PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Design and construction of loudspeakers with low-࡮࢒ drivers for
low-frequency active noise control applications
Marios GIOUVANAKIS1; Konstantinos KASIDAKIS; Christos SEVASTIADIS;
George PAPANIKOLAOU
Aristotle University of Thessaloniki, Greece

ABSTRACTThe design and construction of small sound sources, as the elementary building components of a compoundsource for low-frequency noise control applications, is investigated. The need for small volume cabinet
loudspeakers has led to a vented design with low force factor (‫ )݈ܤ‬drivers, exploiting their high complianceand low-frequency resonance. The combination of such small powerful sources can lead to dipole ormultipole compound set-ups, in which the radiation directivity pattern in open spaces and modal coupling inclosed spaces can be controlled via the parameters of the distinct driving signals. The construction constraintsof the cabinet are accounted for, considering the loudspeaker’s design optimization. A cabinet with two ports
tuned close to the driver’s resonance frequency is constructed to extend the loudspeaker’s output at low
frequencies. The impact of the position, dimension, and number of the tubes to the loudspeaker’s frequency
response are examined through measurements. The design analysis and experiments show the direction to the
optimum construction. A pair of these sources combined in dipole configurations is measured in terms ofpolar radiation pattern. The implementation of such loudspeakers is advantageous in small rooms, where theavailable space is of great concern.
Keywords: Low-Bl Drivers, Low-Frequencies, Active Noise Control

1. INTRODUCTION
Low-frequency sound reproduction with small transducers is quite inefficient. There areapplications in small enclosures where the available space is limited and the volume of control sources
is of great importance, as the active noise control (ANC) of low-frequency sound which demands usualbulky secondary sources. ANC techniques have been proposed with secondary compound sources (1).
The necessity that secondary sources should be efficient in the whole audible range is not a demand. Inorder for a driver to be efficient in the low-frequency range (< 100 Hz) or to a more restrictedfrequency area, it must be included in a large enclosure. So, the dilemma is for the designer to choose
either high efficiency or a small enclosure.
Aarts (2) offered a partial solution to this by using the low-‫ ݈ܤ‬concept in cases that small cabinets
and high efficiency are important. The so-called low force factor driver has a high and maximum
efficiency in a limited frequency region tuned at its low resonance frequency, fS. The force factor Bl,
which is the product of the flux density B [Wb] in the air gap and the effective length l [m] of the voicecoil wire, has a very important role in the loudspeaker’s design as it determines the efficiency,
impedance, SPL response, weight, and cost. As this factor is a measure of the power ofelectromechanical conversion, low values typically show low damping of the loudspeaker’s
mechanical parts at fS . In the same study, the optimum value of the factor to obtain the maximum
sensitivity at the driver’s fS is given. This design offers higher power efficiency and voltage sensitivity
than usual bass drivers. The loudspeaker may be of the moving-magnet type with a stationary coilbecause the magnet can be considerably smaller; therefore the cabinet’s volume may be smaller. The
efficient pass-band range of the driver decreases considerably. Such loudspeakers offer a potential
optimization of the low-frequency reproduction because they behave as narrow bandwidth filters withhigh-quality factors and can be tuned at low resonance frequencies (3).
1

mgiouvan@ece.auth.gr

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In this work, two identical monopole sources of small volume are constructed and measured, afteran optimum design. The concept of vented-box loudspeakers is utilized for the successful extension in
the low-frequency range comparing to sealed-box ones. As in ordinary rooms the first axial modes
extend up to about 80 Hz and below the Schroeder frequency, the aim is to construct monopole sourcesthat have both small volume and high efficiency at a narrow range of about 20 Hz. The construction
constraints are accounted via measurements. Two monopole sources are configured in several dipole
set-ups and their polar patterns are measured in a fully anechoic chamber. The following sources
present the potential of the adaptive directivity sources application to low-frequency ANC in small
enclosures (4-8).

