1 LUMPED ELEMENT MULTIMODE MODELING FOR A SIM- PLIFIED BALANCED-ARMATURE RECEIVER Wei Sun, Wenxiang Hu Institute of Acoustics, Tongji University, Shanghai, China 200092 e-mail: [email protected]Lumped element (LE) models normally used for balanced-armature receiver (BAR) are mainly based on the fundamental mode of its mechanical structure. For the lack of higher order modes, they may be insufficient to predict the system of BAR. Here we propose a LE multimode model aiming at improving the modeling of BAR. The model is developed based on the techniques of mode decomposition, truncation, and selection. Firstly, the mechanical structure of BAR is de- coupled into a set of SDOF systems. Then the modes are truncated using the criterion of energy norm. Afterwards, the dominant modes are selected with DC gains of the concerned modes in the frequency range of interest. Finally, the LE multimode model is built for BAR using the deter- mined dominant modes. The validation is made by comparing with both the corresponding com- bined FE-LE model and the full FE model. Simulation results prove the developed model is not only almost as effective as the combined FE-LE model, and also much more efficient. 1. Introduction Balanced-armature receiver (BAR) has been popular in hearing aids for decades. In recent years, it also rapidly trends in high-end innovative consumer electronics like earbuds, glasses earphone, and earphone integrated with moving-coil and balanced-armature receivers, etc. This mainly attrib- utes to its tiny size, excellent resolution, high sensitivity, and good dynamic performance. For the increasing demand in high-fidelity music sound, it is necessary to develop wideband BAR in which multiple modes are mostly involved. The known literatures on BAR are mainly the lumped element (LE) models considering only the fundamental mode of the vibration system [1-7]. They are obvi- ously incapable of being used to develop the needed BAR. Though there are some possible solu- tions such as finite element (FE) aided LE modeling method [8], full FE modeling method [9], and combined FE-LE modeling method [10], they are either too complicated or still rather time- consuming. Therefore, the wide range of engineering application is infeasible. The study on the distributed-parameter system of electromechnical transducers provides a feasible solution for the multimode modeling of transducers [11]. But it does not apply to the transducer with arbitrary struc- ture as well as lacks the discussion about the mode truncation and selection. We therefore propose a LE multimode model in the frequency domain for BAR, which is capa- ble of incorporating the multiple modes of the mechanical system and efficiently computing as well. On the basis of modal analysis, we first decouple the discretized multi-degree-of-freedom (MDOF) system into a set of single-degree-of-freedom (SDOF) systems [12]. Then the mode truncation is performed with the criterion of energy norm [13, 14]. And the mode selection using DC gains is further followed to determine merely the dominant modes among the truncated modes [15]. With the selected modes, we finally derive the equations in the frequency domain that are used for mod- eling BAR. The developed model is validated with the corresponding full FE model, and the highly improved efficiency is proved by comparing to our earlier combined FE-LE model [10].
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LUMPED ELEMENT MULTIMODE MODELING FOR A SIM-PLIFIED BALANCED-ARMATURE RECEIVER Wei Sun, Wenxiang Hu Institute of Acoustics, Tongji University, Shanghai, China 200092 e-mail: [email protected]
Lumped element (LE) models normally used for balanced-armature receiver (BAR) are mainly based on the fundamental mode of its mechanical structure. For the lack of higher order modes, they may be insufficient to predict the system of BAR. Here we propose a LE multimode model aiming at improving the modeling of BAR. The model is developed based on the techniques of mode decomposition, truncation, and selection. Firstly, the mechanical structure of BAR is de-coupled into a set of SDOF systems. Then the modes are truncated using the criterion of energy norm. Afterwards, the dominant modes are selected with DC gains of the concerned modes in the frequency range of interest. Finally, the LE multimode model is built for BAR using the deter-mined dominant modes. The validation is made by comparing with both the corresponding com-bined FE-LE model and the full FE model. Simulation results prove the developed model is not only almost as effective as the combined FE-LE model, and also much more efficient.
1. Introduction Balanced-armature receiver (BAR) has been popular in hearing aids for decades. In recent years,
it also rapidly trends in high-end innovative consumer electronics like earbuds, glasses earphone, and earphone integrated with moving-coil and balanced-armature receivers, etc. This mainly attrib-utes to its tiny size, excellent resolution, high sensitivity, and good dynamic performance. For the increasing demand in high-fidelity music sound, it is necessary to develop wideband BAR in which multiple modes are mostly involved. The known literatures on BAR are mainly the lumped element (LE) models considering only the fundamental mode of the vibration system [1-7]. They are obvi-ously incapable of being used to develop the needed BAR. Though there are some possible solu-tions such as finite element (FE) aided LE modeling method [8], full FE modeling method [9], and combined FE-LE modeling method [10], they are either too complicated or still rather time-consuming. Therefore, the wide range of engineering application is infeasible. The study on the distributed-parameter system of electromechnical transducers provides a feasible solution for the multimode modeling of transducers [11]. But it does not apply to the transducer with arbitrary struc-ture as well as lacks the discussion about the mode truncation and selection.
