Optimally sensitive and efficient compact loudspeakers

Ronald M. Aarts

Philips Research, High Tech Campus 36, NL-5656 AE Eindhoven, The Netherlands

共

Received 3 August 2005; revised 7 November 2005; accepted 18 November 2005

兲

In conventional loudspeaker system design, the force factor

is chosen in relation to enclosure

volume, cone diameter, and moving mass to yield a flat response over a specified frequency range.
For small-cabinet loudspeakers such a design is quite inefficient. This is shown by calculating the
efficiency and voltage sensitivity. The frequency response is manipulated electronically in a strong
nonlinear fashion, which has consequences for the sound quality, but it then turns out that systems
using much lower force factors can provide greater usable efficiency, at least over a limited
frequency range. For these low-force-factor loudspeakers, a practically relevant and analytically
tractable optimality criterion, involving the loudspeaker parameters, will be defined. This can be
especially valuable in designing very compact loudspeaker systems. An experimental example of
such a design is described. This new, optimal design has a much higher power efficiency as well as
a higher voltage sensitivity than current bass drivers, while the cabinet can be much smaller. ©

2006

Acoustical Society of America.

关

DOI: 10.1121/1.2151694

兴

PACS number

共

兲

: 43.38.Ja, 43.38.Dv

关

AJZ

兴

Pages: 890–896

I. INTRODUCTION

There is a longstanding interest in obtaining a high

sound output from compact loudspeaker arrangements. Com-
pact relates here to both the volume of the cabinet in which
the loudspeaker is mounted as well as to the cone area of the
loudspeaker. Loudspeakers can be built such that they prop-
erly reproduce the entire audible frequency spectrum, down
to 20 Hz; but such systems would be both expensive and
very bulky. In many sound reproduction applications it is not
possible to use large loudspeaker systems because of size or
cost constraints. Typical applications are portable audio, mul-
timedia, and

共

flat

兲

TV sets. Various signal-processing

schemes have been proposed to equalize the response of
small loudspeakers or to use psychoacoustic enhancement
methods; see Larsen and Aarts

共

2004

兲

for some overview.

The aim of this present paper is to discuss a method to ma-
nipulate electronically, in a strong nonlinear fashion, a spe-
cial loudspeaker with a high acoustical output. The depen-
dence of the transducer’s behavior on various parameters, in
particular the force factor

, is investigated. For electrody-

namic loudspeakers the perceived quality is important, but
also the sensitivity

关

Pa/V

兴

and the efficiency are of impor-

tance. Therefore, in the following section the sensitivity and
the efficiency of electrodynamic loudspeakers in general are
discussed. It appears that drivers with very high efficiency
have poor sensitivity at low frequencies. It is not possible to
combine a very high efficiency and a high sensitivity in a
wide frequency range with a compact arrangement. In Sec.
III special drivers with a very low—but optimal—

value

will be discussed. They have an optimal sensitivity and are
only 3 dB less efficient than an infinite-force-factor loud-
speaker, but in a limited frequency range only. These char-
acteristics are obtained at the expense of decreased sound
quality and the requirement of some additional electronics.
Due to the low-

value, the magnet can be considerably

smaller than usual and the loudspeaker can be of the moving-
magnet type with a stationary coil, instead of vice versa. In

Sec. IV it is discussed how such a low-

driver can be

made. It appears to be very cost-efficient, low-weight, flat,
and requires a low-volume cabinet.

II. SENSITIVITY AND EFFICIENCY CALCULATIONS

For low frequencies a loudspeaker can be modeled using

some simple elements, allowing the formulation of approxi-
mate analytical expressions for the loudspeaker sound radia-
tion

共

Beranek, 1954; Thiele, 1971; Small, 1972

兲

. Neglecting

the self-inductance

of the driver’s voice coil, the transfer

function at distance

from voltage

共

兲

to pressure

共

兲

also known as the sensitivity, can be written

共

Aarts, 2005

兲

共

兲

共

兲

共

兲

␳

共

␲

兲

共

兲

with all used variables as listed in Table I. It appears to be
convenient to use the following dimensionless quality factors

, the dimensionless frequency detuning

␯

, and resonance

frequency

␻

冑

共

兲

冑

共

兲

共

␻

兲

␯

␻

−

␻

, and

␻

冑

共

兲

Using these equations, Eq.

