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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}were placed in the corners of the rectangular room with dimensions of 3 × 5.6 × 3.53 m

^{3}to reduce the sound pressure level between 70 Hz and 100 Hz [16]. Thirty 50 × 50 mm shunted loudspeaker cells were assembled as liners inside a pipeline with air flow, and the obtained insertion loss was 16 dB at the target frequency [17]. A surface array of shunted loudspeakers could control the refracted direction of the incident sound around 350 Hz [18]. However, the sound absorption performance of the surface array of shunted loudspeakers has not been investigated.

## 2. Theory

_{0}(ω). Each shunt loudspeaker (named as SL

_{1}to SL

_{4}) consists of a closed-box loudspeaker and a shunt circuit connected to the loudspeaker’s terminals. The length of the duct cross section is a, and the length of the front face of each shunted loudspeaker unit is a/2 respectively. The shunt circuit is shown in Figure 1b, where the negative resistance −R

_{e}and negative inductance −L

_{e}are used to counteract the D.C. resistance R

_{E}and voice coil inductance L

_{E}of the loudspeaker respectively. The capacitance C

_{s}and inductance L

_{s}can be switched according to the design targets.

_{ms}is the mechanical resistance of the driver suspension losses of the loudspeaker, S

_{0}is the effective surface area of the driver cone of the loudspeaker, M

_{ms}is the mechanical mass of the driver cone (including reactive air load) of the loudspeaker, C

_{ms}is the mechanical compliance of the driver suspension of the loudspeaker, C

_{ab}= V/ρ

_{0}c

_{0}

^{2}is the equivalent acoustic capacitance due to the back cavity of the loudspeaker, V is the volume of the back cavity, ρ

_{0}and c

_{0}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 Z

_{s}is the electrical impedance of the shunt circuit.

_{0}= ω/c

_{0}and k

_{q}= qπ/a are the total wavenumber and the lateral wavenumber of the qth mode respectively, ${\psi}_{q}(x,y,{k}_{q})=\mathrm{cos}({k}_{q}x\left)\mathrm{cos}\right({k}_{q}y)/\sqrt{S}$ is the qth mode function, S = a

^{2}is the section area of the duct, A

_{q}and B

_{q}are the qth mode coefficients of the incident wave and reflected wave respectively.

_{z}(x, y, z, ω) is the velocity in the z direction, and the specific acoustic impedance is

_{0q}is the Kronecker delta function. Exploiting the orthogonality of the normal modes, Equation (7) can be simplified as

_{μ}= μπ/a and k

_{λ}= λπ/a are the lateral wavenumbers of the μth and λth modes respectively, A

_{μ}and B

_{μ}are the μth mode coefficients of the incident wave and reflected wave respectively, k

_{μ}= μπ/a is the lateral wavenumber of the μth mode, ${k}_{z}^{\mu}=\sqrt{{k}_{0}^{2}-{k}_{\mu}^{2}}$ is the wavenumber of the μth mode along z direction, ${\psi}_{\mu}\left(x,y,{k}_{\mu}\right)=\mathrm{cos}\left({k}_{\mu}x\right)\mathrm{cos}\left({k}_{\mu}y\right)/\sqrt{S}$ is the μth mode function. The factor Z

_{0}

^{qμ}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.

_{0}(x,y,k

_{0}) is preserved and the coefficients A

_{0}and B

_{0}are derived as

_{0}

^{00}is calculated in Equation (9) when q and μ are taken as zero.

_{0}

^{μμ}and Z

_{0}

^{0μ}are calculated in Equation (9) when q is taken as μ and 0 respectively.

_{0}, B

_{0}in Equations (10) and (11) into Equation (12), the coefficients of high-order modes can be calculated as

_{r}and p

_{in}are reflected pressure and incident pressure at z = 0. It is clear that the more coupling measures Z

_{0}

^{qμ}that are adopted; the more precise can be the results obtained.

## 3. Simulations

_{1}–SL

_{4}can be calculated by the finite element model built in the commercial software (Comsol Multiphysics v5.3) as shown in Figure 2. 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 Table 1. Set the impedance at (0 < x < a/2, a/2 < y < a) of SL

_{1}unit boundary as Z

_{1}, the impedance at (a/2 < x < a, a/2 < y < a) of SL

_{2}unit boundary as Z

_{2,}the impedance at (0 < x < a/2, 0 < y < a/2) of SL

_{3}unit boundary as Z

_{3}, and the impedance at (a/2 < x < a, 0 < y < a/2) of SL

_{4}unit boundary as Z

_{4}. 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

_{r}is the reflected energy and E

_{in}is the incident energy at the surface z = 0.

