Bullock on Boxes - Bullock Romert M - 1993, Elektronika q
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1
I
BULLOCK
BOXES
byRobertM. Bulfock
lll
ON
assistedby
RobertWhite
I
t
I
I
I
I
Chapter
One
THIELE,SMALL,
AND
VENTED
I.,OUDSPEAKER
DESIGN
'l'he
name Thiele and
the term Thiele
I alignment
areknown bY manYwith
an intirest in loudsoeakers.
What is a
Thiele alignment? How is
it obtained?
How is it used?
To answer these
questions,
I shall
describe the
landma* work of A-N.
Thiele
and R.H. Small'ssubsequent
re-
finementsin thedesign
of vented
(bass
reflex)loudsoeakers.-l
haveused
their
work in designing.myown syJtems;
it
provides
a method
wherebythe nome
builder
can constructloudspeaker
sy-
stemsof truly impressive
quality
Before
Thiele, vented loudsPeaker
desienwassimplebut, more
oftenthan
A model's purposeis as follows. The
system of interest
(in
this cas€a vented
loudspeaker) is converted
(in
an ab-
stract sense)into the model,
a system
chosenbecause
its behavior can be an-
alvzed bv some
known theorv
(circuit
thlory in our case).Having inalyzed
themodel.we convertour conclusions
back to conclusions about the system
which interestsus. This analysis
meth-
od is much favored by scientists.
Roughly, they discover new
facts bY
interpretingold facts
in a new settinS.
The model of a ventedloudspeaker
system,the hish
pass
filter, is i well
understood
eleitriial circuit.Any lacet
of its behavior can be determined
by
aoolvineknown
theoryto its schema(-
ii iiisrim. By usingthismodel,Thiele
and S-mallidintifieii
those elementsof
the loudspeaker
system that signifi-
cantlvaffecteditsbehaviorand
quanti-
fied lhese elements
and effects.-They
then based their design
procedureson
this information.
FREQUENCY
RESPON5E
The most significant aspect
of loud-
thatallinput.signals
havethe
samestranSth, the device.may
produce
an outpu[
that vanes wlth lrequency
and this variation is
the lrequency
response.
We can
visualize this most
simply in
terms of a
graph
called_
a frequencyre-
sponse.curye,.si8nal
frequency,belng
Dlottedon thehorizontalaxis
and rela-
iive sisnalstrenqth
on theverticalaxis.
Sisnal'frequenci
is measuredin
Hertz
(H-z)
andrilativL signal
stren8thin dec-
ibels
(dB).
Figure 1 is a frequency
re-
soonse
curve. Think
of the 0 dB level
on the vertical axis
as a reference
level,
i.e.,
a
positive
decibelmeasure
at a
gi-
ven fiequency
means the outPut
strength
at
.that
frequercy is
greater
than the reterence,
whlle a negatlve
sound.
The
earlier Drocedure
dtd not allow
ror a
ventediystem'scriticaldependence
on
certainamplifier
and driver character-
istics.ThieleandSmall's
procedures
do
allow for thesecharacteriitics
and
yield
accurate
and
predictable
results-in
other
words, they
will
Produce
good
sound,
I shall
describe the derivation
of
these
procedures
and
their underlying
a.sumptions. I
shall also include the
necessiry
data for designing
your own
systemand
provide someexamples
to
illustrate
the design
procedures.
BACKGROUND
A loudspeaker
svstemis a complicated
combinition o['electrical,
meihanical
and acoustical
components and
this
mixture of different components
make
analvzinsits
behavior
quite
difficult.
Ho-everl vearsof research
have led to
a usableth-eory,that a
loudspeakersy-
stemcan be visualizedas an electrical
circuit for
purposesof analysis- i.e.,
we can use an
electrical circuit as a
model of a loudspeakersystem.
Speci-
ficallv, Novak,
has shown that a hiSh
Dass
iilter electrical circuit
can be the
model for a
vented loudsPeakersY-
stem.
od8
-3dB
FREAU€NCY
(HT)
BuIIochon Boxes 3
speaker
behavior for design
purposesis
freouencv response.Think
ol a loud-
speiker iysteir and its model,
the high
pass
filtei, as
deviceswhich accept an
input signal and operate on
it to pro-
duce an output signal.
Assume that
neitherdevici will;lter the signal
fre-
auencv;
what
we
can
alter is the rela-
tive sienal strensth.
ln other words,
assumi-ng
not
-produced.poor.quality
I
T
I
I
T
I
I
t
t
T
T
I
t
I
t
t
I
t
I
t
decibel measuremeansit is smaller.
Referringto Fg, l,
f3 is calledthe
cutoff frequency, frequenciesabove f"
are said to be in the passband;
those
below,
in thestopband.ln
practice,
the
stopband outpuf is at
such a low ievel
it
is insisnificant,so. onlv frequencres
abovef, are
"passed."
We canihink of
Fiq. 1 as a typical
,
responsecurve for
either a loudspeaker
svstem or its
model, the high
pass
filtdr.
To decidewhether a oarticular loud-
oda
-3da
'i
'
,.,
r
:
.t,.1
t'
speaker system will.
-produce
good
souno,we mustexamlneIt5rlequency
responsecurve.
To find it usethehigh
pass
filter modelandfind its frequency
response
by studying its schematicdia-
gram
and applying electrical circuit
theorv.
The resultineresponsewill be
that
6f the loudsoeafersvstem-
The key to suicessin using a model
is to establish a relationshiD between
the partsof themodel
andthe
parts
of
the systemin sucha way that onecan
translate
a
requirement
in the model in-
to a requirement on the
system being
modelled. Thiele accomplishedthis by
identifying various parimeters
of thi
loudspeakersystemwith parameters
A 84 response cunte.
in
the
model,thusidentifvinstheDartsof
the loudspeaker svsfem"which in-
tluence resPonse.
Electricalengineerscall the
partrcu-
Iar parameterielationships
neided to
obtaina
given
response
odB
-3d g
in themodelan
alignment.Thiele
adoptedthis term to
describethecorresponding
loudspeak-
er systemparameter
relationships;
this
is the origin of the term,
'Thiele
align-
ment.
