[Last modified 07Feb07]
SPEAKER DESIGN EQUATIONS 3.6
Bill McFadden 1993
[billmc@rdrop.com]
The most current version of this file can be found at:
http://www.rdrop.com/users/billmc
1. Introduction
This is a library of equations for designing ported and closedbox
speaker enclosures. The equations were taken from speaker design books
and technical papers by Richard Small and Neville Thiele (see references
in tutorial section). They are designed for unstuffed enclosures.
Refer to the references for more information on stuffing.
The equations are intended to be used with the HP48GX/SX multiple
equation solver but can also be run with the solver built into the
HP48SX. 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 initial default speaker parameters 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 compliance as driver
suspension
Qts Total driver Q at Fs
Fs Resonant frequency of driver
PEmax Thermallylimited 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 Halfpower (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 displacementlimited acoustic power rating
Per Estimated displacementlimited electrical power rating
\Gno Driver efficiency (\Gn is Greek character eta)
PeakSPL Thermallylimited 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. Closedbox systems are the easiest to design and build and
have the advantages of smaller box size, good lowfrequency power
handling, and superior transient response. Portedbox 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 ClosedBox Systems
Closedbox 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 closedbox system, enter the CB subdirectory and run the
multiple equation solver. Alternatively, run the builtin HP48SX
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 to solve for all
the unknowns. If using the builtin HP48SX solver, you will need to
solve for each unknown individually, using NXEQ to sequence through
the equations.
3.2 PortedBox Systems
Portedbox 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 portedbox 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 ClosedBox 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 handlingwatch out!) See [11] 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 [11]).
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 nondirectional.
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 highfrequency
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 builtin HP48SX 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 displacementlimited acoustic power rating. For the
closed box, Par is the worstcase 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 displacementlimited electrical RMS power rating based on
Par.
Because displacementlimited 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 builtin HP48SX 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 thermallylimited maximum input power,
PEmax.
Like power handling, displacementlimited SPL is a function of
frequency. To examine displacementlimited SPL in detail, enter the
CB or PORTED subdirectory and run the plotter or builtin HP48SX
solver. Select SPLresp from the equations catalog. F is the
frequency variable and SPLmax is the displacementlimited SPL at that
frequency.
SPLmax is plotted first, followed by the thermallylimited RMS sound
pressure level. As before, for frequencies where SPLmax exceeds the
thermallylimited SPL, the maximum SPL may be limited to a value in
between, depending on the peaktoaverage power ratio of the input
signal.
Again, ISECT can be used to find the frequency or frequencies at
which the displacement and thermallylimited SPL ratings are equal.
4.4 Analysis of Equalized ClosedBox System
Using an equalizer to extend the bass response of a closedbox 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 subdirectories, but take into
account the effects of the equalizer.
Because I took the equations right out of the article [11] 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.
5. Design Equations
Here are the equations used by the speaker design library. All values have
SI (mks) units. ^ denotes exponentiation. LOG() is base 10.
5.1 Constants
pi = 3.14159265359
c = speed of sound in air (345 m/s)
Ro = density of air (1.18 kg/m^3)
5.2 ClosedBox Systems
Vb = Vas/Vr
Fb = Qr*Fs
F3dB = Qr*Fs*((1/Qb^22+((1/Qb^22)^2+4)^0.5)/2)^0.5
Fmax = c/(pi*0.83*Dia)
dBpeak = 20*LOG(Amax)
Par = K1/Amax^2
Per = Par/(\Gno)
\Gno = 10^((SPL112)/10)
PeakSPL = SPL+10*LOG(PEmax)
Sd = pi*(Dia*0.83)^2/4
Vd = Sd*xmax
Amax = Qb^2/(Qb^20.25)^0.5 for Qb >(1/2)^0.5, 1 otherwise
K1 = (4*pi^3*Ro/c)*Fb^4*Vd^2
K2 = 112+10*LOG(K1)
Vr = Qr^21
Qr = (1/Qts)/(1/Qb0.1)
Frequencydependent equations:
Fr = (F/Fb)^2
dBmag = 10*LOG(Fr^2/((Fr1)^2+Fr/Qb^2))
Pmax = K1*((Fr1)^2+(Fr/Qb^2))/(\Gno)
SPLmax = K2+40*LOG(F/Fb)
Thermallylimited RMS SPL = PeakSPL+dBmag
5.3 Ported Box Systems
Vb = 20*Qts^3.3*Vas
Fb = (Vas/Vb)^0.31*Fs
F3dB = (Vas/Vb)^0.44*Fs
Fmax = c/(pi*0.83*Dia)
dBpeak = 20*LOG(Qts*(Vas/Vb)^0.3/0.4)
Par = 3*F3dB^4*Vd^2
Per = Par/(\Gno)
\Gno = 10^((SPL112)/10)
PeakSPL = SPL+10*LOG(PEmax)
Dmin = (Fb*Vd)^0.5
Lv = 2362*Dv^2/(Fb^2*Vb)0.73*Dv
Sd = pi*(Dia*0.83)^2/4
Vd = Sd*xmax
K1 = (4*pi^3*Ro/c)*Fs^4*Vd^2
K2 = 112+10*LOG(K1)
Frequencydependent equations:
Fn2 = (F/Fs)^2
Fn4 = Fn2^2
A = (Fb/Fs)^2
B = A/Qts+Fb/(7*Fs)
C = 1+A+(Vas/Vb)+Fb/(7*Fs*Qts)
D = 1/Qts+Fb/(7*Fs)
E = (97/49)*A
dBmag = 10*LOG(Fn4^2/((Fn4C*Fn2+A)^2+Fn2*(D*Fn2B)^2))
Pmax = (K1/\Gno)*((Fn4C*Fn2+A)^2+Fn2*(D*Fn2B)^2)/(Fn4E*Fn2+A^2)
SPLmax = K2+10*LOG(Fn4^2/(Fn4E*Fn2+A^2))
Thermallylimited RMS SPL = PeakSPL+dBmag