Analysis and optimisation of an active noise control system as a potential
acoustic metamaterial building block
Joe Tan1(1) , Jordan Cheer(2) and Stephen Daley(3)(1) University of Southampton, United Kingdom, j.tan@soton.ac.uk(2) University of Southampton, United Kingdom,(3) University of Southampton, United Kingdom,

ABSTRACT
Active Noise Control (ANC) systems have been employed in a variety of applications to provide high performance noise control at low frequencies, within a lightweight and compact package. Recently, acoustic metamaterials (AMM) have been proposed and demonstrated as an alternative approach to achieving high levels ofnoise control. AMM are engineered structures that consist of an array of subwavelength unit cells, which exhibit
behaviour not seen in conventional materials. For example, passive resonators have been designed to achieve
negative effective material properties and manipulate wave propagation. These AMMs, however, typically havenarrow band gaps, where wave propagation is suppressed. When acting as an AMM, ANC systems have beenshown to improve performance, tuneability and adaptability, but physical insights have generally been limited.
Therefore, this paper investigates the effects of traditional ANC on the effective material properties and shows
physical insight through an analysis of the wave propagation within a one-dimensional duct system. The active unit cell has then been optimised to directly minimise the effective material properties and a corresponding
physical analysis has been presented.
Keywords: Active Noise Control, Acoustic Metamaterials, negative density, negative bulk modulus.

1

INTRODUCTION

Active noise control (ANC) is an advanced technology that provides high levels of noise control at low
frequencies whilst maintaining a lightweight and compact package, which is difficult to achieve using traditionalpassive control treatments. ANC also has the capability to adapt to changes in the acoustic environment, as wellas the ability to shape and manipulate the sound quality. In recent years, an alternative approach to achieve highlevels of noise control at low frequencies within a lightweight and compact package has been developed throughthe design of acoustic metamaterials (AMMs). AMMs achieve high performance noise control by designing anengineered structure consisting of an array of subwavelength unit cells. Since the unit cells are subwavelength,
the entire structure is considered to be a homogeneous medium with an effective density and bulk modulus
that can take on positive, negative and even zero values. These dynamic material properties allow acoustic
metamaterials to achieve behaviour that is not seen in conventional materials, such as near-zero refractive index[1, 2] and negative effective density [3, 4] or bulk modulus [5, 6] or even both [7, 8]. The greater range ofmaterial properties that are available to acoustic metamaterials allow these materials to not only control thetransmission of sound, but, they have also shown the potential to be used in more novel technologies, forexample, superlenses [9, 10] and acoustic cloaking [11, 12]. Passive acoustic resonators have shown the ability
to achieve negative effective material properties, however, these materials typically have narrow band gaps andmoderate performance due to losses associated with the passive resonators [4, 13]. To overcome the limitationsof passive AMMs, active control has been combined with passive resonators to increase the level of performanceand width of the band gaps whilst increasing the adaptability and tuneability of these materials [14, 13].
Although a variety of active AMMs have been proposed, the active control systems employed within thesematerials typically differ from traditional ANC systems. That said traditional ANC strategies have been usedin recent years in the context of AMM to control the transmission of sound using active Helmholtz resonators4862

[15] and have also been used to minimise the scattered sound field to create an active acoustic cloak [16]. Furthermore, [17] also identified the links between traditional active control systems and the behaviour of AMMs.
The work presented in this paper will further investigate the effects of traditional ANC systems on the effective material properties whilst providing physical insight how the active control systems manipulate wave
propagation within a one-dimensional anechoically terminated duct. Using the same one-dimensional duct system, this paper will also investigate whether active control can be used to directly control the effective material
properties using an optimisation procedure. Section 2 presents the one-dimensional system used in this study.
Section 3 describes the theoretical formulations of the traditional active control strategy considered in this study
and presents a simulation-based investigation of its performance resulting effective material properties and physical behaviour. Section 4 describes the optimisation procedure used to minimise the effective material properties
using the active control systems and also presents the results of simulations following the same form as in
Section 3. Finally, Section 5 presents the conclusions.

