Dept. for Speech, Music and Hearing

Quarterly Progress andStatus Report

On the dimensioning ofbass-reflex enclosures

Liljencrants, J.

journal: STL-QPSRvolume: 7number: 3year: 1966pages: 007-022

http://www.speech.kth.se/qpsr

A. O N TH.E E ~ ~ v i E I ; T S , T O > C L ~ 3ASS -PaFLEX ENC:LOSURES

J. Liljencrants

Abstract --- A thec~et ic2 : analysis shows that the bass-ref lex loudspeaker

system exhibits the characterin",ics of a minimum-phase highpass fi l ter.

It is thus possible to apply the well-.known Butterworth and Chebyshev

approximations t o ideal filter:: on t l ~ e lcudspzalcer sys tcm dimensioning.

A number of such dimensionings z r c examined for the i r implications on

frequency and t ransient Tesponoe, !.oudspeakcr diaphragm excursion and

enclosure volume. A method of finding the phyoicnl conditions is given

that will implement an a rb i t r a ry system behzvior s ~ e c i f i e d in t e r m s of

the poles of the t r ans fe r function.

Introduction --.- -- --

Since the bass- ref lex enclosure was invented by Thuras (5) in 1930

a number ~f recornmendat io~s have clrcu!-ated in the l i terature together

with indications of pnssihie pe r iormance improvements, A bass -reflex

sys tem has a considerab?e xiumber of degrees of freedom, It is thus

natural that the analysis mostly is confined to special cases , This may

imply that interesting phe-omcna a r e overlooked, Often it i s a l so dif-

ficult t o see the influence of different paraxlzeters on the sys tem pe r -

formance,

The problem of system optimization was t reated by Iieibs (2) who

a lso gave a commented review of the l i terature , T3e optimal solution

discussed below is r c t very different f rom t52t of Keibs, but some m o r e

of the practically interesting p3rarneters a r e kept in the t reatment , We

will a l so use the terrrinology of I-nnc?e;.n circci t theory a n d see how well-

known ideal f i l ter approximztions may he applied t o the problem,

A s sumpti~xn~g

To,- the trttztmct?:: of tl1f.s probl-em we of ccurse res t r ic t ourselves

t o the l o ~ ~ ~ - f r e q u e i ~ c y ranee, The loudspeaker cone is thus assumed t o

mp-re 2-s a y-igjd p i a t ~ n an4 the vlhnle sys tem is regarded as a Feint source

o r t i n ? i i e r wc=.c!n, f:l,.?, mech~.r.!.ca! 3Smens,i.o,?r z;.e assumed t o Ee smafl

compared t~ the sound .tvz7.~e length,

F o r the analysis we use the impedance analogy of Fig. 111-A-1.

All the elements a r e t ransfer red to acoustical quantities. The detailed

derivation of the analogy is omitted since it is easily found in the hand- (1) book li terature, see e. g. Beranek .

The acoustical elements of the analogy have the following meaning:

- - e B1 is the pressure from the generator/amplifier

(Rg+Re)sd having an electromotive force e and a n internal electrical resistance R B, 1, and Re a r e the

g' field strength of the speaker magnet, and the length and electrical resiotance of the voice coil. S is the effective a r e a of the diaphragm. d

- R1 - ( ~ 1 ) ~ + R + R r is the total acoustical resistance of

s the amplifier and the loudspeaker. It consists of the reduced electrical

resistance, the acoustical resistance of the diaphragm suspension Rs, and the radiation resistance Rr f rom both s ides of the diaphragm.

2 hil= ml/sd is the acoustical m a s s of the loudspeaker

element including the air load, ml is the corre- sponding mechanical mass.

2 C1 = clSd is the acoustical compliance of the diaphragm

suspension and cl is its mechanical compliance.

Rb is the acoustical resistance of the box and

C2 = v / y L is its acoustical compliance. V is i ts volume and P O and c a r e the density and speed of sound of the air .

is the acoustical m a s s of the a i r in the bass- reflex port, and finally

is the acoustical resistance of the port including radiation.

