Subject: v01i024: spkr - Speaker Design Equations v3.0, Part01/01
BEGIN_DOC spkr.doc
[Last modified 21-Aug-91]
[This version differs from 2.1 mainly in that I have speeded up the
plotting equations by rewriting them as programs and added plotting for
power handling vs. frequency and SPL vs. frequency. Also, I have
rewritten the documentation.]
--------
SPEAKER DESIGN EQUATIONS 3.0
Bill McFadden 1991
All Rights Reserved
1. Introduction
This is a library of equations for designing ported and closed-box
speaker enclosures. The equations were taken from speaker design
books and technical papers by Richard Small and Neville Thiele
(see references below). They are designed for unstuffed
enclosures. Refer to the references for more information on
stuffing.
The equations are intended to be used with the HP48SX multiple
equation solver in the equation library ROM card, but can also be
run with the built-in HP solver. The binaries are provided in
uuencoded and ->ASC form. An RPL version is also provided, but
does not include the binary variable Mpar needed by the multiple
equation solver.
The default speaker parameters when you first download the file
are for the Eminence 18029 18" driver.
I welcome any comments or refinements.
2. Variables
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
PEmax Thermally-limited maximum RMS input power
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
Fmax Upper frequency limit of driver's piston range
dBpeak Maximum peak or dip of loudspeaker system response
Par Estimated displacement-limited acoustic power rating
Per Estimated displacement-limited electrical power rating
\Gno Percent driver efficiency (\Gn is greek character eta)
PeakSPL Thermally-limited RMS sound pressure level in passband
Sd Estimated effective projected surface area of driver diaphragm
Vd Peak displacement volume of driver diaphragm
K1 Power rating constant
K2 SPL rating constant
The following additional variables are defined for the closed box
case:
Qb Total Q of system at Fb
Amax Maximum amplitude of loudspeaker frequency response
Vr Ratio of Vas to Vb
Qr Ratio of Qb to Qts and Fb to Fs
The following additional variables are defined for the ported box
case:
Dmin Minimum diameter of tubular vent to prevent excessive vent
noise
Dv Diameter of tubular vent
Lv Length of tubular vent
For the ported box case, the following apply:
1. Fb is the tuning frequency for the vent.
2. To use a square vent, enter the vent width times 1.13 or
[2/SQRT(pi)] for Dv.
3. Design
When designing a loudspeaker, two approaches may be followed. The
easiest is to select a driver and design an enclosure for it. The
other is to design the enclosure first, then select or build a
driver that matches it.
The choice between a closed box and ported box depends on several
factors. Closed-box systems are the easiest to design and build
and have the advantages of smaller box size, good low-frequency
power handling, and superior transient response. Ported-box
systems are more difficult to design because they require precise
duct tuning. However, ported boxes have the advantages of
superior bass response, good efficiency, and superior peak power
handling in the passband.
3.1 Closed-Box Systems
Closed-box systems are designed around one variable, box
volume. Box volume is a function of the driver parameters and
the system Q, Qb. To design a system with minimum peak or
droop in the passband, Qb must be 0.707.
The designer has the choice of setting Qb and solving for the
box volume, or setting the box volume and solving for Qb.
There is also the choice of assigning values to both of these
variables and solving for one of the driver parameters.
To design a closed-box system, enter the CB subdirectory and
run the multiple equation solver. Alternatively, run the HP
solver and select DESIGN.EQ as the current equation. Choose
one of the following variables to solve for and assign values
to the rest: Vas, Qts, Fs, SPL, Dia, xmax, Qb, and Vb.
If you don't have all of the parameters available, purge the
ones you don't know, so they'll be undefined and the solver
won't attempt to use them. At a minimum, you will need to
supply all but one of Vas, Qts, Fs, Qb, and Vb.
Next, press <- ALL in the multiple equation solver for solve
for all the unknowns. If using the HP solver, you will need to
solve for each unknown individually, using NXEQ to sequence
through the equations.
3.2 Ported-Box Systems
Ported-box systems are a little more difficult than closed box
systems because there is an additional variable, tuning
frequency. The optimum tuning frequency depends on the driver
parameters and box volume.
