A Multi-Tone Sound Absorber Based on an Array of Shunted Loudspeakers

applied sciences

Article A Multi-Tone Sound Absorber Based on an Array of Shunted Loudspeakers

Chaonan Cong 1, Jiancheng Tao 1,* and Xiaojun Qiu 2

1 Key Laboratory of Modern Acoustics (MOE), Institute of Acoustics, Nanjing University, Nanjing 210093, China; [email protected] 2 Centre for Audio, Acoustics and Vibration, Faculty of Engineering and IT, University of Technology, Sydney, NSW 2007, Australia; [email protected] * Correspondence: [email protected]

Received: 1 November 2018; Accepted: 23 November 2018; Published: 4 December 2018

Abstract: It has been demonstrated that a single shunted loudspeaker can be used as an effective low frequency sound absorber in a duct, but many shunted loudspeakers have to be used in practice for noise reduction or reverberation control in rooms, thus it is necessary to understand the performance of an array of shunted loudspeakers. In this paper, a model for the parallel shunted loudspeaker array for multi-tone sound absorption is proposed based on a modal solution, and then the acoustic properties of a shunted loudspeaker array under normal incidence are investigated using both the modal solution and the ﬁnite element method. It was found that each shunted loudspeaker can work almost independently where each unit resonates. Based on the interaction analysis, multi-tone absorbers in low frequency can be achieved by designing multiple shunted loudspeakers with different shunt circuits respectively. The simulation and experimental results show that the normal incidence sound absorption coefﬁcient of the designed absorber has four absorption peaks with values of 0.42, 0.58, 0.80, and 0.84 around 100 Hz, 200 Hz, 300 Hz, and 400 Hz respectively.

Keywords: shunted loudspeaker array; modal solution; multi-tone sound absorption

1. Introduction The shunted loudspeaker, which consists of a closed-box loudspeaker and a shunt circuit, was first proposed for resonant acoustic field control in 2007 [1]. The acoustic impedance at the loudspeaker diaphragm and the resulting sound absorption coefficient can be adjusted by alternating the electric parameters in the shunt circuit, so the shunted loudspeaker can be implemented for noise control especially at low frequencies [2–4]. Early studies on the shunted loudspeaker mainly focus on normal sound absorption performance. Cernˇ ík et al. studied the effect of the shunt circuit on the sound reflecting performance of a shunted loudspeaker in an impedance duct [2]. Zhang et al. employed the negative impedance converter to counter the D.C. resistance and voice coil inductance of the loudspeaker to broaden the sound absorption bandwidth [5]. Tao et al. proposed constituting a thin compound broadband absorber by placing the shunted loudspeaker at the back of a micro-perforated panel [6]. Jing et al. designed a shunt speaker with a sound absorption coefficient above 0.9 at both 100 Hz and 200 Hz to control the noise of the power transformer [7]. Cho et al. replaced the closed box with a vented enclosure to enhance the sound absorption performance at low frequency [8]. Boulandet et al. adopted the response surface method to optimize the parameters of the shunted loudspeaker (such as the moving mass, the enclosure volume, the filling density of mineral fiber, and the electrical load value) [9]. In the studies above, analog components were used in the shunt circuit, and it was difficult to adjust the electrical impedance precisely. Boulandet et al. employed the real-time field programmable gate array

Appl. Sci. 2018, 8, 2484; doi:10.3390/app8122484 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, x FOR PEER REVIEW 2 of 12

to adjust the electrical impedance precisely. Boulandet et al. employed the real-time field programmable gate array module and the voltage-controlled current source to constitute the shunt Appl. Sci. 2018, 8, 2484 2 of 12 circuits [10]. Rivet et al. proposed a hybrid impedance control architecture for an electroacoustic absorber that combines a microphone-based feedforward control with a current driven electro moduledynamic and loudspeaker the voltage-controlled system [11]. current Considering source the to constitute resonance the characteristics shunt circuits of [ the10]. mechanical Rivet et al. proposedsystem of a the hybrid loudspeaker, impedance the control shunted architecture loudspeaker for an is electroacoustica typical resonant absorber sound that absorber combines with a microphone-basedadjustable feedback feedforward impedance control[12–14]. with a current driven electro dynamic loudspeaker system [11]. ConsideringFor implementation, the resonance the characteristics performance of of the the mechanical array of shunted system loudspeakers of the loudspeaker, needs to the be shunted further loudspeakerinvestigated. isA a linear typical array resonant of 10 soundshunted absorber loudspeakers with adjustable was placed feedback in the reverberant impedance [chamber,12–14]. and a soundFor pressure implementation, attenuation the of performance 14 dB was achieved of the array at 34.9 of shunted Hz [15]. loudspeakersFour shunted needsloudspeakers to be further with investigated.surface area of A 0.05 linear m2 array were of placed 10 shunted in the loudspeakerscorners of the was rectangular placed in room the reverberant with dimensions chamber, of 3 and × 5.6 a sound× 3.53 m pressure3 to reduce attenuation the sound of14 pressure dB was level achieved between at 34.9 70 HzHz [ 15and]. Four100 Hz shunted [16]. Thirty loudspeakers 50 × 50 withmm surfaceshunted area loudspeaker of 0.05 m2 cellswere were placed assembled in the corners as liners of the inside rectangular a pipeline room with with air dimensions flow, and the of 3obtained× 5.6 × 3.53insertion m3 to loss reduce was the 16 sounddB at the pressure target level frequency between [17]. 70 A Hz surface and 100 array Hz [ 16of ].shunted Thirty 50loudspeakers× 50 mm shunted could loudspeakercontrol the refracted cells were direction assembled of as the liners inci insidedent sound a pipeline around with 350 air flow, Hz [18]. and the However, obtained the insertion sound lossabsorption was 16 performance dB at the target of the frequency surface array [17]. Aof surfaceshunted array loudspeakers of shunted has loudspeakers not been investigated. could control the refractedIn this paper, direction a modal of the expansion incident method sound aroundis proposed 350 Hzto calculate [18]. However, the normal the soundsound absorptionabsorption performancecoefficient of of the the array surface of array shunted of shunted loudspeakers loudspeakers with different has not been acoustic investigated. impedances. The finite elementIn this model paper, is further a modal employed expansion to method validate is the proposed accuracy to calculateof the proposed the normal method. sound Simulations absorption coefficientshow that each of the shunted array of loudspeaker shunted loudspeakers can work almost with different independently. acoustic impedances.An experiment The was finite conducted element modelin the is impedance further employed duct with to validate four different the accuracy shunted of the loudspeakers proposed method. to validate Simulations the feasibility show that of eachachieving shunted multi loudspeaker-tone sound can absorption. work almost independently. An experiment was conducted in the impedance duct with four different shunted loudspeakers to validate the feasibility of achieving multi-tone2. Theory sound absorption.