2. LOUDSPEAKER DESIGN
2.1 Theory
The Thiele/Small parameters specify the low-frequency performance of a loudspeaker driver andcontain the information to calculate the speaker response in any box (9, 10). The resonancefrequency of the moving system of a driver is fS given by Eq. (1), in which the input impedance ismaximum. The quality factors ܳ୉ୗ and ܳ୑ୗ of a driver at fS considering electrical and non-electricalresistances only respectively, lead to ܳ୘ୗ , the total Q of the driver at fS. Also, ܸ୅ୗ is the volume of air
in liters having the same acoustic compliance as the driver suspension and is given by Eq. (2).
 ݂ୗ ൌ ͳ൘
ʹߨඥ‫୑ܥ‬ୗ ߊ஄ୗ(1)ܸ୅ୗ ൌ ߩܿ ଶ ܵୈ ଶ ‫୑ܥ‬ୗ (2)where ‫୑ܯ‬ୗ [g] is the mass assembly including air load, ܵୈ [m2 ] the effective projected surface areaof the driver’s diaphragm, ‫୑ܥ‬ୗ the suspension compliance, ρ the air density [kg/m 3 ] and c the speed ofsound [m/s].
The closed box significantly improves the loudspeaker response at low frequencies andintroduces the parameter of the enclosure’s air ܸ୅୆ , i.e. the addition of a second compliance ‫ܥ‬୅୆ , asgiven in Eq. 3 along with that of the speaker’s suspension. The ratio of the two volumes orcompliances denoted by α in Eq. 4, determines the degree of coupling of the box with theloudspeaker (11).
‫ܥ‬୅୆ ൌܸ୅୆൘ ଶ ଶ Ȁܵୈ ߩܿ

(3)

௏



஼

ߙ ൌ ୚ఽ౏ ൌ ஼ఽ౏
ఽా

ఽా

(4)

When inserting the diaphragm into a sealed enclosure, the resonant frequency ݂େ and the quality factor
ܳ୘େ are changed accordingly to α. Because of the small volume of the speaker under development(ߙ ൐ ͵), it is an "air suspension" model, in which the compliance of the enclosed air is smaller thanthat of the suspension and the damping power of the diaphragm comes mainly from the air of thebox. While ݂େ ൐ ݂ୗ in this case, if it is necessary to extend the efficient operating frequency range
lower than or even as same as fS, another action must be taken. In case that the ‫୑ܥ‬ୗ is increased, thefS is reduced and so does the ݂େ . Hence, for a fixed enclosure size the ratio α and, therefore the ݂େ , isincreased. One technique for extending the loudspeaker’s function to lower frequencies is to add
extra mass to the diaphragm, as it is observed by Eq. 1.
With a bass-reflex box (also vented box or reflex-port system) the loudspeaker’s efficiency is
increased from the sound of the diaphragm’s rear side comparing to a closed box at low frequencies.
An extra degree of freedom is inserted through the port in the speaker’s transfer function as it is a ductof variable dimensions and operates as a Helmholtz resonator since the mass of air inside reacts withthe air in the box. There are two resonant frequencies; fS and that of the port-box resonance, fB. The
ratio ݄ ൌ ݂୆ Ȁ݂ୗ is the coupling measure of the loudspeaker with the vented box. The contribution of
the port is significant for one to two octaves above the fB. The output of the port can be combined
properly with that of the loudspeaker in a reciprocal way by enhancing the system ’s response in thelower frequencies. An example of the effect of the parameters h and α on the speaker’s response isdepicted in Figure 3.
Based on the Small’s analysis (11), it is shown that in the acoustical analogous circuit of a vented
box loudspeaker system, an induction Μ AP expressing the acoustic mass of port is added in parallel tothe box compliance of air, C AB. Therefore, the LC-loop created by the box-port system has a tuning
frequency ݂୆  ൌ ͳȀξ‫ܥ‬୅୆ ‫ܯ‬୅୔ and a factor ܳ୐ ൌ  ߱୆ ‫ܥ‬୅୆ ܴ୅୐ that represents the Q of the resonant6922

