Research ArticleResearch on Array Structures of Acoustic Directional Transducer

Guozhu Zhao,1 Kaibo Shi ,2 and Shouming Zhong3

1Simulator Training Center Civil Aviation Flight University of China, Guanghan, Sichuan 618300, China2School of Information Science and Engineering, Chengdu University, Chengdu, Sichuan 610106, China3School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Correspondence should be addressed to Kaibo Shi; skbs111@163.com

Received 3 October 2020; Revised 8 December 2020; Accepted 17 December 2020; Published 4 January 2021

Academic Editor: Shihong Ding

Copyright © 2021 Guozhu Zhao et al.,is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

,is paper focuses on the directivity design of array structures of acoustic directional transducers. Based on Huygens principles,the directivity formula of transducer arrays under random distribution in xyz space is derived when the circular piston transducersare used as the array element, which is used to analyze the directivity and acoustic pressure of conical transducer arrangements. Inaddition, a practical approach to analyze the directivity and acoustic pressure of transducer arrays under random arrangements isproposed. Findings. ,e conical transducer arrays show side lobes at higher frequency. Below the frequency of 2 kHz, arraydirectivity shows rapid changes. Above the frequency of 2 kHz, array directivity varies slowly with frequency. Besides, the beamwidth is Θ−3 dB ≤ 29.85°.

1. Theoretical Calculation of TransducerArray Directivity

For the single transducer, its directivity is decided by theratio of sound wave length λ to size a. Take the circularpiston source on the baffle as an example, as shown Figure 1.As for the single transducer, its directivity is expressed as thefollowing formula [1–3]:

D(θ) �2J1(ka sin θ)

ka sin θ

�

2J1(π d/λ)sin θ(π d/λ)sin θ

. (1)

In this formula, J1 is first-order Bessel function; wavenumber is k � 2π/λ; a represents sound source radius; andd � 2a represents sound source diameter.

For linear arrays of point acoustic sources that consist ofmultiple transducers, the directivity is not decided by theratio of sound wave length λ to transducer size a, but byarray element arrangement. It is important to obtain theformula of directivity for transducer arrays in any randomarrangements. ,e following shows the analysis of tworelevant formulas [4–6].

2. Comparison of Directivity of TransducerArrays and Random Array Configurations

According to Huygens principles, the linear array directivityfunction of n point acoustic sources is derived, as shown inFigure 1. It should be noted that at this point, each unittransducer is regarded as point with no radius, assumingd � 2a. It is possible to derive the directivity function oftransducer arrays in N lines and M rows, as shown inFigure 2 [1]:

D(α, θ) �sin kMd1/2( cos α sin θ( M sin kd1/2( cos α sin θ(

·sin kNd2/2( sin α sin θ( N sin kd2/2( sin α sin θ(

.

(2)

In this formula, d1 and d2 represent line space and spacebetween columns; α is angle between the projection of vectorOP on XOY and positive x axis; and θ is the angle betweenvector OP and z axis. ,e above formula obtains the fol-lowing conclusion: for this function, it is necessary todemonstrate equidistant distribution in a certain direction(set the spacing distance in x axis or y axis); it needs to be a

HindawiMathematical Problems in EngineeringVolume 2021, Article ID 6670277, 5 pageshttps://doi.org/10.1155/2021/6670277

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rectangular distribution instead of random arrays (such aspolygon or circular array). When M or N is 1, this formulacan calculate the array directivity. When M�N� 1, theformula can calculate directivity of single transducer [7–9].

