Issue 
Acta Acust.
Volume 8, 2024



Article Number  16  
Number of page(s)  13  
Section  Audio Signal Processing and Transducers  
DOI  https://doi.org/10.1051/aacus/2024004  
Published online  22 March 2024 
Scientific Article
Characterization and modeling of the internal vibrations of a balanced armature receiver
^{1}
Department of Electrical and Photonics Engineering, Technical University of Denmark (DTU), Building 352, Ørsteds Pl., Kgs. Lyngby, 2800, Denmark
^{2}
Center of AcousticMechanical Microsystems (CAMM), Building 352, Ørsteds Pl., Kgs. Lyngby, 2800, Denmark
^{3}
Acoustic Technology Group (ACT), Building 352, Ørsteds Pl., Kgs. Lyngby, 2800, Denmark
^{*} Correspoding author: birbak@dtu.dk
Received:
22
September
2023
Accepted:
30
January
2024
The balanced armature receiver is a miniature loudspeaker used in hearing aids. When the receiver produces sound, it also exhibits structural vibrations that propagate as both sound and vibrations within the hearing aid structure. These are picked up by the microphones, resulting in feedback. For the first time, a multimass lumped element model, representing the mechanical vibrations of the balanced armature receiver, is presented in the literature. The model distinguishes the internal, moving components to capture how their movement propagates to the chassis structure. Component mass is assigned to model their movement, while their mechanical couplings are captured by springdamper connections. The (ideally) static components (chassis etc.) are lumped into one mass, whose movement represents the structural vibrations that propagate to the hearing aid shell. The model captures the overall behavior of the mechanical system, which means that with physically sound parameter input, we can predict how parameter changes will impact the response. The thorough treatment of the interplay between the components provides a detailed model response that is completed by parameter properties determined through measurements and analytical methods. Validation against the measured electrical and mechanical response, shows that all main features including resonances are explained by the model.
Key words: Vibrations / Balanced armature receiver / Lumped element model / Laser vibrometry
© The Author(s), Published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The balanced armature receiver is a miniature loudspeaker used in hearing aids and highend hearables, chosen for its small size, power efficiency and highfidelity. One of the challenges faced in the design process of hearing aids is feedback reduction. The close proximity between the speaker unit and microphones in hearing aids means that feedback is an inherent challenge that limits the obtainable gain, cf. Figure 1. The receiver works by an electromagnetic excitation of an armature that is mechanically connected to the membrane, thus determining the vibroacoutic behavior. Since the armature is mechanically fixed to the receiver chassis, its vibrating motion results in structural vibrations of the entire receiver unit. When placed in a hearing aid, these receiver vibrations propagate to the rest of the hearing aid structure, both as sound and vibrations. The microphones can pick up these vibrations creating a feedback loop, experienced as a squealing sound by the user. The vibration pattern of the receiver chassis is governed by the internal vibrations and subsequently impacted by the boundary conditions posed by the hearing aid. To capture the resulting vibration pattern, we must start by characterizing the internal vibrations under fixed boundary conditions to have a welldefined system to capture.
Figure 1 Illustration of a generic hearing aid, where the receiver is placed in the hearing aid shell that sits behind the ear. The feedback loop (green) highlights the receiver vibrations that are picked up by the microphone (purple) and sent back to receiver to be reproduced as sound. Another common configuration is to place the receiver in the “ear dome” that sits in the ear canal. While feedback is reduced by this configuration, it is not eliminated. Here, the vibrations propagate via the sound tube (dotdashed) and back to the hearing aid shell (dashed) and microphones. 
Previous academic work addressing structural vibrations of the receiver is limited. The work includes two works by Friis [1] and Varanda [2], respectively. Varanda [2] addresses the rigid body dynamics of the receiver using Newtonian mechanics in the LagrangeHamilton formalism and experimental measurements. The vibration pattern of freebody motion has been found to be dominated by translational movement normal to the membrane and rotation about the armature axis of rotation. The earlier work by Friis examines the vibroacoustic mechanisms responsible for hearing aid feedback using the theory of fuzzy structures. Work by Sun and Hu [3, 4] includes hybrid methods, using an integrated Finite ElementLumped Element method as well as modal analysis to capture the higher order modes. There, the receiver is treated as a single massspringdamper system.
We surpass the detail level of existing literature by presenting an equivalent circuit model that distinguishes the internal, moving masses of the receiver with element properties determined by measurements. This means that we have lumped the ideally static components and attributed mass to the moving components of nonnegligible mass. This allows us to predict the movement of the individual components inside the receiver and capture how much it propagates to the receiver chassis, resulting in structural vibrations. The masses are connected by springs and dampers to capture their internal couplings, such as the pin connection between membrane and armature. Comparison with our measurements of the electrical impedance and the membrane velocity shows that the model captures the main features of the response, including all main resonances
As the working principle and general topology is preserved across different types of receivers, across different manufacturers, the model is generally applicable to other types of single receivers. The model can also be expanded to model dual receivers, which consists of two single receivers oriented to work in antiphase and thus produce a higher output, while minimizing the vibrations. Variation in component material, shape and size is captured in the calculations of the model parameters. This means that the components and their relative connections in the circuit are generally applicable to any type of balanced armature receiver, but also capture more distinct geometries through the parameter calculations.