POSSIBLERESPONSECURVES
ln order to know what
kinds of re-
sponsecurvesIoudspeakersystemscan
have, let us consider
some of the mod-
els known responsecurves.High pass
filters
can exhibit a wide ra-neeof
curyes, so we need
to make sorie as-
sumptions as to what kinds of re-
sponses
are desirable for loudspeaker
systems.
We want the responsecurve to be as
"flat"
as
possible
FIG5
in the
passband.
With
this constraint,Thiele
and Smalliden-
tified
three tvpes of filter response
curves.The fiist is
that of a fourih ,r,-
der Butterworthfilter, whoseresponse
curve is
shown in Fic.2. The filter's
name is abbreviated-to
84, and we
shall use this designationfor the
re-
sponse
curve aswell. A responsecurve
generally
takes the
same name as the
filter from
which it derives.
Becausedrivers have
different char-
acteristics,we
cannotalwavs
set
a 84
responsefrom a loudspealier
-system.
I o cover
a rangeol driver
charactens-
tics, Thiele
and Snall also used a
quasi-third
order Butterworth
filter,
QB3,
and a
Chebyshevfilter,Ca. Figs.
3 and 4 show typical responses
ior
A
C4
oaiant response,
filters
of
thesetwo types.The main dif-
ferencesbetween a 84 and a
QB3
are
that
a
QB3
has
more
"droop"
in the
passband
and its cut-off frequency is
hisher. C4 resDonsesare distinsuished
by- a ripple i'n the
passband-and
a
smaller cut-off frequency than a 84.
You need
to know
the
sizeof
this
ripple
before
deciding
whether this response
curve would be desirable for a
loudspeaker
system.
Both Thiele and Small considered
primarily
thesethreeresponsesasa ba-
sis for their alignments, but there ar€
many other
possibilities.
Two are
shown in Figs.5 and 6. The Fig. 5 curve
is a fourth order boom box response,
denoted BB4; alignments
yielding
this
responseappear in an article by Hoger.
In another articlel Hoge
gives
align-
mentswhich are variations
of
Small's
alignments;Fig. 6 is a variantof a C4.
ror rmproveo
translent response,
Small
recommends Bessel
(Be4)
and
sub-Chebyshev
(SC4),
both of which
resemblea
QB3
(F g
3) but have more
droop in the passband.Thiele alsocon-
siders what are called higher order re-
sponses.
I have built a subwoofer
sys-
tem using
a sixth
order Butterworth
response
(86)
and the sound is out-
standing.ln this article, however,
we
will consider only the
QB3-84-C4
seriesof responses.
You may-
wonder about the use of
the word
"order"
in the namesof filters
and resDonses.This term
is
used
to
classifyhigh
pass
filtersand
therateat
which the resDonsecuryes decteaseat
low frequenciis. First order fiiters de-
crease
at
the rate of
6dB
per octave, se-
cond order at 12dB
p€r
octav€, etc,
Vented loudspeaker
responsecurves
corresoond to fourth
order filters and
so their resoonsecurvesdecreaseat the
rateof 24dil
per
octaveat low frequen-
cies. Closed box loudsoeakersvstems
are modelled by secorid order- fitters
and so their respons€s
fall off at 12dB
per
octave. This-decrease
the advertiseddiarneterof the driver in
inches.This meansthe driver will cov-
er a frequency rangefrom 0 to 500O/d
Hz. Depending on its
properties,
it
may be
usedat evenhigher frequencies
and neednot necessarilybe usedall the
way down to OHz; for example, Elec-
troVoice sellsa unit with a vented mid-
range
driver. Nevertheless,vented
de-
signs are usually associatedwith the
bass
section of a multi-dliver system,
Five parameters determine the dri-
ver's
influence on resDonse.First is
the
dc-resistanceof the vbice coil, R,. Sec-
ond is the resonant frequency of the
driver, f". The next two are
Q
numbers
and measure how effectively the
driver's resistive
parts
damp it a( its
resonant frequency: the smaller the
Q
number, the more effective the damp-
ing.
Qs5
is the
Q
due to electrical resis-
tance;
Qrs
is that due to mechanical
reslslance.
The final driver
parameter
Sives
the
compliance, or springiness,of the dia-
phrasm
mountinq. exDressedin terms
bf a iolume of ai"i wliich would have
the samecomplianceand denoted Vrr.
R6 is given in ohms, f" in
Hertz,
V,s in
cubic inches, centimeters,feet etc; the
Q
numbers are dimensionless.We ar-
rive at these figures by measuring
driver impedance.A driver's voice coil
inductancecan be sizeable,but we can
ignore it since it usually has negligible
effectsin the driver's
piston
range.
The amplifier, crossover network,
and conneiting
cablealso contribute
to
driver damping. To take this into ac-
count. w€ must modifv the driver's
electrical
Q.
Supposethecrossoverand
connectins cableshave a resistanceof
TABLE
I
SMALLALIGNMENTS
FOR
Q.:5
TABLEII
SMALLALIGNMENTS
for
Q":7
Ripple
Ripple
h
c(
t1/Is
(dB)
7.9393 7.7n5 2.5289
.n
1.8494
6.9524 2.3968
7.7678 6.2372 2.2759
'1..6935
5.6132 2.1&7
1..62545.0655
2.0620
7.5629 4.5822 7.9667
1.5054 4.7535 1..8778
1.4522
3.77t4
1.7946
1..40293.4295 7.71.65
1-3577 3.-i223 1.6429
1..37452.8452 1.5732
1,.27482.5944 7.5070
:
7.2376 2.3667 1..4439
7.2028 2.7594 7.3836
1.1702 1..96997.32s8
1.1395' 1.,7964' t.2702
1..1\06 1.6377 t.2167
1.0834 1.4905 1.1651
1.05781.3552 1. .53
1.0335 1.2300 1.0674
1.01031.11461.0215
.9886
7.@70 .9777
.9662 .9113 .9373
.9436 .E266 .9001.