The electr ical inductance of the voice coil may be neglected in

the frequency range of interest.

Fig. 111-A-1. Analogy of a bass reflex system. All elements are transformed to acoustical quantities (MKSA).

\/ -\ C\ > < Generator Loudspeaker element Box Bass reflex port

\A e\

Derivation of the fmquency response of the system

We make use of complex frequency

normalized to w o which is defined a s the geometric mean to the

eigenfrequencie s of the loudspeaker and the box resonator:

Vle also use the following loss factors:

Loudspeaker element: a = l/Gl = w1R1C1 = R 1 9 / M 1

Box:

Port: v = 1 hp = R ~ / w ~ M2

Total reso- nat o r : b = l/a2 = u + v

The acoustical resistances a r e not well behaved. The value of

the resistance io of importance mainly at frequencies in the vicinity of

the resonance of the pertinent system, ViTe assume these resonance

values of the resistances in the formulas a s constants, since the e r r o r s

at other frequencies a r e negligible. To understand this one may recall

that a normally used loudspeaker element is mass controlled in the

frequency range just above the low-frequency cutoff.

Finally we introduce a couple of auxiliary quantities:

and

g tells us to which extent the tuning of the loudspeaker is different

from that of the box resonator, and h will relate the box volume t o the

loudspeaker compliance.

With these normalized quantities we get the following expressions

for the impedances of the three branches of the analog diagram:

Under the point source assumption the sound pressure p at the

distance r will be

The two first factors imply that the pressure is proportional to the time

derivative of the volume velocity (U ) out of the box while the rest of the b formula gives the distance dependence which is of no concern at present.

The box volume velocity is easily obtained directly from the analog

diagram a s

The insertion of (8) and (6) into (7) will render the deoired expre~oion of

sound pressure as a function of frequency:

where P(x) stands for the fourth degree polynomial

4 3 2 P(x)= x + k x + k2x + klx + 1 3

with the coefficients

On the f i r s t par t of the problem the re is room fo r a number of

subjective and objective viewpoints, and a c loser discussion may be

justified. VJe look a t the following:

1. Frequency response. It is desirable with a n even shape of the response curve. It i s a l so well-known that the frequency range of the sys tem may be extended below the loudspeaker resonance.

2. Power handling capacity. The classical advantage of the bass- reflex enclosure over other sys tems is that the diaphragm excursions a r e diminished in the vicinity of the cutoff frequency. This will give a lower distortion at high output power,

3. Transient response. The resonances of the sys tem should be sufficiently damped. The co r re spondence of f IE quency and t ime responses conveyed by the Eaplace t ransform should be borne in mind.

4. Size. F r o m pract ical reasons a smal l volume is advantageous,

Frequency response

The response (12) is that of a minimum-phase cyotem, This makes

it possible to use the classical approximations to ideal f i l ters . Nearest

at hand a r e the Butterworth approximation (maximally flat) and the

Chebyshev (constant ripple in the pass band).

Our highpass function has a quadruple ze ro in the origin and two

conjugate pa i r s of poles, the roots of P(x) = 0. One such pa i r will

together with a dual ze ro in the origin give r i se to a n elementary high-

pass resonance curve. A family of such curves is shown in Fig.

111-A-2.

To go f rom a known lowpass pole configuration t o the co r re - sponding dimensioning pa ramete r s we d o a s follows. F i r s t the poles

a r e normalized. This means that we introduce a scale factor render-

ing the pole product a unity magnitu.de, The lowpass function will then

be

R is the magnitude of a pole in one of the conjugate pa i r s having the

los s factor d l , and 1 /R and d2 are corre sponding for the other pair.

If we execute a lowpass/highpass t ransformation by means of the

frequency substitution X = 1 /x we get the analogous highpass function

Fig. 111-A-2. Normalized highpase resonance curves. The insert illustrates the definition of Q and loss factor.

The only difference a s compared t o the lowpass function is the

quadruple zero and that the polynomial coefficients change order.