2. TheoryThe schematic of the shunted loudspeaker array is presented in Figure 1a, where four shunted loudspeakers with equal areas are installed at the left terminal z = 0 of a square impedance duct. A planeThe wave schematic is incident of the normally shunted on loudspeaker the right terminal array isat presentedz = L with ina constant Figure1a, particle where velocity four shunted v0(ω). loudspeakersEach shunt loudspeaker with equal (named areas areas SL installed1 to SL4) atconsists the left of terminala closed-boxz = loudspeaker 0 of a square and impedance a shunt circuit duct. Aconnected plane wave to the is incident loudspeaker’s normally terminals. on the right The terminallength of at thez = ductL with cross a constant section particleis a, and velocity the lengthv0(ω of). Eachthe front shunt face loudspeaker of each shunted (named loudspeaker as SL1 to SL4 )unit consists is a/2 of respectively. a closed-box loudspeakerThe shunt circuit and a shuntis show circuitn in connectedFigure 1b, where to the loudspeaker’sthe negative resistance terminals. −R Thee and length negative of the inductance duct cross section−Le are isuseda, and to counteract the length ofthe the D.C. front resistance face of each RE shunted and voice loudspeaker coil inductance unit is a/2 LE respectively. of the loudspeaker The shunt respectively.circuit is shown The in Figurecapacitance1b, where Cs and the inductance negative resistance Ls can be− Rswitchede and negative according inductance to the −designLe are targets. used to counteract the D.C. resistance RE and voice coil inductance LE of the loudspeaker respectively. The capacitance Cs and inductance Ls can be switched according to the design targets.

y 0 a r r a

SL3 r

e SL4 k

a a e p s SL1 d v0()

u x o l

a d e

t SL2 n u

h y S

z 0 L (a)

Figure 1. Cont. Appl. Sci. 2018, 8, 2484 3 of 12 Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 12

air cavity Zs

Cs −Re −Le

(b)

FigureFigure 1. (a) A schematic of the shunted loudspeaker array set at the end of an impedance duct;duct; (b) thethe equivalentequivalent circuitcircuit ofof aa shuntshunt loudspeakerloudspeaker unit.unit.

TheThe speciﬁcspecific acousticacoustic impedanceimpedance ofof thethe iithth shuntedshunted loudspeakerloudspeaker ((ii= = 1,1, 2,2, 3,3, 4)4) isis [[44,,1919]] 22 RMSms ms1 0 Bl2 2 Z = +Rmsj M +ms 1 +S0 + B l , i Zi = + jω + + + ,(1) (1) SSCSCSRLZ0S0 0S0 jjω ms0CmsS0 j jω ab0ECab S( 0(R++E + j jω EsLE + Z)s)

