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.EN
.hd "R. Pierce" "15-Mar-1987" "Loudspeaker Prototyping"

.ft B
.ps 16
.vs 20
.ce 4
A Novel Approach
to
Rapid Loudspeaker Design
and Prototyping
.ft
.ps
.vs
.sp 2
.ce 2
.ft B
.ps 12
.vs 14
Richard Pierce
.sp
.vs
.ps
.ft I
Precision Loudspeakers, Inc., Amherst, NH, 03031
.ft R
.sp 2
.(q
A new approach to designing and simulating closed and vented box loudspeaker
systems is presented. Unlike other software solutions such as custom-written
personal calculator or computer programs, this method uses a paradigm which
relieves designer and user from much of the software design and implementation
hassles and also presents the data in a form much more appropriate to
optimization of many separate parameters. The supporting software tool is
available on most personal computers, and the technique has been implemented
on over a half dozen different systems.
.)q
.sp 1.0i


\fB0.\ INTRODUCTION
.pp
Because of the work presented by Thiele [1] and Small [2]-[4] describing
the systematic analysis of loudspeakers and enclosures, the correct design
of direct-radiator loudspeaker systems has become far more a science than
the personal art of a few self-appointed experts. The further efforts
of Small and Margolis [5], Bullock  and White [6] and others [7], have
resulted in specific software methods for solving driver-enclosure designs. 
These and other articles have all used either general purpose computer systems
programmed in languages such as BASIC or FORTRAN, or personal programmable
calculators sporting their own peculiar languages. The task of designing
and writing such a program, one quickly discovers, is divided such that
about 20% of the effort goes to solving the actual problem at hand
(designing direct-radiator loudspeaker systems) and the balance goes to
hassling with the details of the programming language, user interface,
operating system, and so forth. As a result, many of the published software
packages are either limited in scope, difficult to use, or such a rats' nest
of poor programming that they are nearly impossible to adapt to different
applications or programming environments. We frequently encounter these
problems in attempting to port many such programs from environments like
Apple computers to DEC mini-computers. One supposes that you can't expect a
good loudspeaker engineer to be a great software engineer, and vice-versa.
.pp
In attempting to come up with an easy to use software system for doing driver
and enclosure prototyping, we thought of using a software tool originally
intended for business use, the "spreadsheet calculator". The problems that
needed to be solved by such a system were many: designing enclosures given
specified drivers, designing drivers given specified enclosures and parts
inventory, designing the individual components of drivers (voice coils,
magnets, and so on) based on the requirements of the final system, prototyping
new design ideas, and so forth. All this we wanted done by a single,
integrated, simple-to-use tool. Most programs we reviewed would, at best
accomplish but one of these objectives, and were not very interactive, in
the sense that any major change in parameter setup (such as the some of the
driver's Thiele-Small parameters) required us to re-run the program from
scratch.
.pp
On the other hand, a spreadsheet calculator allows us to have all of the
needed parameters in view at once, permitting us to change any combination
of these parameters, and view the resulting new system almost immediately.
.pp
In this article, we will not investigate, in any great detail, the actual
mathematical models involved in loudspeaker design, since they are already
well established. Instead, we will explore the implementation of these
algorithms on a typical spreadsheet calculator, showing the significant
advantages over other methods.
.sh 1 "THE SPREADSHEET PARADIGM"
.pp
Spreadsheet calculators were originally designed to provide business people
with a method of exploring various business scenarios. They are so named
because, in layout, they look much like an accountant's spreadsheet of
rows and columns of figures, all related in an organized fashion specific
to the problem being described. Some examples of commercial spreadsheets
included "Visicalc" and "Lotus 1-2-3"\**.
.(f
\**A colleague has, half jokingly, refered to the tools described here as
"Speaker 1-2-3".
.)f
.pp
A spreadsheet calculator contains an array (usually 2 dimensional) of "cells"
arranged in columns and rows. Each "cell" is addressed by its column and
row number (often, columns are lettered and rows are numbered) and may 
contain a constant, a formula whose values can be either constants or 
references to the values in other cells, labels or simple strings to be
printed on the screen, commands, or simply nothing. The contents of a cell
are changed by moving a cursor to that cell and entering the new contents.