3. Theoretical Calculation of RandomTransducer Array Directivity

Figure 3 shows the rectangular coordinate system of direc-tional acoustic transducer arrays. ,e central point of trans-ducer arrays O is the origin of coordinate. Assumingtransducer array within the xoy plane of the three-dimensionalcoordinate system, any single transducer is at the position Q(x0, y0). In the sound field, the distance between any singleobservation point P (x0, y0, z0) and the origin of coordinate is r,the intersection angle between the point and z axis is θ, androtation angle is φ. It is likely to obtain normalized directivityfunction of n circular piston transducer arrays with the radiusa and random placements:

D(θ, ϕ) �pM(θ, ϕ)θ

pM(θ, ϕ)θ�0

�2J1(ka sin θ)

ka sin θ

n

i�1e

jk xi sin θ sinϕ+yi sin θ cos ϕ| |

.

(3)

From formula (3), it can be concluded that, for thisfunction, there is no need to set the distance betweentransducer array elements or rectangular arrangements toobtain the directivity of random array arrangements. It isonly necessary to identify the coordinate of each array el-ement. When P (x0, y0) is determined, the directivity ofrandom transducer plane layouts can be obtained. Besides,

this formula is relatively complex and difficult to obtaindirectivity patterns of arrays by conducting research on two-dimensional directivity. Hence, a research method of three-dimensional directivity is used to identify directivity [4–6].

4. Directivity Analysis of Three-DimensionalTransducer Arrangement

As shown in Figure 4, the three-dimensional space isestablished in the three-dimensional coordinate system withrandom transducers. O is the central point of the three-dimensional coordinate system. Assume transducer arraysare within the xyz plane of the three-dimensional coordinatesystem.

If the circular piston transducer with the radius a vi-brates at the velocity, u � u0ejωt, where u0 is the velocityamplitude. Assume it is right at the origin of coordinate O.,en, its sound pressure is generated at observation point P(x0, y0, z0) (the distance from origin of coordinate is r. ,eintersection angle with z axis is θ. ,e rotation angle is theposition of φ). ,e resulting sound pressure is

p � jωρ0uaa

2

2r2J1(ka sin θ)

ka sin θ

e

j(ωt− kr). (4)

In this formula, ρ0 is the density of medium. k � 2π/λ isthe wave number (λ is the sound wave length). Frequencyf� 10000Hz. Sound speed is C0 � 340m/s, λ�C0/f.

When circular piston transducer is at any point of xyzspace, set r1 as the sonic path distance between Q and P.Likewise, for at any point of xyz space Q, the sound pressuregenerated by circular piston transducer with the radium a atpoint P is

p1 � jωρ0uaa

2

2r1

2J1 ka sin θQ ka sin θQ

e

j ωt− kr1( ). (5)

In this formula, θQ is the included angle between vectorQP⇀and z. In the far field, it is approximately assumed as

vector OP⇀//QP⇀.,e connection between r1 and r is shown in

formula (4). |OQ⇀

|cos θ1 represents the projection of vectorOQ⇀on vector OP

⇀. From the projection relationship, it can be

inferred

θ

P (r, θ)

2a b

A

Figure 1: Transducer line arrays.

θ

P

M

Ny

x

Z

O

d1

d2

Figure 2: Transducer square arrays.

θ

θ1

Oy

x

z

M

P′

φ

Q (x1, y1, 0)

P (r, θ, φ)

Figure 3: Coordinate system of transducer array.

2 Mathematical Problems in Engineering

r � r1 +|OQ⇀

|cos θ1 � r1 + rQ cos θ1. (6)

In this formula, rQ is the module of vector OQ⇀and θ1 is

the included angle between vectors OQ⇀and OP

⇀. It should be

noted that θ1 may be an acute angle or an obtuse angle. ,erectangular coordinate of point P is (x0, y0, z0). ,e rect-angular coordinate of point Q is (x1, y1, z1). Vectors OP

⇀and

OQ⇀

are represented as x0, y0, z0 and x1, y1, z1 . ,en,included angle cosine in formula (4) is

cos θ1 �x0x1 + y0y1 + z0z1����������

x20 + y

20 + z

20

����������

x21 + y

21 + z

21

�x0x1 + y0y1 + z0z1

rrQ.