To design a ported-box system, enter the PORTED subdirectory.
Run the equation solver of your choice as described above and
enter the driver parameters. Notice there is no Qb variable.
At this point solving for the unknowns will automatically
create a system with optimum passband response. Alternatively,
you can specify values for Vb and/or Fb to see what effect they
have on the system response.
To find the minimum recommended diameter of a tubular vent for
the enclosure, solve for Dmin. This is smallest diameter
permissible to keep the air velocity below 5% of the speed of
sound. Higher velocities can produce audible noise. To
calculate the vent dimensions, enter either of Dv and Lv and
solve for the other, keeping in mind the minimum recommended
value of Dv.
3.3 Cabinet Design
In the CST menu of the CB and PORTED subdirectories is a key
labeled BCALC. Pressing this key runs the box calculator
program. Don't run it directly from the SPKR subdirectory, or
it will not work properly. The program is rather crude, and
does not handle dual woofers, but is adequate for most
designs. It works as illustrated by modeling the driver as a
segment of a solid cone:
_____
/| ^
/ | |
/ | |
/ | |
_____ / | |
^ | | |
| | | |
Rdia | | Dia
| | | |
__v__ | | |
\ | |
| \ | |
| \ | |
| \ | |
| \| __v__
|
| |
|<-Depth->|
| |
To use, enter the driver's depth (distance from front of driver
to back of magnet) and press DEPTH. Enter the rear (magnet)
diameter of the driver and press RDIA. If you want the program
to account for any extra volume taken up by bracing and other
drivers, enter this volume and press XVOL. The program uses
the driver's diameter as entered previously in the equation
solver.
The dimensions default to English units. The program will only
accept real numbers as input; unit objects will cause an
error. (I said it was crude.) To change units, store a value
containing the new unit by typing 'name' STO, where name is one
of Depth, Rdia, or Xvol. The units of the results should make
sense based on the units of the data, but I won't guarantee
it.
You can also change the ratio of Height:Width:Depth used in the
box calculation by pressing GOLD, 1.25:1, or CUST. GOLD
selects the golden mean, 1.62:1:0.62 ((sqrt(5)+1)/2), which is
the most common ratio. 1.25:1 selects another common ratio,
1.25:1:0.8. If you wish to use a custom ratio, enter it and
press CUST.
Each time you change a parameter using a menu key, the results
will be recalculated and redisplayed. The display shows, from
top to bottom, the driver's front diameter, the driver's rear
diameter, the driver's depth, the extra volume taken up by
other objects inside the cabinet, the total internal volume of
the cabinet (including driver and extra volume), the ratio used
to calculate the box dimensions, and the inside height, width,
and depth of the cabinet. FIX 2 is the best display format to
use with the default units.
3.4 Equalization of Closed-Box Systems
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
[9] for theory and circuit details.
First, use the equation solver in the CB subdirectory to solve
for the system as shown above. Next, enter the EQUALIZER
subdirectory. Store 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 [9]).
4. Analysis
4.1 Frequency Response
The equation solver generates three values related to frequency
response, F3dB, Fmax, and dBpeak.
F3dB is the frequency at which the acoustic output power of the
speaker drops by half. Below this frequency, the response will
drop 12 dB per octave for the closed box and 24 dB per octave
for the ported box.
Fmax is the upper limit of the driver's piston range. Piston
range is defined as the range of frequencies for which the
wavelength of sound is greater than the circumference of the
driver's diaphragm. In this range, the driver's output is
non-directional.
Since this package models the driver as a piston, it is
important to note that the equations are only accurate up to
Fmax. In addition, because it is difficult to predict the
driver's high-frequency behavior, it is a good idea to cross
over to a smaller driver at or below Fmax.
dBpeak is the magnitude of the frequency response peak or dip.
For an optimal design, this value will be zero.
To examine the frequency response in detail, enter the CB or
PORTED subdirectory and run the plotter or HP solver. Select
FREQresp from the equations catalog. F is the frequency
variable, and dBmag is the response at that frequency. Using
the solver you can solve for one in terms of the other.