The resultant values of cells that contain equations dependent on the changed
value are then updated. Certain cells can be targeted as "goals", meaning that
the spreadsheet will iterate through the cells that the goals cells are
dependent on until the goals are reached (if possible), displaying the
resultant model. 
.pp
As an example of a simple model, let's look at the formula for the resultant
resonant frequency of an inductor-capacitor tank circuit:
.EQ C (1.1)
f~=~1 over {2 pi sqrt{LC}}
.EN
.lp
A simple spreadsheet based on this equation, showing the formulas, is
illustrated in figure 1.1:
.sp
.(b
.ps -2
.vs -2
.TS
center tab (!) box;
cb cb cb cb
cb r r l.
!A!B!C 
1!L:!\fB0.0\fP!henries
2!C:!\fB0.0\fP!farads 
3!f:!1/(2*pi*sqrt(B1*B2))!hertz
.TE
.ps +2
.vs +2
.(c
Figure 1.1
.)c
.)b
.lp
In cell [B1] we enter the value of the inductor, and in [B2] the value of
the capacitor. The spreadsheet will look at the formula in [B3], note the
references to [B1] and [B2] and use those values to solve the equation.
Cells [A1], [A2], [A3], [C1], [C2], and [C3] contain labels that make the
spreadsheet easier to use and read. The actual appearance of the cells on
the spreadsheet would only show the resultant values, not the equations.
Figure 1.2 shows the same spreadsheet as it would appear with some
reasonable values used for the calculations (in this example, as in all that
follow, cells that can be modified by the user are highlighted).
.sp
.(b
.ps -2
.vs -2
.TS
center tab (!) box;
cb cb cb cb
cb r r l.
!A!B!C 
1!L:!\fB0.001\fP!henries
2!C:!\fB3.60e-3\fP!farads 
3!f:!83.88!hertz
.TE
.ps +2
.vs +2
.(c
Figure 1.2
.)c
.)b
.pp
Often, the user has the option to specify the exact format of numerical
presentation. For example,  cell [B3] displays its value in fixed point
notation with 2 decimal places, whereas cell [B2] is in scientific or "e"
notation with 3 significant digits.
.pp
In more complex applications, cells containing formulas can refer to other
cells that themselves contain formulas, and so on describing potentially
complex models. To illustrate with an example, here is a small model
intended to calculate the resonant frequency of a loudspeaker, given its
effective mass and compliance. In this example, the mechanical units are
first converted to acoustical units, then those results are used to compute
the resonant frequency. The relationships being solved here are:
.sp
.EQ C (1.2)
f sub S~=~1 over { 2 pi sqrt { M sub {AS} C sub {AS} }}
.EN
.lp
where the effective acoustic mass is related to the mechanical mass by:
.sp
.EQ C (1.3)
M sub AS~=~M sub D over { {S sub D} sup 2 }
.EN
.lp
the effective acoustical compliance is related to the mechanical
compliance by:
.sp
.EQ C (1.4)
C sub AS~=~C sub MS {S sub D} sup 2
.EN
.lp
and $C sub MS$ is the mechanical compliance, $M sub D$ is the mass and 
$S sub D$ is the effective projected area of the diaphragm. 
The resultant spreadsheet, with the formulas, would look like:
.sp
.(b
.ps -2
.vs -2
.TS
center tab (!) box;
cb cb cb cb cb
cb r r r l.
!A!B!C!D
1!Effective diameter!d:!\fB0.0\fP!cm
2!Effective area!Sd:!(pi*(C1/200)^2)!sq. m
3!Mechanical mass!Md:!\fB0.0\fP!kg
4!Mechanical compliance!Cms:!\fB0.0\fP!m/N
5!!!!
6!Acoustical mass!Mas:!(C3/(C2^2))!
7!Acoustical compliance!Cas:!(C4*(C2^2))!
8!!!!