circuit at ω B, with R AL to be the acoustic resistance of enclosure losses caused by leakage, along withthe other T-S parameters given by Eq. (4)-(9) from the same work. One more resonance is inserted tothe system in addition to that of the closed box. The loop circuit offers an inverse resonance to the
input impedance with its minimum at fB and combined with the fS results in the impedance response
shown in Figure 1. The vent dimensions in accordance with the fB are calculated by:‫ܮ‬୚ ൌ

ଵǤସ଺ଷଵ଴ళ ோమ௙ా మ ௏ౘ െ ͳǤͶ͸͵ܴ(5)with the length ‫ܮ‬୚ [in], the box volume ܸୠ [cu in] and the radius ܴ of vent [in]. The minimum
diameter to avoid a power loss is ݀୚ ൌ ͵ͻǤ͵͹ඥ݂୆ ܸୈ with ܸୈ ൌ ୈ ‫ݔ‬୫ୟ୶ the cone displacement
volume [cu m] (12).

Figure 1 - Typical input impedance curve of a

Figure 2 - Design steps of a closed box loudspeaker.

vented box loudspeaker system.
2.2 System design decision
The increase in moving mass reduces the resonance frequency and increases the ܳ୘େ. Moreover,
the reference sensitivity is decreased (10). However, there is no confirmation that the loudspeakersuspension will withstand the extra mass, or that the diaphragm motion -axis will not be displaced evenslightly. The probability of such failures is greater in the case of low-‫ ݈ܤ‬drivers because of the "weak"magnet. Moreover, a possible addition of mass must be done very precisely so as not to change theweight distribution on the diaphragm resulting in the non-linear motion of the cone. In the case of avented box, the port must be tuned at a low-frequency and the speaker driver must also have a low
resonant frequency. The parameters that are at the discretion of the designer to handle are the box
volume and the possible addition of mass. Depending on the application, a suitable combination ofthese parameters leads to different responses.
In the present work, small box volume is the major concern. However, as this is decreased, the fB
increases with the overall quality factor, with the latter to be desirable for greater efficiency near the
resonant frequency. It is, therefore, preferable to choose a loudspeaker with as low force factor as
possible rather than adding a moving-mass to extend the response to lower frequencies. The 4-inch
loudspeaker driver that is used fairly meets the above criteria with ݂ୗ ൌ ͵͹  and ‫ ݈ܤ‬ൌ ͵ǤͶ͵ ή .
The volume of implementation to meet the limits and optimize the loudspeaker’s low-frequency
response must be studied. In order to visualize the changes made by each system modification to the
speaker response, a software was developed in Matlab, in which the user can enter the fundamental, theThiele/Small parameters and the box properties and calculate the transfer function of the loudspeaker.
Multiple parameters can be altered and the changes are observed graphically. An example is depicted
for the case of a sealed cabinet in Figure 2. The effect of mass-adding is also shown, which is replacedby the port-mass in the final vented cabinets.
An advantage of a vented box is the low cone excursion comparing to a sealed enclosure, i.e. lower
distortion, above the tuning frequency. However, this does not stand below the fB as the excursion rateincreases rapidly, meaning a potential rumble noise and cone’s destruction, so care must be taken not
to drive the speaker in this range (12).

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Figure 3 - Effect of α and h in the response of a vented box loudspeaker.