where ω is the angular frequency, Rms is the mechanical resistance of the driver suspension losses of where ω is the angular frequency, Rms is the mechanical resistance of the driver suspension losses the loudspeaker, S0 is the effective surface area of the driver cone of the loudspeaker, Mms is the of the loudspeaker, S0 is the effective surface area of the driver cone of the loudspeaker, Mms is mechanical mass of the driver cone (including reactive air load) of the loudspeaker, Cms is the the mechanical mass of the driver cone (including reactive air load) of the loudspeaker, C is the 2 ms mechanical compliance of the driver suspension of the loudspeaker, Cab = V/ρ0c02 is the equivalent mechanical compliance of the driver suspension of the loudspeaker, Cab = V/ρ0c0 is the equivalent acoustic capacitance due to the back cavity of the loudspeaker, V is the volume of the back cavity, ρ0 acoustic capacitance due to the back cavity of the loudspeaker, V is the volume of the back cavity, and c0 are the air density and sound velocity respectively, B is the magnetic flux density of the ρ0 and c0 are the air density and sound velocity respectively, B is the magnetic flux density of the loudspeaker driver, l is the voice coil length, and Zs is the electrical impedance of the shunt circuit. loudspeaker driver, l is the voice coil length, and Zs is the electrical impedance of the shunt circuit. Omitting the time harmonic factor 푒j휔푡, the acoustic field inside the impedance duct shown in Omitting the time harmonic factor ejωt, the acoustic field inside the impedance duct shown in Figure 1a can be expanded using the modal solution as Figure1a can be expanded using the modal solution as  2 2 2 2 jk00−− kqq z -j k k z p(,,,)= x y zq (,,) x y k q A q eq + B q e q ∞ 2 2 2 2 (2) q=0 j k0−kqz −j k0−kqz p(x, y, z, ω) = ∑ ψq(x, y, kq) Aqe + Bqe , , (2) q=0 where q is the mode index, k0 = ω/c0 and kq = qπ/a are the total wavenumber and the lateral wherewavenumberq is the of mode the q index,th modek0 respectively,= ω/c0 and kq q=(xq ,π y,/ ka qare )= cos(the total k q x )cos( wavenumber k q y )/ S is and the the qthlateral mode √ wavenumberfunction, S = a of2 is the the qsectionth mode area respectively, of the duct,ψ Aq q xand, y, kBqq are= thecos q(thkq xmode) cos( kcoefficientsqy)/ S is of the theqth incident mode 2 function,wave andS reflected = a is the wave section respectively area of the. duct, Aq and Bq are the qth mode coefficients of the incident waveThe and boundary reflected waveconditions respectively. at z = 0 and z = L are The boundary conditions at z = 0 and1z = Larep ( x , y , z , ) vz( x , y , z , )| z=0 L == v ( ) , −j (z ) (3) 01 ∂p x, y, z, ωzL= vz(x, y, z, ω )|z=L = = v0(ω), (3) −jωρ0 ∂z z=L p(,,,) x y z   ∂p(x, y, z,−=ω)jk00 Z (,,)(,,,) x y p x y z 0 and− jk Z (x, y, ω)p(x, y, z, ω) = 0,(4 (4)) z 0 0 z= 0 and ∂z z=0 , wherewhere vvzz((x,, y,, zz,, ω) is the velocity in thethe z direction, andand thethe specificspecific acousticacoustic impedanceimpedance isis  Z1 x(0, a 2), y ( a 2, a ) Z1 x ∈ (0, a/2), y ∈ (a/2, a) Z x( a 2, a ), y ( a 2, a )  2 ∈ ( ) ∈ ( ) Z0 (,,) x y  = Z2 x a/2, a , y a/2, a. (5) Z0(x, y, ω) = Z x(0, a 2), y (0, a 2) . (5) Z 3 x ∈ (0, a/2), y ∈ (0, a/2) Z3 x( a 2, a ), y (0, a 2)  4 Z4 x ∈ (a/2, a), y ∈ (0, a/2) Substituting p(x, y, z, ω) in Equation (2) into the boundary condition in Equations (3) and (4), the following is obtained: Appl. Sci. 2018, 8, 2484 4 of 12

Substituting p(x, y, z, ω) in Equation (2) into the boundary condition in Equations (3) and (4), the following is obtained: q q √ 1 q jkz L −jkz L kz(−Aqe + Bqe ) = Sv0(ω)δ0q, (6) ωρ0 ∞ n q q o ∑ ψq x, y, kq [−kz + k0Z0(x, y, ω)]Aq + [kz + k0Z0(x, y, ω)]Bq = 0, (7) q=0 q q 2 2 where kz = k0 − kq is the wavenumber of the qth mode along the z direction and δ0q is the Kronecker delta function. Exploiting the orthogonality of the normal modes, Equation (7) can be simpliﬁed as

∞ µ qµ kz (−Aµ + Bµ) + k0 ∑ Z0 (Aq + Bq) = 0, (8) q=0

qµ ∗ Z0 ≡ x Z0(x, y, ω)ψq(x, y, kλ)ψµ x, y, kµ dS, ∀µ ≥ 0, (9) S where µ and λ are the mode indexes, kµ = µπ/a and kλ = λπ/a are the lateral wavenumbers of the µth and λth modes respectively, Aµ and Bµ are the µth mode coefficients of the incident wave and q µ 2 2 reflected wave respectively, kµ = µπ/a is the lateral wavenumber of the µth mode, kz = k0 − kµ is √ the wavenumber of the µth mode along z direction, ψµ x, y, kµ = cos kµx cos kµy / S is the µth qµ mode function. The factor Z0 mathematically measures the coupling of different modes caused by the inhomogeneity of the acoustical impedance at z = 0. The mode coupling makes it hard to obtain an analytical solution in a compact closed form. However, it is possible to approach the exact solution through iterations. Assume the inhomogeneity is quite weak such that all the high-order modes with q not being zero can be neglected, the plane-wave mode ψ0(x,y,k0) is preserved and the coefficients A0 and B0 are derived as √ 00 00 A0 ≈ ρ0c0 S 1 + Z0 v0(ω)/2/ −jsin(k0L) − Z0 cos(k0L) , (10)

00 00 B0 ≈ A0 1 − Z0 / 1 + Z0 , (11)