9!Resonant frequency!Fs:!(1/(2*pi*sqrt(C6*C7)))!Hz
.TE
.ps +2
.vs +2
.(c
Figure 1.3
.)c
.)b
.lp
"Plugging in" values from a real driver for $d$, $M sub D$ and $C sub MS$
results in:
.sp
.(b
.ps -2
.vs -2
.TS
center tab (!) box;
cb cb cb cb cb
cb r r r l.
!A!B!C!D
1!Effective diameter!d:!\fB13.2\fP!cm
2!Effective area!Sd:!13.68e-3!sq. m
3!Mechanical mass!Md:!\fB15.3e-3\fP!kg
4!Mechanical compliance!Cms:!\fB1.52e-3\fP!m/N
5!!!!
6!Acoustical mass!Mas:!81.7!kg/m^2
7!Acoustical compliance!Cas:!0.285e-6!m^3/N
8!!!!
9!Resonant frequency!Fs:!33.0!Hz
.TE
.ps +2
.vs +2
.(c
Figure 1.4
.)c
.)b
.pp
The only limitation to the complexity of a model is the maximum size of the
spreadsheet, which, even in the smallest version examined, is several
thousand cells.
.pp
The basic requirements of a spreadsheet calculator for loudspeaker design
use include:
.ip \(bu
Calculations performed in floating-point arithmetic,
.ip \(bu
Output formats in fixed-point and floating-point scientific notation,
.ip \(bu
Availability of standard math operators (including exponentiation)
and transcendental functions, such as $log sub 10 (x)$, $sqrt x$, etc..
.pp
The further availability of graphing functions, customizable menus, database
access and so forth, while not necessary, can only but enhance the versatility
of the final tool.
.sh 1 "ENCLOSURE APPLICATIONS"
.pp
In many ways, the spreadsheet model for closed and vented box systems is the
most impressive and useful. The model in use at Precision allows us to quickly
design both types of enclosures given the Thiele-Small parameters of a driver.
It becomes obvious that there are some very strict limitations to the
bandwidth, power handling capacity and cabinet dimensions for direct radiator
loudspeakers when all these parameters are juggled about. More than once has
a customer been disappointed to learn of the physical impossibility of
building a system consisting of a 6 inch woofer in a 10 liter enclosure
that's 3 db down at 20 Hz with 5% efficiency capable of playing at levels
exceeding 110 dB!
.pp
Let's take, as an example, a section from the closed box system design
spreadsheet: 
.sp
.(b
.ps -2
.vs -2
.TS
center tab(!) box;
cb cb cb cb cb
cb r r r l.
!A!B!C!D
1!Effective diameter!d:!\fB13.30\fP!cm
2!Maximum excursion!Xmax:!\fB0.70\fP!cm
3!Resonant frequency!Fs:!\fB33.0\fP!Hz
4!Equivalent volume!Vas:!\fB40.0\fP!L
5!Mechanical Q!Qms:!\fB2.04\fP!
6!Electrical Q!Qes:!\fB0.39\fP!
7!Total Q!Qts:!0.33!
8!Reference efficiency!n0:!0.36!%
9!Output level (1 watt)!SPL:!87.5!dB
10!Effective area!Sd:!138.9!sq. cm
11!Maximum displacement!Vd:!97.3!cu. cm
12!!!!
13!Target system Q!Qtc:!\fB0.707\fP!
14!Enclosure damping!!\fB2\fP!(0-5)
15!Recommended volume!Vb:!8.03!L
16!Tuning ratio!a:!4.61!
17!-3dB frequency!F3:!78.2!Hz
18!Peak response ripple!Rh:!0.00!dB
19!Maximum acoustic output!Par:!0.29!W
20!Maximum SPL!!103.7!dB SPL
21!Maximum input power!Per:!82.9!W
.TE
.ps +2
.vs +2
.(c
Figure 2.1
.)c
.)b
.pp
In this example, we have entered the basic Thiele-Small parameters  $d$,
$X sub MAX$, $f sub S$, $V sub AS$, $Q sub MS$ and $Q sub ES$ in cells
[C1] through [C6]. The spreadsheet calculates the remainder of the
driver parameters $Q sub TS$, $eta sub 0$, $SPL$, $S sub d$ and $V sub d$
automatically, as the needed parameters became available. Next, we enter
the target system Q $Q sub TC$ and the enclosure damping coefficient
(an empirically derived number describing cabinet stuffing) into cells
[C13] and [C14].