Used in conjunction with experimental measurements, this lumped element model provides an intuitive and relatively simple framework for understanding the vibration pattern of the receiver. Within the hearing aid industry, there is a strong interest in computationally light models that are able to predict the vibrations of the receiver. A physics based model maintains a true picture of the parameter couplings, which means it can be used to investigate how changing one parameter affects the others and the output. This paper presents a physics based, mechanical lumped element model of the translational motion of the receiver. All model parameters, except losses, are analytically estimated and adjusted according to experimental measurements.
2 Materials and methods
The working principles of the balanced armature receiver rely on the coupling between the electrical, magnetic, mechanical and acoustic domain. We begin by presenting the necessary background knowledge for the system with a description of the receiver construction and working principles. In Section 2.2, we present the lumped element model of the mechanical system.
2.1 Construction and working principle
The receiver is an electromagnetic transducer, which implies a fixed coil position that does not follow the dynamic mass of the system. This makes the receiver highly responsive. A generalized receiver unit is depicted in Figure 2. The system consists of an armature (green) balanced between a set of permanent magnets (yellow). A magnet housing (dark grey), also called the stack, shortcircuits the flux path in the magnetic circuit to reduce the flux in the armature and thereby limit the nonlinear effects. The armature is free to move within the coil (orange), which is held suspended by a glued, mechanical attachment to the magnet housing and chassis (light grey).
Figure 2 Schematic representation of the balanced armature receiver. The receiver type shown here features an Eshaped armature. The static arm closest to the viewer has been removed from the illustration to make the coil visible. 
An alternating current in the coil induces a magnetic field along the length of the armature (Faraday’s/Lenz’ law). The resulting net flux in the armature unbalances the magnetic potentials in the air gaps, creating a force on the armature that deflects it according to the polarity of the AC signal (Lorentz force law), cf. Figure 3. The motion of the armature is propagated to the membrane (blue) via a drive pin connection (white).
Figure 3 Flux path in simplified receiver geometry for zero (left) and nonzero (right) input current. Geometry is exaggerated in the interest of clarity. 
The membrane is not merely a flat plate, but bevelled at the center, to add stiffness to the structure to make it move as a stiff plate, cf. Figure 4 (right). A metal ring is mounted along the walls of the chassis and holds the Mylar foil suspended. The membrane is adhered to the suspension and mechanically “hinged” by virtue of being placed flush with the metal ring at the short axis opposite the drive point.
Figure 4 Disassembled receiver unit illustrating the membrane shape and internal structure. The outer chassis dimensions are (7.9 × 4.1 × 2.8) mm^{3} in length, width and height, respectively. (Left) shows the back volume of the receiver with the coil, stack, armature and pin. (Right) shows the front volume with the membrane and suspension. The membrane area between the slits, we denote the membrane hinge. 
The receiver type investigated in this paper is the Knowles TEC30087 single receiver [5]. To investigate the internal vibrations, we made a special receiver unit by mechanically removing the top lid to expose the membrane, cf. Figure 4 (right). The resulting sample features an open front volume to expose the membrane at the pin connection point, while the back volume remains closed. The mechanical alteration of the receiver structure introduces a risk of creating cracks in the chassis and the suspension foil. This means that there could be leaks in the back volume that impact the response in ways we did not intend. In the Knowles TEC receiver, two parallel slits run on either side of the pin point, creating a hinge for the membrane, as seen in Figure 4 (right). This will be accounted for in the Lumped Element Model (LEM) presented in Section 2.2.
2.2 Mechanical lumped element model
To model the internal movements of the receiver components, we separate the dynamic components, i.e. intentionally moving, from the ideally static components [2]. The translational mechanical behavior of the receiver is thus modelled through five rigid masses, connected by springs and dampers, cf. Figure 5. We have lumped together the ideally “static” masses that is the chassis, coil, magnets and stack, along with the mechanically fixed parts of the armature; the static arms. For short, we denote this mass M_{c} as it is ultimately the movement of the chassis that we are interested in. The dynamic masses are the moving arm of the armature M_{a}, the membrane M_{m} and the membrane hinge M_{h}, while the pin and suspension are only represented as springdamper connections due to their negligible mass. As shown in Figure 4 the membrane of the Knowles TEC30087 receiver has two slits running on either side of the pin connection point, creating a mechanical crossover filter that takes over for higher frequencies. We denote this part of the membrane, the hinge M_{h}, due to the way it dynamically couples the armature and membrane movement. As the hinge is partially decoupled from the remaining membrane, we model it as a separate mass. In the experimental characterization, it is the hinge response we measure. Not all receiver types have this division of the membrane. In that case the hinge springdamper connection is stiff and lossless. The relative component connections inside the receiver are the same across different receiver types, making this model generally applicable across different single receiver designs. The final mass M_{b} is not part of the receiver, but represents the experimental boundary conditions. The masses are connected by springdamper connections, representing the compliance and associated losses of the hinge, drive pin, armature, suspension and boundary, respectively. A Mylar foil suspension covers the entire backside of the membrane and seals the back and front volume from each other.
