Full Multiphysics Electro-Vibroacoustic Analysis of a Balanced Arma- ture Transducer Mads Herring Jensen 1 COMSOL Diplomvej 373, DK-2800 Kgs. Lyngby, Denmark

ABSTRACT A balanced armature transducer is a high-performance miniature loudspeaker that is often used in hearing aids but also in other in-ear audio products like earbuds. The electrical and vibroacoustic performance of a balanced armature transducer is analyzed using a full mul- tiphysics analysis. The model includes acoustics and associated thermoviscous losses, structural components such as the diaphragm and the armature (made of magnetic steel), and also the electromagnetic components including the permanent magnet and the coil. The finite element modeling of the transducer is performed in COMSOL Multiphysics and predicts the response of the transducer. 1. INTRODUCTION

The balanced armature transducer or balanced armature speaker (BAS) is an electromagnetic trans- ducer where the movement of the transducer is induced by the Maxwell stresses that exist between magnetized bodies (the magnetic forces). A balanced armature transducer is sometimes referred to as a moving-iron speaker or a receiver. A schematic representation of the transducer is depicted in Figure 1. The armature (or small arm) is placed inside the coil and balanced between two permanent magnets. The vibrating armature (made of a magnetic steel) enters a magnetic circuit with alternating magnetic flux generated by the AC coil (the electromagnet). The AC magnetic field has a strong interaction with the DC magnetic field of the permanent magnet and the pole piece circuit. This causes an alter- nating push-pull force that acts on the armature in the region of the magnetic gap.

Diaphragm Oriveroa/pin Magnet coi

Figure 1: Schematic representation of the balanced armature transducer and its components: (left) a

longitudinal cut in the xz-plane, and (right) a transverse cut in the yz-plane (with one symmetry).

1 mads@comsol.com

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The movement of the armature is transferred to the diaphragm via the driving rod (or pin). The move- ment of the diaphragm is coupled to the air in the back volume and the front volume. The front volume is further coupled to the spout that represents the acoustics connection/port. All these components represent an electro-vibroacoustics multiphysics model. Traditionally, numerical simulation models of the balanced armature transducer have consisted of lumped element models or a combination of lumped element and finite element models (FEM), see [1] and [2]. Here, the transducer is modeled using a full FEM model that couples the electromagnetic fields, the elastic structures, and the acous- tics, see [4]. Only the acoustic load at the spout, that represents the acoustic test setup, is modeled using a lumped transfer matrix representation, see [5]. A typical measurement setup is depicted in Figure 2.

Ear simulator (coupler volume) condition Bolancedsemature Earmeldtube speaker microphone

Figure 2: Typical measurement setup when testing the balanced armature speaker in hearing aid ap-

plications. The speaker spout is connected to a piece of earmold tube, an ear simulator, and a mea-

surement microphone (measurement corresponding to the ear drum). 2. GOVERNING EQUATONS

The electromagnetic field in the air and solids are described with Ampère’s law ∇× 𝐇= 𝐉, 𝐄= −𝜕𝐀/𝜕𝑡, 𝐁= ∇× 𝐀, 𝐉= σ𝐄 (1) where constitutive relations are defined between the B and H fields (magnetization model), the J and E fields (conduction model), and the D and E fields (the dielectric model). For example, in the pole piece a nonlinear H-B curve is used to describe the soft iron properties, in the other domains a linear relation is used. In the coil domain, a homogenized current density J c is prescribed. The dependent variable solved for is the A field. The dynamics in the structural domains are described by Navier’s equation

𝜕 2 𝐮

1

2 [ሺ∇𝐮ሻ 𝑇 + ∇𝐮+ ሺ∇𝐮ሻ 𝑇 ∇𝐮] , (2) where u is the displacement vector,   is the density, C is the stiffness tensor,  is the strain tensor, and F is a possible body load. Note that in the frequency domain the strain tensor is linearized.