The spreadsheet can then calculate the optimum box volume, the tuning ratio,
the -3dB frequency, and so forth. If, in this example, the driver's $Q sub TS$
is higher than the target $Q sub TC$, the spreadsheet reports "alignment not
possible" in the recommended volume cell, and the rest of the enclosure data
is left blank. Alternatively, the spreadsheet could be asked to find the
lowest Q possible, or driver parameters needed for a given enclosure size
and response function.
.pp
Spreadsheet models for both closed- and vented-box systems have been
combined to allow comparison of two configurations using the same driver.
In this version, a table comparing the frequency responses at select
frequencies gives an easily visible comparison bewteen the two alignments:
.sp
.(b
.ps -2
.vs -2
.TS
center tab(!) box;
cb cb cb cb cb cb.
!A!B!C!D!E
.T&
cb c s s s s.
31!PERFORMANCE COMPARISON
.T&
cb r r r r l.
32!!!CLOSED!VENTED!
33!Enclosure volume!Vb:!8.03!19.86!L
34!-3dB frequency!F3:!78.2!44.9!Hz
35!Response ripple!Rh:!0.00!0.14!dB
36!Maximum output!Par:!0.29!0.12!W
37!Maximum SPL!!103.7!100.1!dB SPL
38!Maximum input!Per:!82.9!35.3!W
.T&
cb c s s s s.
39!FREQUENCY RESPONSE
.T&
cb l r r r l.
40!!!CLOSED!VENTED!
.T&
cb r r r r l.
41!!15.9!-27.70!-31.60!dB
42!!20.0!-23.70!-24.46!dB
43!!25.2!-19.72!-17.47!dB
44!!31.7!-15.77!-10.75!dB
45!!40.0!-11.93!-4.90!dB
46!!50.4!-8.32!-1.36!dB
47!!63.5!-5.18!-0.24!dB
48!!80.0!-2.82!-0.05!dB
49!!100.8!-1.34!-0.03!dB
50!!127.0!-0.58!-0.03!dB
51!!160.0!-0.24!-0.03!dB
52!!201.6!-0.10!-0.02!dB
.TE
.ps +2
.vs +2
.(c
Figure 2.2
.)c
.)b
.pp
Further information can be derived, such as the parameters for equalization
parameters in higher-order systems, inclusion of temperature-limited power
handling  (all of the above power figures are based on excursion limiting)
and so on.
.pp
What we have not shown in the above examples is much of the underlying
intermediate mathematics. While obviously vital to the final solutions, 
displaying these intermediate results would only serve to confuse the user.
Items such as enclosure losses, conversion from mechanical to acoustical
units, and so forth, can be done "off screen", or in "hidden cells". This
also allows the user to incrementally build the application, as these
intermediate results aid in debugging the model as it grows. There may,
however, be a slight performance penalty in using many simple steps rather
than a few complex ones.
.pp
The value of a spreadsheet implementation becomes apparent when you consider
that the time it takes to update the entire spreadsheet when a parameter is
changed is often less than 2 seconds, and the update can be deferred,
if desired, until several parameter changes have been made, thus saving time 
in recalculating for each entry. Additionally, judicious setup of the model
allows viewing and changing \fIall\fP the relevant parameters in a given
application.
.sh 1 "DRIVER DESIGN APPLICATIONS"
.pp
At Precision, we receive requests to design new drivers for specific
applications, and this requires the ability to specify drivers and driver
components quickly and accurately to respond to the needs of a customer.
