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Design and construction of loudspeakers with low-Bl drivers for low-frequency active noise control applications

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The design and construction of small sound sources, as the elementary building components of a compound source 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 (Bl) drivers, exploiting their high compliance and low-frequency resonance. The combination of such small powerful sources can lead to dipole or multipole compound setups , in which the radiation directivity pattern in open spaces and modal coupling in closed spaces can be controlled via the parameters of the distinct driving signals. The construction constraints of 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 of polar radiation pattern. The implementation of such loudspeakers is advantageous in small rooms, where the available space is of great concern.
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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 GIOUVANAKIS
1
; Konstantinos KASIDAKIS; Christos SEVASTIADIS;
George PAPANIKOLAOU
Aristotle University of Thessaloniki, Greece
ABSTRACT
The design and construction of small sound sources, as the elementary building components of a compound
source 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 () drivers, exploiting their high compliance
and low-frequency resonance. The combination of such small powerful sources can lead to dipole or
multipole compound set-ups, in which the radiation directivity pattern in open spaces and modal coupling in
closed spaces can be controlled via the parameters of the distinct driving signals. The construction constraints
of 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 of
polar radiation pattern. The implementation of such loudspeakers is advantageous in small rooms, where the
available 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 are
applications 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 usual
bulky 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. In
order for a driver to be efficient in the low-frequency range (< 100 Hz) or to a more restricted
frequency 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- concept 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 voice
coil 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 of
electromechanical conversion, low values typically show low damping of the loudspeakers
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 coil
because 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 with
high-quality factors and can be tuned at low resonance frequencies (3).
1
mgiouvan@ece.auth.gr
In this work, two identical monopole sources of small volume are constructed and measured, after
an 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 sources
that 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 and
contain the information to calculate the speaker response in any box (9, 10). The resonance
frequency of the moving system of a driver is fS given by Eq. (1), in which the input impedance is
maximum. The quality factors  and  of a driver at fS considering electrical and non-electrical
resistances 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).

 (1)    (2)
where  [g] is the mass assembly including air load, [m2] the effective projected surface area
of the driver’s diaphragm,  the suspension compliance, ρ the air density [kg/m3] and c the speed of
sound [m/s].
The closed box significantly improves the loudspeaker response at low frequencies and
introduces the parameter of the enclosure’s air , i.e. the addition of a second compliance , as
given in Eq. 3 along with that of the speaker’s suspension. The ratio of the two volumes or
compliances denoted by α in Eq. 4, determines the degree of coupling of the box with the
loudspeaker (11).
  
 (3)   
 
 (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
(  ), it is an "air suspension" model, in which the compliance of the enclosed air is smaller than
that of the suspension and the damping power of the diaphragm comes mainly from the air of the
box. 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  is increased, the
fS is reduced and so does the . Hence, for a fixed enclosure size the ratio α and, therefore the , is
increased. 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 loudspeakers 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 duct
of variable dimensions and operates as a Helmholtz resonator since the mass of air inside reacts with
the air in the box. There are two resonant frequencies; fS and that of the port-box resonance, fB. The
ratio    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 the
lower frequencies. An example of the effect of the parameters h and α on the speaker’s response is
depicted 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 ΜAP expressing the acoustic mass of port is added in parallel to
the box compliance of air, CAB. Therefore, the LC-loop created by the box-port system has a tuning
frequency  and a factor  that represents the Q of the resonant
circuit at ωB, with RAL to be the acoustic resistance of enclosure losses caused by leakage, along with
the other T-S parameters given by Eq. (4)-(9) from the same work. One more resonance is inserted to
the 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:

