstanding, but I thought the questions were which values of Q are resonant and does a resonant circuit have to have 0^o phase at resonance. I've already given my opinion on these. Short version: Q>0.5, and no it can be -90^o, 0^o, or +90^o, depending on what your I/O definitions are, as in my state variable filter example.

In this conext, I like to mention that an universally accepted and logical definition is helpful and necessary to enable a technical/scientific communication without misunderstandings.

OK, go for it. Math will be useful in this case, IMO. Universally accepted will be hard work, primarily because of the confusion between ω_o and ω_r.

 

[DaveE]

OK, I think I've beaten this one to near death. I'll leave with my definition of resonance and some reading if y'all still care. Nope, don't ask for a reference, I've just created it from memory (although, it is very much in line with the book I'll mention below).

An LTI, SISO, network can be represented with a transfer function $H(s) = \frac{N(s)}{D(s)}$ where $N(s)$ and $D(s)$ are polynomials in $s$. For engineering purposes, we can nearly always factor each of these polynomials into a product of 1st order, and/or quadratic terms¹. If either approximately factored polynomial $N(s)$ or $D(s)$ contains a quadratic factor $[1 + (\frac{1}{Q}) (\frac{s}{\omega_o}) + (\frac{s}{\omega_o})^2]$ and if Q>0.5, then that network contains a resonance at the frequency $\omega_o$ .

1- It is possible that there are very unusual cases where this factoring approximation isn't accurate enough. In that case you will need to use higher order factors (cubics or greater) and this definition doesn't apply.

This factoring is explained well in the text "Fundamentals of Power Electronics" 3rd edition by Robert W. Erickson and Dragan Maksimovic´. This is an excellent text and faithful exposition of the work of R. D. Middlebrook at Caltech 40 years ago, or so, when we were students there. I can't give a link here, but it is findable on the web. In particular, section 8.1.6 describes the quadratic response and is essentially the same as I have proposed in this thread.

 

[LvW]

For engineering purposes, we can nearly always factor each of these polynomials into a product of 1st order, and/or quadratic terms¹. If either approximately factored polynomial $N(s)$ or $D(s)$ contains a quadratic factor $[1 + (\frac{1}{Q}) (\frac{s}{\omega_o}) + (\frac{s}{\omega_o})^2]$ and if Q>0.5, then that network contains a resonance at the frequency $\omega_o$ .

DaveE, with all respect - do you agree with me that the last sentence above contains just an assertion without any explanation? If that is your own definition of the term “resonance” - OK, no problem.
But I suspect that the questioner (post#1) prefers an answer based on a generally accepted definition of “resonance” and “resonant frequency”.
That's all I was trying to do in this thread.

 

[DaveE]