Figure 5 Mechanical sketch of the equivalent electrical circuit in Figure 6, consisting of five masses M connected by springdamper connections represented by C and R respectively. The subscripts a, b, c, h, m, p, s denote the armature, boundary, chassis, hinge, membrane, pin and suspension, respectively. 
The mechanical sketch in Figure 5 is translated into an equivalent electrical circuit, cf. Figure 6. A simple, electrical RLcircuit generates the input current that controls the input force on the armature, related by a constant transduction factor. A controlled voltage source V_{b} of opposite polarity in the RLcircuit accounts for the mechanically induced reduction of coil current. The alternating current and the subsequent mechanical motion of the armature, results in changes to the flux through the coil.
Figure 6 (Left) Equivalent circuit derived from the mechanical sketch in Figure 5, using the impedance analogy. The subscripts a, b, c, h, m, p, s of the mechanical circuit parameters denote the armature, boundary, chassis, hinge, membrane, pin and suspension, respectively. The subscripts c, b of the electrical circuit parameters refer to the coil and backemf, respectively. (Right) For low frequencies the kinetic energy of the system will primarily follow the yellow. 
Through Kirchhoff’s equations, we can derive expressions for the total electrical impedance Z_{TE} and the component velocities. The effect of the coupling between the electrical and mechanical domain is included by the coupling of the blocked impedance of the respective domains linked by a transduction factor T [6],
$${Z}_{\mathrm{TE}}={Z}_{E}+\frac{{T}^{2}}{{Z}_{M}+{Z}_{L}}.$$(1)
The mechanical impedance due to the acoustic load Z_{L} is accounted for through simple additions to the mechanical circuit elements, i.e. the membrane mass, suspension compliance and suspension damping. The acoustic load on the membrane corresponds to having an added mass on top of the membrane. We can account for this by adding another mass in series with the membrane mass, which is equivalent to increasing the parameter M_{m}. The compliance of the back volume can be accounted for by adding another compliance in parallel with the suspension compliance C_{s}. An estimate of the mechanical back volume compliance is found using the relation S^{2}V/(c^{2}ρ_{0}), where S is the membrane area and the acoustic compliance given in terms of the volume V, the speed of sound c and the density of air ρ_{0} [7]. This yields 1.7 × 10^{−13} m/N. The stiffness of the back volume thus decreases our estimate of the suspension compliance, which is found empirically by fitting our model to the measured response, cf. Section 3. The effect of acoustic damping can be added as a parallel resistor to the suspension damping R_{s}.
The blocked mechanical impedance Z_{M} is given by the system of equations that govern the circuit in Figure 6. The frequency dependent reduction of the impedance of the inductor is accounted for in the blocked electrical impedance ${Z}_{e}={R}_{c}+(\mathrm{j\omega}{L}_{c}{)}^{{n}_{L}}$ through the factor n_{L} that usually ranges between 0.5 and 0.7 [8]. The parameters R_{c} and L_{c} are the coil resistance and inductance, respectively. The component velocities u_{a}, u_{c}, u_{m}, u_{h} are given by the system of equations that govern the loop flow of kinetic energy in the circuit, Figure 6.
2.3 Model response
The initial parameter estimates were based on mechanical calculations of the isolated components. To capture the measured behavior, we need to analyze how the components work in the final system that is the receiver. The purpose of this analysis is to optimize the parameter estimates based on an understanding of the physical system. To do this successfully, we need to keep in mind the response impact of the simplifying assumptions we make. The parameter estimates presented in this study are single values, thus leaving out their frequency dependency. We begin in the electrical domain by following the current flow in the electrical circuit in Figure 6 (left). The applied input voltage V_{i} induces a current in the coil i_{c}, whose DC resistance R_{c} influences the starting point of the response curves, creating an offset that lasts until the second resonance peak. The coil inductance L_{c} determines the slope of the electrical impedance as well as the amplitude of the transfer function past the first resonance. A higher inductance yields a higher electrical impedance, which ties to a lower mechanical output. The coil inductance is a sensitive parameter and its magnitude is decreased by the backemf due to the mechanical motion of the armature. As the armature is magnetized by the coil induced alternating magnetic field, its vibrating motion also implies that the magnetic field at each spatial point is changing in time. As predicted by Faraday’s law, this creates surface currents in the soft magnetic material, often denoted eddy currents. According to Lenz’ law, the eddy currents create magnetic fields themselves in a direction that counteracts changes to the magnetic field that caused them, i.e. that of the coil. Effectively, this reduces the coil induction at higher frequencies. In the model this is accounted for by the eddy current exponent n_{L}.
