bill@thd.tv.tek.com (William K. McFadden) (06/13/91)
[Last modified 12-Jun-91]
[This version differs from 2.0 in that I have improved the accuracy of
some of the equations and have added a program that computes the inside
dimensions of the box. In the future, I plan to add plotting for power
handling vs. frequency and SPL vs. frequency.]
--------
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 and are provided in uuencoded
form. (I'll have to buy more memory for my HP48 before I can provide
an ->ASC version.) A special ASCII subset is provided for those
without the equation card. 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
The following variables are defined for the closed box case:
Qb Total Q of system at Fb
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
The following 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. Some of the results are approximate.
3. To use a square vent, enter the vent width times 1.13 [2/SQRT(pi)] for Dv.
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 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.) (For
the closed-box case, enter either 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 two minutes for the closed box and
1.5 minutes 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 a 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 find the minimum recommended diameter of a tubular vent for the
ported 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.
To run a frequency response plot, press -> PLOT. The horizontal axis
is frequency, and the vertical axis is the magnitude of the response in
dB. Change the ranges, if desired, and press ERASE followed by 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.
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 directory, 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 section 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
different ratio, press CUST and the program will prompt you for it.
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.
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 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 [9]).
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.
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, 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 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 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.
Here are the speaker design equations in text form, as downloaded from
the HP-48SX.
The special sequences \v/, \Gn , and \pi represent the square root
operation and the greek characters eta, and pi, respectively. SQ() is
the x^2 operation. INV() is the 1/x operation. LOG() is the base 10
logarithm. UVAL(UBASE(Par)) is a fancy way of saying "return the
numerical value of Par in watts." You can omit UVAL(UBASE()) as long
as Par is in watts. In the equations below, IFTE(a,b,c) means "IF a,
THEN return b ELSE return c."
_m/s, _kg/m^3, _kg*s/m^4, _m^2/s^2 and _s/m are used to reconcile the
units in some of the equations. If you are not using an HP calculator
that supports unit conversion, you can ignore the unit objects as long
as you use SI units throughout. This means all lengths, areas, and
volumes are in units of meters, square meters, and cubic meters,
respectively; all frequencies are in Hz; and all powers are in watts.
CLOSED BOX DESIGN
Design Equations
DESIGN.EQ BYTES #4F82h 764
%%HP: T(3)A(R)F(.);
{ 'Vb=Vas/Vr' 'Fb=
Qr*Fs' 'F3dB=Qr*Fs*
\v/((1/Qb^2-2+\v/((1/Qb
^2-2)^2+4))/2)' '
dBPEAK=20*LOG(AMAX)
' 'AMAX=IFTE(Qb>INV
(\v/2),SQ(Qb)/\v/(SQ(Qb
)-.25),1)' 'Vr=Qr^2
-1' 'Qr=1/Qts/(1/Qb
-.1)' 'Par=(4*\pi^3*
1.18_kg/m^3)*Fb^4*
Vd^2/((345_m/s)*
AMAX^2)' 'PeakSPL=
112_dB+10*LOG(UVAL(
UBASE(Par)))' 'Per=
Par/\Gno' '\Gno=10^((
SPL-112_dB)/10)' '
Vd=Sd*xmax' 'Sd=\pi*(
DIA*.83)^2/4' }
Frequency Response
RESPONSE BYTES #6FFFh 112
%%HP: T(3)A(R)F(.);
'dBMAG=20*LOG(SQ(F/
Fb)/\v/(SQ(SQ(F/Fb)-1
)+SQ(F/Fb/Qb)))'
PORTED BOX DESIGN
Design Equations:
DESIGN.EQ BYTES #804Bh 744
%%HP: T(3)A(R)F(.);
{ 'Vb=20*Qts^3.3*
Vas' 'Fb=(Vas/Vb)^
.31*Fs' 'F3dB=(Vas/
Vb)^.44*Fs' 'dBPEAK
=20*LOG(Qts*(Vas/Vb
)^.3/.4)' '\Gno=10^((
SPL-112)/10)' 'Sd=\pi
*(DIA*.83)^2/4' 'Vd
=Sd*xmax' 'Par=(3_
kg*s/m^4)*F3dB^4*Vd
^2' 'PeakSPL=112_dB
+10*LOG(UVAL(UBASE(
Par)))' 'Per=Par/\Gno
' 'Lv=(2362_m^2/s^2
)*Dv^2/(Fb^2*Vb)-
.73*Dv' 'Dmin=\v/(Fb*
Vd*1_s/m)' }
Frequency Response:
RESPONSE BYTES #D7AFh 314
%%HP: T(3)A(R)F(.);
'dBMAG=20*LOG(SQ(SQ
(F/Fs))/\v/(SQ(SQ(SQ(
F/Fs))-(1+SQ(Fb/Fs)
+Fb/(7*Fs*Qts)+Vas/
Vb)*SQ(F/Fs)+SQ(Fb/
Fs))+SQ((SQ(Fb/Fs)/
Qts+Fb/(7*Fs))*(F/
Fs)-(1/Qts+Fb/(7*Fs
))*(F/Fs)^3)))'
Here is a list of values that can be used to verify the equations:
CLOSED BOX DESIGN:
Vas 0.3030_m^3
Qts 0.3300 (unitless)
Fs 30_Hz
SPL 95_dB (unitless)
DIA 0.4572_m
xmax 0.005486_m
Vb 0.07019_m^3
Qb 0.7071 (unitless)
Fb 69.17_Hz
F3dB 69.17_Hz
dBPEAK 0_dB
Par 3.740_W
PeakSPL 117.7_dB (unitless)
Per 187.4_W
\Gno 0.01995 (unitless, 1.995%)
Sd 0.1131_m^2
Vd 0.0006205_m^3
AMAX 1 (unitless)
Vr 4.317 (unitless)
Qr 2.306 (unitless)
PORTED BOX DESIGN:
Vas 0.3030_m^3
Qts 0.3300 (unitless)
Fs 30_Hz
SPL 95_dB (unitless)
DIA 0.4572_m
xmax 0.005486_m
Vb 0.1562_m^3
Fb 36.84_Hz
F3dB 40.16_Hz
dBPEAK 0.0563_dB (unitless)
Par 3.004_W
PeakSPL 116.8_dB (unitless)
Per 150.6_W
\Gno 0.01995 (unitless, 1.995%)
Dmin 0.1512_m
Dv 0.1524_m
Lv 0.1476_m
Sd 0.1131_m^2
Vd 0.0006205_m^3
Here is the uuencoded version of the complete speaker design library.
After uudecoding and downloading into the HP-48SX, BYTES 'SPKR' returns
#88C9h 8930.5.
---cut here---
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end
--
Bill McFadden Tektronix, Inc. P.O. Box 500 MS 58-639 Beaverton, OR 97077
bill@tv.tv.tek.com, ...!tektronix!thd!bill Phone: (503) 627-6920
"How can I prove I am not crazy to people who are?"HCLIMER%UTCVM.BITNET@CUNYVM.CUNY.EDU (harold Climer) (06/14/91)
I am glad to see 48 programs about this topic. I have built severl of
my own speakers using 3/4" plywood,2X2" stiffeners ,a 15" woofer ,midrange
horn, and a 3.5" dome tweeter all from Radio Shack . The thing I must
complement you on most is including References. Many people might become
interested in a topic that has been mentioned in a message or program listed
in this group but might not know where to start. Giving references gives them a
idea where to start or where to find equations ect used in the programs.
I can tell you it is most helpful to a person not well versed in a
subject,but who would like to know more about it.
Harold Climer
Physics Department
U.Tennessee at Chattanooga '82'.'88'