Lv [cm]
16
18
20
38

fB [Hz]
52.1
47.1
44.4
44.0

100
95
90
85
80
75
70
65
60
55
50

case 1: d=18mm, fB=41Hz
case 2: d=27mm, fB=58Hz
case 3: d=32mm, fB=64Hz

dB [SPL]

Table 1 - Effect of ‫ܮ‬୚ on fB in a one-port
test box with d = 3.2 cm

40
50
63
80
100
1/3 Octave Bands - Center Frequency [Hz]Figure 4 - Maximum SPL of a low-‫ ݈ܤ‬speakerin vented box with two tubes of LV = 19 cm

3. LOW-࡮࢒ LOUDSPEAKER IMPLEMENTATION
3.1 Loudspeaker design
In a test box, aluminum pipes of 32, 27 and 18 mm diameter were used. According to Eq. 6 and as
can be seen from Table 1, for reducing the fB in a given enclosure the tube’s length must be increased.
However, this can be achieved as the length increases up to a certain point, in which the air inside thevent resonates and the fB is reduced. It seems that an impractical length of a pipe must be used withina one-port box for the fB to be at 44 Hz. Moreover, the loudspeaker was tested under increasing the
input voltage from 2 V (=1 W for 4 Ω drivers) at a step of 0.1 V until the driver started showing
non-linear function or fluid noise from the tube was coming. The sound pressure level was tracked
with a calibrated sound level meter.
After not so satisfactory results, it was decided to create a second port in the box which wasmeasured with several pipe diameters having the same length, with some results given in Figure 4. It isobserved that a lower fB of 41Hz can be achieved with two vents of much shorter length and diameter.
Especially at the center frequencies of 1/3-octave bands of 40 and 50 Hz, the only case with asatisfactory output is the first one, i.e. two pipes of ‫ܮ‬୚ = 19 cm and d = 18 mm. Ιn Figures 5 and 6, theinput impedance with the tuned-box frequencies are depicted and transfer function for fourcombinations of ‫ܮ‬୚ and d are shown respectively for a double-port box, in order to get the optimum
low-frequency response. The effect upon the fB is apparent. Thus, the choice of the above dimensions
for the ports is considered to be suitable, aiming at greater sensitivity to frequencies in the range 40 –60 Hz.

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Finally, two low-Bl loudspeakers with outer dimensions of ʹͲ ൈ ͳͺǤͷ ൈ ͳͺǤͷ  were constructedin double-port vented boxes (fB = 41 Hz) with 15mm-thick plywood as shown in Figure 7. The total
volume is 5.8 l. Without the driver and ports, the approximate internal volume is 3.5 l.

Figure 5 – Measured input impedance withtwo ports of different lengths and diameters.

Figure 6 – Measured transfer function withtwo ports of different lengths and diameters.

3.2 Constructed loudspeakers acoustic measurements
The measurements to obtain the response results and polar patterns of the constructed speakers
were conducted in a fully anechoic chamber. The input signal to measure the transfer functions was apink noise filtered from 35 Hz to protect the loudspeaker from over-displacement. The speakers werealso driven by sine waves of certain frequencies to obtain their polar pattern.
The two loudspeakers were measured independently first and then combined in various dipole
set-ups to obtain their radiation pattern. The microphone was stationary throughout the measurements,
positioned 1.5 m from the sources’ acoustic center. As the latter, the center of the box was chosen forthe case of one source and the middle of the distance of the two centers for the case of two sources. In
Table 2, it is shown that the two sources have the same sensitivity at 1 kHz. The sound level wasmeasured also with sine waves of 40, 80 and 200 Hz. From the monopoles’ polar patterns in Figure 8,
it seems that their behavior at low frequencies is essentially that of an omnidirectional source with asmall deviation (less than 1-2 dB). As these monopoles present an almost identical response with only
slight deviations, the next step was to be combined in dipole configurations.
Several dipole set-ups, with the two loudspeakers driven by inverse signals in a distance muchsmaller than the radiated wavelength, were measured regarding their directivity pattern as seen in
Figures 9-10. For example, the FB set-up is the usual dipole with 15 cm distance between the twomonopoles. The microphone corresponds to 0 o in the patterns. The purpose of the several set-ups wastheir directivity behavior to be tested for different driver diaphragm and ports orientations of its
comprising monopoles.
Table 2 - Loudspeakers responses
f [Hz]
40
80
200
1000

Loudspeaker 1[dB] (1W@1m)
66.3
80.9
79.2
81.3

Loudspeaker 2[dB] (1W@1m)
66.7
81.1
79
81.3

Figure 7 - The constructed loudspeakers.