_{1}–SL

_{4}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.

_{1}at 100 Hz, which is at the resonance frequency of SL

_{1}. Similarly, most of the acoustic energy is “attracted” toward SL

_{2}–SL

_{4}at the resonance frequencies of SL

_{2}–SL

_{4}respectively. It is observed that at the resonant frequency of a particular shunted loudspeaker unit, the acoustic energy flows towards the unit at resonance. However, the other shunted loudspeaker units are also working as absorbers although the sound intensity near them is rather weak. This is intuitive evidence to demonstrate that all the shunted loudspeaker units work cooperatively as an entire piece.

_{s}or L

_{s}in Figure 1b, and the peak value of its sound absorption coefficient can be adjusted by choosing the loudspeaker with the proper mechanical resistance R

_{ms}.

## 4. Experimental

_{ms}is not equal to the optimal value ρ

_{0}c

_{0}S

_{0}

^{2}/S for the maximal sound absorption peak, because the number of the loudspeaker samples we have is limited. The shunt circuit configuration for the four loudspeakers is listed in Table 7. Theoretically, the resonance of each shunted loudspeaker unit could be adjusted to any frequency. However, considering the nonlinearity of the loudspeaker [21], a more accurate formulation of the impedance Z

_{e}is B

^{2}l

^{2}/S

_{0}/[R

_{E}+ jωL

_{E}+ 1/(1/R

_{2}+ 1/L

_{2}) + Z

_{s}], where R

_{2}is the electrical resistance due to the eddy current losses and L

_{2}is the para-inductance of the voice coil. Therefore, the adjustment of the resonance frequency of the shunted loudspeaker has constraints. In practical, the inductance element in the shunt circuit generates parasitic resistance. By using the ohmmeter to measure the value, negative resistance is employed to cancel the parasitic resistance.