00 where the factor Z0 is calculated in Equation (9) when q and µ are taken as zero. Assume that the inhomogeneity of the shunted loudspeaker array causes the plane-wave mode to be coupled to high-order modes while the high-order modes themselves are not coupled with each other, Equation (8) can be simpliﬁed as

µ µµ µ µµ 0µ −kz + k0Z0 Aµ + kz + k0Z0 Bµ = −k0Z0 (A0 + B0), ∀µ ≥ 0, (12)

µµ 0µ where the factors Z0 and Z0 are calculated in Equation (9) when q is taken as µ and 0 respectively. Substituting A0, B0 in Equations (10) and (11) into Equation (12), the coefﬁcients of high-order modes can be calculated as

µ 0µ −jkz L 00 µ µ µµ µ Aµ ≈ Z0 A0e /(1 + Z0 )/[−j(kz /k0) sin(kz L) − Z0 cos(kz L)], (13)

µ j2kz L Bµ = Aµe . (14) Appl. Sci. 2018, 8, 2484 5 of 12

The normal incidence sound absorption coefﬁcient α can be calculated by

!2 ∞ ∑ ψq x, y, kq Aq p2 q=0 α = 1 − r = 1 − , (15) p2 !2 in ∞ ∑ ψq x, y, kq Bq q=0 where pr and pin are reﬂected pressure and incident pressure at z = 0. It is clear that the more coupling qµ measures Z0 that are adopted; the more precise can be the results obtained.

3. Simulations In this section, the acoustic properties of four parallel arranged shunted loudspeakers are investigated by both the proposed method and the ﬁnite element method. The same loudspeaker is adopted in each shunted loudspeaker, and the measured Thiele–Small (TS) parameters of the loudspeaker are listed in Table1. The resonance frequencies of the shunted loudspeakers are chosen as 100 Hz, 200 Hz, 300 Hz, and 400 Hz, and the designed shunt circuit parameters are listed in Table2.

Table 1. Measured Thiele–Small (TS) parameters and dimensions of a closed-box loudspeaker unit.

Parameter Notation Value Unit

DC resistance RE 32.00 Ω Inductance of coil LE 7.24 mH Moving mass Mms 15.25 g Mechanical resistance Rms 1.57 kg/s Mechanical compliance Cms 0.67 mm/N Force factor Bl 17.12 T·m -2 2 Effective area S0 1.51 × 10 m Back cavity volume V 2.2 × 10-3 m3 Cavity depth D 7.5 cm

Table 2. Shunt circuit parameters used in the simulation of the shunted loudspeaker (SL) array.

SL1 SL2 SL3 SL4 R (Ω) −31.95 −31.95 −31.95 −31.95 L (mH) −7.23 −7.23 −7.23 −7.23 Ls (mH) - 37.19 7.69 3.62 Cs (µF) 87 - - -

The sound absorption coefficient of the array composed of SL1–SL4 can be calculated by the finite element model built in the commercial software (Comsol Multiphysics v5.3) as shown in Figure2. The plane at z = 1.8 m is set as a plane wave incident surface and e incident sound pressure is 1 Pa. The bottom surface at z = 0 m is uniformly divided into four parts, which are defined as impedance boundaries. The size of each impedance boundary is the same as the effective area in Table1. Set the impedance at (0 < x < a/2, a/2 < y < a) of SL1 unit boundary as Z1, the impedance at (a/2 < x < a, a/2 < y < a) of SL2 unit boundary as Z2, the impedance at (0 < x < a/2, 0 < y < a/2) of SL3 unit boundary as Z3, and the impedance at (a/2 < x < a, 0 < y < a/2) of SL4 unit boundary as Z4. The element size is selected as an extremely fine mesh size. The free tetrahedral mesh consists of 24,973 domain elements, 3344 boundary elements, and 315 edge elements, with the number of degrees of freedom as 36536. The absorption coefficient is calculated by E α = 1 − r , (16) Ein where Er is the reflected energy and Ein is the incident energy at the surface z = 0. Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 12

Plane wave incident surface zz

Appl.Appl. Sci. Sci.2018 2018, 8,, 24848, x FOR PEER REVIEW SL3 66 of of 12 12

Plane wave incident surface x 0 zz SL4

SL2

yySSLL31

Figure 2. The finite element model of four shunted loudspeakersx in parallel. 0 The solid curve in Figure 3 shows the normal incidence absorption coefficient of the shunted SL4 loudspeaker array by using the proposed method in Section 2. Four resonance frequencies occur at 100 Hz, 200 Hz, 300 Hz, 400 Hz, where the sound absorption coefficientsSL2 are all 1.00. The dashed curve in Figure 3 reveals the result by using the Finite Element Method (FEM), where the frequency y deviations at the resonance frequencies are 0 Hz, 5 Hz, 6y Hz,SL and1 4 Hz, respective due to a simulation

