Bullock on Boxes - Robert M. Bullock III.pdf - Elektronika Audio - megasite

BULLOCK

BOXES

byRobertM. Bulfock lll

assistedby RobertWhite

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

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

' ,., r

.t,.1

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.

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

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)

.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)

7/Qrs:7/Q'Es + 7/Q,s.

14)

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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