Notion of Q factor for abstract Two-Port Network

Consider an abstract two-port network/quadripole electrical network with input voltage $$\U_{in}\$$, output voltage $$\U_{out}\$$, current at input port $$\I_{1}\$$ and current at output port $$\I_{2}\$$: My question is rather general: Is there a mathematical criterion how to decide if it is possible to associate a "meaningful" notion of the Q factor to this network?

What I mean by "mathematical criterion"? Well, if we think about (driven) harmonic oscillator from classical mechanics then in mechanical case one could say that one can associate a Q factor to the mechanical system if and only if the system can be modeled by differential equation of the form

$$\frac{d^2x}{dt^2} + 2 a \omega_0 \frac{dx}{dt}+\omega_0^2x= F_{ext}$$

where $$\a, \omega_0 \$$ are a priori constants ($$\omega_0\$$ will be recognized as frequency of the system) and $$\ F_{ext} \$$ external ("driving") force. Then one sets $$\ Q:= \frac{1}{2a} \$$ and interpret physically Q of this oscillating system as the quotient of the enegy stored and energy lost per cycle.

If we go back to electronics then usually one studies a given network instead by modeling it as a differential equation, by analyzing the so called transfer function $$\H(s)\$$ of this network (essentially this is an "equivalent" approach to the modeling via differential equation; indeed the Laplace transformation mediate between them).

And indeed taking a closer look to networks like the high/low/bandpass filters (which all are elemetary examples for networks, which "have" a well defined Q-factor), the structure of their transfer functions (see eg here: https://en.wikipedia.org/wiki/Q_factor#Physical_interpretation ) includes rather naturally notion of the Q factor.

Conjecture/Question 1: If we have a general two-port network electrical network as above and assume we know it's transfer function, are we able only from careful inspectation of it's shape to decide if this network allows a meaningful notion of of a Q factor?
A naive guess: Can it be said that a network allows a meaningful notion of a Q factor if and only if it's denominator is a quadratic polynomial with non zero linear factor?

Question 2: In light of my first question there arises quite naturally a point which confuses me. Recall how the transfer function of a system with input signal $$\ x(t) \$$ and output signal $$\ y(t) \$$ was definedabstract. Then the transfer function is by definition

$$H(s) := \frac{Y(s)}{X(s)}$$

where $$\ X(s) \$$ and $$\ Y(s) \$$ are the Laplace transforms of $$\ x(t) \$$ and output $$\ y(t) \$$. So seemingly the shape of the denominator of the transfer function $$\ H(s) \$$ depends strongly of the input signal.

At first glance that's not really surprising, but on the other hand if my conjecture in Question 1 is true, then it strongly depends if the network allows a meaningful notion of Q on which kind of input signal is given.
So in other words it may happen that for one input signal the Q of the network is meaningful defined, for another input signal not.

That's strange if we try to think about the the mechanical driven harmonic oscillator I introduced above. The analogon of the input signal there is the external driving force $$\F_{ext} \$$. But nevertheless there the notion of Q factor exist independing on which external driving force stimulates the system.

So it's strange in the sense that seemingly in case of electronic network it depends strongly if there if a notion of Q factor on "what kind of input signal" pass to the network, while in case of machanical system the notion of Q factor not depends on the external driving force, the pendant of the input signal.

Can this confusion/ thinking error of mine be coherently resolved?

• $Q$, in one definition (there isn't just one such) is a ratio related to the energy lost per radian, not full cycle. There are $2\pi$ radians in a full cycle. But not all systems are 2nd order. And $Q$ has a clear meaning (the way you write it, anyway) only in the case of 2nd order. So that comports with one of your questions, well. Yet I also see the confusion. But it sounds like only a two-way dialog will resolve it. An answer would need to be long, otherwise. (The 'denominator' you mentioned is also called the characteristic equation, by the way.)
– jonk
Jan 6 at 5:00
• @jonk: I probably see your point. So the point you want to emphasize is that the way I introduced the definition of $Q$ above, suggest almost tautologically that my $Q$ can be only defined for 2nd order systems, ie those having a quadratic poynomial as denominator/characteristic equation in the transfer function? So in light of definition of $Q$ I gave the Question 1 is rather self-responsive, because this definition of $Q$ was a priori established for 2nd order system? That's your point, right? Jan 7 at 14:59
• If yes, then I should admit that unfortunately, I have awkwardly expressed my real concern, because originally I wanted to find a criterion how to decide if a given system allows a meaningful notion of $Q$ factor, based on the inspection of structure of it's transfer function. Of course, if a priori $Q$ as done by mine only defined for 2nd order networks, then I'm tautologically just seeking for 2nd order networks, the dog bites its tail :) Question 1 solves itself. But originally I intended to analyze it for a broader class of networks, not only the of 2nd order. Jan 7 at 15:00
• In order to express my original concern more precise to avoid such misunderstandings, I think the key point is to change the definition of $Q$ factor to that one which is admissible for bigger class of systems, not only the 2nd order systems. Yes, as you mentioned there are more than one definitions of $Q$. The three I know are the stored energy per cycle definition (yes you right, I missed indeed the $\pi$ factor there),the bandwidth definition (as the reciprocial) and the damping definition (see en.wikipedia.org/wiki/Q_factor#Definition). Jan 7 at 15:00
• Wow! You wrote almost as much as in your question, above. The concept of $Q$ has a long history and not everyone agrees upon the same definition. There are good reasons for that fact, too. But I don't want to write as much as you did, trying to pass along what's in my head about it. Suffice it that within the context of electronics, there is a well-defined meaning which applies only to 2nd order. 1st order cannot possibly have such a concept and everything 3rd order and higher cannot be entirely made to support the idea, either, though in specific cases it can be approximated, with caveats.
– jonk
Jan 7 at 21:56

• Yes, I should have added that the networks I consider are always assumed to be linear, thanks. But nevertheless I'm still a bit confused about the statemet you wrote that the transfer function associated with the differential equation does NOT depend upon any particular choice of "input" function. Could you elaborate it (it seems indeed to be to core of my confusion)? I thought that the input function is $H(s) := \frac{Y(s)}{X(s)}$ and since $X(s)$ is as Jan 6 at 0:57
• Laplace trafo of input signal, clearly it depends on it. But then it seems to be logical that then $H(s)$ also depends on the imput signal, since by definition it depends on $X(s)$. What is the error in my considerations? Jan 6 at 0:57
• Or is your point that the ratio of the both Laplace trafos (=transfer function) itself not depend which signal arives as input. So the output "kills" all special features of the input signal and $H(s)$ reflects after this "cancelation" of input by output really only the intrinsic (!) structure of the circuit? That's the point, right? Jan 6 at 1:06
• So essentially what you want to say is that if $x(t)$ and $x'(t)$ are any two different and arbitrary input signals (and $y(t)$ resp $y'(t)$ the associated output) then we have always $\frac{Y(s)}{X(s)} = \frac{Y'(s)}{X'(s)} (=H(s))$ (where$X(s), Y(s), X'(s), Y'(s)$ are Laplace trafos of $x(t), y(t), x'(t), y'(t)$? Jan 7 at 15:12