The mechanical input force F on the armature is estimated by a transduction factor T that scales the coil current and controls the backemf to the electrical circuit. For this study, we have empirically estimated the transduction factor to be T = 3 by manually fitting the model to measurements, but in reality it is a function of armature displacement [9]. The compliance C_{a} and damping R_{a} of the armature creates a voltage drop that lowers the effective force on the armature. From Newton’s third law, the action force on the armature causes an equal, opposite reaction force on the chassis. The reaction force of the chassis acts on the armature with the result of decelerating the armature velocity, hence u_{i }= u_{a} − u_{c}. Past the first resonance frequency, the armature M_{a} and hinge M_{h} begin move independently, as the impedance of the pin compliance decreases and leads the flow away from M_{h} and back to the controlled source, F. For higher frequencies, the impedance of the hinge compliance C_{h} decreases. This means that part of the kinetic energy is then drawn through C_{h} and R_{h} and back to the source, effectively decreasing the velocity of the membrane M_{m}. The hinge thus works as a mechanical cross over filter that takes over for the high frequencies, as seen in Figure 7 where the lumped element velocity response is shown for the armature, hinge and membrane. At node N5 the kinetic energy of the membrane is partly transferred to the boundary and chassis, while the remaining is transferred to the suspension. For increasing frequency the armature deflection and impedance of the suspension compliance C_{s} decreases. This results in less kinetic energy being drawn to the chassis M_{c} and boundary M_{b}.
Figure 7 Velocity response of the lumped element model in Figure 6 for the armature, membrane and hinge. The input voltage is set to 1 V to get the voltage to velocity transfer function, which makes the response comparable to the measured transfer functions. After the first resonance, the phase shifts between the components create constructive interference, such that the membrane velocity exceeds that of the armature. 
We compare our multimass lumped element model to a simplified twomass model in Figure 8. The mechanical twomass model is obtained by removing the branches that represent the chassis and the boundary, as well as the hinge components in Figure 6. In the simplified circuit the first resonance peak is still primarily governed by the armature compliance. The second resonance peak is now primarily governed by the pin compliance. Therefore, we assign the hinge parameter values to the pin in the response seen in Figure 8. The comparison shows that while the behavior of the total membrane is quite similar, the behavior hinge response captured by the multimass model follows the measured response more accurately. The comparison shows that the behavior of the membrane is somewhat different, especially after the first resonance where the hinge starts to move independently.
Figure 8 (Left) Simple 2mass circuit. The topology of the multimass circuit in Figure 6 is preserved for easy comparison. (Right) The velocity transfer of the multimass lumped element model in Figure 6 is compared to the response of a simplified 2mass lumped element model of the receiver. 
2.4 Parameter analysis
While detailed modeling of the individual model parameters is beyond the scope of this paper, we will now make a qualitative discussion of how the components behave and the effects on the model parameters. The most sensitive parameters in the lumped element model were found to be the mass of the armature M_{a} and the membrane M_{m}, the compliance of the armature C_{a} and hinge C_{h}, as well as the inductance L_{c} and the associated eddy current exponent n_{L}.
2.4.1 Armature
The estimated mass of the armature listed in Table 1 is based on the full length of the armature. In reality and especially at higher frequencies it is mostly the tip of arm that moves and so the effective mass is reduced. In the lumped element model, reducing the armature mass moves both resonance peaks up in frequency and increases the amplitude of the second resonance. A reduction in the effective armature length will also reduce the armature compliance. For the receiver used in this study, the armature width tapers towards the armature tip, in the section that lies between the permanent magnets. We presume that this is to keep the mechanical compliance of the armature fairly constant and decrease nonlinear effects. In the mechanical parameter estimate we also assumed a point force. In reality, the armature is subject to a nonuniform, distributed force. In addition to the frequency dependency of the mechanical stiffness, the armature compliance is enhanced by a magnetic stiffness compensation that results in a nonlinear relationship between the magnetically induced force on the armature and the armature displacement from the balanced position [9, 10]. In the model response, the armature compliance has a high impact on the position of the first resonance peak. Based on these reflections, the parameter M_{a} is decreased and C_{a} is increased to obtain a more physically accurate model response.
The membrane hinge was too light to be detected by the scale. An estimate was therefore made based on the size ratio between membrane and hinge according to calculated mass of 16.8. The primary discrepancy between calculated and weighed values is found in the mass of the membrane.
In addition to variations in the effective length and consequently mass, the armature is also subject to higher order modes [11], which breaks down the assumption of a springboard motion. An interesting study for future work, would be to investigate the armature modes and frequency dependency of the mass and compliance, using Finite Element modeling.
2.4.2 Membrane, hinge and suspension
The membrane mass is another sensitive parameter and its exact value is difficult to estimate accurately. The thin structure makes it hard to estimate the exact mass. In addition, this parameter is impacted by the acoustic boundary conditions. The membrane hinge mass is very small compared to the whole membrane and is therefore not a sensitive parameter in the model. The hinge compliance C_{h} impacts the position of the second resonance peak. To estimate C_{h} the hinge was also initially treated as a bending cantilever beam that was fixed in one end. In reality, the hinge geometry is quite thin and short and its motion is constrained by the suspension foil. Considering these complications, the accuracy of this method may be limited. It is therefore reasonable to let this parameter vary quite a bit, when fitting the model response to our experimental measurements.
2.5 Measurement setup
The experimental measurements were performed on a Knowles TEC30087 with an open front volume that exposes entire membrane, cf. Figure 4 (right). The back volume of the sample is closed. The electrical impedance and velocity response were measured under fixed boundary conditions. The receiver was fixed to the edge of an iron plate of 1.6 kg using doublesided tape (Advance, Leicester England). The iron plate, which is denoted M_{b} in the lumped model, was placed on a compliant foam to isolate the system from external vibrations.
