How to Prove Equivalence of Quality Factor Expressions for High Q Resonators?

[Xoxo]

Hello all,

It's known that the Quality factor Q is defined as :

Q = 2*pi*(Energy stored at resonance) / (Energy loss per cycle)​

and for high Q resonators, It's known that Q can be also given by :

Q = Resonance frequency / 3dB Bandwidth​

My question is, How can i prove that ? that both expressions of Q are equivalent if Q is large enough ?
I can prove it for many circuits, but i want a general rigorous proof of that.

Thank you in advance

 

[Baluncore]

Welcome to PF.
Q is a quality factor and not a precision measure.
There are situations where the different definitions converge.

See; https://en.wikipedia.org/wiki/Q_factor#Physical_interpretation
“ The factors Q, damping ratio ζ, attenuation rate α, and exponential time constant τ are related such that: [12] ”
Reference [12]. Siebert, William McC. Circuits, Signals, and Systems. MIT Press.

 

[NascentOxygen]

How can i prove that ? that both expressions of Q are equivalent if Q is large enough ?

I believe I have shown this holds for a second-order system, with series R- L-C.

Its TF involves the term $\dfrac 1 {s^2\ +\ \frac {\omega_o} Q s\ +\ {\omega_o}^2}$[/color]

 

[Xoxo]

I believe I have shown this holds for a second-order system, with series R- L-C.

Its TF involves the term $\dfrac 1 {s^2\ +\ \frac {\omega_o} Q s\ +\ {\omega_o}^2}$

Where did you show this holds for 2nd order RLC circuits ?

 

[NascentOxygen]

Where did you show this holds for 2nd order RLC circuits ?

Sorry. I meant that I believe it can be shown to be true.

I'd start like this:
Apply a voltage to the circuit at the resonant frequency and see what current flows. All losses occur in the resistance.

 

[Xoxo]

Sorry. I meant that I believe it can be shown to be true.

I'd start like this:
Apply a voltage to the circuit at the resonant frequency and see what current flows. All losses occur in the resistance.

I know, i can prove that, my question is that i want a general proof assuming the circuit is a black box

 

[NascentOxygen]

I know, i can prove that, my question is that i want a general proof assuming the circuit is a black box

A black box containing any general n^th order linear system, do you mean?

 

[Xoxo]

A black box containing any general n^th order linear system, do you mean?

yes

 

[f95toli]

yes

The most "general" way of doing this would be to start with say the ABCD matix for a two-port systems and then derive an expression for S21 near resonance; this would then give you the results you want.
However, in order to do so you STILL need to make some assumptions about the circuit; and these assumptions basically amount to assuming that the circuit can be described as an effective series- or parallell LCR-resonant circuit (near resonance). Note that this does NOT mean that the resonator is made up of discrete component; the same procedure works for e.g. cavity resonators or lambda/2 and lambda/4 resonators.

Moreover, the 3-dB "rule" for the Q value of a resonance is not a general result; it ONLY works for circuits that can be described as a 2nd order RCL circuit (which fortunately includes most systems of interest). It does not work for systems which e.g. include non-linear elements (which skews the resonance) or systems where the resonance is heavily distorted for some other reason (because it e.g. is coupling to other spurious modes in the circuit), For any real circuit it is an approximation at best. I never use it for any "serious" measurements.

 

[Xoxo]

The most "general" way of doing this would be to start with say the ABCD matix for a two-port systems and then derive an expression for S21 near resonance; this would then give you the results you want.
However, in order to do so you STILL need to make some assumptions about the circuit; and these assumptions basically amount to assuming that the circuit can be described as an effective series- or parallell LCR-resonant circuit (near resonance). Note that this does NOT mean that the resonator is made up of discrete component; the same procedure works for e.g. cavity resonators or lambda/2 and lambda/4 resonators.

Moreover, the 3-dB "rule" for the Q value of a resonance is not a general result; it ONLY works for circuits that can be described as a 2nd order RCL circuit (which fortunately includes most systems of interest). It does not work for systems which e.g. include non-linear elements (which skews the resonance) or systems where the resonance is heavily distorted for some other reason (because it e.g. is coupling to other spurious modes in the circuit), For any real circuit it is an approximation at best. I never use it for any "serious" measurements.

You're my hero :D
That's what i want, thank you