共

兲

can be written as

共

␻

兲

共

␻

兲

共

␻

兲

共

␻

兲

−1

+ 1

冉

␳

冊

共

兲

The first fraction on the right-hand side of Eq.

共

兲

expresses

the typical high-pass characteristic of a loudspeaker, while
the second fraction gives the value for high frequencies

共

␻

Ⰷ

␻

兲

. At the resonance frequency

共

␻

兲

Eq.

共

兲

becomes

890

J. Acoust. Soc. Am.

119

共

兲

, February 2006

0001-4966/2006/119

共

兲

/890/7/$22.50

共

␻

兲

共

␻

兲

共

␻

兲

␳

␻

共

兲

Equation

共

兲

shows that the sensitivity at the resonance fre-

quency depends on the mass

via

␻

. Section IV will

elaborate on this, and it is shown that at the resonance fre-
quency it is beneficial to have a low-

value.

The electrical input impedance can be written

共

Aarts,

2005

兲

共

␻

兲

冋

1 +

␯

册

共

兲

From this it appears that—via

—

plays an important

role in the electrical impedance, which is most pronounced at
the resonance frequency. By neglecting

rad

, which is very

small at low frequencies—in particular at

␻

—the electri-

cal input impedance at

␻

can be approximated as

共

␻

兲 ⬇

共

兲

共

兲

In order to calculate the power efficiency of loudspeak-

ers, it is required to calculate the electrical power delivered
to the driver as well as the acoustical power radiated by the
loudspeaker. The latter depends on the radiation impedance
of the driver. Below, expressions for these three quantities
are derived.

Assuming a sinusoidal driving signal, the time-averaged

electrical power

delivered to the driver can be written as

= 0.5

兩

兵

其

= 0.5

兩

冋

1 +

␯

册

共

兲

where

兵

其

is the real

共

resistive

兲

part of the input imped-

ance

. The radiation impedance

rad

of a plane circular

rigid piston with a radius

in an infinite baffle can be

derived as

共

Morse and Ingard, 1968, p. 384

兲

rad

␲

␳

关

1 − 2

共

兲

共

兲

共

兲

共

兲兴

共

兲

where

is a Struve function and

is a Bessel function

共

Abramowitz and Stegun, 1972, Secs. 12.1.7 and 9, respec-

tively

兲

, and

is the wave number

␻

. An accurate, full-

range approximation of

is given in Aarts and Janssen

共

2003

兲

共

兲 ⬇

␲

−

共

兲

冉

␲

− 5

冊

sin

冉

12 −

␲

冊

1 − cos

共

兲

For low frequencies

共

␻

Ⰶ

␻

= 1.4

兲

the damping influ-

ence of

rad

can either be neglected, or the following ap-

proximation

共

Aarts, 2005

兲

can be used:

兵

rad

其 ⬇

␲

␳

共

兲

/2.

共

兲

Assuming

= 343 m / s,

␳

= 1.21 kg/ m

, Eq.

共

兲

yields

⬇ 共

0.15

兲

共

兲

where

␻

/ 2

␲

The time-averaged acoustically radiated power can be

calculated as

= 0.5

兩

兵

rad

其

共

兲

which can be written

共

Aarts, 2005

兲

0.5

共

兲兲

1 +

␯

共

兲

Using Eqs.

共

兲

and

共

兲

, the power efficiency can now be

calculated as

␩

共

␯

兲

关

共

␯

+ 1/

兲

兴

−1

共

兲

This function depends on all loudspeaker parameters and the
frequency. In classical loudspeaker design theory the param-
eters are chosen such that the sensitivity function

共

␻

兲

given by Eq.

共

兲

has a flat characteristic for

␻

⬎

␻

, which

implies that

⬇

1 /

冑

2. This gives little freedom in the de-

sign parameters. Furthermore, one wants a reasonable ef-
ficiency. Recently, Vanderkooy

et al.

共

2003

兲

investigated

the use of high-

drivers. The aim of that study was to

obtain efficient loudspeakers; however, they have a poor
sensitivity at low frequencies. In the following section the
use of low-

drivers is discussed; those drivers appeared

to be highly sensitive and exhibit a good efficiency, but
only around the resonance frequency.