The electrical impedance was measured with an impedance analyzer (MFIA, Zurich Instruments) in four terminal mode measuring both voltage and current for increased accuracy. The velocity response was measured with a scanning laser vibrometer (PSV400, Polytec). To focus the laser beam onto the pin point connection of the membrane, we used a closeup lens with 4x magnification and a working distance of 3 cm. To investigate the vibroacoustic coupling of the receiver, additional velocity measurements were performed in a vacuum chamber (Gugliotta). These measurements were performed with a point laser (OFV353, Polytec).
2.6 Initial parameter estimates
The receiver components form a coupled system, which means that changes in one parameter will accompany changes to other parameters as well. To create a robust and physically sound model, these mutual parameter dependencies must be controlled by the model structure. Therefore, for the initial parameter estimates, each component is treated a as a separate entity and based on purely mechanical considerations. To estimate the mass and compliance parameters, we extract the internal receiver dimensions from CT scans, cf. Table 2 and identify the component materials and corresponding properties, cf. Table 3. The values are listed in Tables 1, 4 and 5. Further qualitative analysis of the mutual impacts from domain couplings and experimental boundary conditions will form the basis for subsequent adjustment of the circuit parameters, when matching the measured and modelled system response, cf. Section 3.
Component materials and associated values used to calculate the initial parameter estimates. The values for nickel Ni are the mean values of maximum and minimum values provided by the citation, assuming a uniform distribution.
2.6.1 Mass
The mass parameters are found using two different methods. First, we weigh each part by sequentially adding them on the scale (LPC313, VWR) to reduce measurement uncertainties. Then, we calculate the masses from dimensions and material densities listed in Table 3. Each method is associated with uncertainties, but the level of deviations provide an indicator of the accuracy of the estimates, while yielding the order of magnitude and relative scaling between each component. The most difficult component to estimate is the membrane, because its thin and lightweight structure adds uncertainty to both mass and dimension measurements. The membrane was weighed with the suspension foil still attached. We left the foil on, because it adds to the effective weight of the membrane in use. From the CT scans it is not possible to see the suspension foil. Therefore, we use the weighted estimate as our initial parameter input. The membrane hinge was too light to be detected by the scale, and was therefore calculated by scaling the membrane mass to the corresponding surface area of the hinge. For the armature and chassis, the results of both methods are in agreement. The sample used for the measurements presented in Section 2.5 has a lower chassis weight due to the removed top cover. However, since the measurements are performed for fixed boundary conditions, this difference in chassis weight has no practical influence.
2.6.2 Compliance
The compliance parameters are estimated by treating the armature and membrane hinge as bending cantilevers. The large membrane structure is bevelled at the center, presumably to add stiffness to the structure. Therefore, we assume that it moves as a stiff plate and attribute no compliance to it. The armature deflection δ that translate into membrane motion is at the drive point of action. The input distributed input force is simplified to a single point load. The deflection of the armature at a distance x from the pivot for an input force F at position a is
$$C=\frac{\delta}{F}=\frac{{a}^{2}}{6\mathrm{EI}}\left(3xa\right)\mathrm{at}\hspace{0.5em}a\le x\le L$$(2)
where E is Young’s modulus and I = bh^{3}/12 is the area moment of inertia, where b is the base width and h is the height of the cantilever crosssection. For the membrane, the input force position a is at the position of the drive pin and we are interested in the deflection at the same point. For the armature, the input force is generated by the effective magnetic field and is therefore not a point force, albeit in the lumped element model it is treated as such. The armature compliance will also vary with respect to the effective length and width. We expect the effective length to decrease for increasing frequency. The effective width also decreases as the effective length decreases, because the armature shape tapers towards the tip in this receiver model. This may contribute to a less frequency dependent mechanical compliance. However, we still expect the mechanical stiffness to decrease with increasing frequency. For the sake of simplicity, we neglect the tapered shape and approximate the armature as a uniform beam. The drive pin compliance is estimated using the simple relation between Young’s modulus and a straight rod: C = 2l/(EA), where l and A are the length and crosssectional area, respectively.
2.6.3 Damping
The primary mechanical damping parameter is expected to originate from the Mylar foil suspension. Therefore the remaining damping parameters are initially set to a very low value, cf. Table 4. The suspension damping is estimated by how much the membrane movement is damped. For a massspringdamper system, the damping can be expressed by the following,
$$R=\zeta 2\sqrt{M/C}=\zeta {c}_{c}$$(3)
where M is the mass, C is the compliance, c_{c} is the critical damping coefficient, ζ = (2Q)^{−1} is the damping ratio. The quality factor Q represents the rate of damping relative to the oscillation, i.e. high Q factors indicate slowly damped system. We find the Qfactor from the first resonance peak of the velocity measurement in air, presented in Section 3 and Figure 9. At the first resonance peak, the armature and membrane move with the same velocity [16]. This means that the total mass to displace is M_{a }+ M_{m} and the spring compliance armature compliance, C_{a}.
Figure 9 Measured velocity transfer function at the hinge in air and vacuum, respectively. The resonance frequencies are (2 kHz, 6.5 kHz) for air (blue, solid) and (2.1 kHz, 6.3 kHz) for vacuum (gray, dashed). 
