# Q factor for an LRC series circuit

Trying to prove to myself that the Q factor for an LRC series circuit is $\sqrt{L/C}/R$, but I am struggling with it...

My reasoning runs that the half power upper frequency limit is (ignoring the $2\pi$ factor) R/L, so half the frequency delta is $R/L-1/(\sqrt{LC})$ and the full bandwidth (assuming symmetry) at half-power is $2(R/L-1/\sqrt{LC})$

Then given $Q=f_{res}/BW$, we have $Q = (1/\sqrt{LC})/(2(R/L - 1/\sqrt{LC}))$,

which can be simplified to $L/(2(R\sqrt{LC} - L))$ which is similar to the correct answer, but not correct, with an errant factor of 2 and a stray -L kicking around.

Where has my reasoning gone wrong?

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The resonant frequency is the geometric mean of the upper and lower half power frequencies, not the arithmetic mean you seem to be using here.

$\omega_0 = \sqrt{\omega_l \omega_h} \ne \dfrac{\omega_l + \omega_h}{2}$

That's probably all you need but here's some additional information just in case.

[UPDATE: upon further reflection, there's something else you need. The relationship between Q and BW you are using is a high-Q approximation. Deriving the genuine Q from this approximation requires a bit more than I've given below.]

For a series RLC, we have:

$Z = \dfrac{1 + j \omega RC - \omega^2LC}{j \omega C}$

The numerator is of the form:

$1 + j\frac{1}{Q} \frac{\omega}{\omega_0} - (\frac{\omega}{\omega_0})^2$

Comparing these two forms, see that:

$\omega_0 = \frac{1}{\sqrt{LC}}$

$Q = \dfrac{\sqrt{\frac{L}{C}}}{R}$

How to see that the resonant frequency is the geometric mean?

Consider the product of two first order functions:

$(1 + j\dfrac{\omega}{\omega_l})(1 + j\dfrac{\omega}{\omega_h}) = 1 + j \omega (\dfrac{1}{\omega_l} + \dfrac{1}{\omega_h}) - \dfrac{\omega^2}{\omega_l \omega_h}$

From this, we deduce:

$\omega_0 = \sqrt{\omega_l \omega_h}$

$Q \omega_0 = \omega_l || \omega_h =\dfrac{1}{\dfrac{1}{\omega_l} + \dfrac{1}{\omega_h}} \rightarrow Q = \dfrac{\omega_0}{\omega_l + \omega_h}$

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I thought your point about the geometric mean would make it clear so i accepted the answer, but I am still at a loss. I don't understand what the Z in the above answer represents - or rather how you have derived it. If Z = $\sqrt{R^2 + (X_L - X_C)^2}$ - how does that relate to what you have here? –  adrianmcmenamin Jul 1 '12 at 21:55
Right, I get it now! Many thanks. –  adrianmcmenamin Jul 1 '12 at 23:38
Great! I knew you'd see it! –  Alfred Centauri Jul 1 '12 at 23:47