bill@flutter.tv.tek.com (William K. McFadden) (02/08/91)
I have put together a library of equations for designing ported and
closed-box speaker enclosures. The basic design equations were taken
from a couple of hobbyist-oriented speaker design books. The power
rating equations were taken from papers by Richard Small (see
references below).
The equations are intended to be used with the multiple equation solver
in the equation library ROM card. The uuencoded binary source for
these equations is in a companion posting. After uudecoding, the size
is about 7K. I welcome any comments or refinements.
The main directory is called SPKR and consists of two subdirectories:
CB Closed Box Design
PORTED Ported Box Design
Running the multiple equation solver from either subdirectory will
produce a menu of variables:
Vas Volume of air having same acoustic complaince as driver suspension
Qts Total driver Q at Fs
Fs Resonant frequency of driver
SPL Efficiency of driver in dB SPL at 1W/1m
DIA Diameter of driver
xmax Peak displacement limit of driver diaphragm (1/2 of "throw")
Vb Inside volume of enclosure
Fb Resonance frequency of enclosure
F3dB Half-power (-3 dB) frequency of loudspeaker system response
dBPEAK Maximum peak or dip of loudspeaker system response
Par Acoustic power rating
PeakSPL Equivalent sound pressure level (at 1m) of acoustic power rating
Per Electrical power rating (worst case)
\Gno Percent driver efficiency (\Gn is greek character eta)
Sd Effective projected surface area of driver diaphragm (approximated)
Vd Peak displacement volume of driver diaphragm
In addition, the following variables are defined for the closed box
case:
Qb Total Q of system at Fs
AMAX Maximum amplitude of loudspeaker system response: 10^(dBPEAK/20)
Vr Ratio of Vas to Vb
Qr Ratio of Qb to Qts and Fb to Fs
For the ported box case, the following apply:
1. Fb is the tuning frequency for the vent.
2. Most of the results are approximate.
To use, run MSOLVR in either directory. Enter the speaker parameters
into the variables Vas, Qts, Fs, SPL, DIA, xmax. (If you don't have
all the parameters available, purge the ones you don't know, so they'll
be undefined and the solver won't attempt to use them.) For the
closed-box case, define one of Vb or Qb and solve for the other (or
make it a calculated value with MCALC). Pressing <- ALL will solve for
all the unknowns for which a solution exists (indicated by a small box
in the menu). This takes about 2.5 minutes for the closed box and
about half as long for the ported box.
To find the optimum box size for the closed box system, set Qb=0.707
(e.g., 1/sqrt(2)) and solve for Vb. Solving for Vb for the ported box
always finds the optimum box size. The optimum box size is defined as
the size which produces no peak or dip in the frequency response (e.g.,
dBPEAK=0). (A B2 response is used for the optimum closed box, and B4
for the ported box.)
To solve for given box size, for the closed box system, enter a value
for Vb, type 'Qb' MCALC, and solve for any or all unknowns. For the
ported box, enter a value for Vb and solve for the unknowns. To return
to the optimum enclosure, for the closed box, set Qb = 0.707 and type
'Vb' MCALC. For the ported box, type 'Vb' MCALC.
To run a frequency response plot, press -> PLOT. The X axis is
frequency, and the Y axis is the magnitude of the response in dB.
Change the ranges, if desired, and press ERASE and DRAW. It takes
about a minute for the closed box, and four minutes for the ported
box.
You can also use the built-in solver to locate points of interest in
the frequency response by pressing -> SOLVE.
If you get curious, the design equations are in a list called
DESIGN.EQ, and the frequency response equation is in a variable called
RESPONSE.
There is a subdirectory in CB called EQUALIZER that will find the
component values for an active equalizer that can extend F3dB of any
closed box system to any desired lower limit (at the expense of
efficiency and power handling--watch out!) See pp. 142 of the March
1990 AES Journal for theory and circuit details.
First, use the multiple equation solver in the CB directory to solve
for the system as shown above. Next, enter the EQUALIZER
subdirectory. Enter the new desired cutoff frequency into F3dB, and
press CIRCUIT. The component values will appear in the display. The
values of R, C, N are chosen by the user to make the remaining
component values realistic (see article).
You can run a response plot of the equalizer with -> PLOT. It's pretty
interesting, but takes FOREVER (like 20 min.). The reason is I copied
the equations right out of the article without any optimization for
speed. (If anybody wants to tackle this, be my guest.) Wherever
possible, I left out the units so it would run faster. You can also
solve for points of interest with -> SOLVE. The point where maximum
boost occurs is at F3dB. If you put this in for f and solve for dB,
you will see how much boost is needed without having to wait all day.
(Don't enter values for Fb and Qb; they are defined in the parent
directory, and entering values will redefine them locally. If you do
this by mistake, purge Fb and Qb.) Efficiency and power handling of
the system at this frequency will be degraded by this amount if the
equalizer is used. This gives a pretty good worse case scenario.
Don't be surprised if more than 20 dB of boost is needed to get down to
20 Hz, even for large drivers. "There ain't no such thing as a free
lunch." If you don't need the equalizer program, just PGDIR the
EQUALIZER subdirectory. Doing so will save about 1.6K.
By the way, the default speaker parameters when you first download the
file are for the Eminence 18029 18" driver.
The following is a small tutorial on speaker enclosures:
An optimum enclosure is defined as one that has no peak or droop in the
passband.