  (5)
with the length [in], the box volume [cu in] and the radius of vent [in]. The minimum
diameter to avoid a power loss is  with   the cone displacement
volume [cu m] (12).
Figure 1 - Typical input impedance curve of a
vented box loudspeaker system.
Figure 2 - Design steps of a closed box loudspeaker.
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 loudspeaker
suspension will withstand the extra mass, or that the diaphragm motion-axis will not be displaced even
slightly. The probability of such failures is greater in the case of low- drivers because of the "weak"
magnet. Moreover, a possible addition of mass must be done very precisely so as not to change the
weight distribution on the diaphragm resulting in the non-linear motion of the cone. In the case of a
vented 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 of
these 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    .
The volume of implementation to meet the limits and optimize the loudspeakers 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, the
Thiele/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 replaced
by 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 rate
increases rapidly, meaning a potential rumble noise and cones destruction, so care must be taken not
to drive the speaker in this range (12).
Figure 3 - Effect of α and h in the response of a vented box loudspeaker.
Table 1 - Effect of on fB in a one-port
test box with d = 3.2 cm
fB [Hz]
52.1
47.1
44.4
44.0
Figure 4 - Maximum SPL of a low- speaker
in 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 the
vent resonates and the fB is reduced. It seems that an impractical length of a pipe must be used within
a 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 was
measured with several pipe diameters having the same length, with some results given in Figure 4. It is
observed 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 a
satisfactory output is the first one, i.e. two pipes of = 19 cm and d = 18 mm. Ιn Figures 5 and 6, the
input impedance with the tuned-box frequencies are depicted and transfer function for four
combinations of 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.
50
55
60
65
70
75
80
85
90
95
100
40 50 63 80 100
case 1: d=18mm, fB=41Hz
case 2: d=27mm, fB=58Hz
case 3: d=32mm, fB=64Hz
1/3 Octave Bands - Center Frequency [Hz]
dB [SPL]
Finally, two low-Bl loudspeakers with outer dimensions of   were constructed
in 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 with
two ports of different lengths and diameters.
Figure 6 Measured transfer function with
two 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 a
pink noise filtered from 35 Hz to protect the loudspeaker from over-displacement. The speakers were
also 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 for
the 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 was
measured 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 a
small 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 much
smaller 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 two
monopoles. The microphone corresponds to 0o in the patterns. The purpose of the several set-ups was
their directivity behavior to be tested for different driver diaphragm and ports orientations of its
comprising monopoles.
Table 2 - Loudspeakers responses
f [Hz]
Loudspeaker 1
[dB] (1W@1m)
Loudspeaker 2
[dB] (1W@1m)
40
66.3
66.7
80
80.9
81.1
200
79.2
79
1000
81.3
81.3
Figure 7 - The constructed loudspeakers.
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 9 Dipole measurement in a fully
anechoic chamber
Figure 10 - Dipole configurations. FB:
front-back, FS: front-side, SIS: side-inverse
side.
4. DISCUSSION
In 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 the
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 is
presented. Especially in the two lower frequencies, the dips in the dipole shape are weaker than in the
case 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 centers
in addition to the diaphragms; therefore more radiation axes in different directions are defined from
their combinations. These result in the polar patterns not to have the figure-of-eight shape.
Figure 11 - The measured polar patterns of three dipole configurations (a: FB, b: FS, c: SIS) for the
frequencies of 40 Hz (blue), 63 Hz (green) and 100 Hz (red).
5. CONCLUSIONS
This work presents the study and construction of loudspeakers with low-  drivers for
low-frequency applications such as ANC in small closed spaces. Utilizing a low force factor driver
with 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 region
(a)
(b)
(c)
where the loudspeakers radiated through the driver diaphragm, but when the radiation was through the
port tubes, it deviated from the typical figure-of-eight response.
The potential implementation of such loudspeakers in combining dipoles or more compound
sources is the next step, as they offer the flexibility to configure numerous topologies and provide an
alternative to the common bulky monopole sources, especially in small rooms with big restriction in
positioning. 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 the
enclosure’s size. Nonetheless, further investigations including combinations of more-than-one dipoles
in different topologies should be made for their radiation patterns, as the design of such compound
sources 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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... The local noise control is attained by reducing the sound pressure at the microphone position. This work extends our previous studies on low-frequency noise control simulations with quadrupole sources [23,24], and the development of small lowfrequency compound sources with low force factor (low-Bl) drivers [25,26], suitable for control application in small common or factory control rooms. In the study of [24], the successful attenuation of all the investigated modal frequencies would not be possible if monopoles or dipoles were solely used. ...
... Each monopole source, with outer dimensions of 20 cm × 18.5 cm × 18.5 cm , includes a 4"(⌀ = 101.6 mm, 4 Ohm) Visaton (Haan, Germany) KT 100 V driver with low resonance frequency, = 37 Hz, and low force factor, = 3.43 T • m , in an optimal bass-reflex box design [25]. The developed lightweight subwoofers were arranged in two quadrupole configurations for noise control (Figure 8). ...
... Each monopole source, with outer dimensions of 20 cm × 18.5 cm × 18.5 cm, includes a 4 ( = 101.6 mm, 4 Ohm) Visaton (Haan, Germany) KT 100 V driver with low resonance frequency, f S = 37 Hz, and low force factor, Bl = 3.43 T·m, in an optimal bass-reflex box design [25]. The developed lightweight subwoofers were arranged in two quadrupole configurations for noise control (Figure 8). ...
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The low-frequency performance of a vented-box loudspeaker system is directly related to a small number of easily measured system parameters. This system is a fourth-order (24-db per octave cutoff) high-pass filter which can be adjusted to have a wide variety of response characteristics. Enclosure losses have a significant effect on system performance and should be taken into account when assessing or adjusting vented-box system. The efficiency of a vented-box loudspeaker system is shown to be quantitatively related to system frequency response, internal losses, and enclosure size.
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An investigation of the equivalent circuits of loudspeakers in vented boxes shows that it is possible to make the low-frequency acoustic response equivalent to an ideal high-pass filter or as close an approximation as is desired. The simplifying assumptions appear justified in practice and the techniques involved are simple.