An identification with (1 1) will give

= kl = gb + a/g + ghv

The left members of the equations a r e thus given quantities and the

dimensioning parameters a r e found by solving the system. This is

of course rather intricate to do in a general manner. VJe observe

that we have five independent variables (since b - u + v) and two

of them can thus be chosen with a certain degree of freedom.

There is a duality between some t e r m s in the left and right

members of (1 7), the similarity in position of the factors K - g,

dl - b, and d2 - a being apparent. This might provoke a temptation

t o put some entirely unjustified equality signs here and thus identify

a "speaker resonance" and a "box resonance" in the compound system.

In reality such an identification would not give any non-trivial solu-

tion t o (1 7).

Table 111-A-1 l is ts the charac te r i~ t i co of some different cases

t o compare. The f i r s t alternatives a r e the Butterworth and a

Chebyshev (see a lso Fig. 111-A-3) and the case quoted by Keibs

a s optimal. There is also one case where one of the pole pa i rs has

been given a very large loss factor making the system resistance

controlled in the low frequency range. The frequency response will

then have a + 6 d ~ / o c t a v e slope. F o r a wider comparison we finally

examine the special cases with a n infinite baffle and a closed box.

In these cases the loudspeaker resonance frequency is used for the

normalizing, g = 1, and the loss factor is set t o unity giving a

reasonably flat response. The acoustical compliance of the closed

box is chosen equal t o that sf the loudspeaker element, h = 1.

Table 111-A-1. C h a r a c t e r i s t i c s of the di f ferent s y s t e m s considered.

1 B a s s re f l ex 0.131 1.462 0.061 0.678 0.681 0.180 1.29 0.31 1.15 0 0 Chebyshev 110. 893 2.783 1.482

0.317 0.604 0.680 1.300 1.469 0.925 1.365 0.148 1 .02 0.05 0.05

2 B a s s ref lex 0.383 0.925 0.925 0.383 1 1.41 2.62 0 0 Butterworth

3 B a s s re f l ex Optimal acc. t o Keibs

4 B a s s re f l ex Highly damped

5 Infinite baffle L o s s fac to r = 1

6 Closed box L o s s fac to r = 1 h = 1

0.925 0. 383 0. 383

0.417

0.417

: 1 0. 34

0 .5

0.707

A

0.925

0.909

0.909

3 0. 59

0.866

1.225

1

1

1

1.469

0.681

1

1.414

0.766 2.613 1 3.414 2.613 .

1 . 2 .400

1 . 2

3 . 5 0

3.440

5.52

-

1

1

1

1

1

1.414

i

2.400

5.09

-

-

0.94

1

1

1.21 .

1.23

-

-

2.31

1.44

1.01

2.33

1.69

0

1

0. 1

2.40

2.1

4.24

3.75

1

1

0. 1

0

0.1

0

0.1

-

-

0

0.1

0

0.1

-

Fig. 111-A-3. a. The poles of a lowpass Butterworth sys tem a r e equally spaced on a semicircle. If the r ea l par t s of the poles a r e scaled down with the factor aT/cug the poles will l ie

on a n ell ipse and the system will get a Chebyshev behavior.

b. A normalizing of the pole product will give a root locus fo r the lowpass Chebyshev case ( - - - - - - - - ). A reflexion in the unit c i rcle will render the corresponding locus for the highpass case ( ). The LP/HP transformation a l so gives a quadruple ze ro in the origin.

The scaling parameter is arb i t ra r i ly chosen f rom pract ical reasons. Two cases of Table 111-A-1 a r e indicated with dots.

In analogy with (1 2) and (1 4) the closed box will have

with the idini te baffle h 04

Fig, 111-A-3 shows the loci of the poles for normalized

~ h e b ~ s h e v configurationst It is seen here that the cutoff frequency

is lowered when the loss factoi. is decreasedr This means that a

more extreme Chebyshev design should provide an extended frequency

range. If one on top of this can find a design with g > 1 the gain

will be even larger with given cul.

