sbhattac@rnd.gba.nyu.edu (Shankar Bhattacharyya) (09/18/90)
As before, this is not aimed at people who are on their forty-seventh
woofer. However, comment from them would be nice. I would not want to
mislead people by making errors.
Formulae for closed boxes:
This is the minimum set of formulae, just sufficient to calculate box
volume, system Q, and F3.
If you find any errors, please let me know. I have not checked things very
thoroughly. A little program I wrote, which implements soem of it,
generates numbers which look fairly decent, but it sometimes differs from
published results in modest ways.
If you get interested in power handling, and output spl, I suggest that you
look at the literature, to which I provided references earlier. For most
amateur applications, I believe that this is plenty.
NOTATION:
Fs = driver free air resonance frequency
Qes = driver Q at Fs, counting electrical resistance only, free air
Qms = driver Q at Fs, counting non-electrical resistance only, free air
Qts = total driver Q at Fs, free air
(let's not pick nits about the "at Fs")
EBP = efficiency bandwidth product = Fs / Qes
Vas = compliance of unbaffled driver, expressed as volume of air with
same compliance
Sd = effective piston area of driver
Xmax = maximum linear excursion
Vd = maximum linear displacement = Sd * Xmax (Small's usage)
parameters for a closed box system:
Vb = net internal volume of enclosure
Fc = system resonance frequency, with driver mounted in the enclosure
F3 = half power frequency, cutoff frequency, -3 db point
Qtc = Q of complete loudspeaker system, i.e. of driver "plus" enclosure
alpha = compliance ratio = Vas / Vb (approximation)
The approximation holds reasonably closely if the box is unfilled.
I'm going to assume that it is good. Stuffing the box makes the enclosure
seem larger, and you would have to determine what the effective internal
volume is (compliance, actually).
Cautions:
Xmax is intended to be given as an amplitude, i.e. one way. Dynaudio
expresses it as peak-to-peak, or twice the maximum amplitude.
Small seems to want Vd expressed as a one way number. Some manufacturers
seem to use it for the peak-to-peak number. Since my main speakers don't
have much spl capability, I've never worried about this anyway. Any decent
woofer produces more output than they do.
It is entirely possible that some manufacturers may specify Xmax as the
one-way displacement, but Vd as a peak-to-peak number. I don't know.
So check before you compare these things.
SPL CALCULATIONS:
This information is straight out of one of Linkwitz's articles. I presume
it is correct. I include this just in case someone is interested. If you
work through it, it is instructive to see why woofers are expensive. It
assumes a pulsating sphere, i.e. a speaker small compared to the
wavelength, and radiation into 4*pi stearadians.
SPL in dB = 94.3 + 20logx + 40logf + 40logd - 20logr
where x = displacement from mean position
d = effective diameter of woofer
r = distance between loudspeaker and measuring point
f = frequency
frequency in hertz, x, d, r in meters.
CLOSED BOX FORMULAE:
EBP = Fs / Qes
Small says that drivers for which EBP is about 50 are good for closed boxes,
and about 100 for vented boxes.
If you don't have Qes, assume it is about 1.1 X Qts.
alpha = Vas / Vb
= (Fc / Fs) ** 2 - 1
= (Qtc / Qts) ** 2 - 1
term = - 2 + 1/(Qtc *8 2) (just for simplicity)
F3 = Fc * sqrt { 0.5 * [ term + sqrt(term * term + 4) ] }
On the whole, I think this means that for a closed box woofer you want Fs
in the neighbourhood of 20 hz, Qes and Qts about 0.4-0.5. Qtc < Qts
cannot be realized.
If Qts is only a little less than 0.7, Qtc = 0.7 will require a huge box.
For many drivers you will find that the required volume declines very,
very fast as you shoot for slightly higher Qtc. Qtc = 0.8 has about 0.25 dB
ripple, 0.9 has about 0.5 dB, 1.0 has about 1 dB. I think that 0.5 dB
ripple is quite tolerable, so, if dramatic size reductions can be realized
by accepting slightly higher Qtc, I would accept it happily.
Next article: numbers on a bunch of drivers.
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Shankar Bhattacharyya, Information Systems, New York University
sbhattac@rnd.gba.nyu.edu
----------------------------------------------------------------------sbhattac@rnd.gba.nyu.edu (Shankar Bhattacharyya) (09/25/90)
I made a typographical error in part 3 of "making woofers".
The error was brought to my attention by a most helpful individual, who also
included some verifying calculations on Dynaudio and Audio Concepts
drivers. I appear to have deleted the message accidentally, in a rush to
get to a class, and I would appreciate another message so that I can get in
touch. I only got a casual look before I took off, and I cannot even
provide a userid to acknowledge the help, or to pursue the calculations.
In any case, my thanks.
The error was in the line for the section "CLOSED BOX FORMULAE".
Incorrect: term = - 2 + 1/(Qtc *8 2)
Correct: term = - 2 + 1/(Qtc ** 2)
Sorry about that. I'm a miserable typist.
Since this gets used to calculate F3, it is obviously of some interest to
get it right. Then:
F3 = Fc * sqrt { 0.5 * [ term + sqrt(term * term + 4) ] }
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Shankar Bhattacharyya, Information Systems, New York University
sbhattac@rnd.gba.nyu.edu
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