We therefore propose a LE multimode model in the frequency domain for BAR, which is capa-ble of incorporating the multiple modes of the mechanical system and efficiently computing as well. On the basis of modal analysis, we first decouple the discretized multi-degree-of-freedom (MDOF) system into a set of single-degree-of-freedom (SDOF) systems [12]. Then the mode truncation is performed with the criterion of energy norm [13, 14]. And the mode selection using DC gains is further followed to determine merely the dominant modes among the truncated modes [15]. With the selected modes, we finally derive the equations in the frequency domain that are used for mod-eling BAR. The developed model is validated with the corresponding full FE model, and the highly improved efficiency is proved by comparing to our earlier combined FE-LE model [10].
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2. Theory review
2.1 Decomposition of an undamped forced vibration system An arbitrary continuous undamped vibration system can be discretized into a MDOF system with
the aid of finite element method. Equation of the system in the frequency domain takes the form [12]
−휔 퐌퐗(휔) + 퐊퐗(휔) = 퐅(휔) (1)
where 퐌, 퐊 are respectively the mass, stiffness 푁푁 real symmetric matrices, 푁 is the number of degrees of freedom; displacement vector 퐗 and excitation force vector 퐅 are both 푁1 vectors.
The corresponding characteristic equation of the system can be written as
|퐊 − 휔 퐌| = 0. (2)
After solving Eq. (2), the system satisfies the relation
퐊훗 = 휔 퐌훗 (3)
where 훗 = [휙 휙 ⋯ 휙 ] is the eigenvector corresponding to the eigenvalue 휔 . With the orthogonality properties of the system, Eq. (1) can be diagonalized with mode vector 횽
as
([푘 ] − 휔 [푚 ])퐐 = 횽 퐅(휔) (4)
where [푘 ] = 횽 퐊횽 and [푚 ] = 횽 퐌횽 are all the 푁푁 diagonalized matrices; and 퐐 =[푞 (휔) 푞 (휔) ⋯ 푞 (휔)] is the modal coordinate vector, 푞 is the modal coordinate of the 푟th mode, 횽 = [ 훗 훗 … 훗 ] is the mode vector set for all the 푁 modes.
Equation (4) now is an additive combination of a set of single-degree-of-freedom (SDOF) equa-tions. In other words, the discretized highly coupled MDOF system now is decomposed into a col-lection of 푁 independent SDOF systems. This results in simplifying the complex MDOF system into a set of simple SDOF systems, which makes the complicated or unsolvable continuous system convenient to be analytically calculated.
Considering a single-input and single-output (SISO) system, e.g. an external excitation force 푓 (휔) at point 푝 and a measuring position at point 푙, we can derive the modal coordinate of the 푟th mode from Eq. (4) as
푞 =( )
(5)
where 휙 is the mode vector at point 푝 of the 푟th mode. Then the total displacement response of system at point 푙 will be
푥 (휔) = ∑ ( ) (6)
where 휙 is the mode vector at point 푙 of the 푟th mode. For the sake of being applied conveniently in modeling BAR, the equivalent parameters for each
SDOF system from Eq. (6), e.g. the 푟th SDOF system, can be expressed as
푘 = , 푚 = , 푓 = 푓 (휔). (7)
2.2 Determination of dominant modes Since there could be numerous decoupled SDOF systems from a MDOF system, it is still a bit
impractical in applying. Therefore, we first truncate the concerned modes using the criterion of en-ergy norm [13, 14] and then select merely the dominant modes with DC gains [15] in the frequency range of interest.
Mode truncation can be implemented based on the criterion of energy norm, which is defined as
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퐸 = 퐉 + 퐉∗ (8)
where 푁 is the truncated modal number, 퐸 is the total energy of the system with all the truncated
푁 modes, 퐉 = ∑ 푘 푞 is the potential energy related norm and 퐉∗ = ∑ 푚 푞
is the kinetic energy related norm. Energy of system 퐸 in Eq. (8) is getting closer to the total energy of system 퐸 with increase of
푁 . To reduce the system as much as possible, 푁 is an optimal value making 퐸 close 퐸 to an acceptable level.