.9272
.7527 .8660
-8992 .6868 .8348
.01
.8780 .6297
.8064 .01
.8578 .5798 .7ffi4 .02
.8385 .5361
.7567 .03
.8203
.4978 .73st .05
.8031 .4642 .nss .07
.7870 .434s .6975
.09
.7719 .4083
.6810 .12
.757E
.3849 .66s9 .1s
a'"
.2000
.2100
.22co
.2300
.24co
.2500
.2&O
.2700
.2800
.29oo
.3000
.3100
.3200
.3300
.3400
.3500
.3600
.3700
.3800
.3900
.4000
.4100
.aoo
.4300
.4400
.4500
.4ffi
.47@
.4800
.4900
.5000
.5100
.5200
.5300
.5400
.5500
.5600
.5700
.5800
.5900
.6000
.6100
.62co
.6300
.64co
.6500
Ir/Is
(dB)
4,,
.2W0
.2100
.2200
.2300
.2400
.2500
.2@O
.2700
.2800
.29oo
.3000
.3100
.3200
.3300
.3400
.3500
.3600
.3700
.3800
.3900
.4000
.4100
.42co
.4300
.4400
.4500
.4600
.47co
.4800
.4900
.5000
.5100
.s200
.5300
.5400
.5500
.5600
.5700
.5800
.5900
.6000
.6100
.6200
.6300
.6400
.6500
z.@1.4 7.5746 2.5914
1. ao 6.7702 2.4566
't.4232
6.0730 2.3332
7.7459 5.4646 2.2798
7.6757 4.9346 2.77s7
7.6701.4.4594 2.0780
7.5502 4.0415 7.9276
7.4948 3.6697 7.8430
7.4434 3.3358 7.7637
7.3957 3.0364 7,6889
7.351.22.7663 7.61.83
1.3@7 2.5220 7.5574
1..27082.3@7 1..4877
't.2344
2.0980 1.4269
1..2m3 1.9134 136a7
L16at 1.7444 1.3129
1..1378L.5893 7.2592
L1493 1.4464 \.2074
1..08237.3147 1..7576
1.0s68
1.1929 1.109s
1.0326 1.0801 1.0632
1.0095 .9757 1,.01.90
.9877 .8785 .9767
.9552 .7920 .9377
.9425 .7754 .90 6
.9200 .6480 .8684
.8979 .5888 .8379 .01
.8766 .5370 .E100 .01
.8560 .4915 .7844 .O2
.8364 .451.6 .7609 .03
.at7E .41.66 .7395 .04
.8002 .3857 .7198 .06
.7836 .3583 .70t7 .08
.7680 .3340 .6852 .11
.7s33 .3122 .6699 .13
.7394 .2927 .6558 .16
.7263 .2752 .6428
.20
.7140 .2592 .6307 .23
.7024 .2447 .679s .27
.6915 .2374 .6W7 .31
.6811 .2192 .5994 .35
.671.3 .2080 .5903 .40
.6620 .1975 .5818 .44
.6531 .1.878 .5738 .49
.6447 .7787 .5663 .54
.6367 .7707 .5592 .s9
in responseat
iow freouencies iust reflects the
well-
known fict that loudspeakersytems
do
not provide significant output at very
low freouencies.One of our designob-
iectives
is to obtain thelowest
pdssible
cut-off frequency consistentwiih a flat
Passband.
ALIGNMENT PARAMETERS
We need to know which oarametersof
the loudspeaker svstem
determine its
,"sponse i,r-", Thiele found the fre-
quency
response
is
completely
deter-
mined by several amplifier. driver, en-
closure, and vent oarameters
which re-
flect those
parts
of the
loudspeakersys-
tem related to the model's electrical
comDonenrs.
Aisume
that the driver is a moving
coil diaphragm type operaling in its
piston range.
According
to Thiele, this
is from OHz to SOOO/dHz where d is
.744s .3640 .6sm .79
'
.7327 .3453 .6393
.23
t
.7205
.3284 .627s .27
"
.7096 .3131 .6166
.31
.6993
.2992 .6065 .36
.6896 .2865 .5971 .41
.6805 .2749
.5883 .46
.6719 .2641 .5ffi2 .51
.6638 .2542 .5726 .57
-6s67 .2449
.5654 .63
.6488 .2363 .5587 .6E
.6418 .2283 .5524 .74
't
Bullochon Eoxes 5
R, ohms in serieswith the driver, and
the damping
effectof the amplifier is
the sameas a resistorof R, ohms in
serieswith
the
driver;
then we change
Q*
to
Q'*
by theformula:
and amplifier combination.Thieleand
Small based their alignments on the
QB3-84-C4
seriesof responsesso that
only onepossiblealignmentisobtained
for each
value
of
Q'".
We
will cover
only thesealignments.
Tablesl, II, and.f/1list Small align-
ments.To us€a table,find thevalueof
Qrs
in the first
column
and
use the val-
uesof h, a, f:/fr found in that row. We
can also find alignmentsby
various
formulas-for
example, Saffran's for-
mulas in Speaker
Builder, lssue L/80,
p.
35
("Mailbox"
section).With these
formulas, you can compute a Thiele
alignment from a
given
value of
Q'(:Q'").
Such formulas are
handy
but provide only approximate values,
whereasaccurately
prepared
tables
will
provide
exact
values.
THIELE ALIGNMENTS
Thieleswas the first to analyze
the elec-
tricalcircuit modelin order to
provide
specificvented speakersystem
align-
ments.ln this landmarkpaper he
pre-
senteda large tableof alignments,
the
first nineof
which werein the
QB3-84-
C4 series.These
nine are
just
samples
from a continuumof
QB3-84-C4
align-
ments
that can be obtainedby varying
the value ot
Qrs
. If
Q.s
--
.383,
the
alignmentis a 84;
if
Q^<.383,
it is a
QB3,
andit
Q,r
> .383.it is a C4.
There are usuallv
practical
bounds
on the values
ol
Q,,.