III. SPECIAL DRIVERS FOR LOW FREQUENCIES

Two options are described whereby modifying a conven-

tional loudspeaker driver can lead to enhanced bass perfor-
mance. This is achieved by modifying the force factor of the
driver, in particular by employing either a very strong or very
weak magnet compared to what is commonly used in typical
drivers. Both these approaches also require some preprocess-
ing of the signal before it is applied to the modified loud-

TABLE I. System parameters of the model.

radius of the cone

flux density in the air gap

force factor

voice coil voltage

BlI

is the Lorentz force acting on the voice coil

冑

−1

voice coil current

total spring constant=

共

driver alone

兲

共

box

兲

effective length of the voice coil wire

total moving mass, including the air load mass

␻

resonance frequency

␻

= 1.4

transient frequency

sound pressure

electrical resistance of the voice coil

mechanical damping

electrodynamic damping=

共

兲

real part of

rad

兵

其

total damping=

␳

density of the air

Laplace variable

surface of the cone with radius

velocity of the voice coil

rad

mechanical radiation impedance=

J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006

Ronald M. Aarts: Optimally sensitive and efficient compact loudspeakers

891

speaker. In the remaining section the influence of the force
factor on the performance of the loudspeaker is reviewed.

Direct-radiator loudspeakers typically have a very low

efficiency, because the acoustic load on the diaphragm or
cone is relatively low compared to the mechanical load. In
addition, the driving mechanism of a voice coil is quite in-
efficient in converting electrical energy into mechanical mo-
tion. The force factor

is deliberately kept at an intermedi-

ate level so that the typical response is sufficiently flat to use
the device without significant equalization. It was already
shown in Sec. II that the force factor

plays an important

role in loudspeaker design. It determines among others the
frequency response and its related transient response, the
electrical input impedance, and the weight of a loudspeaker;
the following will discuss these various characteristics.

To show the influence on the frequency response, the

sound-pressure level

共

SPL

兲

of a driver with three

values

共

low, medium, and high

兲

is plotted in Fig. 1, while all other

parameters are kept the same.

It is seen that the curves change drastically for varying

. The most prominent difference is the shape, but also

apparent is the difference in level at high frequencies. While
the low-

driver has the highest response at the resonance

frequency, it has a poor response beyond resonance, so in use
this loudspeaker requires special treatment, as discussed in
Sec. IV. The high-

driver has a good response at higher

frequencies, but a poor response at lower frequencies, which
requires special equalization. In between, there is the well-
known curve for a medium-

driver. The influence of

the sensitivity at the resonance frequency is further clarified
by plotting the SPL at the resonance frequency versus the
normalized

, as is shown in Fig. 2.

It appears that at the resonance frequency there is an

optimal value for the voltage sensitivity at

= 1, where

is the optimal-

value discussed in Sec. IV. The under-

lying reason for the importance of

is that, besides deter-

mining the driving force, it also provides

共

electrodynamic

兲

damping to the system. The total damping

is equal to the

sum of the real part of the radiation impedance

, the me-

chanical damping

, and the electrodynamic damping

共

兲

, where the electrodynamic damping dominates for

medium- and high-

loudspeakers, and is most prominent

around the resonance frequency. The variables in this elec-
trodynamic damping term cannot be selected independently.
This can be seen as follows. The voice coil resistance can be
written as

␳

共

兲

where

␳

and

are the electric conductivity and area of the

voice coil wire, respectively. The volume occupied by the
voice coil is equal to

共

兲

Combining these two equations yields the electrodynamic
damping

共

兲

␳

共

兲

which shows that the volume occupied by the voice coil, and
the material used for the magnet and voice coil wire, deter-
mines the electrodynamic damping, and not the length

the voice coil’s wire.

The power efficiency given in Eq.

共

兲

can be written as

␩

共

␻

兲

共

兲

兵共

兲

共

兲共

兲

共

␻

␯

兲

其

共

兲

共

␻

␯

兲

Ⰷ

关共

兲

共

兲共

兲

兴

, and

is ap-

proximated by Eq.

共

兲

, then Eq.

共

兲

can be written as

FIG. 1.