For example, one application might require a driver with the same diameter,
$f sub S$ and $V sub AS$ as a standard model, but a lower $Q sub TS$. One
way of accomplishing this is by changing the motor assembly. Changing the
magnet structure itself is difficult because of the limited selection of
magnet materials and configurations and the time and expense necessary in
retooling for new front plates and pole pieces. But the voice coil itself
can be changed readily. A portion of the driver and enclosure design
spreadsheet allows the user to configure new voice coils, selecting amongst
a wide variety of standard wire sizes and cross-sections, winding lengths,
number of layers and so forth. All the resultant parameters that might affect
the final performance of the system are calculated, including DC resistance,
coil mass, $X sub MAX$, $Bl$ product, and so on. It is most enlightening
to change the size of wire from, say 28 gauge to 30 gauge, or from copper
to aluminum wire, leaving everything else constant, and watch what happens
to the final system response as a result of the changes in $Bl$ product,
mass and DC resistance. Again, all of the answers are available within a few 
seconds of a parameter change. An extract from the voice coil spreadsheet
looks like:
.sp
.(b
.ps -2
.vs -2
.TS
center tab(!) box;
cb cb cb cb cb
cb r  r  r  l.
!A!B!C!D
78!Wire type:!
79!Copper!1!!
80!Aluminum!2!!
81!Phosphor/Bronze!3!!
82!Silver!4!!
83!Select wire type:!\fB1\fP!!
.T&
cb c s s s.
84!WIRE PARAMETERS!!!
.T&
cb r r r l.
85!Select wire guage:!!\fB31\fP!
86!Material!!Copper!
87!Specific gravity!!8.96!
88!Resistivity!!1.72!
.T&
cb c s s s.
89!COIL FORM PARAMETERS!!!
.T&
cb r r r l.
90!Coil former diameter!df:!\fB3.25\fP!cm
91!Coil former length!Lf:!\fB1.30\fP!cm
92!Number of layers!!\fB2\fP!
.T&
cb c s s s.
93!MAGNET PARAMETERS!!!
.T&
cb r r r l.
94!Top plate thickness!Tfp:!\fB0.79\fP!cm
95!Flux density in gap!B:!\fB9500.0\fP!gauss
.T&
cb c s s s.
96!VOICE COIL DATA!!!
.T&
cb r r r l.
97!Number of turns!N:!111!
98!Coil outside diameter!!3.34!cm
99!Total wire length!Lw:!11.28!m
100!Length of wire in gap!l:!6.86!m
101!Bl product!Bl:!7.03!N/A
102!Coil mass!Mvc:!3.24!g
103!DC resistance!Re:!6.05!Ohms
.TE
.ps +2
.vs +2
.(c
Figure 3.1
.)c
.)b
.pp
Another spreadsheet model facilitates quick measurement of Thiele-Small
and dynamic parameters of drivers using the normal free-air and reference
volume tests (see the example in the appendix).
.sh 1 "FURTHER\ APPLICATIONS\ AND\ ENHANCEMENTS"
.pp
We have developed standard crossover configurations, allowing near-automatic
optimization of crossovers and impedance equalization networks to the
specific characteristics of the driver. Part of the spreadsheet for driver
and enclosure design generates the component values for the electrical
equivalent circuit for the completed system, along with an accurate model
for the voice coil inductance which includes the effective shunting resistance
due to eddy currents in the magnet circuit. This enables us to accurately
design and simulate of crossovers loaded with realistic terminal impedances.
.pp
Currently, we are also building an application that would integrate the
spreadsheet models with the engineering database. This would allow us
to specify a given target response (either a set of driver parameters or
a completed system specification) and have the spreadsheet either recommend
the closest matching standard driver, or a plausible set of off-the-shelf
components along with any needed changes.
.pp
Finally, we are exploring the possibility of interfacing the spreadsheet
directly with data acquisition hardware and software, allowing real-time
data to be available appropriately scaled to engineering units in selectable
cells within the spreadsheet. The models may then be used in conjunction
with a very sophisticated driver quality control program for production use.
A hardcopy of the resulting spreadsheet may be used either as a repair ticket
for out-of-spec drivers or as an assurance to the customer of the driver's
performance.
.sh 1 CONCLUSIONS
.pp
The spreadsheet implementation has proven itself to be a rapid, easy to use
and versatile tool for modeling the performance of drivers and systems in
engineering and environment. It has shown itself especially valuable in
determining the feasability of otherwise marginal configurations. The
ability to generate new voice coil and magnet configurations has been
especially useful in designing drivers to fit very specific and well defined
application areas.