_{i}is corrected by the area ratio factor a

^{2}/4S

_{0}. Substituting the corrected impedance Z

_{i}(a

^{2}/4S

_{0}) into Equation (5), the sound absorption coefficient can be calculated according to Equation (15) and shown as the dashed line in Figure 6. The resonance 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 dotted curve, where the resonance frequencies are at 109 Hz, 219 Hz, 308 Hz, 399 Hz and the sound absorption coefficients are 1.00, 0.77, 0.86, and 0.93 respectively.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Fleming, A.J.; Niederberger, D.; Moheimani, S.R.; Morari, M. Control of resonant acoustic sound fields by electrical shunting of a loudspeaker. IEEE Trans. Control Syst. Technol.
**2007**, 15, 689–703. [Google Scholar] [CrossRef] - Cerník, M.; Mokrý, P. Sound reflection in an acoustic impedance tube terminated with a loudspeaker shunted by a negative impedance converter. Smart Mater. Struct.
**2012**, 21, 115016–115024. [Google Scholar] [CrossRef] - Tang, J.; Wang, K.W. Active-passive hybrid piezoelectric networks for vibration control: Comparisons and improvement. Smart Mater. Struct.
**2001**, 10, 794–806. [Google Scholar] [CrossRef] - Zhang, Y.M. Dynamic Mass Modification by Electric Circuits. Master’s Thesis, The University of Hong Kong, Hong Kong, China, 2012. [Google Scholar]
- Zhang, Y.; Chan, Y.J.; Huang, L. Thin broadband noise absorption through acoustic reactance control by electro-mechanical coupling without sensor. J. Acoust. Soc. Am.
**2014**, 135, 2738–2745. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tao, J.; Jing, R.; Qiu, X. Sound absorption of a finite micro-perforated panel backed by a shunted loudspeaker. J. Acoust. Soc. Am.
**2014**, 135, 231–238. [Google Scholar] [CrossRef] [PubMed] - Tao, J.; Cong, C.; Qiu, X. Thin low frequency sound absorbers based on shunted loudspeakers. J. Acoust. Soc. Am.
**2016**, 140, 2993. [Google Scholar] [CrossRef] - Cho, Y.; Wang, S.; Hyun, J.; Oh, S.; Goo, S. Analysis of sound absorption performance of an electroacoustic absorber using a vented enclosure. J. Sound Vib.
**2018**, 417, 110–131. [Google Scholar] [CrossRef] - Boulandet, R.; Lissek, H. Optimization of electroacoustic absorbers by means of designed experiments. Appl. Acoust.
**2010**, 71, 830–842. [Google Scholar] [CrossRef] [Green Version] - Boulandet, R.; Rivet, E.; Lissek, H. Sensorless electroacoustic absorbers through synthesized impedance control for damping low-frequency modes in cavities. Acta Acust. United Acust.
**2016**, 102, 696–704. [Google Scholar] [CrossRef] - Rivet, E.; Karkar, S.; Lissek, H. Broadband low-frequency electroacoustic absorbers through hybrid sensor-/shunt-based impedance control. IEEE Trans. Control Syst. Technol.
**2017**, 25, 63–72. [Google Scholar] [CrossRef] - Lissek, H.; Boulandet, R.; Fleury, R. Electroacoustic absorbers: Bridging the gap between shunt loudspeakers and active sound absorption. J. Acoust. Soc. Am.
**2011**, 129, 2968–2978. [Google Scholar] [CrossRef] [PubMed] - Boulandet, R. Tunable Electroacoustic Resonators through Active impedance Control of Loudspeakers. Ph.D. Thesis, Swiss Federal Institute of Technology in Lausanne, Zurich, Switzerland, 2012. [Google Scholar]
- Boulandet, R.; Lissek, H. Toward broadband electroacoustic resonators through optimized feedback control strategies. J. Sound Vib.
**2014**, 333, 4810–4825. [Google Scholar] [CrossRef] [Green Version] - Lissek, H.; Boulandet, R.; Moreau, A.S. Practical active and semi-active strategies for the control of room acoustics in the low frequency range. In Proceedings of the INTER-NOISE 2010, Lisbon, Portugal, 13–16 June 2010. [Google Scholar]
- Rivet, E.; Boulandet, R.; Lissek, H.; Rigas, I. Study on room modal equalization at low frequencies with electroacoustic absorbers. In Proceedings of the Acoustics 2012, Nantes, France, 23–27 April 2012. [Google Scholar]
- Matten, G.; Ouisse, M.; Collet, M.; Karkar, S.; Lissek, H.; Boulandet, R.; Versaevel, M. Design and experimental validation of an active acoustic liner for aircraft engine noise reduction. In Proceedings of the MEDYNA 2017, Sevilla, Spain, 25–27 April 2017. [Google Scholar]
- Lissek, H.; Rivet, E.; Laurence, T.; Fleury, R. Toward wideband steerable acoustic metasurfaces with arrays of active electroacoustic resonators. J. Appl. Phys.
**2018**, 123, 091714. [Google Scholar] [CrossRef] [Green Version] - Eargle, J. Loudspeaker Handbook; Kluwer Academic Publishers: Norwell, MA, USA, 2003. [Google Scholar]
- International Organization for Standardization (ISO). Acoustics—Determination of Sound Absorption Coefficient and Impedance in Impedance tubes. Part 2: Transfer Function Method; ISO 10534-2; ISO: Geneva, Switzerland, 1998. [Google Scholar]
- Vanderkooy, J. A Model of Loudspeaker Driver Impedance Incorporating Eddy Currents in the Pole Structure; University of Waterloo: Waterloo, ON, Canada, 1988. [Google Scholar]

**Figure 1.**(

**a**) A schematic of the shunted loudspeaker array set at the end of an impedance duct; (

**b**) the equivalent circuit of a shunt loudspeaker unit.

**Figure 3.**The normal incidence sound absorption coefficients by analytical solution and the finite element model method: SL

_{1}unit (magenta dotted line marked with plus sign), SL

_{2}unit (green dotted line marked with upward-pointing triangle), SL

_{3}unit (red dotted line marked with solid circle), SL

_{4}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).

**Figure 4.**The acoustic intensity of parallel shunted loudspeaker array (

**a**) at 100 Hz; (

**b**) at 200 Hz; (

**c**) at 300 Hz; (

**d**) at 400 Hz.

Parameter | Notation | Value | Unit |
---|---|---|---|

DC resistance | R_{E} | 32.00 | Ω |

Inductance of coil | L_{E} | 7.24 | mH |

Moving mass | M_{ms} | 15.25 | g |

Mechanical resistance | R_{ms} | 1.57 | kg/s |

Mechanical compliance | C_{ms} | 0.67 | mm/N |

Force factor | Bl | 17.12 | T·m |

Effective area | S_{0} | 1.51 × 10^{-2} | m^{2} |

Back cavity volume | V | 2.2 × 10^{-}^{3} | m^{3} |

Cavity depth | D | 7.5 | cm |

SL_{1} | SL_{2} | SL_{3} | SL_{4} | |
---|---|---|---|---|

R (Ω) | −31.95 | −31.95 | −31.95 | −31.95 |

L (mH) | −7.23 | −7.23 | −7.23 | −7.23 |

L_{s} (mH) | - | 37.19 | 7.69 | 3.62 |

C_{s} (μF) | 87 | - | - | - |

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

_{1}unit).