error in Comsol.Figure 2. The finite element model of four shunted loudspeakers in parallel. The dottedFigure curves 2. reflect The finite the elementsound absorption model of four coefficients shunted loudspeakers of SL1–SL4 units,in parallel. where the resonance frequenciesThe solid arecurve 100 in Hz, Figure 2003 Hz,shows 300 the Hz, normal 400 Hz incidence of each absorption unit respectively. coefficient Comparing of the shunted with the loudspeakerresonanceThe solid array frequencies curve by using in ofFigure the the proposed array,3 shows it the method is revealednormal in Section incidence that each2. Four absorption shunted resonance loudspeaker coefficient frequencies of almost the occur shunted works at 100loudspeakerindependently. Hz, 200 Hz, array 300 Therefore, by Hz, using 400 aHz, the multi whereproposed-tone the noise method sound absorber absorption in Section can be 2. coefficients designedFour resonance by are using all frequencies 1.00. multiple The dashed occurshunted at curve100loudspeakers Hz, in Figure 200 Hz,3 withreveals 300 differentHz, the 400 result Hz, resonance by where using th frequencies thee sound Finite Elementabsorption in the Method same coefficients plane, (FEM), whereare where all each1.00. the frequencyThe unit dashed can be deviationscurvedesigned in Figure at independently. the resonance 3 reveals the frequencies result by are using 0 Hz, the 5 Finite Hz, 6 Hz,Element and 4 M Hz,ethod respective (FEM), duewhere to athe simulation frequency errordeviations in Comsol. at the resonance frequencies are 0 Hz, 5 Hz, 6 Hz, and 4 Hz, respective due to a simulation error in Comsol. The dotted curves reflect the sound absorption coefficients of SL1–SL4 units, where the resonance frequencies are 100 Hz, 200 Hz, 300 Hz, 400 Hz of each unit respectively. Comparing with the resonance frequencies of the array, it is revealed that each shunted loudspeaker almost works independently. Therefore, a multi-tone noise absorber can be designed by using multiple shunted loudspeakers with different resonance frequencies in the same plane, where each unit can be designed independently.

FigureFigure 3. 3.The The normal normal incidence incidence sound sound absorption absorption coefﬁcients coefficients by by analytical analytical solution solution and and the the ﬁnite finite

elementelement model model method: method SL: 1SLunit1 unit (magenta (magenta dotted dotted line line marked marked with with plus plus sign), sign), SL 2SLunit2 unit (green (green dotted dotted lineline marked marked with with upward-pointing upward-pointing triangle), triangle), SL 3SLunit3 unit (red (red dotted dotted line line marked marked with with solid solid circle), circle), SL 4SL4 unitunit (cyanine (cyanine dotted dotted line line marked marked with with blank blank circle), circle), shunted shunted loudspeaker loudspeaker array array by by analytical analytical solution solution (black(black solid solid line), line), shunted shunted loudspeaker loudspeaker array array by by FEM FEM (blue (blue dashed dashed line). line).

The dotted curves reﬂect the sound absorption coefﬁcients of SL1–SL4 units, where the resonance frequencies are 100 Hz, 200 Hz, 300 Hz, 400 Hz of each unit respectively. Comparing with the resonance frequenciesFigure of 3. theThe array, normal it incidence is revealed sound that absorption each shunted coefficients loudspeaker by analytical almost solution works and independently. the finite Therefore,element a multi-tone model method noise: SL absorber1 unit (magenta can be dotted designed line bymarked using with multiple plus sign), shunted SL2 unit loudspeakers (green dotted with differentline resonance marked with frequencies upward-pointing in the same triangle), plane, SL where3 unit (red each dotted unit line can marked be designed with solid independently. circle), SL4 unit (cyanine dotted line marked with blank circle), shunted loudspeaker array by analytical solution (black solid line), shunted loudspeaker array by FEM (blue dashed line). Appl. Sci. 2018, 8, 2484 7 of 12 Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 12

The acoustic intensity of the shunted loudspeaker array at four resonance frequencies is The acoustic intensity of the shunted loudspeaker array at four resonance frequencies is shown in shown in Figure 4, where the volume density (directions) of the red arrows shows the magnitude Figure4, where the volume density (directions) of the red arrows shows the magnitude (directions) of (directions) of the sound intensity. Most part of the acoustic energy is “attracted” toward SL 1 at the sound intensity. Most part of the acoustic energy is “attracted” toward SL1 at 100 Hz, which is at 100 Hz, which is at the resonance frequency of SL 1. Similarly, most of the acoustic energy is the resonance frequency of SL1. Similarly, most of the acoustic energy is “attracted” toward SL2–SL4 “attracted” toward SL2–SL4 at the resonance frequencies of SL2–SL4 respectively. It is observed at the resonance frequencies of SL2–SL4 respectively. It is observed that at the resonant frequency that at the resonant frequency of a particular shunted loudspeaker unit, the acoustic energy flows of a particular shunted loudspeaker unit, the acoustic energy ﬂows towards the unit at resonance. towards the unit at resonance. However, the other shunted loudspeaker units are also working as However, the other shunted loudspeaker units are also working as absorbers although the sound absorbers although the sound intensity near them is rather weak. This is intuitive evidence to intensity near them is rather weak. This is intuitive evidence to demonstrate that all the shunted demonstrate that all the shunted loudspeaker units work cooperatively as an entire piece. loudspeaker units work cooperatively as an entire piece.