.pp
In spite of the seeming lack of a language standard for spreadsheets, we
have successfully implemented these models on a wide variety of implementations
***>,
including Access Technology's \fISUPERCOMP-20\fP running on a DEC PDP-11/23,
Applix' \fIALIS\fP, an integrated office-automation system on Sun Microsystems
workstations running Unix, and a puplic domain spreadsheet calculator \fIsc\fP.
Only minor changes were required when porting the models from one system to
another.
.pp
These models suffer, like all other simulation and prototyping systems,
in that the results are only as dependable as the accuracy of the model
Much effort has been expended in researching the literature to verify
the models, and further hard prototyping has been done to verify that the
models, no matter how theoretically accurate, do have some resemblance to
practical reality. All in all, though, the results are gratifying.
.pp
Probably the most important advantage is that the spreadsheet approach frees
the model developer from many of the subtle annoyances of programming. Most
of the work is spent developing the model from the applications standpoint.
As a software engineer, it has long been my opinion that one of the worst
things to have happened to engineering is the original birth and continued
resuscitation of languages such as BASIC and FORTRAN. To have to learn the
silly idiosyncracies of languages like these is tantamount to cruel and
unusual punishment, and a spreadsheet goes a long way to eliminating these
problems.
.sh 1 ACKNOWLEDGEMENTS
.pp
Many of the actual algorithms were borrowed from the work by Small and
Margolis done for programmable calculators [5], but the general concepts,
of course, owe their existance to Thiele, Small, and Ashley and all the rest.
The actual time to develop both the spreadsheet models and this article was
ruthlessly stolen from my family.
.sh 1 "APPENDIX - EXAMPLES"
.pp
The model presented here calculates the Thiele-Small parameters for a driver,
given a simple set of measurements of the driver in free air and in a sealed
test chamber. The algorithms used are from Small in [2] and [3]. The driver's
DC resistance ($R sub E$), resonant frequency ($f sub S$) and impedance at
resonance ($R sub ES + R sub E$) is measured. Using these figures, the two
frequencies $f sub L$ and $f sub H$ are determined where the impedance is:
.EQ C (7.1)
r sub 0~=~{{R sub ES} {R sub E}} over {{R sub ES} + {R sub E}}
.EN
.(b L F
The driver's $Q$ due to mechanical losses $Q sub MS$ is calculated:
.EQ C (7.2)
Q sub MS~=~{ f sub S sqrt {r sub 0}} over {f sub H - f sub L} 
.EN
.)b
.(b L F
along with the $Q$ due to electrical losses $Q sub ES$ is:
.EQ C (7.3)
Q sub ES~=~Q sub MS over {r sub 0 - 1}
.EN
.)b
.(b L F
and the Q due to all losses $Q sub TS$ is:
.EQ C (7.4)
Q sub TS~=~{Q sub MS Q sub ES} over {Q sub MS + Q sub ES}
.EN
.)b
.pp
A similar set of measurements is made, but with the driver mounted in a sealed
box of volume $V sub CT$. These measurements will yield the values for
$f sub CT$, $Q sub MCT$ and $Q sub ECT$. We can derive the rest of the needed
Thiele-Small and the electromechanical parameters of the driver as follows:
.(b L F
The equivalent volume of compliance $V sub AS$:
.EQ C (7.5)
V sub AS~=~V sub CT left ( {{f sub CT Q sub ECT} over {f sub S Q sub ES}}~-~1 r
***>ight )
.EN
.)b
.(b L F
the reference efficiency $eta sub 0$ (for a $V sub AS$ stated in liters):
.EQ C (7.6)
eta sub 0~=~9.64 times 10 sup -10 {{f sub S} sup 3 V sub AS} over {Q sub ES}
.EN
.)b
.(b L F
the mechanical compliance $C sub MS$:
.EQ C (7.7)
C sub MS~=~V sub AS over { rho c sup 2 {S sub D} sup 2}
.EN
.sp
where $rho$ is the density of air ($1.18 kg / {m sup 3}$) and $c$ is the
velocity of propogation of sound in air ($345 m / s $)
.)b
.(b L F
The mechanical mass $M sub D$ is derived by:
.EQ C (7.8)
M sub D~=~1 over { 2 pi {f sub S} sup 2 C sub MS}
.EN
.)b
.(b L F
the force factor, or $Bl$ product:
.EQ C (7.9)
Bl~=~{ left ( {2 pi f sub S R sub E M sub D}
over Q sub TS right ) } sup {1 over 2}
.EN
.)b
.(b L F
and the mechanical resistance due to suspension losses $R sub MS$:
.EQ C (7.10)
R sub MS~=~{ B sup 2 l sup 2 } over {R sub ES - R sub E}
.EN
.)b
.(b L F
One further calculation derives the sound pressure level for the driver at
1 meter stimulated by 1 watt in our measurement environment:
.EQ C (7.11)
SPL~=~112~+~10~log sub 10 ( eta sub 0 )
.EN
.)b
.pp
Of course, care must be taken throughout all the calculations in the
spreadsheet model to ensure consistancy of basic units.