Parameter | Notation | Value | Unit |
---|---|---|---|

DC resistance | R_{E} | 32.12 | Ω |

Inductance of coil | L_{E} | 7.24 | mH |

Moving mass | M_{ms} | 15.25 | g |

Mechanical resistance | R_{ms} | 1.15 | kg/s |

Mechanical compliance | C_{ms} | 0.67 | mm/N |

Electrical resistance due to eddy current losses | R_{2} | 39.50 | Ω |

Para-inductance of voice coil | L_{2} | 6.75 | mH |

Force factor | Bl | 17.12 | T·m |

Effective area | S_{0} | 1.51 × 10^{-2} | m^{2} |

Back cavity volume | V | 2.2 × 10^{-}^{3} | m^{3} |

Cavity depth | D | 7.5 | cm |

Parameter | Notation | Value | Unit |
---|---|---|---|

DC resistance | R_{E} | 5.71 | Ω |

Inductance of coil | L_{E} | 0.37 | mH |

Moving mass | M_{ms} | 15.19 | g |

Mechanical resistance | R_{ms} | 1.31 | kg/s |

Mechanical compliance | C_{ms} | 0.83 | mm/N |

Electrical resistance due to eddy current losses | R_{2} | 1.11 | Ω |

Para-inductance of voice coil | L_{2} | 0.33 | mH |

Force factor | Bl | 5.05 | T·m |

Effective area | S_{0} | 1.51 × 10^{-2} | m^{2} |

Back cavity volume | V | 1.2 × 10^{-}^{3} | m^{3} |

Cavity depth | D | 7.5 | cm |

Parameter | Notation | Value | Unit |
---|---|---|---|

DC resistance | R_{E} | 6.98 | Ω |

Inductance of coil | L_{E} | 0.24 | mH |

Moving mass | M_{ms} | 7.72 | g |

Mechanical resistance | R_{ms} | 0.94 | kg/s |

Mechanical compliance | C_{ms} | 0.27 | mm/N |

Electrical resistance due to eddy current losses | R_{2} | 1.57 | Ω |

Para-inductance of voice coil | L_{2} | 0.31 | mH |

Force factor | Bl | 5.21 | T·m |

Effective area | S_{0} | 1.51 × 10^{-2} | m^{2} |

Back cavity volume | V | 1.4 × 10^{-}^{3} | m^{3} |

Cavity depth | D | 7.5 | cm |

Parameter | Notation | Value | Unit |
---|---|---|---|

DC resistance | R_{E} | 7.31 | Ω |

Inductance of coil | L_{E} | 0.33 | mH |

Moving mass | M_{ms} | 10.38 | g |

Mechanical resistance | R_{ms} | 1.02 | kg/s |

Mechanical compliance | C_{ms} | 0.29 | mm/N |

Electrical resistance due to eddy current losses | R_{2} | 1.41 | Ω |

Para-inductance of voice coil | L_{2} | 0.25 | mH |

Force factor | Bl | 5.60 | T·m |

Effective area | S_{0} | 1.51 × 10^{-2} | m^{2} |

Back cavity volume | V | 1.4 × 10^{-}^{3} | m^{3} |

Cavity depth | D | 7.5 | cm |

SL_{1} | SL_{2} | SL_{3} | SL_{4} | |
---|---|---|---|---|

R (Ω) | −32.00 | - | - | −7.00 |

L (mH) | −7.00 | - | - | - |

L_{s} (mH) | 5 | - | - | 0.2 |

C_{s} (μF) | 47 | - | - | - |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cong, C.; Tao, J.; Qiu, X.
A Multi-Tone Sound Absorber Based on an Array of Shunted Loudspeakers. *Appl. Sci.* **2018**, *8*, 2484.
https://doi.org/10.3390/app8122484

**AMA Style**

Cong C, Tao J, Qiu X.
A Multi-Tone Sound Absorber Based on an Array of Shunted Loudspeakers. *Applied Sciences*. 2018; 8(12):2484.
https://doi.org/10.3390/app8122484

**Chicago/Turabian Style**

Cong, Chaonan, Jiancheng Tao, and Xiaojun Qiu.
2018. "A Multi-Tone Sound Absorber Based on an Array of Shunted Loudspeakers" *Applied Sciences* 8, no. 12: 2484.
https://doi.org/10.3390/app8122484