(a) (b)

Figure 4. The acoustic intensity of parallel shunted loudspeaker array (a) at 100 Hz; (b) at 200 Hz; (c) at 300Figure Hz; 4. (d The) at 400acoustic Hz. intensity of parallel shunted loudspeaker array (a) at 100 Hz; (b) at 200 Hz; (c) at 300 Hz; (d) at 400 Hz. From the analysis above, the normal incidence absorption coefficients of the shunted loudspeaker arrayFrom calculated the analysiswith the analytical above, the solution normal agree incidence reasonably absorption well with coefficients the results of obtained the shunted with theloudspeaker finite element array method. calculated The with most the part analytical of the acoustic solution energy agree is reasonably “attracted” well toward with the the shunted results loudspeakerobtained with unit the at finite its resonance element frequency.method. The Therefore, most part a multi-toneof the acoustic noise energy absorber is “attracted” can be realized toward by designingthe shunted multiple loudspeaker shunted unit loudspeakers at its resonance with frequency. different resonance Therefore, frequencies. a multi-tone noise absorber can be realizedThe optimal by designing system canmultiple be achieved shunted by loudspeakers designing each with shunt different loudspeaker resonance unit frequencies. first and then combingThe alloptimal the units system together. can Inbe theachieved design by of eachdesigning shunt loudspeakereach shunt loudspeaker the peak frequency unit first of itsand sound then absorptioncombing all coefficient the units cantogether. be adjusted In the by design choosing of each the propershunt loudspeaker value of Cs or theLs peakin Figure frequency1b, and of the its peaksound value absorption of its sound coefficient absorption can be coefficient adjusted canby choosing be adjusted the by proper choosing value the of loudspeaker Cs or Ls in Figure withthe 1b, properand the mechanical peak value resistance of its soundRms absorption. coefficient can be adjusted by choosing the loudspeaker with the proper mechanical resistance Rms.

4. Experimental The experimental setup in an acoustic impedance duct is shown in Figure 5, where the array of four shunted loudspeakers is placed at the right end. The cross section of the duct is about 0.40 m × Appl. Sci. 2018, 8, 2484 8 of 12

4. Experimental Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 12 The experimental setup in an acoustic impedance duct is shown in Figure5, where the array of four 0.40shunted m and loudspeakers the cut-off frequency is placed at is the430 right Hz. A end. source The loudspeaker cross section is of located the duct at is the about left end0.40 mof ×the0.40 duct m (Iandt is theabout cut-off 7 m frequencyaway from is the 430 shouted Hz. A source loudspeaker loudspeaker array isand located is not at seen the leftin Figure end of 5 the) and duct driven (It is throughabout 7 m the away amplifier. from the The shouted DC voltage loudspeaker source is array used and to supply is not seen the power in Figure for5 )the and negative driven through impedance the convertersampliﬁer. The in theDC voltage shunt circuits. source is The used sound to supply absorption the power coef forficient the negative was measured impedance using converters the two in- microphonethe shunt circuits. transfer The function sound absorptionmethod according coefﬁcient to wasISO measured10534-2 with using a B&K the two-microphone PULSE 3560B analyzer transfer [function20]. The method parameters according to set to up ISO are 10534-2 as follows: with a the B&K sampling PULSE 3560B rate is analyzer 25.6 kHz, [20 ]. and The the parameters frequency to resolutionset up are asis set follows: as 1 Hz; the the sampling microphone rate is spacing 25.6 kHz, is 0.35 and m; the the frequency distance resolution from the right is set microphone as 1 Hz; the tomicrophone the samplespacing is 1.32 m; is the 0.35 distance m; the distancefrom the fromleft microphone the right microphone to the source to is the 3.64 sample m; a random is 1.32 m;signal the isdistance set as the from input the to left the microphone source with to the the voltage source level is 3.64 of m; 0.07 a randomVrms to signalensure is tha sett asthe the signal input to tonoise the ratiosource is withabove the 10 voltagedB. level of 0.07 Vrms to ensure that the signal to noise ratio is above 10 dB.

FigureFigure 5. 5. MeasurementMeasurement setup setup of of the the shunted shunted loudspeaker loudspeaker array. array.

AnAn optimal optimal system system was was designed designed based based on the on analysis the analysis in Section in3 . Section The Thiele–Small 3. The Thiele parameters–Small parametersand dimensions and dimensions of each loudspeaker of each loudspeaker are listed in are Tables listed3– in6, Tables where the3–6, cavity where depth the cavity is optimized depth is 2 optimizedat 7.5 cm. at However, 7.5 cm. However, the mechanical the mechanical resistance resistanceRms is not Rms equal is not toequal the t optimalo the optimal value valueρ0c0S ρ0 0c/0S0for2/S forthe the maximal maximal sound sound absorption absorption peak, peak, because because the the number number of of the the loudspeaker loudspeaker samplessamples wewe have is is limited.limited. The The shunt shunt circuit circuit configuration conﬁguration for for the the four four loudspeakers loudspeakers is is listed listed in in Table Table 77.. Theoretically, thethe resonance resonance of of each shunted loudspeaker unit unit could could be adjusted to any frequency. However, However, consideringconsidering the nonlinearitynonlinearity ofof thethe loudspeakerloudspeaker [ 21[21],], a a more more accurate accurate formulation formulation of of the the impedance impedanceZe 2 2 Zise Bis Bl 2/l2S/S00/[/[RE ++ jωL jωLE E+ +1/(1/ 1/(1/R2 +R 1/2 L+2) 1/ + ZL2s],) +whereZs], where R2 is theR2 electricalis the electrical resistance resistance due to the due eddy to the current eddy lossescurrent a lossesnd L2 andis theL2 parais the-inductance para-inductance of the of voice the voice coil. coil. Therefore, Therefore, the the adjustment adjustment of of the the resonance resonance frequencyfrequency ofof thethe shunted shunted loudspeaker loudspeaker has has constraints. constraints. In practical,In practical, the inductancethe inductance element element in the in shunt the shuntcircuit circuit generates generates parasitic parasitic resistance. resistance. By using By the using ohmmeter the ohmmeter to measure to measure the value, the negative value, resistancenegative resistanceis employed is employed to cancel the to cancel parasitic the resistance. parasitic resistance.