2

SYSTEM SETUP

Figure 1 shows the one-dimensional duct system used throughout this study. This figure shows that the onedimensional system contains a primary source at one end and an anechoic termination at the other end and theactive control unit is placed at the centre of the duct with two pressure microphones located either side of theactive control unit. The traditional ANC strategy considered in this study is the minimisation of downstreampressure using three different source configurations; a single monopole, dipole and pair of monopole control
sources. Also, using the system shown in Figure 1, a single monopole control source has been optimised
to minimise the effective bulk modulus and a dipole control source has also been optimised to minimise theeffective density.
L
p1
A

d

p2
B

P rimary Disturbance

p3

p4

C
D
Anechoic T ermination
Active Control U nit

Figure 1: Diagram showing the rigid terminated duct setup, with the four micro- phone positions andFigure 1. The anechoicallyterminated duct setup showing the primary disturbance source, the four pressure
the location of the active control unit
microphone positions and the location of the active control unit.

The frequency range of interest has been restricted to low frequencies, where the wavelength is much larger
than the diameter of the duct and, thus, plane wave propagation can be assumed. The total sound pressure at apoint, x, in the duct can be represented by the linear superposition of the pressures due to the primary, p p (x),and secondary, ps (x), sources and can be written asp(x) = p p (x) + ps (x)(1)

where x is the coordinate position in the duct. Following the assumption of plane wave propagation, the soundpressure due to the primary source can be expressed as
p p (x) =ρ0 c0q p e−jk|x+L|
2S(2)

where ρ0 and c0 are the density and speed of sound in air, S is the cross-sectional area of the duct, L isthe distance between the primary source and active control unit, k is the wavenumber and q p is the primarysource strength. The following sections will describe the formulation to obtain the secondary source sound fieldrequired to control the downstream pressure and the effective material properties of the fluid medium.

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3

MINIMISATION OF DOWNSTREAM PRESSURE

There has been a wide variety of active control strategies developed for various applications, however, for the
one-dimensional duct control problem the most fundamental active controller aims to minimise the downstream
propagation of the acoustic wave. In the following sections, this control strategy will be formulated for different
secondary source arrangements before the behaviour of these arrangements is evaluated.
3.1 Control Strategy Formulation
The first secondary source arrangement that will be considered here is a single monopole control source
located at x = 0 in the duct and driven to minimise the downstream pressure. Using the formulation describedin [17], the optimal secondary source strength in this case is
qs = −q p e−jkL .

(3)The second case considered in this study is a dipole control source and the sound pressure produced by thissource in the upstream and downstream sections of the duct can be written as [17]ps (x) = −f −jk|x|
e
,
2S

for

x<0

and

ps (x) =f −jk|x|
e
,
2S

for

x>0

(4)

where f is the dipole force. Using the formulation described in [17] in this case, the optimal dipole force is
f = ρ0 c0 e−jkL .

(5)

When minimising the downstream pressure, the single monopole and dipole control sources increase thesound pressure in the upstream section of duct. To overcome this limitation, a pair of monopole control sources
can be used to minimise the downstream pressure in the duct whilst producing zero upstream sound radiation.
In this case, the first monopole source is located at x = 0 and the second monopole source is located at x = d,
as shown in Figure 1. To achieve zero upstream sound radiation, the first monopole source strength, qs1 , must
be related to the second monopole source strength, qs2 , by
qs1 = −qs2 e−jkd .

(6)

Using the formulation described in [17] in this case, the second monopole control source strength that minimises
the downstream pressure is then given by
qs2 = −q p

e−jkL.
2j sin(kd)(7)

3.2 Performance Metrics
The performance of each secondary source configuration optimised using the downstream pressure minimisation control strategy has been assessed using the transmission and reflection coefficients and dissipated energy.
Additionally, the effective material properties have been calculated using the retrieval method described in [18]so that the behaviour of the ANC strategies can be linked to the metamaterial literature. Using the pressure
at the four microphones shown in Figure 1, the complex wave amplitudes of the four travelling waves can becalculated, which are indicated by the coefficients A to D shown in Figure 1. The performance metrics can thenbe obtained using the complex wave amplitudes and they can be expressed as
T=

B
C
, R=
and Ed = 1 − (|R|2 + |T |2 )AA

where T is the transmission coefficient, R is the reflection coefficient and Ed is the dissipated energy.