It
Q,,
< .2, the
cut-off
frequencyis usuallyundesirab-
ly high. At the otherextreme,
for large
Qrs
lhe alignments
are C4 and so the
responsehasa ripplewhich
increases
TABLEIII
SMALLALIGNMENTS
for
Q.:19
Ripple
h
qF
t'/t"
(dB)
1,.89607.9232 2.4845
1.8085 7.0834 2.3s43
7.7292 6.3554
2.2357
1.64O9 5.tZU2 t.LZ)5
1,.59085.7627
2.0241
7.5307 4.6706 1.9299
7.4742 4.2342 1.8421
1.4225 3.U52 1.7599
7.3747 3.4971 7.6826
1.3303 3.1843
1.6097
7.2890 2. 22 1.5406
7.2505 2.6469 1.4748
7.2746 2.4750
1.4721-
1.1809 2.2038 1.3521
1..14932.07@ r.2945
1..11977.8342 7.2390
1.0918
1.6719 1.1855
1.0656 1.5225 1.1339
1.0109
1.3846 1.0841
7.0\75 1,.25717.0363
.9954 1.1390 .9907
.9732 7.03?J .9482
.9507
.9381 . 92
.9282 .85s0 .8736
.9062 .7a22 .8410 .01
.8848 .7187
.8114 .01
.8644 .6632 .78/.4 .02
.8451 .6148 .7ffi .03
.8269 .s725
.7377 .05
.&97 .535s .7775 .O7
.7937 .5029 .6997
.10
.77a7 .4742 .6823 .13
.7648 .4487 .6670 .1.6
.7577 .4261 .6529
.20
.7396
.4059 .6407 .24
.7282 .3877 .6282 .29
.7776 .3714 .6773
.34
.7077 .3s65
.@72 .39
.6983 .3431. .s979 .44
.6896 .3308 .s892 .50
.6874 .3195
.s472 .55
.6736 .3@2 .s737 .67
.6663 .2996 .s667 .68
.6s94 .2ffi7 .5601
.74
.6529 .2825 .5s40 .80
.6467 .2748 .s482
.87
a'"
.2.AOO
.2100
.22ffi
.2300
.24co
.2500
.26co
.27cD
.2800
.z9oo
.3000
.3100
.3200
.3300
.3400
.3500
Q'85
:
[(RE+&+R,)/R.]Q.'.
(1)
One way to determine R, is to hook
everything
up as
it would be in the
finishedsystemand measurethe resis-
tance between the amplifier connec-
tions.
Subtract
R" from thisandthere-
sult is
R.
Small" describes the most accurate
method for finding
&;
or, if we know
the damping factor D of the amplifier,
we can find R" from the formula
&
:
R"/(D-1)
Q)
.3700
.3800
.3900
.4000
.4100
.4200
.4300
.4400
.4500
.4600
.47co
.4800
.49co
.5000
.5100
.5200
.s300
.5400
.5500
.5600
.57co
.s800
.5900
.6000
.6100
.62@
.6300
.6400
.6500
where R" is the nominal impedance
of
the
driver. Typical
damping factors
varv from 15 to 500; 25-30seemsto
be
fairiy common. Thiele'
says that if
R, * R, is lessthan 5 percentof R", then
we can use the unmodified
value
Qss
if
responsevariations
of up to .4dB are
tolerable.
The actual
Q
number used
to specify
an alignment is called the total
Q,
denoted
Q,
or
Q,",
and
is tound from
the formula
a^: a'," Q.,/(Q'"
+
Q""),
(3)
or
7/Qrs:7/Q'Es
+ 7/Q,s.
14)
in
ma8nitude
t\ ith-Q,s.By
Q,.:
.7 this
rippleexceeds1.5d8and an audiophile
would
probably find it obiectionable.
Exact alignment values are
quite
comolicatedto calculateand are
best
found usinga computer.I will not pro-
vide Thiele alisnmentsfor reasonsto
Assuming we know
the values for
bemadeclear
below.lf you would like
Q'",
f", and V,.s,an alignment
is a set to usethem anyway, Saffran'sformu-
of relationshipswhich mustbesatisfied lasshouldbe sufficient.Error canbeas
between
Q'",
f", Vrr, Vr, and
f,. The high
as5
percent.Besureto correcthis
relations
for
given
Qrs
are that the tormula for h to h: .38/Q,.
3,111f
TABLE
IV
ALIGNMENT FORMULAS
j,',1|)l;rrot'ALrcNMENrs
analyzins the model; Small denotes Small" observedthat vented
loud-
thesetw; ratiosby
h ando
{alpha)
re-
speaker
systemsdesignedaccordingto
spectively.Thus,
an alignmentis equi- Thiele alignmentsdid not always ex-
valent to a list of the three numbers hibit the freq'-rency
,."f
",l];fi
\lr-J
h: .aL9/et"l,'
q.:.07s3/els'.1
f3lfr
:
.31s/QlrrlJ
response
predicted
Q',,
h, e: for example,a Thiele 84 by
the model. He determinedthat the
alignment is
Q'":.363,
5:1,
q:./2.
box and vent wereexertinga damping
lf you have a driver with
Q6:,383,
effectwhichalteredtheresponse
curve.
fs: 25H2, and V"":1gggg;nj, then a He
referredto thesedampingeffectsas
84 responsewill be obtainedif f":
lossesbecausethey usuallyresulted
in
hf":
i;25:25112 and
Vr:V,"zc: decreasedoutPut at certain
f.equen-
\zL-t
h:.420lQrtr
e:.0s69lQiJ5J
frlfr
:
.3os/Q+ir-
1OOOO/J2:7071inr.An alignmsnlu5-
cies,and accountedfor them by
"ob-
ually
includesa value of f7f" also so
serving that theselossesmay
be ade-
you
can seewhat thecut-offfrequency
quately
approximated
for design
pur-
will
be with the.alignment;however,
posesby a singlefrequencyinvariant
vou do not need it to make vour de- leakageloss.'
sign.
ln otherwords, heintroducedanoth-
An alignmentuniquelydeterminesa
er
parameterrvhich represented
Q':ro
h:.421lQr?r
o.:.0689/Qi"",'
fJfs
:
'2q619it""
losses
Thevalueof h
is
usually
within29o.
thevalueof f,i f, within67o,andthe
value of
o.
between
_177o
and
+25 C".
responsecurve for
j
paiticular driver dui to box and
driver leaks,soundab-
6 Bullock on Boxes
which is sometimesmore convenient.
The remaining influences
on re-
sDonseare the volume V, of
thebox on
*hi.h th" driver is mounted
and the
resonant freguency f, of the box
resultingfrom the
presence
of the
vent.