Table 3. Measured Thiele–Small (TS) parameters and dimensions of a closed-box loudspeaker (SL1 unit).

Parameter Notation Value Unit DC resistance RE 32.12 Ω Inductance of coil LE 7.24 mH Moving mass Mms 15.25 g Mechanical resistance Rms 1.15 kg/s Mechanical compliance Cms 0.67 mm/N Electrical resistance due to eddy current losses R2 39.50 Ω Para-inductance of voice coil L2 6.75 mH Force factor Bl 17.12 T·m Effective area S0 1.51 × 10-2 m2 Back cavity volume V 2.2 × 10-3 m3 Cavity depth D 7.5 cm Appl. Sci. 2018, 8, 2484 9 of 12

Table 3. Measured Thiele–Small (TS) parameters and dimensions of a closed-box loudspeaker

(SL1 unit).

Parameter Notation Value Unit

DC resistance RE 32.12 Ω Inductance of coil LE 7.24 mH Moving mass Mms 15.25 g Mechanical resistance Rms 1.15 kg/s Mechanical compliance Cms 0.67 mm/N Electrical resistance due to eddy current losses R2 39.50 Ω Para-inductance of voice coil L2 6.75 mH Force factor Bl 17.12 T·m -2 2 Effective area S0 1.51 × 10 m Back cavity volume V 2.2 × 10-3 m3 Cavity depth D 7.5 cm

Table 4. Measured TS parameters and dimensions of a closed-box loudspeaker (SL2 unit).

Parameter Notation Value Unit

DC resistance RE 5.71 Ω Inductance of coil LE 0.37 mH Moving mass Mms 15.19 g Mechanical resistance Rms 1.31 kg/s Mechanical compliance Cms 0.83 mm/N Electrical resistance due to eddy current losses R2 1.11 Ω Para-inductance of voice coil L2 0.33 mH Force factor Bl 5.05 T·m -2 2 Effective area S0 1.51 × 10 m Back cavity volume V 1.2 × 10-3 m3 Cavity depth D 7.5 cm

Table 5. Measured TS parameters and dimensions of a closed-box loudspeaker (SL3 unit).

Parameter Notation Value Unit

DC resistance RE 6.98 Ω Inductance of coil LE 0.24 mH Moving mass Mms 7.72 g Mechanical resistance Rms 0.94 kg/s Mechanical compliance Cms 0.27 mm/N Electrical resistance due to eddy current losses R2 1.57 Ω Para-inductance of voice coil L2 0.31 mH Force factor Bl 5.21 T·m -2 2 Effective area S0 1.51 × 10 m Back cavity volume V 1.4 × 10-3 m3 Cavity depth D 7.5 cm Appl. Sci. 2018, 8, 2484 10 of 12

Table 6. Measured TS parameters and dimensions of a closed-box loudspeaker (SL4 unit).