2.6.4 Electrical
The coil resistance R_{e} = 50 Ω is based on the initial value of the measured electrical impedance in Figure 10, and matches quite well with the data sheet value of 46 Ω ± 10% [5]. The initial estimate of inductance was based on the data sheet value of 10 mH ±15%. The initial estimates are listed in Table 5.
Figure 10 Measured total electrical impedance. The resonance frequencies are (1.6 kHz, 6.3 kHz) and the antiresonance (6.6 kHz). 
3 Results
3.1 Experimental results
The receiver was characterized through measurements of the electrical impedance and the mechanical velocity of the membrane hinge, as described in Section 2.5.
3.1.1 Vibroacoustic coupling
Since our circuit in Figure 6 does not include the acoustic domain yet, we have investigated the impact of acoustics on the measurements. The measured transfer functions are shown in Figure 9. The transfer functions relate the membrane hinge velocity to the input voltage, making the measurement independent of the signal type. The main impact is on the damping, while the resonance frequencies remain fairly unchanged. This verifies that it is physically sound to fit our mechanical lumped model to measurements, keeping in mind that the added acoustic load will impact the damping of the system and increase the effective mass of the membrane and membrane hinge. The difference between the curves in Figure 9 is likely the effect of a reduction of electrical impedance, caused by the fact that the membrane is easier to move. The impact of the acoustic radiation impedance of the membrane is estimated by calculating the associated acoustic mass, which will effectively add to the membrane mass and suspension damping. An estimate of the acoustic mass is found using the expression for a parallel acoustic mass and resistor, ${M}_{A}=\frac{8{\rho}_{0}}{3{\pi}^{2}{r}_{p}}$, where ρ_{0} is the density of air and r_{p} is the radius of a circular piston in an infinite baffle^{1}, yielding an area corresponding to that of the rectangular receiver membrane [7]. For the receiver membrane M_{A} = 132 kg/m^{4} and the equivalent mechanical mass is then ${M}_{A}(\pi {r}_{p}^{2}{)}^{2}=0.4\times 1{0}^{7}$ kg. Compared to the overall membrane mass estimate and its uncertainty, the added mass of the acoustic load is quite small for the open membrane configuration.
3.1.2 Electrical impedance
The electrical domain of the receiver is characterized by measuring the total electrical impedance. The mechanical resonances of the system are visible in the electrical response, as a high electrical impedance results in less mechanical movement and vice versa. The electrical domain can thus provide information about the mechanical resonances of a receiver with a closed chassis. To investigate the impact of having a completely exposed membrane, yet closed back volume, we have also measured the electrical impedance of a standard receiver with a closed chassis, cf. Figure 10. The open front volume decreases the acoustic load on membrane, making the membrane easier to move. In the closed receiver, the acoustic load on the membrane will correspond to having an added mass on top of the membrane, thus increasing the effective membrane mass in the mechanical circuit.
The first resonance peak is moved down in frequency with respect to the closed sample. This suggests that the back volume has become more compliant. To investigate this we have studied the sample with a microscope (DinoLite) and found that one of the chassis walls has disconnected from the mounting ring of the suspension. The result is a leak in the back volume, measuring 1.7 mm long and 0.1 mm at the widest point. Mechanically, the increased back volume compliance is equivalent to an increased suspension compliance, as it makes the membrane easier to move. In addition to the shift in frequency, the first resonance is significantly damped. Considering the leak in the back volume, it is possible that there are other leaks or types of damage to the receiver that has impaired the mechanical system. It is possible that the pin connection has partially detached from the membrane and is no longer be stiff. This would introduce a lot of damping as we see in the first resonance peak. The second resonance peak, which is primarily governed by the membrane/hinge, is moved up in frequency by approximately 20% and has a slightly larger band width and smaller amplitude, compared to the closed chassis response. This suggests that the hinge compliance is been reduced, which could be tied to a poor pin connection.
3.2 Parameter fitting
Based on the reflections presented in Section 2.4, we have adjusted the parameter estimates to account for their mutual couplings and measurement boundary conditions, cf. Tables 6 and 7. Based on the reflections stated in Section 2.4, the values are manually fitted to the measured response of the receiver. Modeling the frequency dependency and couplings between parameters is an extensive study in itself that is beyond the scope of this paper. Nevertheless, it is certainly an important expansion of the model work presented here that will add to the completeness of the predictive, physics based model.
Adjusted mechanical parameters.
The electrical parameters of the coil remain within the data sheet values. However, the Eddy current loss factor n_{L} is closer to one than expected, which means that the electrical impedance is less reduced at higher frequencies. For the mechanical parameters, we have adjusted the mass of the armature, as well as the compliance of the armature and hinge. To account for the reduced effective length of the armature as well as the magnetic stiffness compensation, the estimated armature mass has been reduced by half, while its compliance has increased by a factor of 5. Changing these values, also changed the calculation for the expected damping that we attribute to the suspension R_{s}, reducing the value by 27%. Due to the minimal acoustic load, the membrane and hinge mass have not been changed. However, because of the anticipated inaccuracy of the method used to estimate the hinge compliance, we have increased this parameter by a factor of 3.