Subsequently, the system can be further reduced by using DC gain to select the dominant modes among the truncated modes. DC gain of the 푟th SDOF system can be formulated into [15]
(DC gain) = . (9)
3. Determination of lumped multimode parameters for BAR
(a) (b)
Figure 1: Geometry of the concerned simplified BAR: (a) mechanical domain; (b) acoustic domain.
To make this work consistent with our earlier study [10], we use the same simplified BAR model (Fig. 5) to validate the developed LE multimode model. The membrane usually consists of metal and foil layers. Here we concern the dominant metal layer only, but the two volumes are completely isolated over the hole in this study. Table 1 lists the dimensions for all the contained mechanical and acoustic components. The corresponding properties of materials are listed in Table 2．A constant force with amplitude 1 10-3 N is vertically loaded at drive point (DP). Variables to be evaluated are the velocity at MP (measure point, centralized at the connection area between drive pin and membrane) and the sound pressure level (SPL) in the 2cc coupler, which is obtained by averaging acoustic pressure over the bottom surface of the 2cc coupler (i.e. right side of the 2cc coupler).
Table 1: Dimensions of contained mechanical and acoustic components.
Components of the simplified BAR LengthWidthThickness (mmmmmm)
Mechanical domain
Armature 51.500.10
Drive pin 0.70.150.025 Membrane with
hole Membrane: 5.2020.04
(hole 1.600.200.025, 0.20 mm to fixed side)
Acoustic domain
Rear volume 5.2020.96 Front volume 5.2020.30
Tube Length 5 mm, diameter 1 mm
2cc coupler Length 10 mm, diameter 16 mm
Tube
2cc coupler
Rear volume Front volume
DP Armature
Membrane
Fixed bounda-
MP
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Table 2: Material properties.
Component Material type Density (kg/m3)
Young’s modulus (Pa) Poisson’s ratio
Membrane Aluminium 2,700 0.70 1011 0.35
Drive pin Copper 8,960 1.20 1011 0.34
Armature Soft iron 7,850 2.00 1011 0.33 Volumes, tube,
2cc coupler Air 1.20 344 (m/s) (sound speed) -
3.1 Lumped parameters based on the fundamental mode As the literature [10], lumped parameters for acoustic components (Table 3) are determined
without visco-thermal effects. But other than this literature, we halve the acoustic mass of front vol-ume because of the one fixed end of membrane and add the acoustic mass of the small slit from front volume to tube. Meanwhile, lumped parameters for the mechanical domain are obtained by treating the entire mechanical structure as a whole. With these improved approximations, we expect the better modeling in the LE relevant models.
Table 3: Lumped parameters for mechanical and acoustic components.
3.2 Lumped multimode parameters for mechanical domain For the earlier LE model of BAR created based on the fundamental mode of the mechanical sys-
tem only [10], it is insufficient to predict the system especially in the frequency range influenced by the higher order modes. We resolve this problem by using the above-discussed techniques of mode decomposition, truncation, and selection. With the technique of mode decomposition, the mechani-cal domain of BAR can be decoupled into a set of SDOF systems with the aid of finite element method. If 20 kHz is the maximum frequency of interest, the first twelve modes up to 130 kHz are used for the following mode truncation and mode selection.
3.2.1 Mode truncation and mode selection After decoupling the mechanical structure of BAR, we firstly employ the criterion of energy
norm in Eq. (8) to get the energy norms with respect to MP at all the concerned frequencies. Fig-ure 2(a) indicates the contour plot of ratio 퐸 퐸⁄ as a function of both the truncated modal number and frequency over the entire frequency range of interest, where 퐸 is supposed to be the total en-ergy of the system. The energy norms are getting reasonably close to 퐸 when the truncated modal number is up to six. We therefore assume that the first six modes are sufficient to represent the whole system in the frequency range of interest.
To reduce the modal number even more, we select only the dominant modes using DC gains (Eq. (9)) among the truncated six modes. As demonstrated in Fig. 2(b), the third and sixth modes
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are at least three orders smaller in amplitude than the others. Therefore, we can hold that the four modes (1st, 2nd, 4th, and 5th) are the dominant modes, which are capable of representing the whole system in the concerned frequency range.
(a) (b)
Figure 2: Energy norms and DC gains of the system: (a) contour plot of energy norms over the interested frequency range as function of the truncated modal number and frequency; (b) DC gains of the first six
modes.
3.2.2 Equivalent parameters of selected modes at MP According to Eq. (7), the equivalent parameters with respect to MP (Table 4) can respectively be
figured out for the four dominant modes. Table 4: Equivalent parameters of the four dominant modes with respect to MP.