共

Color online

兲

Sound-pressure level

共

SPL

兲

for the driver MM3c with

three

values: low

= 1.2

共

solid

兲

, medium

= 5

共

dash-dot

兲

, and high

= 22 N / A

共

dash

兲

, while all other parameters are kept the same as given in

Table II, all with 1-W input power and at 1 m distance. At the resonance
frequency, the highest SPL is obtained by the low-

driver, while the high-

driver has at low frequencies—in particular at the resonance

frequency—a poor response.

FIG. 2.

共

Color online

兲

The SPL at the resonance frequency versus the nor-

malized force factor

共

兲

for the driver MM3c, where

共

兲

is the opti-

mal force factor given in Eq.

共

兲

, in this present case

共

兲

= 1.19. The other

parameters are given in Table II.

892

J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006

Ronald M. Aarts: Optimally sensitive and efficient compact loudspeakers

␩

共

␻

Ⰶ

␻

Ⰶ

␻

兲 ⬇

共

兲

␳

␲

共

兲

This is a well-known result in the literature

共

Beranek, 1954

兲

and clearly shows the influence of

, however, is valid in a

limited frequency range only.

Using Eq.

共

兲

, the power efficiency

␩

is plotted in Fig.

3, which clearly shows the dependency on frequency.

Figure 3 shows the efficiency function

␩

as function of

the frequency for various values of

, while all other param-

eters are kept the same. It appears that the curves change
drastically for varying

, but only very modestly around the

resonance frequency. This can further be clarified by using
Eq.

共

兲

and calculating the limit

lim

→

⬁

␩

共

␻

兲

共

␻

兲

共

兲

The curve for

␩

共

␻

兲

for this infinite-

value is the thick-solid

curve in Fig. 3. Assuming that

Ⰶ

and

␻

, and using

Eqs.

共

兲

and

共

兲

, this yields at the resonance frequency

lim

→

⬁

␩

共

␻

兲 ⬇

共

␻

兲

⬇

␳

共

␻

兲

␲

共

兲

Equation

共

兲

shows the approximate value of the power

efficiency at the resonance frequency for infinite

. It ap-

pears that the four curves of Fig. 3 are almost coincident at
the point given by Eq.

共

兲

, even for the low-

curve. This

is further elucidated in Fig. 4. This graph shows the power
efficiency at the resonance frequency versus

, where

is the optimal-

value discussed in Sec. IV.

Figure 4 shows an s curve, where the part for very low-

values exhibits a very poor efficiency. There, the Lorentz

force acting on the driver’s voice coil is small with respect to
the damping. Then, a rather steep part of the curve follows,

and finally a plateau exists, which is given by Eq.

共

兲

. The

importance of

is further elucidated in the following sec-

tion.

IV. LOW-FORCE-FACTOR DRIVERS

As explained before, normally low-frequency sound re-

production with small transducers is quite inefficient. Two
measures are proposed to increase the efficiency. First, a spe-
cial transducer is used with a low-

value, attaining a high

efficiency and the highest possible sensitivity at that particu-
lar frequency. Second, nonlinear processing essentially com-
presses the bandwidth of a 20- to 120-Hz bass signal down
to a much narrower span. This span is centered at the reso-
nance of the low-

driver where its efficiency is maximum.

These drivers are only useful for subwoofers. In the follow-
ing an optimal force factor is derived to obtain such a result.

The proposed solution, to obtain a high sound output

from a compact loudspeaker arrangement with a good effi-
ciency, consists of two steps. First, the requirement that the
frequency response must be flat is relaxed. By making the
magnet considerably smaller and lighter

共

see Fig. 5, right

FIG. 3.

共

Color online

兲

The power efficiency

␩

for the driver MM3c with

four

values: low

= 1.2

共

solid

兲

, medium

= 5

共

dash-dot

兲

, high

= 22 N / A

共

dash

兲

, and lim

→

⬁

共

thick solid

兲

, while all other parameters are

kept the same as given in Table II. Note that the efficiency is strongly
dependent on

at all frequencies except at resonance, where the efficiency

is affected only modestly by

FIG. 4.

共

Color online

兲

The power efficiency

␩

共

␻

兲

versus the normal-

ized force factor

共

兲

for the driver MM3c, where

共

兲

is the optimal

force factor given in Eq.

共

兲

, in this present case

共

兲

= 1.19. The other

parameters are given in Table II.

FIG. 5.