𝜌 0

𝜕𝑡 2 = ∇⋅ሺ𝑪: 𝜖ሻ+ 𝐅, 𝜖=

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In the acoustic domain, two different representations are used. For the frequency domain model (see below), the Helmholtz equation is solved and an equivalent fluid model is used to introduce the ther- moviscous boundary layer losses of the slit-like region in the front volume. The losses are introduced through frequency dependent complex valued speed of sound and density expressions. The model used is based on the so-called low reduced frequency (LRF) model, see [6-8]. For the time domain simulation, the equivalent fluid formulation is not possible. In this case, the full set of linearized Navier-Stokes equations are solved, that is the governing equations of thermoviscous acoustics:

𝜕𝜌

𝜕𝑡 + ∇⋅ሺ𝜌 0 𝐯ሻ= 0, 𝜌= 𝜌 0 ሺ𝛽 𝑇 𝑝−𝛼 𝑝 𝑇ሻ

𝜕𝐯

2

𝜕𝑡 = ∇⋅𝜎, 𝜎= −𝑝𝐈+ 𝜇ሺ∇𝐯+ ሺ∇𝐯ሻ 𝑇 ሻ− ሺ

(3)

𝜌 0

3 𝜇−𝜇 𝐵 ሻሺ∇⋅𝐯ሻ𝐈

𝜕𝑇

𝜕𝑝

𝜌 0 𝐶 𝑝

𝜕𝑡 −𝛼 𝑝 𝑇 0

𝜕𝑡 = ∇⋅ሺ𝜅∇𝑇ሻ

where p , v , and T represent the acoustic fluctuations in pressure, velocity, and temperature, respec- tively.  T is isothermal compressibility,  p is the (isobaric) coefficient of thermal expansion,  is the dynamic viscosity,  B is the bulk viscosity,  is the coefficient of thermal conductivity, C p is the (specific) heat capacity at constant pressure, and T 0 is the quiescent temperature. It is assumed that the quiescent reference temperature and pressure are constant. At the fluid-structure interface, the kinematic condition prescribes continuity in the velocity/displace- ment and the dynamic condition prescribes continuity in the traction. The electromagnetic forces acting on the armature are included through volumetric and surface con- tributions from the Maxwell stress. The conditions read:

1

2 ሺ𝐇⨂𝐇ሻ: ∇𝜒

𝐅= 𝐉× 𝐁−

(4)

1

2 𝜇 0 ሺ𝐇⋅𝐇ሻ𝐧+ 𝜇 0 ሺ𝐧⋅𝐇ሻ𝐧

𝜎𝐧= −

where F is the body force in Equation 2,  is the magnetic susceptibility, and  n is the surface traction on a surface with normal n , (see [9] for more details).

2.1. Frequency Domain The frequency domain analysis corresponds to a small-signal type of analysis where the governing equations are linearized. Because the model contains DC fields, that is, the permanent magnetic field and a possible DC deformation of the elastic structures (due to magnetic forces), the equations are linearized around the DC component. This type of analysis is referred to as “Frequency Domain Per- turbation” in the COMSOL Multiphysics software, see [3]. All the AC (perturbation) field including the acoustic fields (which are always perturbation quantities) are assumed to have the time harmonic form: 𝑝ሺ𝒙, 𝑡ሻ= 𝑝Ƹሺ𝐱ሻ exp ሺ𝑖𝜔𝑡ሻ . (5)

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The linearization of the equations also ensures that electromagnetic forces (Equation 4) are handled correctly. They are inherently nonlinear as they are given by product of the dependent variables. As an example, the surface Maxwell stress will in the frequency domain be of the type: −𝜇 0 ሺ𝐇⋅𝐇 𝟎 ሻ𝐧+ 𝜇 0 ሺ𝐧⋅𝐇ሻ𝐧 , (4) where the subscript 0 refers to the DC field.

2.2. Time Domain In the time domain, the full set of governing equations are solved including the nonlinear terms in the strain relation and the electromagnetic forces. For large deformations of the armature and diaphragm, the acoustic equations are solved in a moving mesh setup. The formulation is based on an arbitrary Lagrange-Euler (ALE) type of analysis. This allows the inclusion of effects like nonlinear thin-film damping and nonlinear magnetic force contributions captured using a moving mesh.