The pole configurations and frequency response curves for

the different alternatives of Table III-A-1 a r e shown in Figs.

1114-4 and III-A-5. In Figs. III-A-5 - III-k-7 the frequency

scale has been re -normalized to the resonance frequency of the

loudspeaker. This will make the curves bring out the performance

of the different box design types with reepect to one and the same

loudspeaker element. To make the re-scaling possible the value of

g must be known which implies that the Eqs (17) have to be solved

first. The actual solutions used a r e indicated in Table III-A-1 and

could be regarded a s near optimal.

If some special frequency response is wanted it can be con-

structed by adding two template curves of Fig. III-A-2. The nor-

malizing will then involve that the unit frequency of these templates

should be held with geometric symmetry around the normalizing

frequency. The displacement will then determine R in Eq (1 6 ) .

When appropriate values of R and the 10s s factors have been

chosen they can be inserted into the left member of Eq (17). It

should then be remembered that the poles of magnitude R have t h

loss factor d in the high-pass case, Eq (16), a s opposite to the 2 low-paee caee of Eq (1 5.j.

Fig. 111-A-4. ~ o l e / z e r o d iagrams for the different sys tem types of Table 111-A-1.

~ i ~ . 111-A-5. Frequency responses for the alternatives of Table 111-A-1. The frequency scale i s normal- ized to the resonance frequency iol of the loud- speaker element.

Fig. 111-A-6. Diaphragm excursions for the alternatives of Table 111-A - 1 .

Fig. 111-A-7. Step responses for the alternatives of Table 111-A-1. The time scale i s normalized to the resonance frequency of the loudspeaker element.

Diaphragm excursion

The variation of diaphragm excursions with frequency differs from

the sound pressure response in a 12 d ~ / o c t a v e slope and an antiresonance.

This is directly seen from a comparison between Eqs (12) and (14). It

is also seen that the frequency and loss factor of the antiresonance depend

only of the properties of the box and port. The diaphragm excursions

for the different alternatives a r e shown in Fig. 111-A-6.

It is now directly seen that the combined losses in the box and port

have to be rather small if the antiresonance is to have any significance

for the diaphragm excursions. T h i ~ justifies the assumption above that

v / ~ may be neglected in the numerator of Eq (9) which will give r ise to

Eq (121,

The electrical input impedance of the system is the sum of the

essentially constant (at low frequencies) blocked impedance and the

motional impedance. The variation with frequency of the motional

impedance is identical in shape with the diaphragm velocity which in

turn is obtained from the curves of Fig. 111-A-6 by superimposing a

tilt of t 6 d~ /oc t ave . It i s now easily understood that the two peaks in

the impedance curve do not give a very useful information on the system.

The interesting thing is rather the dip between them that tells the reso-

nance frequency w and loss factor of the box resonator. 2

Transient properties

A complete analysis of the transient behavior of the system is

rather involved both theoretically ancl practically. The following dis-

cussion will thus only briefly review the general method and we then

proceed to some experimental measurements.

In general, when a system with the transfer function H(G) is excited

with a signal having the Laplace transform G(s), then the transform of

the output will be

The response in the time domain will then be the inverse transformation

of this, o r

To study the transient behavior of the system only, one should use a

signal with a transform G(s) a s simple as possible. The natural choice

would then be the unit impulse having the transform 1. The impulse

response is then

A practical problem when this is applied to a highpass system as our

loudspeakers is that the infinitely high unit impulse is directly t rans - mitted and then followed by the interesting part of the response. The

unit step has the transform G(s) = l/s, o r with normalized frequency

G(x) = l/x. If we disregard a proportionality constant the transform of

the 8tep response of our system will be

An inverse transformation will give the general form of the step response

a s

o and w a re the real and imaginary parts of the poles, and A and cp a re

the magnitudes and phases of the residues . The impulse response

differs from this in the values of the residues and an additional t e rm

c?- (t ), the directly transmitted input pulse.

