Talk:Q factor

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[edit] FWHM vs 3dB

The article states:

the bandwidth is defined as the 3 dB change in level besides the center frequency.
The definition of the bandwidth BW as the "full width at half maximum" or FWHM is wrong.

Unless the definition depends on the slight difference between -3dB (1/1.995) and half maximum (1/2), this would seem to be the same thing. Anyone know for sure, here? -- DrBob 17:57, 27 Sep 2004 (UTC)

Strictly, the FWHM is full width at half-max, that is half power. In dB this is -3.01 dB, close enough to 3 dB down to not worry about it.
The definition "In optics..." appears strange to me. In mechanics the Q factor can be shown to be equal to 2 * Pi times the energy stored in the oscillator divided by the energy dissipated per cycle. I suspect this is the correct definition, applicable to optics, mechanics, and any other oscillating system.
24.245.15.183
I suspect you're absolutely correct about the definition applying to all oscillating systems. "Q-switching", though, is a very real phenomenon and used to great advantage in pulsed laser systems.
By the way, you can easily sign your "talk" posts by appending four tildes (~~~~) to the posting. When you "Save changes", this will be replaced by your username in a handy linked form and a timestamp of your edit.
Atlant 14:27, 10 August 2005 (UTC)


[edit] Q the cycles for energy to go to zero?

Alison Chaiken 00:00, 23 September 2005 (UTC): I've always thought of the quality factor as the number of cycles that it takes for energy to be dissipated from the system. Thus a Q of 1000 means that the excited oscillation will ring down to zero in 1000 cycles. I would think this article should mention this insight. I would add it except that I can remember if Q is the number of cycles to ring down to 1/2 the original energy or what exactly.

No, that's wrong. It never decays to zero. The article states correctly that it's the number of cycles required to decay to 1/535 of its original energy.--24.52.254.62 01:25, 21 October 2006 (UTC)

[edit] Alternative to 1/535 definition

I'd like to suggest a change to the definition of Q as the number of cycles for the response to decay to 1/535 of the original amplitude. Although this is certainly true, and used in the Crowell book, I've never seen it used elsewhere. The main problem is that as a definition for actually measuring Q (which is fun and instructive for students to do with a pendulum), it is practically useless. If you actually try to count the cycles, by the time the response decays to 1/535 it is so flat that the accuracy is lost. It also doesn't offer much insight into the mathematics.

The usual way of defining Q in engineering texts is that it is 2 \pi \, times the number of cycles for the response to decay to 1/e = 0.368\, which is one time constant of the exponential decay envelope. Although at first sight this looks more complicated, it shows that the Q is just the ratio of the exponential and sinusoidal time constants of the response: if the response is: Ae^{\alpha t}cos{(\omega t + \phi)}\,, then Q = 2\pi f/\alpha = \omega/\alpha\,. And it is also a practical definition for measuring Q, since at 0.368 the response is still sloped enough to determine accurately when the amplitude falls below it.

I suggest changing the definition of Q to 2\pi\, times the number of cycles for the response to decay to 1/e = 0.368\, of the original amplitude. --Chetvorno 21:02, 21 August 2007 (UTC)

I calculated Q = f_0/B\,, the definition of Q\, over the bandwidth and your formula Q = 2 \pi f / \alpha\, for a signal like Ae^{\alpha t}cos{(\omega t + \phi)}\, and find out that there is always missing a factor of 2 between these formula. It has to be Q = \frac{ 2 \pi f} {2\alpha }\, = \frac{\omega}{2 \alpha} like formula E.7 on this webpage http://ccrma.stanford.edu/~jos/filters/Quality_Factor_Q.html 77.10.132.145 (talk) 11:01, 5 December 2007 (UTC)

[edit] Reference for the Q-value equation in mechanical systems?

Where does the equation Q = \frac{\sqrt{M K}}{R} on mechanical systems derive from? I can't find a source and it's not cited. 130.233.189.53 14:07, 26 October 2007 (UTC)


Somehow it remained without attention until now, that the article suggests "In a parallel RLC circuit, Q is equal to the reciprocal of the above expression.", referring to the correct formula for Q in a RLC circuit as "the above expression". Which means that a parallel RLC circuit would have a Q-factor that grows with increasing R. Which, in turn, is an obvious nonsense. —Preceding unsigned comment added by 83.79.25.246 (talk) 19:58, 20 November 2007 (UTC)

Ooops, sorry. disregard the above. Of course, a parallel RLC circuit would prefer a higher R, ideally - no R at all (means open circuit istead of R), for an endless oscillation. —Preceding unsigned comment added by 83.78.43.62 (talk) 20:59, 20 November 2007 (UTC)

[edit] Relationship to damping ratio

I'm not sure whether the equations relating Q to ζ are marked "citation needed" because there is some doubt about their accuracy; or whether it's just there because it's good practice to reference these. When I first looked at them I was doubtful about them, but I think that was just because they were in a slightly unfamiliar form. I can't supply a citation as none of the texts I have to hand use the damping ratio as defined here. However, I can show that it follows straightforwardly from the definitions of Q and ζ given in Wikipedia which hopefully is sufficient.

This article says Q = 1 / 2ζ which we would like to verify. The damping ratio defines ζ in terms of the differential equation

\frac{d^2x}{dt^2} + 2\zeta\omega_0\frac{dx}{dt} + \omega_0^2 x = 0.

which has solutions of the form

x = A e^{-\zeta\omega_0t} \sin(\omega_0t + \delta).

This means the energy stored is

E = \tfrac{1}{2} m\omega_0^2 A^2 e^{-2\zeta\omega_0t},

where m is the mass being oscillated (or, if you prefer, an arbitrary constant to make it dimensionally consistent — it cancels out later); and the power loss is

-\frac{dE}{dt} = m\zeta\omega_0^3 A^2 e^{-2\zeta\omega_0t}.

Using the definition of Q, this gives

Q = \omega_0 E / \left(-\frac{dE}{dt}\right) = \frac{1}{2\zeta}

as required. A bit more manipulation gives the more familar (to me, anyway) form

\,Q = \pi f \tau

where f = ω0 / 2π is the frequency, and τ, the time to decay to 1 / e. (I.e. τ = 1 / ζω0.)

Hope that helps. — ras52 (talk) 14:42, 30 December 2007 (UTC)


The equation for the Q-factor of a spring (in the Mechanical systems section) is similarly marked with a "citation needed". This one is even easier to see. The differential equation defining ζ was

\frac{d^2x}{dt^2} + 2\zeta\omega_0\frac{dx}{dt} + \omega_0^2 x = 0.

Multiplying through by M gives the familar "F = ma" equation, and from the definition of R in the article, obviously R = 2ζω0M. Then

Q = \frac{1}{2\zeta} = \frac{\omega_0 M}{R} = \sqrt{\frac{K}{M}} \frac{M}{R} = \frac{\sqrt{KM}}{R}.

I think you're unlikely to find many references for these things as, at least in my experience, Q factors are not greatly used in mechanics. Looking through the indexes of three or four degree-level texts that I have to hand that mention this sort of mechanical oscillations, I can't find a single mention of Q factors.

I also notice that whilst these mechanical relations are marked with "citation needed", equally simple electrical relations are not. For example, the Q factor of RLC circuit is as obvious as the mechanical ones, but that is not marked "citation needed". Clearly citations would be desirable for all of these, but it is seems we're requiring a higher standard of referencing for mechanical examples than electrical examples. — ras52 (talk) 16:04, 30 December 2007 (UTC)