The power rating of each driver is given in watts RMS. This is the
continuous thermal power rating of the speaker. Most speakers can
handle two to four times as much power for brief periods without
overheating.
The efficiency of the speaker is given in decibels of sound pressure
level (SPL). 0 dB SPL is defined as 2.0E-10 bar (2.0E-5 N/m^2), which
is the lowest level of 1 KHz tone the ear can detect. A 10 dB increase
in SPL results in an apparent doubling of the loudness and requires 10
times as much acoustic power. Accordingly, a 10 dB decrease halves the
loudness and reduces the acoustic power by a factor of 10.
Most driver manufacturers specify the SPL of the driver with a one watt
input measured at a distance one meter away. To calculate the SPL at
other power levels, add the following number to the SPL rating:
10*log(POWER), where POWER is in watts, and the log is base 10. This
equation is derived from the fact that a doubling of electrical power
produces an doubling of acoustic power. To calculate the SPL at other
distances, subtract the following number from the SPL rating:
20*log(DISTANCE), where distance is in meters. This equation is
derived from the inverse square law of wave propagation.
One watt of acoustic power is equal to about 112 dB SPL at one meter.
To calculate the efficiency of the speaker in percent, use the
following: %EFFICIENCY = 100*(10^((RATING - 112)/10)), where RATING is
the driver's SPL rating in dB, at one watt, measured at one meter. For
example, a driver with a 92 dB SPL rating @ 1W/1m is 1% efficient.
For the sealed box enclusure, the optimum volume in cubic feet can be
determined. Many designers like to use a 0.62:1:1.62 ratio for the
cabinet dimensions. This is known as the golden ratio. A box designed
to this ratio will be less peaky than one whose dimensions are equal.
Another ratio sometimes used is 0.8:1:1.25. You can determine the
middle dimension by taking the cube root of the enclosure volume.
(Keep in mind this is the inside volume and doesn't take into account
the volume taken up by bracing materials and the driver itself.) The
box will have a resonant frequency and a Q. For an optimum sealed box,
the resonant frequency is equal to the -3dB point, and the Q is 0.707.
The frequency (in Hz) at which the speaker's response is 3 dB down can
be found. This is also known as the half-power point, because it is
the frequency at which the acoustic output power drops by half. Below
this frequency, the response will have a second order roll off, e.g.,
the output decreases 12 dB for every halving of the frequency below the
-3 dB point.
The ported enclosure is a little more complicated. As with the sealed
box, the ported enclosure has an optimum volume (stated in cubic feet)
and -3 dB point (stated in Hz). The speaker also has a tuning
frequency, called Fb. This is the resonant frequency of the
enclosure's duct. The tuning frequency is determined by the cross
sectional area and length of the duct. You may consult a book on
speaker design to determine the proper duct size. Ported enclosures
have a steeper roll off than sealed boxes. The roll off is fourth
order, or 24dB for every halving of the frequency below the -3dB
point. At very low frequencies, the driver will be undamped, hence the
speaker could be damaged by excessive cone movement. It is therefore
wise to roll off the signal below the -3dB frequency to avoid damage.
This constraint does not apply to sealed boxes, which damp cone
movement at all frequencies.
REFERENCES:
[1] Hobbyist speaker building books, such as the one sold at Radio
Shack.
[2] L.L. Beranek, Acoustics (McGraw-Hill, New York, 1954).
[3] J.F. Novak, "Performance of Enclosures for Low-Resonance
High-Compliance Loudspeakers," J. Audio Eng. Soc., vol. 7, p 29 (Jan.
1959).
[4] A.N. Thiele, "Loudspeakers in Vented Boxes, Parts I and II," J.
Audio Eng. Soc., vol. 19, pp. 382-392 (1971 May); pp. 471-483 (1971
June).
[5] R.H. Small, "Direct-Radiator Loudspeaker System Analysis," J.
Audio Eng. Soc., vol. 20, p. 383 (June 1972).
[6] R.H. Small, "Closed-Box Loudspeaker Systems," J. Audio Eng. Soc.,
vol. 20, p. 798 (Dec. 1972), and vol. 21, p. 11 (Jan/Feb 1973).
[7] R.H. Small, "Vented-Box Loudspeaker Systems," J. Audio Eng. Soc.,
vol. 21, (four parts, starting in the June 1973 issue).
[8] W.M. Leach, Jr., "A Generalized Active Equalizer for Closed-Box
Loudspeaker Systems," J. Audio Eng. Soc., Vol. 38, pp. 142-145 (March
1990).
[1] is useful as an introduction and has a lot of construction tips.
[2] is a standard reference text that seems to be the industry bible.
[3] is historically significant, and is the foundation for [4]. [4]
and [6] are the landmark works on loudspeaker systems (you can't
consider youself knowledgeable without having read them). [5] is
background for [6], and [7]. [7] updates the original work of [4].
[8] is a recent paper that shows how to equalize closed-box systems to
any desired low-frequency cutoff. [3], [4], [5], [6], and [7] are
reprinted in the AES two-part "Loudspeakers" anthology.
--
Bill McFadden Tektronix, Inc. P.O. Box 500 MS 58-639 Beaverton, OR 97077
bill@videovax.tv.tek.com, {hplabs,uw-beaver,decvax}!tektronix!videovax!bill
Phone: (503) 627-6920 "SCUD: Shoots Crooked, Usually Destroyed"