.(b
.pp
A simple spreadsheet model implementing these algorithms might look like:
.sp
.ps -2
.vs -2
.TS
center tab(!) box;
cb cb cb cb cb
cb r  r  r  l  l.
!A!B!C!D!E
4!Test volume!Vct:!\fB0.00\fP!Liters!
5!!!!!
6!Effective diam.!D:!\fB0.00\fP!cm!
7!Effective area!Sd:!3.1416*(C6/2)^2!sq. cm!
8!DC resistance!Re:!\fB0.00\fP!Ohms!
.T&
cb r r r r l.
9!!!Free air!Chamber!
10!Resonance!f:!\fB0.00\fP!\fB0.00\fP!Hz
11!Impedance at resonance!Res+Re:!\fB0.00\fP!\fB0.00\fP!ohm
12!Frequencies where z=!!sqrt(C8*C12)!sqrt(C8*D12)!Ohm
13!Low frequency!Fl:!\fB0.00\fP!\fB0.00\fP!Hz
14!High frequency!Fh:!\fB0.00\fP!\fB0.00\fP!Hz
15!Geom. mean of Fl and Fh!!sqrt(C14*C15)!sqrt(D14*D15)!Hz
.T&
cb c s s s s.
16!THIELE-SMALL PARAMETERS!!!!
.T&
cb r r r l l.
17!Resonant frequency!Fs:!C11!Hz!
18!Equivalent volume!Vas:!C4*(((D11*D35)/(C11*C23))-1)!Liters!
19!Mechanical Q!Qms:!C11*sqrt(C12/C8)/(C16-C15)!!
20!Electrical Q!Qes:!C22/(C12/C8-1)!!
21!Total Q!Qms:!(C22*C23)/(C22-C23)!!
22!Reference efficiency!n0:!(9.64e-10*C20^3*C21/C23)*100!%!
23!Output (1 watt in)!SPL:!112.+10*log(C25/100)!dB SPL!
.T&
cb c s s s s.
24!ELECTRO-MECHANICAL PARAMETERS!!!!
.T&
cb r r r l l.
25!Compliance!Cms:!C21/(1.18*345^2*(C7/10000)^2)!m/N!
26!Mass!Md:!1/((6.2832*C20^2)*C29)!Kg!
27!Losses!Rms:!C32^2/(C12-C8)!Kg/s!
28!Force factor!Bl:!sqrt(6.2832*C20*C8*C30/C23)!N/A!
29!!!!!
30!!Qmc:!D11*sqrt(D12/C8)/(D16-D15)!!
31!!Qec:!D34/(D12/C8-1)!
.TE
.ps +2
.vs +2
.(c
Figure 7.1
.)c
.)b
.(b
.pp
A working example, based on the measurements of a real driver, is shown
in figure 7.2:
.sp
.ps -2
.vs -2
.TS
center tab(!) box;
cb cb cb cb cb
cb r  r  r  l  l.
!A!B!C!D!E
4!Test volume!Vct:!\fB3.78\fP!Liters!
5!!!!!
6!Effective diam.!D:!\fB13.30\fP!cm!
7!Effective area!Sd:!138.9!sq. cm!