(7)

Formulas (4) and (5) are combined and arranged toobtain

r1 � r −x0x1 + y0y1 + z0z1

r, (8)

where, x0, y0, and z0 in formula (6) are converted to circularcylindrical coordinates as the following:

x0 � r sin θ sinϕ,y0 � r sin θ cos ϕ,

z0 � r cos θ.

⎧⎪⎪⎨

⎪⎪⎩(9)

Formula (7) is substituted to formula (6):

r1 � r − x1 sin θ sinϕ + y1 sin θ cosϕ + z1 cos θ( . (10)

In the far field, the amplitude of formula (2) r1 ≈ r,θQ ≈ θ. Formula (8) is substituted to formula (2) to obtainthe following: the sound pressure generated by any point Qat point P in the xyz space is

p1 � jωρ0uaa

2

2r2J1(ka sin θ)

ka sin θ

e

j ωt− kr+k x1 sin θ sinϕ+y1 sin θ cos ϕ+z1 cos θ( )[ ]. (11)

If planar transducer arrays consist of n circular pistontransducers, the i transducer is in position (xi, yi, zi) of theplane. Each transducer vibrates at the velocity u � uaejωt.

,e sound pressure generated by n circular piston trans-ducers at point P in the xyz space is shown in the following:

pM � jωρ0uaa

2

2r2J1(ka sin θ)

ka sin θ

e

j(ωt− kr)

n

i�1e

jk x1 sin θ sinϕ+y1 sin θ cosϕ+z1 cos θ( ). (12)

According to Bessel function, when x � 0,J1(x)/x � (1/2). Based on formula (10), the normalized

directivity function of transducers in random space ar-rangement can be obtained:

D(θ, ϕ) �pM(θ, ϕ)θ

pM(θ, ϕ)θ�0

�2J1(ka sin θ)

ka sin θ

M

i�1e

jk x1 sin θ sinϕ+y1 sin θ cos ϕ+z1 cos θ( )

. (13)

According to formula (11), it is feasible to calculate thefeasibility of circular piston transducers with random ar-rangement in three-dimensional space. However, the aboveformula is a function concerned with variables θ and φ,

which are hard to identify the directivity of the array. Hence,three-dimensional directivity research method is used in theprocess. To be more specific, conversion of coordinates iscarried out in formula (11).

θ

φ

Q1 (X1, Y1, 0)

Q1 (X1, Y1, Z1)

P (r, θ, φ)

O

Y

Z

X

Figure 4: ,e three-dimensional coordinate system of transducerarrays.

Mathematical Problems in Engineering 3

x0 � sin θ sinϕ · D( θ, ϕ ),y0 � sin θ cos ϕ · D( θ, ϕ ),

y0 � cos θ · D( θ, ϕ ).

⎧⎪⎪⎨

⎪⎪⎩(14)

Assume the most central transducer is origin of coor-dinate O (0, 0), then the position coordinate of acousticdirectional dispersion system is shown in Figure 4. ,erectangular coordinates of acoustic directional transducerarray are substituted in formula (11) to calculate the di-rectivity angle of transducer. By changing the frequency, it isfeasible to get the result shown in Figure 3.

It is known from Figure 5 that acoustic directionaltransducer array shows weak directivity at the sound wavefrequency level of 500Hz. If the sound wave frequency levelincreases, the level of directivity also rises gradually.When thefrequency level reaches 4 kHz, apparent sidelobe shows up.When the frequency level reaches 6 kHz, directivity becomesfavorable, but sidelobe becomes more apparent as well.

,e figure shows that beam width Θ−3 dB graduallynarrows as frequency increases. Below the beam width of2 kHz, frequency change is apparent. Above the beam widthof 2 kHz, frequency change is slow. ,e frequency of dis-persion sound wave ranges between 2.1 and 3.4 kHz.When itis above 2 kHz, the beam width of acoustic directionaltransducer is Θ−3 dB ≤ 29.85°, directional acute angle(Θ−3 dB/2)≤ ± 14.925°. Favorable directivity is shown.

