1. Mar 29, 2017

### 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.

2. Mar 29, 2017

### 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.

3. Mar 29, 2017

### Staff: Mentor

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

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

4. Mar 30, 2017

### Xoxo

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

5. Mar 30, 2017

### Staff: Mentor

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.

6. Mar 30, 2017

### Xoxo

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

7. Mar 30, 2017

### Staff: Mentor

A black box containing any general nth order linear system, do you mean?

8. Mar 31, 2017

### Xoxo

yes

9. Mar 31, 2017

### f95toli

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.

10. Mar 31, 2017

### Xoxo

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