8!DC resistance!Re:!\fB6.30\fP!Ohms!
.T&
cb r r r r l.
9!!!Free air!Chamber!
10!Resonance!f:!\fB33.0\fP!\fB113.0\fP!Hz
11!Impedance at resonance!Res+Re:!\fB43.5\fP!\fB40.5\fP!Ohms
12!Frequencies where z=!!16.6!15.97!Ohms
13!Low frequency!Fl:!\fB20.0\fP!\fB92.5\fP!Hz
14!High frequency!Fh:!\fB57.7\fP!\fB142.0\fP!Hz
15!Geom. mean of Fl and Fh!!33.97!114.61!Hz
.T&
cb c s s s s.
16!THIELE-SMALL PARAMETERS!!!!
.T&
cb r r r l l.
17!Resonant frequency!Fs:!33.0!Hz!
18!Equivalent volume!Vas:!42.7!L!
19!Mechanical Q!Qms:!2.30!!
20!Electrical Q!Qes:!0.39!!
21!Total Q!Qms:!0.33!!
22!Reference efficiency!n0:!0.38!%!
23!Output (1 watt in)!SPL:!87.8!dB SPL!
.T&
cb c s s s s.
24!ELECTRO-MECHANICAL PARAMETERS!!!!
.T&
cb r r r l l.
25!Compliance!Cms:!1.58e-3!m/N!
26!Mass!Md:!14.8e-3!Kg!
27!Losses!Rms:!1.33!Kg/s!
28!Force factor!Bl:!7.03!N/A!
29!!!!!
30!!Qmc:!7.60!!
31!!Qec:!1.40!!
.TE
.ps +2
.vs +2
.(c
Figure 7.2
.)c
.)b
.pp
A few items are notable. First, the geometric means of the low and high
frequency points on the impedance curves are calculated and displayed on row
15 to provide a check for the accuracy of frequency measurements. Both are 
skewed slightly higher in frequency than the corresponding resonance because
of the relatively greater affect of the voice coil inductance on the higher
frequency impedance measurements. Secondly, rows 30 and 31 are used to
calculate the mechanical and electrical Q's of the driver in the test chamber,
and the resulting data is used further to calculate the drivers equivalent
volume $V sub AS$.

.sh 1 "REFERENCES"
.ip [1]
A. N. Thiele, "Loudspeakers in Vented Boxes,"
\fIJ. Audio Eng. Soc.\fR,
vol 19, pp. 382-392 (1971 May); pp. 471-483 (1971 June).
.ip [2]
R. H. Small, "Direct Radiator Loudspeaker System Analysis,"
\fIJ. Audio Eng. Soc.\fR,
vol. 20, pp. 383-395 (1972 June).
.ip [3]
R. H. Small, "Closed-Box Loudspeaker Systems, Pts. I and II,"
\fIJ. Audio Eng. Soc.\fR,
vol. 20, pp 798-808 (1972 Dec.);
vol 21, pp. 11-18 (1973 Jan.).
.ip [4]
R. H. Small, "Vented-Box Loudspeaker Systems, Pts. I-IV,"
\fIJ. Audio Eng. Soc.\fR,
vol. 21, pp. 363-372 (1973 June);
pp. 438-444 (1973 July/Aug.);
pp. 549-554 (1973 Sept.);
pp. 635-639 (1973 Oct.).
.ip [5]
G. Margolis and R. H. Small,
"Personal Calculator Programs for Approximate
Vented-Box and Closed-Box Loudspeaker System Design,"
\fIJ. Audio Eng. Soc.\fR,
vol. 29, no. 6 pp. 421-441 (1981 June).
.ip [6]
R. Bullock and R. White, 
"BOXRESPONSE: An Apple Program for the Thiele-Small Models,"
\fISpeaker Builder\fR, vol. 5, no. 1 pp. 13-18 (1984 March).
.ip [7]
M. L. Lampton, "Program WOOF: A Numerical Evaluator of
Loudspeaker Systems," \fIIEEE Trans. Audio Electroacoust.\fR,
vol AU-20, pp. 354-366 (1972 Dec.).