4.2 Power Handling
The equation solver generates power ratings called Par and
Per.
Par is the displacement-limited acoustic power rating. For the
closed box, Par is the worst-case value for wideband signals
(all the way down to DC). For the ported box, it is an
estimate based on the characteristics of musical signals.
Per is the displacement-limited electrical RMS power rating
based on Par.
Because displacement-limited power handling is actually a
function of frequency, the values of Par and Per only give
small part of the picture. To examine power handling in
detail, enter the CB or PORTED subdirectory and run the plotter
or HP solver. Select POWresp from the equations catalog. F is
the frequency variable, and Pmax is the maximum electrical
input power at that frequency.
Pmax is plotted first, followed by PEmax, the manufacturer's
thermal RMS power rating. At some frequencies, Pmax will
exceed PEmax. As frequency increases, Pmax can reach thousands
of watts. Exceeding PEmax is permissible for short durations,
but under no circumstances should you exceed Pmax even briefly
or the driver may be physically damaged.
Because Pmax is calculated with sine waves in mind, the peak
power rating at a given frequency will be 2*Pmax.
Using the ISECT function of the plotter, it is possible to
determine the frequency range(s) over which it is safe to apply
the full rated thermal power, PEmax, without damage from
excessive displacement. Just place the cursor near the
intersection of the curves and press ISECT in the FCN submenu.
In the same manner, you can also use ISECT to find frequencies
where the curves approach one another but don't touch.
4.3 Sound Pressure Level
The equation solver generates a value for maximum SPL called
PeakSPL. This is the maximum RMS output level of the system in
the passband when driven by the thermally-limited maximum input
power, PEmax.
Like power handling, displacement-limited SPL is a function of
frequency. To examine displacement-limited SPL in detail,
enter the CB or PORTED subdirectory and run the plotter or HP
solver. Select SPLresp from the equations catalog. F is the
frequency variable and SPLmax is the displacement-limited SPL
at that frequency.
SPLmax is plotted first, followed by the thermally-limited RMS
sound pressure level. As before, for frequencies where SPLmax
exceeds the thermally-limited SPL, the maximum SPL may be
limited to a value in between, depending on the peak-to-average
power ratio of the input signal.
Again, ISECT can be used to find the frequency or frequencies
at which the displacement- and thermally-limited SPL ratings
are equal.
4.4 Analysis of Equalized Closed-Box System
Using an equalizer to extend the bass response of a closed-box
system does not come without costs. For each octave of bass
extension, a 12 dB boost is necessary (and requires 16 times as
much power).
To evaluate these costs, two equations are provided in the
EQUALIZER subdirectory: FREQresp and POWresp. These function
like their counterparts in the CB and PORTED subdiectories, but
take into account the effects of the equalizer.
Because I took the equations right out of the article [9]
without any optimization for speed, these equations run very
slowly. However, I left out the units wherever possible so
the equations would run faster.
FREQresp calculates the response of the equalizer, rather than
the system, to give you an idea of the amount of boost required
to equalize the system. The greatest boost occurs at the new
F3dB.
POWresp calculates the equivalent power handling of the
system. At each frequency, Pmax is reduced by the amount of
boost the equalizer provides. This is useful to see what the
power handling of an equivalent, unequalized system would be.
There is no equation for maximum SPL vs. frequency because it
is the same as the unequalized system.
END_DOC
BEGIN_MISC spkr.txt
[Last modified 21-Aug-91]
LOUDSPEAKERS TUTORIAL
by William K. McFadden
For the purposes of this discussion, an optimum enclosure is defined as
one that has no peak or droop in the passband response.
1. Power Ratings
The power rating of a driver is usually (but not always) specified
in watts RMS by the manufacturer. This is the continuous thermal
power rating of the driver. Exceeding this rating for more than a
moment will cause voice coil overheating, which can result in
warping or burn-out.
Speaker systems also have a displacement-limited power rating
(Per). This is the amount of power the system can take without
exceeding the absolute maximum voice coil displacement. Per is a
function of frequency and depends on the design of the enclosure.