The model response plotted for the values listed in Tables 6 and 7 is presented with error bars that show the spread of the output. The spread is calculated from a data set of the model response obtained for all possible combinations of decreasing or increasing selected parameters by a designated factor. In the following we will show the result of three cases: 1) Variation of the loss parameters; 2) Variation of M_{a}, M_{m}, C_{a} and C_{h}; and 3) Variation the electrical parameters.
3.2.1 Case 1
The loss parameters are associated with a high degree of uncertainty. Damping is not only a material property, but also depends on the geometry of the components as well as the mechanical connections between them. While the drive pin is designed to be a very stiff connection we know that bending of the pin, friction at the attachment points and uncertainty in placement as well as geometry of the pin can all introduce losses. It is thus very difficult to evaluate the damping in the pin joint. In contrast, the thin Mylar foil suspension is designed to be very soft. It has a complicated geometry and can easily be strained, making the associated losses highly uncertain. The small geometries of the system components increases the impact of fabrication tolerances of the different components. A rigorous error analysis will therefore provide important knowledge on which receiver components impact the receiver output the most. However, this is outside the scope of this paper.
Since the losses are difficult to estimate, we have made an educated, qualitative guess for the parameter variances. In Figure 11, we have let the armature and hinge loss vary by ±20%, the pin loss by ±90% and the suspension loss by only ±10%, as this is the most sensitive loss parameter. While the pin loss is a highly uncertain parameter, it is not very sensitive and therefore we let it vary the most. The loss associated with the bending action of armature and hinge are still expected to be the least dominant. The resulting effect is mainly seen at higher frequencies and the dip of the second resonance in particular seems to be captured.
Figure 11 Case 1) Comparison of measurement to model response with error bars, voltage to velocity transfer function (left) and electrical impedance (right), associated with varying the loss parameters by ±20%, ±10% and ±90% for the armature/hinge, suspension and pin, respectively. 
3.2.2 Case 2
The mass and compliance of the armature and membrane/hinge are some of the most sensitive parameters of the system. Furthermore, we know that frequency dependency, geometry uncertainties and compliance approximations introduce uncertainty to these static input parameters. This makes these parameters an interesting subject for further model refinement, which we will cover in future work. The compliance of the armature and hinge largely controls the resonance frequency of the first and second peak, respectively.
The armature and membrane mass affect the positions of both resonance peaks and the amplitude of the second resonance peak in particular. Since the model and measurement response deviate the most at higher frequencies, and the hinge parameters primarily affect the higher frequency range of the response, we let the hinge parameters vary the most. In Figure 12 we see the result of letting the parameters M_{a} and C_{a} are vary by ±20% while M_{m} and C_{h} is varied by ±50%. This is the case that brought the model results closest to the measured response.
Figure 12 Case 2) Comparison of measurement to model response with error bars, voltage to velocity transfer function (left) and electrical impedance (right), associated with varying the mass and compliance parameters for the armature and membrane by ±20% and ±50%, respectively. 
3.2.3 Case 3
Lastly, we have calculated the model response for a series of parameter sets, where the electrical parameters are varied by ±10% with all permutations. The result is shown in Figure 13. The effect is particularly strong in the electrical impedance, but does not compensate the difference between the measured and the modeled transfer function, where the discrepancy between model and measurement is the most prominent.
Figure 13 Case 3) Comparison of measurement to model response with error bars, voltage to velocity transfer function (left) and electrical impedance (right), associated with varying the electrical parameters by ±10%. 
3.2.4 Comparison
These three cases show that the uncertainty of the mechanical mass and compliance of the two main dynamic components have the most widespread impact on both the electrical and mechanical response along with the losses at higher frequencies. Meanwhile, fairly small variations in the electrical parameters result in a fairly large spread at higher frequencies and mostly in the electrical domain. Thus, the uncertainties in the electrical parameters mostly offsets the response at higher frequencies, but does not affect the shape of the curves.
It is important to note that it is the permutations in the parameter sets that make the spread roughly capture the measured response. This means that readjusting the input parameter set will not make it possible to change the output response to fit the measurements. While rigorous modeling of the frequency dependent behavior of the masses and compliances is beyond the scope of this paper, it will enhance the match between the model and the measurements. Finite element modeling is a good tool for such investigations.
4 Conclusion
Our multimass mechanical lumped element model surpass the detail level of existing models and captures the translation normal to the receiver membrane. In the model we lump the ideally static masses, i.e. the chassis, stack, permanent magnets and coil, while assigning mass to each of the intentionally moving components. The components are connected by springs and dampers to represent their mechanical couplings, creating a physically true representation of how the internally moving components impact the movement of the receiver structure, i.e. the chassis, stack, coil and permanent magnets. The model parameters are estimated through measurement data and simple analytical expressions. The model is generally applicable to most common types of single balanced armature receivers. This is also demonstrated in the comparison with the simplified twomass circuit. By twoway comparison with the measured electrical and mechanical response we have demonstrated that the lumped element model is able to capture the overall behavior of the device, including all main resonances. The model even worked satisfactory for a mildly damaged sample. The damage was sustained during mechanical alteration to open the front volume. This created a leakage in the back volume and possibly internal damage as well, which resulted in a deviating response. The results demonstrate the robustness of the model and highlight its applicability for any kind of receiver design. A very good match between model and velocity measurement has already been obtained with a different sample not presented in this paper, equipped with holes in the chassis to expose the membrane and armature at the pin connection. From the parameter fitting error investigation, we have found that the effective mass and compliance of the armature in particular, is important for an accurate model response. To obtain an exact match between the measurements and model response with one set of single value parameters, we are doing further work to refine the model. This work includes rigorous error propagation analysis, as well as detailed modeling of the input parameters to capture their frequency dependency, caused by effects such as higher order bending modes and magnetic saturation, hysteresis and stray fields.