Parameter Notation Value Unit Appl. Sci. 2018, 8, x FOR PEER REVIEW 10 of 12 DC resistance RE 7.31 Ω Inductance of coil LE 0.33 mH Table 7. ShuntMoving circuit massparameters used in the experimentMms of the shunted 10.38loudspeaker (SL) g array. Mechanical resistance Rms 1.02 kg/s Mechanical complianceSL1 SL2 Cms SL3 0.29SL4 mm/N Electrical resistanceR due (Ω) to eddy current−32.00 losses - R2 - 1.41−7.00 Ω Para-inductanceL (mH) of voice− coil7.00 - L2 - 0.25- mH Force factor Bl 5.60 T·m Ls (mH) 5 - - 0.2 -2 2 Effective area S0 1.51 × 10 m BackC cavitys (μF) volume 47 - V - 1.4 × 10- -3 m3 Cavity depth D 7.5 cm Take the shunt circuit parameters in Table 7 and the TS parameters and dimensions in Tables 3–6 into EquationTable 7. Shunt (1), the circuit specific parameters acoustic used impedance in the experiments of each of shunted the shunted loudspeaker loudspeaker are (SL) then array. calculated. It should be noted that the measured sound absorption is contributed to by both the loudspeaker diaphragm and the front surfaceSL of1 the closed-box,SL2 because theSL diaphragm3 areaSL4 is smaller than the cross section of R the(Ω ) duct. Therefore,−32.00 the impedance - Zi is corrected - by the area−7.00 ratio factor a2/4S0. L (mH) −7.00 - - - Substituting the corrected impedance Zi(a2/4S0) into Equation (5), the sound absorption coefficient L (mH) 5 - - 0.2 can be calculateds according to Equation (15) and shown as the dashed line in Figure 6. The resonance Cs (µF) 47 - - - frequencies are at 110 Hz, 222 Hz, 313 Hz, 402 Hz, where the sound absorption coefficients are 1.00, 0.78, 0.91, and 0.90 respectively. Meanwhile, the results by using the FEM method are shown in the dottedTake curve the shunt, where circuit the resonance parameters frequencies in Table7 areand at the 109 TS Hz, parameters 219 Hz, and308 Hz, dimensions 399 Hz and in Tables the sound3–6 intoabsorption Equation coefficients (1), the specific are 1.00, acoustic 0.77, 0.86, impedances and 0.93 of respectively. each shunted loudspeaker are then calculated. It shouldThe bemeasured noted that sound the absorption measured soundcoefficient absorption of the shunted is contributed loudspeaker to by array both is the shown loudspeaker in Figure diaphragm6 (the solid and curve), the frontwhere surface the sound of the absorption closed-box, coefficients because the at 100 diaphragm Hz, 200 areaHz, and is smaller 300 Hz than are the0.42, 2 cross0.58, section0.80, and of 0.84 the duct.respectively, Therefore, with the the impedance resonance peaksZi is corrected being at 107 by theHz, area208 Hz, ratio 302 factor Hz, anda /4 S3950. 2 SubstitutingHz respectively. the corrected The frequency impedance shiftZ i( ofa / the4S0 ) absorption into Equation peaks (5), thebetween sound the absorption simulation coefficient and the canexperiment be calculated results according is not more to Equation than 11 (15) Hz, and which shown may as be the caused dashed by line the in measurement Figure6. The resonanceprecision of frequenciesthe loudspeaker are at TS 110 parameters Hz, 222 Hz, and 313 the Hz, electronic 402 Hz, components where the sound in the absorption shunt circuit coefficients in the experiments. are 1.00, 0.78,The 0.91,differences and 0.90 of respectively. the peak values Meanwhile, of the sound the results absorption by using coefficient the FEM between method areexperimental shown in theand dottedsimulation curve, results where may the resonancebe due to extra frequencies parasitic are resistances at 109 Hz, of 219 the Hz, electronic 308 Hz, components. 399 Hz and the sound absorption coefficients are 1.00, 0.77, 0.86, and 0.93 respectively.

FigureFigure 6. 6.Measured Measured and and calculated calculated sound sound absorption absorption coefﬁcients coefficients of of the the shunted shunted loudspeaker loudspeaker array. array.

5. ConclusionsThe measured sound absorption coefﬁcient of the shunted loudspeaker array is shown in Figure6 (the solid curve), where the sound absorption coefﬁcients at 100 Hz, 200 Hz, and 300 Hz are 0.42, 0.58, This paper investigates the feasibility of designing thin multi-tone sound absorbers based on an analog circuit shunted loudspeaker array. A model for the parallel shunted loudspeaker array for multi-tone sound absorption is proposed based on a modal solution, and then the acoustic properties of a shunted loudspeaker array under normal incidence are investigated using both the modal solution and the finite element method. The physical mechanism in a finite standing wave duct was investigated, and it was found that the acoustic energy of a different frequency is “attracted” toward the corresponding shunted loudspeaker at that specific frequency and converted into electrical Appl. Sci. 2018, 8, 2484 11 of 12

0.80, and 0.84 respectively, with the resonance peaks being at 107 Hz, 208 Hz, 302 Hz, and 395 Hz respectively. The frequency shift of the absorption peaks between the simulation and the experiment results is not more than 11 Hz, which may be caused by the measurement precision of the loudspeaker TS parameters and the electronic components in the shunt circuit in the experiments. The differences of the peak values of the sound absorption coefﬁcient between experimental and simulation results may be due to extra parasitic resistances of the electronic components.

5. Conclusions This paper investigates the feasibility of designing thin multi-tone sound absorbers based on an analog circuit shunted loudspeaker array. A model for the parallel shunted loudspeaker array for multi-tone sound absorption is proposed based on a modal solution, and then the acoustic properties of a shunted loudspeaker array under normal incidence are investigated using both the modal solution and the finite element method. The physical mechanism in a finite standing wave duct was investigated, and it was found that the acoustic energy of a different frequency is “attracted” toward the corresponding shunted loudspeaker at that specific frequency and converted into electrical energy. The measured sound absorption coefficient of the designed prototype has four peaks with values of 0.42, 0.58, 0.80, and 0.84 around 100 Hz, 200 Hz, 300 Hz, and 400 Hz. Further work includes investigation of the sound absorption of the proposed multi-tone absorber array under oblique incidence and in a reverberation room.

Author Contributions: X.-J.Q. initiated and developed the ideas related to this research work; C.-N.C. and J.-C.T. developed novel methods, derived relevant formulations, and carried out numerical analyses and experimental analyses. C.-N.C. wrote the paper draft under J.-C.T. and X.-J.Q.’s guidance; and C.-N.C. finalized the paper. Funding: This research was funded by the National Science Foundation of China with the grant number 11474163 and 11874218, and the Australian Research Council with the grant numberLP140100987. Acknowledgments: The authors would also like to thank Li Ruiqi for the help in the formula derivation. Conflicts of Interest: The authors declare no conflict of interest.

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