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(8)

Using the transmission and reflection coefficients, the effective refractive index, ne f f , and impedance, ze f f ,can also be calculated as [18]ne f f =ρ0 c0 q−j ln(φ ) + 2πm
and ze f f =kd
1 − 2R + R2 − T 2

where

(9)
q
1 − R2 + T 2 + q
q = (R2 − T 2 − 1)2 − 4T 2 , φ =2Tand m is the branch number. The effective refractive index and impedance can then be used to calculate theeffective material properties, which can be expressed as [19]
ρe f f =ne f f Ze f f
c0

and Be f f =Ze f f c0,
ne f f(10)

where ρe f f is the effective density and Be f f is the effective bulk modulus of the fluid medium. From Eq. 9, it
can be seen that the refractive index and impedance contain multiple solutions, which can lead to the incorrect
selection of the effective material properties. In order to obtain the correct refractive index, the algorithm
described in [20] has been used to estimate the correct branch number, which in turn selects the correct effective
material properties.
3.3 Results
The effective material properties and performance metrics for each source configuration have been calculatedusing the procedure outlined above and these results are presented in Figure 2. From these results, it can beseen that the single monopole (blue) and dipole (red) sources achieve perfect reflection, zero transmission andzero dissipated energy, whilst the pair of monopole sources (black) achieves zero transmission and reflection,
and, therefore, perfect dissipation. Although the performance metrics are identical for the single monopole anddipole sources, the effective material properties are completely different, as shown in Figure 2. Both material
properties are zero in the single monopole case, which corresponds to a pressure-release boundary condition.
In the dipole case, both of the effective material properties tend towards infinity, which corresponds to a rigid
boundary condition. The pair of monopole sources creates the perfect absorption condition by achieving aneffective density equal to the ambient air density and zero bulk modulus. To provide further insight, the effective
refractive index has been calculated and is presented in Figure 2f. This plot shows that the three active control
systems create a refractive index that is greater than unity. A physical understanding of what the refractive
index means in terms of the wave propagation can be gained by considering its definition, which is given by
the ratio between the speed of sound and phase velocity, v phase , of the incident waves, which isne f f =c0.
v phase(11)

This indicates that the phase velocity across the secondary sources in all three cases presented in Figure 2
is slower than the speed of sound, but that in the case of the pair of monopole sources, where the refractive
index is extremely large, the incident wave effectively stops.

4

OPTIMISATION PROCEDURES FOR DIRECT CONTROL OF THE EFFECTIVE
MATERIAL PROPERTIES

It has been shown in the previous section how a traditional ANC strategy with various secondary source configurations influence the effective material properties and resulting wave propagation within a one-dimensional
duct. In the context of AMMs there is a general objective to control the effective material properties directly.

4865

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2. The performance of active control systems that are optimised to minimise the downstream pressureusing a single monopole (blue), a dipole (red) and a pair of monopole (black) control sources. (a), (c) and (e)show the transmission and reflection coefficients and dissipated energy respectively and (b), (d) and (f) show
the effective density, bulk modulus and refractive index respectively.
Therefore, this section presents an investigation into whether the secondary source configurations used in theprevious section can be used to minimise the effective material properties directly. This is achieved using an
optimisation procedure to achieve broadband negativity for the effective density and bulk modulus. Two cases
have been considered in this study; a single monopole source that is driven to minimise the effective bulk
modulus and a dipole source that is driven to minimise the effective density. In both cases, the optimisation
procedure used the Globalsearch algorithm [21] in conjunction with the interior-point algorithm [22] to calculate
the optimal source strengths numerically in Matlab, because it is not straightforward to calculate the optimal
source strengths that minimise the effective material properties analytically. The Globalsearch algorithm has
been implemented to increase the likelihood of obtaining the global minimum from the relatively complex costsurfaces associated with the effective material properties.
4.1 Results
The performance of the optimised monopole and dipole secondary sources has been evaluated using thetransmission and reflection coefficients and dissipated energy, as well as the effective material properties andrefractive index of the fluid medium, as described in Section 3.2. These metrics are shown in Figure 3 forthe optimised systems. These results show that the single monopole source (blue) achieves transmission andreflection that both gradually increase over frequency, and negative dissipated energy. This indicates that theactive control system is increasing the energy in the system and this can also be related to the transmission

