Calculating Qec Qmc
In books such as The Loudspeaker Design Cookbook by Dickason and that book by Weems, Designing, Building, and Testing...., I've seen the formula
Qtc = ((Vas/ Vb) + 1)^2 *Qts
However, I was wondering if putting a driver in a closed box only increased Qm or only increased Qe or both. Intuitively it would seem correct to suppose that Qe would remain the same and Qm would increase. That's my intuition, as putting the driver in a closed box isn't changing anything electrical, but only the mechanical side of its operation.
Then I manipulated some equations by Thiele in a paper of his to discover that
Qec = ((Vas/ Vb) + 1)^2 * Qes
Qmc = ((Vas/Vb) +1)^2 * Qms
This being in agreement with the standard equation relating Qts to Qtc that I started with.
Would all of you in the know with T/S calculations agree with my equations for Qec and Qmc?
Why should Qe change when the electrical condition of the driver is unchanged?
correction to equations
Sorry! -to all those who have looked at this. The equations are incorrect, I used the wrong symbolism for square root. The correct equations for Qec and Qmc are:
Qec = sqrt((Vas/Vb) +1) * Qes
Qmc = sqrt((Vas/Vb) +1) * Qms
Rather than reply to my own post, I did try to be able to edit my post, but was unsuccessful for what reason I don't know.
Your second set is correct.
Qxc/Qxs = Fc/Fs , so Q changes in proportion to the resonant frequency
Electrical damping is Bl^2/Re - unchanged
You can't edit because they lock posts for editing after perhaps 1/2 hour. They could leave it open, but choose not to.
In terms of mechanical parameters
Qms = (1/Rms) x sqrt(Mms/Cms)
When placed in a box Cms is replaced by the combination of Cms in parallel with Cb, where Cb is the compliance of the box.
Qes = (Re/Bl)^2 x sqrt(Mms/Cms)
Again, in a box Cms must be replaced with Cms in parallel with Cb.
Since Cb in parallel with Cms is always less that Cms, both Qes and Qms increase by sqrt(Cms/Cab) where
Cab = Cb x Cms /(Cb + Cms)
If you work through the math,
Sqrt(Cms/Cab) = Sqrt( (Cb + Cms)/Cb) = Sqrt(1 + Cms/Cb) = sqrt(1 + alpha).
Alpha = Vas/Vb.
Vas = Sd^2 x Cms x const.
Vb = Sd^2 x Cb x const.
Cms/Cb = Vas/ Vb = alpha.
The bottom line is that both Qe and Qm increase by a factor of
Sqrt(1 + alpha), as does Qt because of the change in compliance.
Excelent! Thanks John.
I can see this in the calculations of Unibox, too.
Okay, so it DOES seem reasonable that decreasing compliance by back loading the driver with a closed box would increase Qm. But it DOES NOT seem reasonable at all that decreased compliance would increase Qe. Is this perhaps a case of Thiele/ Small as a simplication becoming stretched?
Now, I'm not going to lose sleep if this doesn't become reasonable. I'm just exercising my curiosity here. :)
Thanks for confirming my "discovery", anyway.
Thanks to Ron for the info about editing.
I did neglect to edit my first equation relating Qtc to Qts. I'll do that now just to make this more worthy of the archive.
Qtc = sqrt((Vas/ Vb) + 1) * Qts
John proved both Qes and Qms are modified by compliance (the only variable which affects Fs). This is why I put my answer in terms of Fs - most never get past Fs, Qts, Vas modeling...
Qm and Qe come directly out of the equation which governs the motion of the cone.
Yes, I see in John K's post that both Qes and Qms are directly proportional to sqrt(Mms/ Cms). But these are equations devised by Thiele and Small that define what Qes and Qms are. So merely stating their equations doesn't answer my question.
In other words Thiele and Small understood a connection between compliance and electrical Q. The equation for Qes states the connection, but the equation by itself doesn't provide any explanation as to why there is a connection.
In the case of Qms, Qms directly proportional to stiffness is fairly easy to comprehend. A spring with increased tension can store and release more energy as a result of being made more stiff.
But WhY should compliance affect electrical Q? Given the sophistication of the authors of these equations, I would think that there probably is an explainable link between electrical Q and compliance. -That is what I would maybe like to be able to grasp so long as it doesn't require being able to do Calculus.
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