5. Conclusion

Based on the Huygens principle of sound waves, this workhas derived the formula for the directivity of transducerarrays in random arrangement when circular piston

transducers are used as array elements. Based on this for-mula, it studied the directivity and sound pressure of conicaltransducer array arrangements. ,e work provided a way toanalyze directivity and sound pressure of transducer arraysin random arrangements for conical transducer and acousticdirectional transducers. Findings: conical transducer arraysdemonstrate sidelobe at high-frequency levels, but it can beoverlooked compared to the main lobe. Below the frequencylevel of 2 kHz, array directivity changes rapidly. Above thefrequency level of 2 kHz, array directivity changes moreslowly, and the beam width is Θ−3 dB ≤ 29.85°.

,e work expands the formula of calculating the direc-tivity of circular piston transducers with random array ar-rangements. Based on the digital simulation of computers, itresolves the difficult issues in the directivity design of three-dimensional arrays of acoustic directional transducers, pro-viding positive significance for designing acoustic directionaltransducer arrays.

Data Availability

No data used to support the study.

Conflicts of Interest

,e authors declare that they have no conflicts of interest.

Acknowledgments

,is work was financially supported by the Civil AviationFlight University of China fund project: fault diagnosis of theoperating system of the certificate simulator (XM2803).

1

0.5

0

Z

11

0 0

–1 –1

0.590

60

300

330

300

270

1

Y X

(a)

1

0.5

0

Z

11

0 0–1 –1

0.590

60

300

330

300

270

1

Y X

(b)

Figure 5: Directivity pattern of acoustic directional dispersion device. (a) 500Hz. (b) 1 kHz.

4 Mathematical Problems in Engineering

References

[1] J. Moro and C. M. Sprinkle, “,in high performance constantdirectivity waveguide and speaker,” U.S. Patent Application15/236,720, 2017.

[2] G. K. Munch and J. Dyreby, “Loudspeaker transducer ar-rangement for directivity control,” U.S. Patent 9,942,659, 2018.

[3] J. Ahrens and S. Bilbao, “Interpolation and range extrapolationof sound source directivity based on a spherical wave propa-gation model,” in Proceedings of the ICASSP InternationalConference on Acoustics, Speech and Signal Processing,pp. 4662–4666, IEEE, Barcelona, Spain, May 2020.

[4] H. Wang, X. Wang, C. He, and C. Xue, “Directivity theory andanalysis of 2D capacitive micro-machined ultrasonic trans-ducer array,” Journal of Nanoelectronics and Optoelectronics,vol. 12, no. 8, pp. 786–794, 2017.

[5] G. Huang, J. Chen, and J. Benesty, “A flexible high directivitybeamformer with spherical microphone arrays,”9e Journal ofthe Acoustical Society of America, vol. 143, no. 5, pp. 3024–3035,2018.

[6] X. Zhen-yang, W. Xin-peng, and Z. Jing-yuan, “Research onthe directivity of transducer array based on typical array ele-ments,” Journal of Physics: Conference Series, IOP Publishing,vol. 1237, no. 4, , p. 042063, 2019.

[7] J. Ling, Y.-Q. Chen, Y. Chen et al., “Design and characterizationof high-density ultrasonic transducer array,” IEEE SensorsJournal, vol. 18, no. 6, pp. 2285–2290, 2018.

[8] S. Yan, “Robust time-domain wideband modal beamformingwith circular arrays,” IEEE Transactions on Aerospace andElectronic Systems, In press.

[9] Q. Zhao, W. Zheng, and J. Li, “Research on the directionality ofdifference frequency sound field based on circular ultrasonicarray,” Journal of Physics: Conference Series, IOP Publishing,vol. 1314, no. 1, p. 012040, 2019.

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