ALIGNMENTS
illi'"li1,Lilf,
[ Pobierz całość w formacie PDF ]
zanotowane.pl doc.pisz.pl pdf.pisz.pl lemansa.htw.pl
1
I
BULLOCK
BOXES
byRobertM. Bulfock
lll
ON
assistedby
RobertWhite
I
t
I
I
I
I
Chapter
One
THIELE,SMALL,
AND
VENTED
I.,OUDSPEAKER
DESIGN
'l'he
name Thiele and
the term Thiele
I alignment
areknown bY manYwith
an intirest in loudsoeakers.
What is a
Thiele alignment? How is
it obtained?
How is it used?
To answer these
questions,
I shall
describe the
landma* work of A-N.
Thiele
and R.H. Small'ssubsequent
re-
finementsin thedesign
of vented
(bass
reflex)loudsoeakers.-l
haveused
their
work in designing.myown syJtems;
it
provides
a method
wherebythe nome
builder
can constructloudspeaker
sy-
stemsof truly impressive
quality
Before
Thiele, vented loudsPeaker
desienwassimplebut, more
oftenthan
A model's purposeis as follows. The
system of interest
(in
this cas€a vented
loudspeaker) is converted
(in
an ab-
stract sense)into the model,
a system
chosenbecause
its behavior can be an-
alvzed bv some
known theorv
(circuit
thlory in our case).Having inalyzed
themodel.we convertour conclusions
back to conclusions about the system
which interestsus. This analysis
meth-
od is much favored by scientists.
Roughly, they discover new
facts bY
interpretingold facts
in a new settinS.
The model of a ventedloudspeaker
system,the hish
pass
filter, is i well
understood
eleitriial circuit.Any lacet
of its behavior can be determined
by
aoolvineknown
theoryto its schema(-
ii iiisrim. By usingthismodel,Thiele
and S-mallidintifieii
those elementsof
the loudspeaker
system that signifi-
cantlvaffecteditsbehaviorand
quanti-
fied lhese elements
and effects.-They
then based their design
procedureson
this information.
FREQUENCY
RESPON5E
The most significant aspect
of loud-
thatallinput.signals
havethe
samestranSth, the device.may
produce
an outpu[
that vanes wlth lrequency
and this variation is
the lrequency
response.
We can
visualize this most
simply in
terms of a
graph
called_
a frequencyre-
sponse.curye,.si8nal
frequency,belng
Dlottedon thehorizontalaxis
and rela-
iive sisnalstrenqth
on theverticalaxis.
Sisnal'frequenci
is measuredin
Hertz
(H-z)
andrilativL signal
stren8thin dec-
ibels
(dB).
Figure 1 is a frequency
re-
soonse
curve. Think
of the 0 dB level
on the vertical axis
as a reference
level,
i.e.,
a
positive
decibelmeasure
at a
gi-
ven fiequency
means the outPut
strength
at
.that
frequercy is
greater
than the reterence,
whlle a negatlve
sound.
The
earlier Drocedure
dtd not allow
ror a
ventediystem'scriticaldependence
on
certainamplifier
and driver character-
istics.ThieleandSmall's
procedures
do
allow for thesecharacteriitics
and
yield
accurate
and
predictable
results-in
other
words, they
will
Produce
good
sound,
I shall
describe the derivation
of
these
procedures
and
their underlying
a.sumptions. I
shall also include the
necessiry
data for designing
your own
systemand
provide someexamples
to
illustrate
the design
procedures.
BACKGROUND
A loudspeaker
svstemis a complicated
combinition o['electrical,
meihanical
and acoustical
components and
this
mixture of different components
make
analvzinsits
behavior
quite
difficult.
Ho-everl vearsof research
have led to
a usableth-eory,that a
loudspeakersy-
stemcan be visualizedas an electrical
circuit for
purposesof analysis- i.e.,
we can use an
electrical circuit as a
model of a loudspeakersystem.
Speci-
ficallv, Novak,
has shown that a hiSh
Dass
iilter electrical circuit
can be the
model for a
vented loudsPeakersY-
stem.
od8
-3dB
FREAU€NCY
(HT)
BuIIochon Boxes 3
speaker
behavior for design
purposesis
freouencv response.Think
ol a loud-
speiker iysteir and its model,
the high
pass
filtei, as
deviceswhich accept an
input signal and operate on
it to pro-
duce an output signal.
Assume that
neitherdevici will;lter the signal
fre-
auencv;
what
we
can
alter is the rela-
tive sienal strensth.
ln other words,
assumi-ng
not
-produced.poor.quality
I
T
I
I
T
I
I
t
t
T
T
I
t
I
t
t
I
t
I
t
decibel measuremeansit is smaller.
Referringto Fg, l,
f3 is calledthe
cutoff frequency, frequenciesabove f"
are said to be in the passband;
those
below,
in thestopband.ln
practice,
the
stopband outpuf is at
such a low ievel
it
is insisnificant,so. onlv frequencres
abovef, are
"passed."
We canihink of
Fiq. 1 as a typical
,
responsecurve for
either a loudspeaker
svstem or its
model, the high
pass
filtdr.
To decidewhether a oarticular loud-
oda
-3da
'i
'
,.,
r
:
.t,.1
t'
speaker system will.
-produce
good
souno,we mustexamlneIt5rlequency
responsecurve.
To find it usethehigh
pass
filter modelandfind its frequency
response
by studying its schematicdia-
gram
and applying electrical circuit
theorv.
The resultineresponsewill be
that
6f the loudsoeafersvstem-
The key to suicessin using a model
is to establish a relationshiD between
the partsof themodel
andthe
parts
of
the systemin sucha way that onecan
translate
a
requirement
in the model in-
to a requirement on the
system being
modelled. Thiele accomplishedthis by
identifying various parimeters
of thi
loudspeakersystemwith parameters
A 84 response cunte.
in
the
model,thusidentifvinstheDartsof
the loudspeaker svsfem"which in-
tluence resPonse.
Electricalengineerscall the
partrcu-
Iar parameterielationships
neided to
obtaina
given
response
odB
-3d g
in themodelan
alignment.Thiele
adoptedthis term to
describethecorresponding
loudspeak-
er systemparameter
relationships;
this
is the origin of the term,
'Thiele
align-
ment.