共

Color online

兲

Picture of the prototype driver

共

MM3c

兲

with a 10

Euro cents coin. At the position where a normal loudspeaker has its heavy
and expensive magnet, the prototype driver has an almost empty cavity;
only a small moving magnet is necessary, which is shown in the right corner.

J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006

Ronald M. Aarts: Optimally sensitive and efficient compact loudspeakers

893

side

兲

a large peak in the SPL curve

关

see Fig. 1

共

solid curve

兲兴

will appear.

Because the magnet can be considerably smaller than

usual, the loudspeaker can be of the moving magnet type
with a stationary coil

共

see Fig. 5

兲

instead of vice versa. At the

resonance frequency the voltage sensitivity can be a factor of
10 higher than that of a normal loudspeaker. In this case an
SPL of almost 90 dB at 1-W input power at 1-m distance is
achieved at the resonance frequency, even when using a
small cabinet

共

⬍

1 1

兲

. Because it is operating in resonance

mode only, the moving mass can be enlarged—which might
be necessary owing to the small cabinet—to keep the reso-
nance frequency sufficiently low. This is done without de-
grading the efficiency of the system because at the resonance
frequency

␯

= 0 and the product

␻

␯

in Eq.

共

兲

becomes

equal to zero. See Table II.

Due to the high and narrow peak in the frequency re-

sponse, the normal operating range of the driver decreases
considerably. This makes the driver unsuitable for normal
use. To overcome this, a second measure is applied. Nonlin-
ear processing essentially compresses the bandwidth of a
20- to 120-Hz 2.5-octave bass signal down to a much nar-
rower span—which is centered at the resonance of the low-

driver—where its efficiency is maximum. This can be

done with a setup as depicted in Fig. 6 and will be discussed
below.

Without loss of generality, it is assumed that the low-

frequency part of the music can be modeled as a sinusoid
with frequency

␻

which is modulated by a slowly varying

signal

共

兲

艌

0. This yields

共

兲

关

共

兲兴

sin

共

兲

共

兲

where

is a constant, or more precisely

peak value of

共

兲

共

兲

is the modulation index. This model is realistic since

共

兲

is a

bandlimited signal, say between 20 to 120 Hz. The fre-
quency of

␻

can be variable and will lead to a certain

pitch. Taking the Fourier transform of Eq.

共

兲

, the mag-

nitude of the spectrum can be written as

兩

共

␻

兲兩

␲

␦

共

␻

−

␻

兲

1
2

共

␻

−

␻

兲

␲

␦

共

␻

兲

1
2

共

␻

兲

共

兲

where

␦

is a unit impulse and the capital function

共

␻

兲

indicates the Fourier transform of

共

兲

. Equation

共

兲

shows

the well-known amplitude-modulated

共

兲

spectrum, as

known from AM radio broadcasting. In contrast to normal
AM radio, in the present case

= 0, this is to make the

amplitude of

共

兲

proportional to the amplitude of

共

兲

. If the

processing depicted in Fig. 6 is applied to the signal

共

兲

, the

signal

共

兲

is recovered by an envelope detector and is used

to modulate a sinusoid, but now with frequency

␻

, where

␻

is fixed and equal to the resonance frequency of the trans-
ducer. This yields

out

共

兲

共

兲

sin

共

兲

共

兲

with the corresponding spectrum

兩

共

␻

兲兩

1
2

关

共

␻

−

␻

兲

共

␻

兲兴

共

兲

The result is that the coarse structure

共

兲共

the envelope

兲

of the music signal after the compression or “mapping” is the
same as before the mapping; an example is shown in Fig. 7.
Only the fine structure has been changed to a sinusoid of the
same frequency as the driver’s resonance frequency.

The upper panel shows the waveform of a rock-music

excerpt; the thin curve depicts its envelope,

共

兲

. The middle

and lower panels show the spectrograms of the input and
output signals, respectively, clearly showing that the fre-
quency bandwidth of the signal around 60 Hz decreases after
the mapping, yet the temporal modulations remain the same.

Using Eq.

