|13th June 2009, 03:24 AM||#11|
What you are missing is that Q applies or is calculated at resonance. But first, consider the definition of resonance, it is when the positive and negative reactances are of equal magnitude so that they exactly cancel and there is no imaginary part in the total impedance. If you alter the total compliance the resonance will shift to a frequency where the reactance of the mass is equal to that of the new total compliance. The reactances were equal and of one magnitude at the free air resonance, and equal but of a different magnitude at the closed box resonance. Now, Q is the ratio of reactance over resistance at resonance, the reactance changed, the electrical damping did not, but obviously the Q does change.
Remember that for damping purposes you can short the voltage source, then pull the source resistance through the gyrator (representing the driver motor) into the mechanical secondary circuit of the driver for analysis.
I suggest that you look up the equations for the impedance of a mass, and compliance.
Qm is simply expressing the mechanical restance as a ratio to reactance at resonance.
Qe expresses the electrical damping resistance as a ratio.
And Qt combines Qm and Qe to express the combined resistive damping as a ratio.
|13th June 2009, 11:32 AM||#12|
Join Date: Aug 2004
They are not equations derived by Thiele and Small, they come directly from the equation of motion for the cone. Probably too much detail but we start by writing that the sum of the forces on the cone = the acceleration of the cone (Newtons Law)
F - Rms x’ – C x = M x’’
where F is the applied force, Rms is mechanical resistance, C is compliance, M is moving mass. x = displacement, x' = velocity, x" = acceleration of the cone.
F = BL * I, the force generated in the motor,
I = (Es - Eb) /Re
where Es is the (amplifier) appled voltage , Eb is the back EMF and Re is the voice coil resistance. ( If you consider voice coil inductance we need to use Ze, but Le is not a significant contribution of Ze around resonance so we can use Re.)
The back EMF is
Eb = BL * x'
I = (Es - BL*x')/Re
The we can substitute and write,
(B L)(Es / Re) = m x’’ + (Rms + (B L)^2/Re) x’ + x / c
The term [Rms + (BL)^2/Re] is the damping term.
Now, this is recognized as a standard ordinary differential equation
F = Mx" + Bx' + Kx
for a 2nd order system which has a natural frequency,
wn = sqrt(K/M)
1/Q = B / sqrt(MK)
Comparing to out equation for the driver cone,
M = m,
K = 1/c
B = (Rms + (BL)^2/Re)
w = sqrt(mc)
1/Q = (Rms + (BL)^2/Re) / sqrt(m/c) = Rms/sqrt(m/c) + (BL)^2/Re)/ sqrt(m/c)
1/Q = 1/Qm + 1/Qe
From whence we get the relations
Q = Qm * Qe / (Qm + Qe)
Again, the compliance,c, in the equation, is the total compliance (driver suspension plus box air spring).
Anyway, this is the basic analysis of the moving system and you should be able to see how the compliance enters into both Qm and Qe. What I showed before shows how the compliance terms changes when the air spring is included.
If you work through the math for the in box system you can write the equation as
F(t) = x” + Ù (1/Qms + 1/Qes) x’ + Ù^2 (1 + á) x
where Ù is an inverse dimensionless frequency, wn/w and á =Vas / Vbox.
There is nothing magical here. The dependence of Qm and Qe on compliance just comes out in the wash.
The reason to wrie the equation in dimensionless form is that it tells us about the relative effect of the different terms. For example, when w is small (very low frequency), Ù will be large, and the term Ù^2 (1 + á) x will dominate the other terms. This is the well know case where at low frequency the driver is controled by the compliance alone. When w is very large, Ù will be very small (and Ù ^2 even smaller) and the driver motion is dominated by the cone mass. This is the mass controled region where we have flat frequency response. When w is around wn, Ù will be about 1.0 and all three terms must be considered. The damping terms will be on the order of 1/Qm = 1/Qe = 1/Qt. So if Qt is very small, 1/Qt is very large and the damping will dominate around w = wn. If Qt is large, 1/Qt is small and damping drops out and we have an lightly oscillator with natural frequency = wn.
John k.... Music and Design NaO Dipole Loudspeakers.
|13th June 2009, 05:00 PM||#13|
Join Date: Jun 2002
Location: USA, MN
Impedance of a mass and compliance?
Mobility or impedance analogy?
Time for you to do your own homework.
Our species needs, and deserves, a citizenry with minds wide awake and a basic understanding of how the world works. --Carl Sagan
Armaments, universal debt, and planned obsolescence--those are the three pillars of Western prosperity. —Aldous Huxley
|13th June 2009, 10:20 PM||#14|
Edit: Wondering what provoked this response ... My previous post was directed to the OP by the way, it is not all about you.
Are you trying to impress me?
Wow, you know the difference between an impedance and mobility analogy?
Yes, I am impressed with your rudeness! LOL!
I'm the one who answered the question here, does that bother you so much?
You need to do your homework, the Q is the same no matter which analogy you choose, LOL!
Yes, the internet is a nut house!
|13th June 2009, 10:23 PM||#15|
Oh, wait, perhaps it bothers you that this statement by you makes no sense what-so-ever, LOL!
|14th June 2009, 03:49 AM||#16|
People should also understand that T&S theory showed that a sealed box woofer is essentially a 2nd order high pass filter. This ignores the voice coil inductance which if high can result in some slope in the pass band, but this is easily accounted for on top of the basic 2nd order high pass view.
Consider what we call a prototype filter design, which is where we scale Fc to 1 Hz and the amplitude to 0 dB. Then we plot a family of filter responses as a function of Q. This would represent Qtc for a closed box, and Qts for free air operation. This family of curves represents any 2nd order speaker. Simply scale frequency by Fc, and shift the amplitude by making 0 dB the pass band sensitivity and pick the curve for the particular Qtc (or Qts). If you can picture the prototype family of curves then you now know every type of 2nd order or closed box system that can be built. Simple, anyone can do it.
Some Qtc values of particular interest are .5, .707, 1, and 2. I'll let people look them up to understand why.
Think, critically damped (-6dB at Fc), maximally flat (-3dB at Fc), 0dB output at Fc, and +6dB output at Fc.
One could also plot a family of transient response, say step response curves as a function of Qtc.
I'll let those learning look up the characteristics for some interesting Qtc values.
This is why Qts and Qtc are so interesting and important.
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