Selected mode Resonance (Hz) 푚 (kg) 푘 (N/m) Load gain
1st mode: M1 2,958 2.11E-06 727 1
2rd mode: M2 5,568 9.15E-06 11,191 1
4th mode: M4 16,694 4.07E-05 447,796 1
5th mode: M5 20,724 2.05E-06 34,810 1
4. Development of LE multimode model for BAR With the determined dominant SDOF systems of the mechanical domain, we can analytically
build the LE multimode model in which each SDOF mechanical system independently couples with the acoustic domain. Solution of the defined continuous mechanical-acoustic system now turns into the calculations of this set of simple SDOF coupled mechanical-acoustic systems. The response of the entire system correspondingly becomes the superposition of the four SDOF systems.
According to Newton’s second law, the equation for each SDOF system takes the form of the spring-mass system. Then we can formulate the 푟th SDOF system in frequency-domain into
퐹 , = 푗휔푚 +⁄
푣 , (10)
where 퐹 , = 푟 , 퐹 , 푟 , , and 푣 , are respectively the equivalent magnetic force, the load gain, and the velocity at MP for the 푟th SDOF system.
In the acoustic domain, the system satisfies the relation 푃 = 푍 푞 [16]. 푃, 푍 , and 푞 are respectively the lumped acoustic pressure, impedance, and the volume velocity induced by the membrane.
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The mechanical-acoustic coupling can be realized with the two passing variables via membrane. Firstly, membrane transfers its vibrating velocity to the air domain with the induced volume veloc-ity. Then the generated acoustic pressures in the air volumes impose back on membrane with the induced force. These two mode-dependent variables can be expressed as
푞 , = 퐴 , 푣 , (11)
퐹 , = 푃 − 푃 퐴 , (12)
where 푞 , , 퐴 , , and 퐹 , are respectively the volume velocity induced by the moving membrane, the effective area of membrane, and the force produced by the pressures in air volumes for the 푟th SDOF system; 푃 and 푃 are respectively the induced lumped pressures in front and rear volumes.
As the different modes are different in vibration shape, the transduction ratios, i.e. effective areas of membrane, are therefore dependent on modes. These ratios can be determined by
퐴 , = ∬ ,
, (13)
where 푣 , and 푑퐴 are respectively the distributed velocity of membrane and the area element of membrane for the 푟th mode.
According to Eq. (10) and the equations of acoustic system as well as the principles of mechani-cal-acoustical analogy, we can develop the equivalent circuit for the simplified BAR including mul-tiple modes of the mechanical domain (Fig. 3). This circuit is the lumped element multimode model for BAR, which is capable of predicting BAR more efficient than the combined FE-LE modeling method and more reasonable than the commonly LE model.
Figure 3: Equivalent circuit of LE multimode model for BAR.
5. Numerical results and analyses Based on the developed LE multimode model, we perform a frequency analysis using PSpice.
The input is a constant force with amplitude 1 10-3 N at DP. And the interested frequency here ranges from 100 Hz to 10 kHz with 100 frequency points per decade, which covers the main audio range. The variables to be evaluated are the velocity at MP and the SPL response in the 2cc coupler. The validation is made by referring to the corresponding combined FE-LE model (combined FE modeling of mechanical domain with LE modeling of acoustic domain) as well as the full FE mod-el [10]. In addition, we also involve the commonly used LE model into the comparison to point out its obvious difference to the developed model.
5.1 Validation with the full FE model Taking the full FE model as the reference, we compare the developed LE multimode model with
our earlier LE single-mode model in their ability to predict the system of BAR. Results shown in Figs. 4(a) and 4(b) are respectively the velocity frequency response at MP and the SPL response in
푚 1/푘
푚
푣 ,
푣 ,
푀 , 푀 ,
1/푘 퐴 , 퐶 , 퐶 , 퐶 ,
푃 푃
퐹
푟 ,
푟 ,
푞
퐴 ,
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the 2cc coupler. Obviously, the LE multimode model agrees well with the full FE model for the velocity at MP over the concerned frequency range and for the SPL response in the 2cc coupler up to 8 kHz. Slight difference around the first peak and above could be caused by the absence of the higher order modes of the acoustic domain. However, the commonly used LE model is apparently deviated from the FE model for both the velocity and the SPL response. This proves that the devel-oped LE multimode model is more reasonable than the LE model.
(a) (b)
Figure 4: Frequency responses of the system (FE, LE-4modes, and LE represent the full FE model, the LE multimode model, and the LE model, respectively): (a) velocity at MP; (b) SPL in the 2cc coupler.