POSSIBLERESPONSECURVES
ln order to know what
kinds of re-
sponsecurvesIoudspeakersystemscan
have, let us consider
some of the mod-
els known responsecurves.High pass
filters
can exhibit a wide ra-neeof
curyes, so we need
to make sorie as-
sumptions as to what kinds of re-
sponses
are desirable for loudspeaker
systems.
We want the responsecurve to be as
"flat"
as
possible
FIG5
in the
passband.
With
this constraint,Thiele
and Smalliden-
tified
three tvpes of filter response
curves.The fiist is
that of a fourih ,r,-
der Butterworthfilter, whoseresponse
curve is
shown in Fic.2. The filter's
name is abbreviated-to
84, and we
shall use this designationfor the
re-
sponse
curve aswell. A responsecurve
generally
takes the
same name as the
filter from
which it derives.
Becausedrivers have
different char-
acteristics,we
cannotalwavs
set
a 84
responsefrom a loudspealier
-system.
I o cover
a rangeol driver
charactens-
tics, Thiele
and Snall also used a
quasi-third
order Butterworth
filter,
QB3,
and a
Chebyshevfilter,Ca. Figs.
3 and 4 show typical responses
ior
A
C4
oaiant response,
filters
of
thesetwo types.The main dif-
ferencesbetween a 84 and a
QB3
are
that
a
QB3
has
more
"droop"
in the
passband
and its cut-off frequency is
hisher. C4 resDonsesare distinsuished
by- a ripple i'n the
passband-and
a
smaller cut-off frequency than a 84.
You need
to know
the
sizeof
this
ripple
before
deciding
whether this response
curve would be desirable for a
loudspeaker
system.
Both Thiele and Small considered
primarily
thesethreeresponsesasa ba-
sis for their alignments, but there ar€
many other
possibilities.
Two are
shown in Figs.5 and 6. The Fig. 5 curve
is a fourth order boom box response,
denoted BB4; alignments
yielding
this
responseappear in an article by Hoger.
In another articlel Hoge
gives
align-
mentswhich are variations
of
Small's
alignments;Fig. 6 is a variantof a C4.
ror rmproveo
translent response,
Small
recommends Bessel
(Be4)
and
sub-Chebyshev
(SC4),
both of which
resemblea
QB3
(F g
3) but have more
droop in the passband.Thiele alsocon-
siders what are called higher order re-
sponses.
I have built a subwoofer
sys-
tem using
a sixth
order Butterworth
response
(86)
and the sound is out-
standing.ln this article, however,
we
will consider only the
QB3-84-C4
seriesof responses.
You may-
wonder about the use of
the word
"order"
in the namesof filters
and resDonses.This term
is
used
to
classifyhigh
pass
filtersand
therateat
which the resDonsecuryes decteaseat
low frequenciis. First order fiiters de-
crease
at
the rate of
6dB
per octave, se-
cond order at 12dB
p€r
octav€, etc,
Vented loudspeaker
responsecurves
corresoond to fourth
order filters and
so their resoonsecurvesdecreaseat the
rateof 24dil
per
octaveat low frequen-
cies. Closed box loudsoeakersvstems
are modelled by secorid order- fitters
and so their respons€s
fall off at 12dB
per
octave. This-decrease
the advertiseddiarneterof the driver in
inches.This meansthe driver will cov-
er a frequency rangefrom 0 to 500O/d
Hz. Depending on its
properties,
it
may be
usedat evenhigher frequencies
and neednot necessarilybe usedall the
way down to OHz; for example, Elec-
troVoice sellsa unit with a vented mid-
range
driver. Nevertheless,vented
de-
signs are usually associatedwith the
bass
section of a multi-dliver system,
Five parameters determine the dri-
ver's
influence on resDonse.First is
the
dc-resistanceof the vbice coil, R,. Sec-
ond is the resonant frequency of the
driver, f". The next two are
Q
numbers
and measure how effectively the
driver's resistive
parts
damp it a( its
resonant frequency: the smaller the
Q
number, the more effective the damp-
ing.
Qs5
is the
Q
due to electrical resis-
tance;
Qrs
is that due to mechanical
reslslance.
The final driver
parameter
Sives
the
compliance, or springiness,of the dia-
phrasm
mountinq. exDressedin terms
bf a iolume of ai"i wliich would have
the samecomplianceand denoted Vrr.
R6 is given in ohms, f" in
Hertz,
V,s in
cubic inches, centimeters,feet etc; the
Q
numbers are dimensionless.We ar-
rive at these figures by measuring
driver impedance.A driver's voice coil
inductancecan be sizeable,but we can
ignore it since it usually has negligible
effectsin the driver's
piston
range.
The amplifier, crossover network,
and conneiting
cablealso contribute
to
driver damping. To take this into ac-
count. w€ must modifv the driver's
electrical
Q.
Supposethecrossoverand
connectins cableshave a resistanceof
TABLE
I
SMALLALIGNMENTS
FOR
Q.:5
TABLEII
SMALLALIGNMENTS
for
Q":7
Ripple
Ripple
h
c(
t1/Is
(dB)
7.9393 7.7n5 2.5289
.n
1.8494
6.9524 2.3968
7.7678 6.2372 2.2759
'1..6935
5.6132 2.1&7
1..62545.0655
2.0620
7.5629 4.5822 7.9667
1.5054 4.7535 1..8778
1.4522
3.77t4
1.7946
1..40293.4295 7.71.65
1-3577 3.-i223 1.6429
1..37452.8452 1.5732
1,.27482.5944 7.5070
:
7.2376 2.3667 1..4439
7.2028 2.7594 7.3836
1.1702 1..96997.32s8
1.1395' 1.,7964' t.2702
1..1\06 1.6377 t.2167
1.0834 1.4905 1.1651
1.05781.3552 1. .53
1.0335 1.2300 1.0674
1.01031.11461.0215
.9886
7.@70 .9777
.9662 .9113 .9373
.9436 .E266 .9001.