Figure 3: Computational mesh used in the FEM simulation of the transducer. 3. THE MODEL

The balanced armature transducer geometry and material properties used in this study are generic. The coil is modeled as copper, the armature uses a soft iron with losses material, the driving rod and diaphragm are modeled as steel, and the armature uses as generic magnetic steel with a relative per- meability of 100. All the material data is taken from the built-in database in COMSOL Multiphysics. Some characteristic dimensions of the transducer are: the transducer box is 2 mm x 2.7 mm x 4 mm, the spout has a diameter of 1 mm Ø, and the diaphragm has a thickness of 50 µm. A voltage of 0.1 V (rms) is applied to the coil which consists of 50 turns of a copper wire of diameter 150 µm Ø. The test setup (see Figure 2) consists of a 10 mm earmold tube with a diameter of 1 mm. The ear simulator is a generic occluded ear-canal simulator that follows the IEC 60318-4 international standard [10]. The system is terminated by a measurement microphone with representative RCL values of. R = 119e6 Ns/m 5 , C = 0.62e-13 m 5 /N, and L = 710 kg/m 4 . In the frequency domain, the test setup is described by a transfer matrix representation that is coupled to the transducer using the “Lumped Port” functionality

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of COMSOL Multiphysics [3]. For time domain simulations, the full geometry of the test setup is mod- eled. The computational mesh used for the FEM simulation is depicted in Figure 3. 4. RESULTS

Selected results are presented in Figures 4 to 7. The DC component of the model magnetic field and displacement (that is used as the linearization point) is depicted in Figure 4. The displacement of the armature and diaphragm is depicted for 100 Hz and 20 kHz in Figure 5. The results show that higher order modes are excited in the diaphragm (the displacement is exaggerated). Figure 6 shows the cor- responding sound pressure level (SPL) distributions at the two frequencies. Finally, in Figure 6 the linear acoustic and electric response of the transducer is depicted. The response or sensitivity curve at the left and the electric input impedance of the coil at the right.

Figure 4: DC component of the magnetic field (left) as and the displacement of the structure (right)

Figure 5: Displacement at 100 Hz (left) and 20 kHz (right).

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Figure 6: Sound pressure level in the transducer at 100 Hz (left) and 20 kHz (right).

Figure 7: Response curve measured in the ear simulator (left) and the electric input impedance to

the coil in the transducer (right). 5. CONCLUSIONS

In this study, a full multiphysics finite element model of a balanced armature transducer has been presented. The model includes most physical effects of interest for the development of this type of transducers. Certain aspects that can be further investigated relate to the thermal management of the transducer, for example, how the permanent magnet and other components behave when heated. 6. REFERENCES

1. Bai, M. R, You, B.-C., & Lo, Y.-Y. Electroacoustic analysis, design, and implementation of a

small balanced armature transducer. Journal of the Acoustical Society of America , 136(5), 2554- 2560 (2014). 2. Jensen, J. Nonlinear Distortion Mechanisms and Efficiency of Miniature Balanced-Armature

Loudspeaker. PhD thesis, DTU Electrical Engineering, 2014. 3. COMSOL Multiphysics 6.0, Acoustics Module, and AC/DC Module, COMSOL Inc., Burlington

MA, USA, 2022. 4. Balanced Armature Transducer, COMSOL tutorial model, https://www.com- sol.com/model/61741 5. Transfer Matrix of a Tube and Coupler Measurement Setup, COMSOL tutorial model,

https://www.comsol.com/model/104291

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6. Kampinga, R., Viscothermal Acoustics Using Finite Elements, Analysis Tools for Engineers,

PhD thesis, University of Tweente, The Netherlands, 2010. 7. Stinson, M. R., The propagation of plane sound waves in narrow and wide circular tubes, and

generalization to uniform tubes of arbitrary cross-sectional shapes Journal of the Acoustical So- ciety of America , 89, 550- (1990). 8. H. Tijdeman, H, On the propagation of sound waves in cylindrical tubes, Journal of Sound and

Vibration , 39, 1- (1975). 9. Modeling Speaker Drivers: Which Coupling to Use, COMSOL Blog (2022). https://www.com-

sol.com/blogs/modeling-speaker-drivers-which-coupling-feature-to-use/ 10. IEC 60318-4, Electroacoustics — Simulators of human head and ear — Part 4:

Occluded-ear simulator for the measurement of earphones coupled to the ear by means of ear inserts, edition 1.0 (2010).

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