Fo r a closer treatment of the Laplace transform and especially

the residue calculus the reader is referred to the abundant handbook

literature on the topic (see for instance (3)).

In practice the residues a r e preferably calculated by graphical

methods from a plot of the poles in the frequency plane. An alternative

and often eas ier method is however to build an electrical analog to the

system and directly record its performance. Fig. 111-A-7 shows the

results of such a simulation of the system transfer functions described 7'k in Table 1114-1. (See next page. ) In the same diagram there is a

time scale valid for the fairly representative case with wl = 2n . 50 radians/second.

The most striking example is the Chebyshev case which now clearly

shows the price of the extended frequency range. The system has a very

weakly damped resonance near the cutoff frequency which accounts for a

long transient time. If we look at the Butterworth case and the system

It i s usual that the transient properties of a system is tested

with tone bursts a s the excitation signal. The output will then consist

of a stationary part being the input modified t o phase and amplitude

by the t ransfer function H(s). It a lso contains a transient part at the

beginning and end of the bone burst. If we assume a sinusoidal input

of frequency w the t ransform G(s) in Eq (20) will be g

The frequencies and loss factors of the components in the transient

response a r e the same a s in the s tep response (24) but the residues

get other values since (25) will enter a s a factor in the calculation of

these. The transient response contains no components of frequency

w These a r e instead restricted to the stationary response. The dif- g'

ferent appearances of the transient response when w i s varied has g

nothing to do with the system but instead reflects the influence of Eq

(25) on the residues.

The main advantage of the tone burst method is rather that it is

possible to excite parasitic resonances more o r l e s s selectively. To

make the picture complete however the tone burst response for a

Butterworth system i s shown in Fig. 111-A-8.

Resonator volume

The second Eq (1 7) shows that maximal values of h should be

expected with u = v = b = 0, The corresponding values of the other

parameters a r e given in Table 111-A-1. In practice it is however

impossible to have the resonator completely undamped, it is not even

desirable with respect to the detrimental effects of standing waves. The

table therefore also l is ts the more practical solutions having u = v = 0.1.

The Chebyshev case i s an exception in that it has no physical solution

for this magnitude of the loss factors. Instead a solution is given with

u = v = 0.05,

F o r a given loudspeaker element the box volume is inversely

proportional to h. r"+ study of Table 111-A-1 reveals then that the case

with a heavily damped system can give a box volume that is about 60 O/o

of that fo r the Butterworth case. Re a contrast the minimum practical

volume for the exemplified Chebyshev design is about 7 t imes a s large

as for Butterworth,

Fig. 111-A-8. Time response of a fourth degree Butterworth system t o excitation with a sinusoid and a cosinusoid of frequency WO. The stationary part of the response i s attenuated 3 dB and phase shifted 180 degrees. The high frequency components of the s tep in the cosinusoid a r e however t ransmit ted without attenuation and phase shift.

Fig. 111-A-9. Time response of a fourth degree Butterworth system t o excitation with a sinusoid of varying phase. The excitation frequencies a r e iuo and 2 w 0 respectively. Only the envelopes of the responses a r e shown.

Conclusions

The Butterworth behavior could be deemed a s the optimum in the

light of the previous discussion. The frequency range is not extended

a s compared to the closed box of equal volume, but the diaphragm

stiffening effect i s fully utilized in the same time as the transient

time as a rule i s sufficiently short.

Apart from this the heavily damped system is Prominent mostly

because of its small volume, but its frequency response has to be

corrected somewhere else in the total s y s t e m

Design

When system behavior has been decided on the left hand members

of (17) a r e given. The problem is then to solve this equation system to

find the parameter values. Fig. 111-A-1 0 shows the result of an i tera-

tive solution process that was executed on a PDP-7 computer. The loss

factors a r e given as functions of the box volume parameter h and the

tuning parameter g. The diagram directly indicates within which limits

of the various parameters the desired behavior may be realized. It i s

for instance seen that a smaller box necessitates smaller loss factors

in the box and port in the same time a s the demand on proper tuning of

the resonator becomes more proxxinent. It is thus not practical to make

the box a s small a s the theoretical minimum. On the other hand, if the

box is too large the resonator losses have to be so high that the anti-

resonance in the diaphragm excursion characteristic i s flattened out.