Acknowledgments
The authors would like to thank Sina BaierStegmaier from the department of Physics at the Technical University of Denmark, for performing the CT scans used for this study. In addition we would like to thank Yu Luan for facilitating the vacuum measurements at GN Resound in Ballerup, Denmark. Finally, we would like to thank Knowles for providing the sample used for this study and for supporting discussions with Thomas Jensen.
Conflict of interest
The authors declare that they have no conflicts of interest in relation to this article.
Data availability statement
Data are available on request from the authors.
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All Tables
The membrane hinge was too light to be detected by the scale. An estimate was therefore made based on the size ratio between membrane and hinge according to calculated mass of 16.8. The primary discrepancy between calculated and weighed values is found in the mass of the membrane.
Component materials and associated values used to calculate the initial parameter estimates. The values for nickel Ni are the mean values of maximum and minimum values provided by the citation, assuming a uniform distribution.
All Figures
Figure 1 Illustration of a generic hearing aid, where the receiver is placed in the hearing aid shell that sits behind the ear. The feedback loop (green) highlights the receiver vibrations that are picked up by the microphone (purple) and sent back to receiver to be reproduced as sound. Another common configuration is to place the receiver in the “ear dome” that sits in the ear canal. While feedback is reduced by this configuration, it is not eliminated. Here, the vibrations propagate via the sound tube (dotdashed) and back to the hearing aid shell (dashed) and microphones. 

In the text 
Figure 2 Schematic representation of the balanced armature receiver. The receiver type shown here features an Eshaped armature. The static arm closest to the viewer has been removed from the illustration to make the coil visible. 

In the text 
Figure 3 Flux path in simplified receiver geometry for zero (left) and nonzero (right) input current. Geometry is exaggerated in the interest of clarity. 

In the text 
Figure 4 Disassembled receiver unit illustrating the membrane shape and internal structure. The outer chassis dimensions are (7.9 × 4.1 × 2.8) mm^{3} in length, width and height, respectively. (Left) shows the back volume of the receiver with the coil, stack, armature and pin. (Right) shows the front volume with the membrane and suspension. The membrane area between the slits, we denote the membrane hinge. 

In the text 
Figure 5 Mechanical sketch of the equivalent electrical circuit in Figure 6, consisting of five masses M connected by springdamper connections represented by C and R respectively. The subscripts a, b, c, h, m, p, s denote the armature, boundary, chassis, hinge, membrane, pin and suspension, respectively. 

In the text 
Figure 6 (Left) Equivalent circuit derived from the mechanical sketch in Figure 5, using the impedance analogy. The subscripts a, b, c, h, m, p, s of the mechanical circuit parameters denote the armature, boundary, chassis, hinge, membrane, pin and suspension, respectively. The subscripts c, b of the electrical circuit parameters refer to the coil and backemf, respectively. (Right) For low frequencies the kinetic energy of the system will primarily follow the yellow. 

In the text 
Figure 7 Velocity response of the lumped element model in Figure 6 for the armature, membrane and hinge. The input voltage is set to 1 V to get the voltage to velocity transfer function, which makes the response comparable to the measured transfer functions. After the first resonance, the phase shifts between the components create constructive interference, such that the membrane velocity exceeds that of the armature. 

In the text 
Figure 8 (Left) Simple 2mass circuit. The topology of the multimass circuit in Figure 6 is preserved for easy comparison. (Right) The velocity transfer of the multimass lumped element model in Figure 6 is compared to the response of a simplified 2mass lumped element model of the receiver. 

In the text 
Figure 9 Measured velocity transfer function at the hinge in air and vacuum, respectively. The resonance frequencies are (2 kHz, 6.5 kHz) for air (blue, solid) and (2.1 kHz, 6.3 kHz) for vacuum (gray, dashed). 

In the text 
Figure 10 Measured total electrical impedance. The resonance frequencies are (1.6 kHz, 6.3 kHz) and the antiresonance (6.6 kHz). 

In the text 
Figure 11 Case 1) Comparison of measurement to model response with error bars, voltage to velocity transfer function (left) and electrical impedance (right), associated with varying the loss parameters by ±20%, ±10% and ±90% for the armature/hinge, suspension and pin, respectively. 

In the text 
Figure 12 Case 2) Comparison of measurement to model response with error bars, voltage to velocity transfer function (left) and electrical impedance (right), associated with varying the mass and compliance parameters for the armature and membrane by ±20% and ±50%, respectively. 

In the text 
Figure 13 Case 3) Comparison of measurement to model response with error bars, voltage to velocity transfer function (left) and electrical impedance (right), associated with varying the electrical parameters by ±10%. 

In the text 
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