5.2 Comparison with the combined FE-LE model Figures 5(a) and 5(b) display the velocity at MP and SPL in the 2cc coupler, respectively, for the
LE multimode model and the combined FE-LE model. Apparently, the two models are in substan-tial agreement over the concerned frequency range. We therefore conclude that the LE multimode model is nearly as effective as the combined FE-LE model. Whereas the developed model takes definitely less time (79 seconds, including the modal analysis) than the combined FE-LE model (3,409 seconds) under a common personal computer with 4-core 3.3 GHz of CPU and 8 GB of RAM. So we can also hold that the efficiency is highly improved by the developed model.
(a) (b)
Figure 5: Frequency responses of the system (MechaFE-AcoLE represents the combined FE-LE model): (a) velocity at MP; (b) SPL in the 2cc coupler.
6. Conclusion LE multimode model in the frequency domain has been developed for the simplified BAR. It has
been validated that the developed model is not only almost as effective as the corresponding com-
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bined FE-LE model, but also the efficiency of computation is highly improved. Additionally, the developed model is proved to be more reliable than the LE model by referring to the full FE model. With the developed model, engineering application in the multimode modeling of BAR becomes as simple as with the previous LE model, but as reliable as the combined FE-LE model.
The developed model may be improved by adding the electromagnetic field, the dampings from both mechanical and acoustic fields. And even further, the improvement can be achieved by incor-porating lumped multimode modeling of the acoustic domain. In addition, the developed LE multi-mode modeling method is also applicable for developing other types of transducers.
REFERENCES
1 Hunt, F. V., Electroacoustics: The Analysis of Transduction, and Its Historical Background, Acoustical Society of America, New York, 213-235, (1954, 1982).
2 Madsen, H. S., Small balanced armature receiver for electronic telephone sets, Journal of the Audio En-gineering Society, 19(3), 209-212, (1971).
3 Kampinga, W. R., Bosschaart, C., Wijnant Y. H., and Boer, A. de, The coupling of a hearing aid loud-speaker membrane to visco-thermal air layers, Proceedings of the 14th International Congress on Sound and Vibration, Cairns, Australia, 9-12 July, (2007).
4 Jensen, J., Agerkvist, F. T., and Harte, J. M. Nonlinear time-domain modeling of balanced-armature re-ceivers, Journal of the Audio Engineering Society, 59(3), 91-101, (2011).
5 Kim, N. and Allen, J. B. Two-port network analysis and modeling of a balanced armature receiver, Hearing Research, 301, 156-167, (2013).
6 Tsai, Y. T. and Huang, J. H. A study of nonlinear harmonic distortion in a balanced armature actuator with asymmetrical magnetic flux, Sensors and Actuators A, 203, 324-334, (2013).
7 Jensen, J., Nonlinear Distortion Mechanisms and Efficiency of Balanced-Armature Loudspeakers, PhD thesis, Technical University of Denmark, Denmark, (2014).
8 Bai, M. R., Liu, C. Y., and Chen, R. L. Optimization of microspeaker diaphragm pattern using combined finite element-lumped parameter models, IEEE Transactions on Magnetics, 44(8), 2049-2057, (2008).
9 Nguyen, C. H. and Pietrzko, S. J. Vibroacoustic FE analysis of an adaptive plate with PZT actua-tor/sensor pairs connected to a multiple-mode, electric shunt system, Finite Elements in Analysis and Design, 43(15), 1120-1134, (2007).
10 Sun, W. and Hu, W. X. Integrated finite-element and lumped-element modeling of balanced-armature receiver, Proceedings of the 21st International Congress on Sound and Vibration, Beijing, China, 13-17 July, (2014).
11 Tilmans, H. A. C. Equivalent circuit representation of electromechanical transducers: II. Distributed-parameter systems, Journal of Micromechanicals and Microengineering, 7, 285-309, (1997).
12 He, J. M. and Fu, Z. F. Modal Analysis, Butterworth-Heinemann, London, 94-129, (2001).
13 Moore, R. E. and Cloud, M. J. Computational Functional Analysis, Woodhead Publishing, USA, edn. 2, 24-27, (2007).
14 Xie, N. G. and Song, P. Y. An annotation of modal superposition method of linear osillation, Hydro-Science and Engineering, 1, 52–56, (2003), in Chinese.
15 Hatch, M. R., Vibration Simulation Using MATLAB and ANSYS, Chapman & Hall/CRC, USA, 412-414, (2001).
16 Beranek, L. L. and Mellow, T. J. Acoustics – Sound Fields and Transducers, Elsevier – Academic Press, 14, (2012).