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Figure 8 - The measured polar patterns of the constructed speakers for the frequencies of 40 Hz(blue), 63 Hz (green) and 100 Hz (yellow).

Figure 10 - Dipole configurations. FB:
front-back, FS: front-side, SIS: side-inverse
side.

Figure 9 – Dipole measurement in a fully
anechoic chamber

4. DISCUSSIONIn Table 1, it is observed that in the operation range between 44 and 52 Hz, the length of the port
tubes can be changed keeping the loudspeaker volume constant. However, a minimum distance of the
tube end from the loudspeaker inner back side must be considered, because of the expected noise.
Another notable comment regarding the use of vented box loudspeakers is that care must be always
considered not to drive them with high-level signals on frequencies below the fB, as in this frequency
region the diaphragm excursion is increased rapidly and the danger of destroying the driver is severe.
Regarding the FB set-up, it leads to a dipole polar pattern (Figure 11). Sufficient reduction of the6926

radiated sound is observed at perpendicular directions of the dipole’s axis in all frequencies. As far as
the other two set-ups are concerned, an expected turning of the pattern in relation to the microphone ispresented. Especially in the two lower frequencies, the dips in the dipole shape are weaker than in thecase of 100 Hz. This is probably due to the maximized operation of the ports for frequencies close to
the fB, as can be observed for the case of 40 Hz (blue line). The ports function as extra acoustic centersin addition to the diaphragms; therefore more radiation axes in different directions are defined fromtheir combinations. These result in the polar patterns not to have the figure-of-eight shape.
(a)

(b)

(c)

Figure 11 - The measured polar patterns of three dipole configurations (a: FB, b: FS, c: SIS) for thefrequencies of 40 Hz (blue), 63 Hz (green) and 100 Hz (red).

5. CONCLUSIONSThis work presents the study and construction of loudspeakers with low- ‫ ݈ܤ‬drivers for
low-frequency applications such as ANC in small closed spaces. Utilizing a low force factor driverwith higher suspension compliance, lightweight and small vented box loudspeakers are constructed.
Vented box design offers better performance at the lower frequency range than closed boxes, giving
the designer the ability to reduce the volume of the speaker cabinet. By increasing the number of ports,
better performance in the low-frequency region is obtained. Moreover, the fB can be tuned in a narrow
low-frequency range by changing only the vents while keeping the same cabinet. The designed
speakers were tested in dipole set-ups. The polar responses were satisfying in the frequency region6927

where the loudspeakers radiated through the driver diaphragm, but when the radiation was through theport tubes, it deviated from the typical figure-of-eight response.
The potential implementation of such loudspeakers in combining dipoles or more compoundsources is the next step, as they offer the flexibility to configure numerous topologies and provide analternative to the common bulky monopole sources, especially in small rooms with big restriction inpositioning. The requirement that the frequency response to be flat is relaxed as the performance of the
speaker in the pass-band is of no concern. The efficient ANC at a narrow low-frequency band in small
spaces leads to amplify the speaker response around the fS and keep it as low as possible along with theenclosure’s size. Nonetheless, further investigations including combinations of more-than-one dipolesin different topologies should be made for their radiation patterns, as the design of such compoundsources is an advantage for ANC applications.

ACKNOWLEDGEMENTS
The acoustic measurements were taken place in a fully anechoic chamber, built in accordance with
the ISO 3745 standard, of the Hellenic Institute of Metrology, Sindos/Thessaloniki, Greece.

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