Thus, it is meaningless for manufacturers to specify peak power
handling without also specifying the enclosure and the frequency
range.
At some frequencies, Per will exceed the thermal RMS power
rating. For continuous tones, the smaller of the two ratings
applies. For signals with large crest factors or low duty cycles,
Per applies, providing the average power does not exceed the
thermal rating. Per is calculated for sine waves, which have a 3
dB crest factor. The peak power rating at a given frequency is
therefore 2*Per.
2. Efficiency & Loudness
The efficiency of a driver 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 average person can
detect. A 10 dB increase in SPL results in an apparent doubling
of the loudness and requires 10 times as much power. Accordingly,
a 10 dB decrease halves the loudness and reduces the power
requirement by a factor of 10.
Most driver manufacturers specify the SPL of the driver with a one
watt input measured at a distance of one meter. 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 112 dB SPL at one meter.
To calculate the efficiency of the speaker in percent, use the
following: %EFFICIENCY = 100*(10^((SPL - 112)/10)), where SPL 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.
3. Sealed Box Enclosures
For the sealed box enclosure, the optimum volume 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 have smaller resonant peaks 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 drivers.)
The box will have a resonant frequency and a Q. For an optimum
sealed box, the resonant frequency is equal to the -3 dB point,
and the Q is 0.707. The -3 dB frequency 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.
4. Ported Box Enclosures
The ported enclosure is a little more complicated. As with the
sealed box, the ported enclosure has an optimum volume and -3 dB
point.
The enclosure also has an optimum tuning frequency, Fb, which is
the resonant frequency of the enclosure's duct. The tuning
frequency is determined by the cross sectional area and length of
the duct. For a tubular duct, the following equation applies,
LENGTH = 2118*DIAMETER^2/(Fb^2*Vb) - 0.73*DIAMETER, where LENGTH
is the length of the duct in inches, DIAMETER is the inside
diameter of the duct in inches, Fb is the tuning frequency in Hz,
and Vb is the box volume in cubic feet.
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. Below Fb, the
displacement-limited power rating will be very low because the
driver is essentially operating free air. It is therefore wise to
roll off the signal below the -3dB frequency to avoid damage.
This constraint does not usually apply to sealed boxes, which damp
cone movement at all frequencies.
REFERENCES:
[1] Hobbyist speaker building books, such as the one sold by 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, pp. 383-395 (1972 June).
[6] R.H. Small, "Closed-Box Loudspeaker Systems," J. Audio Eng. Soc.,
vol. 20, pp. 798-808 (1972 Dec.); vol. 21, pp. 11-18 (1973 Jan./Feb.).
[7] R.H. Small, "Vented-Box Loudspeaker Systems," J. Audio Eng. Soc.,
vol. 21, pp. 363-372 (1973 June); pp. 438-444 (1973 July/Aug.); pp.
549-554 (1973 Sept.); pp. 635-639 (1973 Oct.).
[8] G. Margolis and R. H. Small, "Personal Calculator Programs for
Approximate Vented-Box and Closed-Box Loudspeaker System Design," J.
Audio Eng. Soc., vol. 29, pp. 421-441 (1981 June); pp. 824 (1981
Nov.).
[9] 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 the industry bible on acoustics. [3] is historically
significant, and is the foundation for [4]. [4] and [6] are the
landmark works on loudspeaker systems (you can't consider yourself
knowledgeable without having read them). [5] is background for [6],
and [7]. [7] updates the original work of [4]. [8] presents versions
of the equations of [4] through [7] suitable for programmable
calculators as well as example programs using them. [9] 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.
END_MISC
BEGIN_RDME spkr.rdm
[Last modified: 21-Aug-91]
Speaker Design 3.0 files:
README: This file
spkr.doc: Instructions for use
spkr.txt: Loudspeakers tutorial
spkr.rpl: RPL Source code
BYTES #70F0h 8515
END_RDME
--
Bill McFadden Tektronix, Inc. P.O. Box 500 MS 58-639 Beaverton, OR 97077
bill@tv.tv.tek.com, ...!tektronix!soul!bill Phone: (503) 627-6920