.9272
.7527 .8660
-8992 .6868 .8348
.01
.8780 .6297
.8064 .01
.8578 .5798 .7ffi4 .02
.8385 .5361
.7567 .03
.8203
.4978 .73st .05
.8031 .4642 .nss .07
.7870 .434s .6975
.09
.7719 .4083
.6810 .12
.757E
.3849 .66s9 .1s
a'"
.2000
.2100
.22co
.2300
.24co
.2500
.2&O
.2700
.2800
.29oo
.3000
.3100
.3200
.3300
.3400
.3500
.3600
.3700
.3800
.3900
.4000
.4100
.aoo
.4300
.4400
.4500
.4ffi
.47@
.4800
.4900
.5000
.5100
.5200
.5300
.5400
.5500
.5600
.5700
.5800
.5900
.6000
.6100
.62co
.6300
.64co
.6500
Ir/Is
(dB)
4,,
.2W0
.2100
.2200
.2300
.2400
.2500
.2@O
.2700
.2800
.29oo
.3000
.3100
.3200
.3300
.3400
.3500
.3600
.3700
.3800
.3900
.4000
.4100
.42co
.4300
.4400
.4500
.4600
.47co
.4800
.4900
.5000
.5100
.s200
.5300
.5400
.5500
.5600
.5700
.5800
.5900
.6000
.6100
.6200
.6300
.6400
.6500
z.@1.4 7.5746 2.5914
1. ao 6.7702 2.4566
't.4232
6.0730 2.3332
7.7459 5.4646 2.2798
7.6757 4.9346 2.77s7
7.6701.4.4594 2.0780
7.5502 4.0415 7.9276
7.4948 3.6697 7.8430
7.4434 3.3358 7.7637
7.3957 3.0364 7,6889
7.351.22.7663 7.61.83
1.3@7 2.5220 7.5574
1..27082.3@7 1..4877
't.2344
2.0980 1.4269
1..2m3 1.9134 136a7
L16at 1.7444 1.3129
1..1378L.5893 7.2592
L1493 1.4464 \.2074
1..08237.3147 1..7576
1.0s68
1.1929 1.109s
1.0326 1.0801 1.0632
1.0095 .9757 1,.01.90
.9877 .8785 .9767
.9552 .7920 .9377
.9425 .7754 .90 6
.9200 .6480 .8684
.8979 .5888 .8379 .01
.8766 .5370 .E100 .01
.8560 .4915 .7844 .O2
.8364 .451.6 .7609 .03
.at7E .41.66 .7395 .04
.8002 .3857 .7198 .06
.7836 .3583 .70t7 .08
.7680 .3340 .6852 .11
.7s33 .3122 .6699 .13
.7394 .2927 .6558 .16
.7263 .2752 .6428
.20
.7140 .2592 .6307 .23
.7024 .2447 .679s .27
.6915 .2374 .6W7 .31
.6811 .2192 .5994 .35
.671.3 .2080 .5903 .40
.6620 .1975 .5818 .44
.6531 .1.878 .5738 .49
.6447 .7787 .5663 .54
.6367 .7707 .5592 .s9
in responseat
iow freouencies iust reflects the
well-
known fict that loudspeakersytems
do
not provide significant output at very
low freouencies.One of our designob-
iectives
is to obtain thelowest
pdssible
cut-off frequency consistentwiih a flat
Passband.
ALIGNMENT PARAMETERS
We need to know which oarametersof
the loudspeaker svstem
determine its
,"sponse i,r-", Thiele found the fre-
quency
response
is
completely
deter-
mined by several amplifier. driver, en-
closure, and vent oarameters
which re-
flect those
parts
of the
loudspeakersys-
tem related to the model's electrical
comDonenrs.
Aisume
that the driver is a moving
coil diaphragm type operaling in its
piston range.
According
to Thiele, this
is from OHz to SOOO/dHz where d is
.744s .3640 .6sm .79
'
.7327 .3453 .6393
.23
t
.7205
.3284 .627s .27
"
.7096 .3131 .6166
.31
.6993
.2992 .6065 .36
.6896 .2865 .5971 .41
.6805 .2749
.5883 .46
.6719 .2641 .5ffi2 .51
.6638 .2542 .5726 .57
-6s67 .2449
.5654 .63
.6488 .2363 .5587 .6E
.6418 .2283 .5524 .74
't
Bullochon Eoxes 5
R, ohms in serieswith the driver, and
the damping
effectof the amplifier is
the sameas a resistorof R, ohms in
serieswith
the
driver;
then we change
Q*
to
Q'*
by theformula:
and amplifier combination.Thieleand
Small based their alignments on the
QB3-84-C4
seriesof responsesso that
only onepossiblealignmentisobtained
for each
value
of
Q'".
We
will cover
only thesealignments.
Tablesl, II, and.f/1list Small align-
ments.To us€a table,find thevalueof
Qrs
in the first
column
and
use the val-
uesof h, a, f:/fr found in that row. We
can also find alignmentsby
various
formulas-for
example, Saffran's for-
mulas in Speaker
Builder, lssue L/80,
p.
35
("Mailbox"
section).With these
formulas, you can compute a Thiele
alignment from a
given
value of
Q'(:Q'").
Such formulas are
handy
but provide only approximate values,
whereasaccurately
prepared
tables
will
provide
exact
values.
THIELE ALIGNMENTS
Thieleswas the first to analyze
the elec-
tricalcircuit modelin order to
provide
specificvented speakersystem
align-
ments.ln this landmarkpaper he
pre-
senteda large tableof alignments,
the
first nineof
which werein the
QB3-84-
C4 series.These
nine are
just
samples
from a continuumof
QB3-84-C4
align-
ments
that can be obtainedby varying
the value ot
Qrs
. If
Q.s
--
.383,
the
alignmentis a 84;
if
Q^<.383,
it is a
QB3,
andit
Q,r
> .383.it is a C4.
There are usuallv
practical
bounds
on the values
ol
Q,,.