共

兲

and neglecting

rad

, the voltage sensitivity

at the resonance frequency can be written as

共

␻

兲

共

␻

兲

共

␻

兲

␻

SBl

␳

␲

共

兲

共

兲

Equation

共

兲

is maximized by adjusting the force factor

by differentiating

共

␻

兲

with respect to

and setting

⳵

共

兲

= 0, resulting in

共

兲

共

兲

It appears that the maximum voltage sensitivity is reached
when the electrodynamic damping term

共

兲

is equal to

the mechanical damping term

; in this case the optimal

force factor is defined as

共

兲

冑

共

兲

The consequences of this optimality criterion are discussed
below. One obvious observation is that the SPL response
becomes, as can be seen in Fig. 1

共

solid curve

兲

, very peaky.

TABLE II. The lumped parameters of the new, and experimental driver with
the

共

optimal

兲

low-

MM3c; see Fig. 5 for its compact magnet system. See

Table I for the abbreviations and the meaning of the variables.

Type

MM3c

⍀

6.4

N/A

1.2

N/m

1022

14.0

Ns/m

0.22

17.2

16.8

FIG. 6. Frequency mapping scheme. The box labeled “BPF” is a bandpass
filter, and “Env. Det.” is an envelope detector. The latter can be a simple
rectifier followed by a low-pass filter. The signal

out

is fed

共

via a power

amplifier

兲

to the driver.

894

J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006

Ronald M. Aarts: Optimally sensitive and efficient compact loudspeakers

The height of the peak is calculated by substituting Eq.

共

兲

into Eq.

共

兲

, which yields the optimal voltage sensitivity

共

␻

兲

␻

␳

␲

共

兲

共

兲

The specific relationship between

共

兲

and both

and

关

Eq.

共

兲兴

causes

to be inversely proportional to

共

兲

共

which may seem counterintuitive

兲

, and thus also inversely

proportional to

冑

. For this particular value of

the

Lorentz force is large enough to get a sufficiently strong
driver with good efficiency, but the electromagnetic
damping is sufficiently low to reach the optimal sensitiv-
ity.

The power efficiency at the resonance frequency under

the optimality condition, obtained by substitution of Eq.

共

兲

into Eq.

共

兲

, yields

␩

共

␻

兲

共

兲

共

兲

共

兲

This can be approximated for

Ⰶ

␩

共

␻

兲 ⬇

共

兲

which clearly shows that for a high power efficiency at the
resonance frequency, the cone area must be large, because

—according to Eq.

共

兲

, or more explicitly Eq.

共

兲

—is

proportional to the squared cone area, and that the mechani-
cal damping must be as small as possible. The damping must
be not too small, however, because the transient response
depends on the damping as well, as is discussed in Larsen
and Aarts

共

2004

兲

. Comparing Eq.

共

兲

with Eq.

共

兲

shows

that the optimally sensitive driver is only 3 dB less efficient
than the infinite

one; however, this is only at the reso-

nance frequency, but this is the working frequency of the
new driver. This can also be seen in Fig. 3 where

␩

共

solid

curve

兲

is close to the infinite

curve, but only at the

resonance frequency.

V. DISCUSSION

Sound reproduction at low frequencies with small driv-

ers in small cabinets is not efficient. Small drivers have a low
radiation impedance with respect to the total damping

关

see

Eq.

共

兲兴

. Small cabinets have a stiff air spring which needs

FIG. 7. The signals before and after the frequency-mapping processing of Fig. 6. The upper panel shows the time signal at

, and the thin curve the output

of the envelope detector. The middle and lower panels show the spectrogram of the input and output signals, respectively.

J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006

Ronald M. Aarts: Optimally sensitive and efficient compact loudspeakers

895

a high moving mass to obtain the desired low resonance
frequency. This will be reiterated more quantitatively below.
For a given volume of the enclosure

, the corresponding

of the “air spring” can be calculated as

␳

共

兲

共

兲

Mounting a loudspeaker in a cabinet will increase the total
spring constant

by an amount given by Eq.

共

兲

, and sub-

sequently increase the resonance frequency of the system. To
compensate for this bass loss, the moving mass has to be
increased; thus,

冑

is increased, which raises

关

see Eq.

共

兲兴

. Then—to obtain a flat frequency characteristic—

must be increased to preserve the original value of

This is at the cost of a more expensive magnet and a loss
in efficiency. This is the designer’s dilemma: high effi-
ciency or small enclosure? To meet the demand for a cer-
tain cutoff frequency, the enclosure volume must be
larger. Alternatively, the efficiency for a given volume
will be less than for a system with a higher cutoff fre-
quency.

