Q factor for R+LC tank circuit: which formula is correct?

[strauser]

TL;DR

Derive Q of R in parallel with tank circuit

I've been experimenting with an LC tank circuit in series with a resistance R, and I've noted that the Q seems to increase with R. I've tried to derive this result via phasor analysis, but I'm not sure if my expression is correct.

To make things clear, I'm talking about the circuit with impedance $Z=R+jX_L || X_C=R+j(\dfrac{\omega L}{1-\omega^2 LC})$

The only thing I've found via google is this:

https://electronics.stackexchange.com/questions/108788/voltage-output-from-a-tank-circuit

where the first answer suggests that $Q=R\sqrt{\dfrac{C}{L}}$ which at least agrees with my measured results. I've found however that $Q=R\sqrt{\dfrac{C}{L+4R^2C}}$

So which result, if either, is right? I note that mine approximates the quoted result if $L \gg 4R^2C$.

 

[LvW]

At first you have to convince yourself about the DEFINITION of the quantity you call "Q".
What is your definition?

 

[strauser]

At first you have to convince yourself about the DEFINITION of the quantity you call "Q".
What is your definition?

Well, nothing too unusual: $Q=\dfrac{\omega_\text{resonance}}{B_\text{3dB}}$ where $B_\text{3dB}$ is the 3dB bandwidth.

I find your question a little odd though; aren't all definitions equivalent up to a factor of $2\pi$. It's been years since I looked at this stuff in detail, but I don't recall any defn that would give a substantially different result. Maybe I forget?

 

[LvW]

Well, may be that my question appears to you "a little odd" - nevertheless, would you mind to tell us HOW you have found the expression for Q you have mentioned?
According to system theory, the Q value of a frequency-dependent netork is defined using the pole position in the compex s-plane - and only for some special cases this value is identical to the ratio "resonant frequency/bandwidth".
More than that, for R approaching infinity your Q expression would be close to Q=0.5.
This is not correct. In contrast, Q must be very large...

 

[eq1]

Maybe I forget?

In addition to what LvW said it’s critical to precisely define what is the input and output variable. For example are you looking at voltage or current? Across or through what? Etc.

 

[strauser]

Well, may be that my question appears to you "a little odd" - nevertheless, would you mind to tell us HOW you have found the expression for Q you have mentioned?
According to system theory, the Q value of a frequency-dependent netork is defined using the pole position in the compex s-plane - and only for some special cases this value is identical to the ratio "resonant frequency/bandwidth".
More than that, for R approaching infinity your Q expression would be close to Q=0.5.
This is not correct. In contrast, Q must be very large...

I don't have time to reply to this fully today, but your final observation suggests that I've indeed effed up somewhere.

I'll put up the details tomorrow.

 

[tech99]

Summary: Derive Q of R in parallel with tank circuit

I've been experimenting with an LC tank circuit in series with a resistance R, and I've noted that the Q seems to increase with R. I've tried to derive this result via phasor analysis, but I'm not sure if my expression is correct.

To make things clear, I'm talking about the circuit with impedance $Z=R+jX_L || X_C=R+j(\dfrac{\omega L}{1-\omega^2 LC})$

The only thing I've found via google is this:

https://electronics.stackexchange.com/questions/108788/voltage-output-from-a-tank-circuit
where the first answer suggests that $Q=R\sqrt{\dfrac{C}{L}}$ which at least agrees with my measured results. I've found however that $Q=R\sqrt{\dfrac{C}{L+4R^2C}}$

So which result, if either, is right? I note that mine approximates the quoted result if $L \gg 4R^2C$.

If you have L, C and R all in parallel, the Q is approx R/X, where X is the reactance of either L or C, which are the same when Q is greater than about 2. For the case when R, L and C are in series, Q= X/R exactly. The topic is actually a bit intricate for parallel circuits, because max voltage and zero phase do not quite coincide for low Q values. It is covered in the very old book Radio Engineering, by Terman.