The essential point i s thus that for a given loudspeaker element

and system behavior the re i s a definite optimal box volume, namely

the smallest practically possible,

When the parameters of Fig. 111-A-10 have been chosen we can

proceed to find the physical dimensions of the resonator with the aid

of some properties of the speaker element. The ones most easily

measured a r e w l , ml, and Sd. The resonance frequency is determined

from the maximum of the electrical impedance. The loudspeaker should

be baffled during this measurement so that the a i r load is included in

the oscillatory system. After this the diaphragm is loaded with a known

mass m and a new resonance frequency w is found. W e then get the 3 3 mechanical mass a s

The effective diaphragm area is approximately

where d is the mean diameter of the compliant edge suspension of the d

diaphragm,

After some manipulating with the definition equations we arr ive

at the box volume

In passing we note that for a given resonance frequency the box will

be smaller for a smaller diaphragm area and a la rger mass. These

quantities also have a direct influence on the loudspeaker efficiency.

If the bass -reflex port is of cylindrical shape the classical

theory yields

biz = p 0 l 0 / s P

( 2 9 )

where S r , and 1 ' a re the area, radius, and effective length of P' P

the port, The interdependence of a rea and radius makes

which shows that we cannot control the acoustical resistance of the

port merely by giving it certain dimensions, After appropriate sub-

stitutions we find the loss factor of the port as -,

In general this value is rather small. If one for some reason wants

a higher loss factor this has to be accomplished with some kind of

damping material in the port.

The conjunction of effective length and a r e a is given by Eq (29).

The physical length of the tunnel is obtained by subtraction of the end

corrections. If we assume both ends t o be baffled the physical length

will be

1 = 1' - 2 0.85 dp/2 (33)

which eventually will give the final formula

which i s shown graphically in Fig. 111-A-1 1.

The a r e a of the port is irrelevant t o the function of the system

as long as i t s length i s correctly chosen in correspondence t o the area.

The most suitable practical solution will thus be t o choose a tunnel

length equal t o the wall thickness of the box and then match the a r e a

t o this.

The following is a possible way of determining the loss factors of

the box and the port. First a microphone is mounted inside the box and

then the mounting hole for the loudspeaker is closed tightly. If the

resonator is now excited with a n external loudspeaker the microphone

output will give information on the resonance frequency w2 a s well a s

the 3 dB relative bandwidth which i s u + v'. F i r s t the port a r e a is

adjusted t o give the correct resonance and then the box is filled with

damping mater ia l until the appropriate value of u is obtained. If nec-

e s sa ry the port is then damped t o give a relative resonance bandwidth

u + v.

If the resonance is badly pronounced before the box is damped

this is a good indicator that the wall construction is too weak o r a l -

ternatively that there a r e leaks in the box.

Finally the loudspeaker element is mounted and the loss factor a

is adjusted with damping material around the loudspeaker basket The

correct value of a is most easily found by checking that the frequency

response is the desired.

Example

On a smal l loudspeaker element the following values were de-

t e rmined:

Fig. 111-A-11. Dimensioning diagram for the bass reflex port. The port i s assumed to be circular of diameter d

P' area S and axial length 1. The curves show 1 as

?' a function of dp. The parameter of the curves i s

1 - 4 hg in MKSA units.

Po Sd

Fig. 111-A-12. a. Step response for a n experimental bass reflex cabinet designed for Butterworth behavior. The fine s t ructure i s due t o parasi t ic resonances and reflexions.

b. Frequency response of the same system measured in a n anechoic chamber a t two different angles t o the loudspeaker axis.

In the low frequency range it i s seen that the assumption of a point source i s justified.

The response a t frequencies above 2.000 c/s i s due t o a pair of t reble loudspeakers mounted closely t o the bass loudspeaker.