This dilemma is

共

partially

兲

solved by using the low-

concept as discussed in Sec. IV, however, at the expense of a
decreased sound quality and the need for some additional
electronics to accomplish the frequency mapping. While the
new driver is not a hi-fi one, many informal listening tests
and demonstrations

confirmed that the decrease of sound

quality appears to be modest, apparently because the audi-
tory system is less sensitive at low frequencies. Also, the
other parts of the audio spectrum have a distracting influence
on this mapping effect, which has been confirmed during
formal listening tests

共

Le Goff

et al.

, 2004

兲

, where the de-

tectability of mistuned fundamental frequencies was deter-
mined for a variety of realistic complex signals. Finally, the
part of the spectrum which is affected is only between, say,
20 and 120 Hz, so the higher harmonics of these low notes
are mostly out of this band and are thus not affected. They
will contribute in their normal unprocessed fashion to the
missing fundamental effect. All these factors support the no-
tion that detuning becomes difficult to detect once the target
complex is embedded in a spectrally and temporally

rich

sound context, as it is typical for applications in modern
multimedia reproduction devices

共

Le Goff

et al.

, 2004

兲

VI. CONCLUSIONS

The force factor

plays a very important role in loud-

speaker design. It determines the efficiency, the sensitivity,
the impedance, the SPL response, the weight, and the cost. It
appears to be not possible to obtain both a high efficiency as
well as a high sensitivity in a wide frequency range. At the
loudspeaker’s resonance frequency, however, it appears to be
possible to meet this criterion. The voltage sensitivity is op-
timal when the electrical damping force is equal to the me-
chanical one, while it is only 3 dB less efficient than an
infinite force-factor loudspeaker. A new low-

driver has

been developed which together with some additional elec-
tronics, yields a low-cost, lightweight, compact, physically
flat, optimally sensitive, and very-high-efficiency loud-
speaker system for low-frequency sound reproduction.

ACKNOWLEDGMENTS

I would like to thank Joris Nieuwendijk

共

Philips Applied

Technologies

兲

and Okke Ouweltjes

共

Philips Research

兲

, who

gave valuable assistance to the low-

project.

Demonstrations and

MATLAB

scripts are on http://www.dse.nl/

⬃

rmaarts

Aarts, R.

共

2005

兲

. “High-efficiency low-

loudspeakers,” J. Audio Eng. Soc.

, 579–592.

Aarts, R., and Janssen, A.

共

2003

兲

. “Approximation of the Struve function H1

occurring in impedance calculations,” J. Acoust. Soc. Am.

113

, 2635–

2637.

Abramowitz, M., and Stegun, I.

共

1972

兲

Handbook of Mathematical Func-

tions

共

Dover, New York

兲

Beranek, L.

共

1954

兲

Acoustics

共

McGraw-Hill, New York

兲共

Reprinted by

ASA 1986

兲

Larsen, E., and Aarts, R.

共

2004

兲

Audio Bandwidth Extension. Application of

Psychoacoustics, Signal Processing and Loudspeaker Design

共

Wiley, New

York

兲

Le Goff, N., Aarts, R., and Kohlrausch, A.

共

2004

兲

. “Thresholds for hearing

mistuning of the fundamental component in a complex sound,” in Pro-
ceedings of the 18th International Congress on Acoustics

共

ICA 2004

兲

Paper Mo.P3.21, p. I-865

共

Kyoto, Japan

兲

Morse, P., and Ingard, K.

共

1968

兲

Theoretical Acoustics

共

McGraw-Hill, New

York

兲

Small, R.

共

1972

兲

. “Closed-box loudspeaker systems. I. Analysis,” J. Audio

Eng. Soc.

, 798–808.

Thiele, A.

共

1971

兲

. “Loudspeakers in vented boxes. I,” J. Audio Eng. Soc.

, 382–392.

Vanderkooy, J., Boers, P., and Aarts, R.

共

2003

兲

. “Direct-radiator loudspeaker

systems with high

,” J. Audio Eng. Soc.

, 625–634.

896

J. Acoust. Soc. Am., Vol. 119, No. 2, February 2006

Ronald M. Aarts: Optimally sensitive and efficient compact loudspeakers