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being greater than unity. Figures 3b and 3d show that the single monopole source achieves negative effective
bulk modulus over the frequency range of interest, demonstrating the effectiveness of the optimisation procedure,
whilst the effective density is close to the ambient air density. Although, this optimised ANC configuration andthe pair of monopole sources that are driven to minimise the downstream pressure generate the same effective
density, the performance of each ANC strategy is completely different due to the bulk modulus in each caseas shown in Figures 2d and 3d. The single monopole achieves an effective bulk modulus of around -2000
whilst the pair of monopole achieves zero bulk modulus. This shows that a large negative and zero bulk
modulus coupled with a density close to the ambient air density will achieve perfect transmission and dissipation
respectively.
To provide further insight into the physical behaviour of the single monopole achieving a negative bulk
modulus and a density that is close to the ambient air density, the effective refractive index has also been
calculated and is presented in Figure 3f. This result shows that the optimised single monopole source produces
a near-zero refractive index. According to Eq. 11, it can be seen that this result is achieved by the phasevelocity across the secondary source tending towards infinity, which not only leads to perfect transmission, but
the transmitted waves will also experience minimal phase change.
From Figure 3, it can be seen that the dipole source optimised to minimise the effective density (red)achieves a transmission coefficient that approaches zero, a reflection coefficient that approaches one and, therefore, close to zero dissipated energy; this is similar to the result presented in Figure 2 when a dipole secondary
source is driven to minimise the downstream pressure. However, it can also be seen from Figure 3 that the optimsed dipole source achieves negative effective density whilst the effective bulk modulus in this case is strictly
positive, with a small positive value; this is distinct from the dipole source driven to minimise the downstreampressure, which exhibits both positive density and bulk modulus. As in the single monopole case, the effective
refractive index has been calculated to provide further insight into the physical behaviour of the dipole source
optimised to achieve negative density and this is presented in Figure 3f. It can be seen that the optimised
dipole source produces a refractive index that is greater than unity, which means that the phase velocity acrossthe control source must be slower than the speed of sound. This is similar to the dipole driven to minimise thedownstream pressure, however, the differences in the effective material properties result in different magnitudesof the real and imaginary parts of the refractive index.

5

CONCLUSIONS

Active control is a well-developed technology that provides high performance noise control at low frequencies whilst maintaining a lightweight and compact package. In recent years, similar levels of noise control canbe achieved using AMMs by achieving negative or zero effective density and bulk modulus. Active control has
been combined with AMMs to improve the performance, tuneability and adaptability of the materials, however,
the effects of active control on the effective material properties and wave propagation have not been extensively
investigated. Therefore, this paper has investigated the effective material properties of a traditional ANC system
in a one-dimensional duct using a single monopole, dipole and pair of monopole sources. The effective refractive index was also calculated to provide insight into how these active systems manipulate wave propagation ina one-dimensional environment.
In addition, this paper has also investigated whether these secondary source configurations can be used todirectly minimise the effective material properties using an optimisation procedure and might thus be used as ametamaterial building block. It has been shown that the single monopole source can be optimised to achieve anegative bulk modulus and the dipole source can be optimised to achieve a negative density over the frequency
range of interest. These results are consistent with prior literature in AMMs that state that a monopole source
can create the monopole resonances required to achieve negative effective bulk modulus and a dipole source
can create the dipole resonances required to achieve negative effective density. It has also been shown that the
optimised single monopole source also achieves near-zero refractive index, which leads to perfect transmission
with minimal phase change. The use of these monopole and dipole sources may, therefore, be utilised as unit

4867

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3. The performance of the single monopole (blue) and the dipole (red) sources that are driven tominimise the effective bulk modulus and density respectively. (a), (c) and (e) show the transmission andreflection coefficients and dissipated energy respectively and (b), (d) and (f) show the effective density, bulk
modulus and refractive index respectively.
cells in the development of broadband single negative active acoustic metamaterials.

ACKNOWLEDGEMENTS
This research was partially supported by an EPRSC iCASE studentship (Voucher number 17000146) and an
EPSRC Prosperity Partnership ( EP/S03661X/1).

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