It
Q,,
< .2, the
cut-off
frequencyis usuallyundesirab-
ly high. At the otherextreme,
for large
Qrs
lhe alignments
are C4 and so the
responsehasa ripplewhich
increases
TABLEIII
SMALLALIGNMENTS
for
Q.:19
Ripple
h
qF
t'/t"
(dB)
1,.89607.9232 2.4845
1.8085 7.0834 2.3s43
7.7292 6.3554
2.2357
1.64O9 5.tZU2 t.LZ)5
1,.59085.7627
2.0241
7.5307 4.6706 1.9299
7.4742 4.2342 1.8421
1.4225 3.U52 1.7599
7.3747 3.4971 7.6826
1.3303 3.1843
1.6097
7.2890 2. 22 1.5406
7.2505 2.6469 1.4748
7.2746 2.4750
1.4721-
1.1809 2.2038 1.3521
1..14932.07@ r.2945
1..11977.8342 7.2390
1.0918
1.6719 1.1855
1.0656 1.5225 1.1339
1.0109
1.3846 1.0841
7.0\75 1,.25717.0363
.9954 1.1390 .9907
.9732 7.03?J .9482
.9507
.9381 . 92
.9282 .85s0 .8736
.9062 .7a22 .8410 .01
.8848 .7187
.8114 .01
.8644 .6632 .78/.4 .02
.8451 .6148 .7ffi .03
.8269 .s725
.7377 .05
.&97 .535s .7775 .O7
.7937 .5029 .6997
.10
.77a7 .4742 .6823 .13
.7648 .4487 .6670 .1.6
.7577 .4261 .6529
.20
.7396
.4059 .6407 .24
.7282 .3877 .6282 .29
.7776 .3714 .6773
.34
.7077 .3s65
.@72 .39
.6983 .3431. .s979 .44
.6896 .3308 .s892 .50
.6874 .3195
.s472 .55
.6736 .3@2 .s737 .67
.6663 .2996 .s667 .68
.6s94 .2ffi7 .5601
.74
.6529 .2825 .5s40 .80
.6467 .2748 .s482
.87
a'"
.2.AOO
.2100
.22ffi
.2300
.24co
.2500
.26co
.27cD
.2800
.z9oo
.3000
.3100
.3200
.3300
.3400
.3500
Q'85
:
[(RE+&+R,)/R.]Q.'.
(1)
One way to determine R, is to hook
everything
up as
it would be in the
finishedsystemand measurethe resis-
tance between the amplifier connec-
tions.
Subtract
R" from thisandthere-
sult is
R.
Small" describes the most accurate
method for finding
&;
or, if we know
the damping factor D of the amplifier,
we can find R" from the formula
&
:
R"/(D-1)
Q)
.3700
.3800
.3900
.4000
.4100
.4200
.4300
.4400
.4500
.4600
.47co
.4800
.49co
.5000
.5100
.5200
.s300
.5400
.5500
.5600
.57co
.s800
.5900
.6000
.6100
.62@
.6300
.6400
.6500
where R" is the nominal impedance
of
the
driver. Typical
damping factors
varv from 15 to 500; 25-30seemsto
be
fairiy common. Thiele'
says that if
R, * R, is lessthan 5 percentof R", then
we can use the unmodified
value
Qss
if
responsevariations
of up to .4dB are
tolerable.
The actual
Q
number used
to specify
an alignment is called the total
Q,
denoted
Q,
or
Q,",
and
is tound from
the formula
a^: a'," Q.,/(Q'"
+
Q""),
(3)
or
7/Qrs:7/Q'Es
+ 7/Q,s.
14)
in
ma8nitude
t\ ith-Q,s.By
Q,.:
.7 this
rippleexceeds1.5d8and an audiophile
would
probably find it obiectionable.
Exact alignment values are
quite
comolicatedto calculateand are
best
found usinga computer.I will not pro-
vide Thiele alisnmentsfor reasonsto
Assuming we know
the values for
bemadeclear
below.lf you would like
Q'",
f", and V,.s,an alignment
is a set to usethem anyway, Saffran'sformu-
of relationshipswhich mustbesatisfied lasshouldbe sufficient.Error canbeas
between
Q'",
f", Vrr, Vr, and
f,. The high
as5
percent.Besureto correcthis
relations
for
given
Qrs
are that the tormula for h to h: .38/Q,.
3,111f
TABLE
IV
ALIGNMENT FORMULAS
j,',1|)l;rrot'ALrcNMENrs
analyzins the model; Small denotes Small" observedthat vented
loud-
thesetw; ratiosby
h ando
{alpha)
re-
speaker
systemsdesignedaccordingto
spectively.Thus,
an alignmentis equi- Thiele alignmentsdid not always ex-
valent to a list of the three numbers hibit the freq'-rency
,."f
",l];fi
\lr-J
h: .aL9/et"l,'
q.:.07s3/els'.1
f3lfr
:
.31s/QlrrlJ
response
predicted
Q',,
h, e: for example,a Thiele 84 by
the model. He determinedthat the
alignment is
Q'":.363,
5:1,
q:./2.
box and vent wereexertinga damping
lf you have a driver with
Q6:,383,
effectwhichalteredtheresponse
curve.
fs: 25H2, and V"":1gggg;nj, then a He
referredto thesedampingeffectsas
84 responsewill be obtainedif f":
lossesbecausethey usuallyresulted
in
hf":
i;25:25112 and
Vr:V,"zc: decreasedoutPut at certain
f.equen-
\zL-t
h:.420lQrtr
e:.0s69lQiJ5J
frlfr
:
.3os/Q+ir-
1OOOO/J2:7071inr.An alignmsnlu5-
cies,and accountedfor them by
"ob-
ually
includesa value of f7f" also so
serving that theselossesmay
be ade-
you
can seewhat thecut-offfrequency
quately
approximated
for design
pur-
will
be with the.alignment;however,
posesby a singlefrequencyinvariant
vou do not need it to make vour de- leakageloss.'
sign.
ln otherwords, heintroducedanoth-
An alignmentuniquelydeterminesa
er
parameterrvhich represented
Q':ro
h:.421lQr?r
o.:.0689/Qi"",'
fJfs
:
'2q619it""
losses
Thevalueof h
is
usually
within29o.
thevalueof f,i f, within67o,andthe
value of
o.
between
_177o
and
+25 C".
responsecurve for
j
paiticular driver dui to box and
driver leaks,soundab-
6 Bullock on Boxes
which is sometimesmore convenient.
The remaining influences
on re-
sDonseare the volume V, of
thebox on
*hi.h th" driver is mounted
and the
resonant freguency f, of the box
resultingfrom the
presence
of the
vent.
ALIGNMENTS
illi'"li1,Lilf,
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