PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Comparison of numerical methods for miniature loudspeakermodeling
Juha BACKMAN1
1Consultant, Finland

ABSTRACTThis work compares the results of obtained using conventional acoustics FEM simulation models to CFD
based results for miniature loudspeakers. The methods discussed here include acoustical models withdifferent approaches to fluid properties (linear, nonlinear, lossy) using a moving geometry, and a stationarygeometry CFD model. The results indicate that even if the conventional acoustical models are highly
successful in predicting the pressure distribution in an enclosure, and in predicting the resonance frequencies,
the full features can be revealed only through a more detailed fluid dynamic analysis.
Keywords: Electroacaoustics, Transducers, Modelling

1. INTRODUCTION
Finite element modelling is a standard tool in the design of any audio transducer. The small-scaledevices, such as headphones, mobile phones, and hearing aids pose challenges not found in large form
factor transducers (such as full-scale loudspeakers):- The distance from almost any point in the interior to the nearest boundary is short enough sothat point is within viscous and thermal boundary layers- The displacement of a microspeaker in normal usage conditions is a significant fraction of thetotal air volume of the system, so the nonlinearity of the air is a possibility- The geometry of the system varies due to the diaphragm displacement- The system contains narrow internal gaps and ports close to boundaries so that the geometry of
the system can lead to the formation of laminar vortices already at low signal levels

2. THE COMPUTATIONAL MODEL
2.1 Basic features and assumptions
The geometry chosen represents a typical microspeaker. The features shared by most designs
include an airspace behind the diaphragm that is coupled to the external space through narrow gaps.
The geometric nonlinearity in the surround is included (or rather, could not be easily excluded without
making the model geometry otherwise unrealistic), but the materials as such are assumed to have linear
elastic properties.
The driver electromagnetic coupling is assumed to be linear, and the electromagnetic damping ismodelled through a velocity-dependent component in the force. The electromagnetic aspects of thedriver are not otherwise included in the FEM model. Although electromagnetic nonlinearity is asignificant factor in real-life drivers, this choice was made in this study to focus more clearly on thenonlinearity features due to the fluid flow and the chosen fluid models. All the examples werecomputed using Comsol Multiphysics, versions 5.3a and 5.4.
The cases studied include:- Harmonic solution for a linear transducer without flow losses (benchmark, represents the mostcommonly used method)
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juha.r.backman@gmail.com

-Time-domain solution of a linear transducer with lossless acoustics (approximated as a very
low signal level simulation)- Time domain solution of a transducer with geometry variation due to large diaphragm
displacement but with linear lossless air inside the cavities- Geometry variation and nonlinear lossless air (no viscosity or thermal conduction), computed
using the ideal gas state equation instead of the linearized equation- Geometry variation and thermal and viscous losses in the fluid, but not on the boundaries. Thepropagation losses in the fluid itself are small over the distances and frequencies in this study,
so this should also be considered more as a benchmark case
- Non-isothermal CFD solution with stationary geometry
The interesting case which unfortunately had to be left outside this study is using acoustic
equations with boundary effects, which should yield better estimates for internal acoustic damping.
This is a topic for further study.
2.2 The challenges and benefits of time-domain modelling
The usual choice for solving finite element problems is to use the frequency domain solutions. Thesolvers can be written efficiently, and the frequency resolution is determined only by the maximum
size of the mesh elements. Time domain solutions, on the other hand, require more memory, are lessefficient to compute, and due to the numerical behavior of the finite element solvers the time
discretization has usually to be much finer than the 1/2fs (where fs is the sampling frequency) that theNyquist theorem would suggest. Thus, time-domain solutions are usually avoided in acoustics.
However, capturing nonlinearity in frequency domain computations removes to a large extent thenumerical efficiency advantage, since frequency-domain nonlinearity methods, such as harmonicbalance, are iterative and work best only with weak nonlinearities. The explicit time domain
formulation, on the other hand, can be basically extended to any level of nonlinearity, althoughpractical considerations limit the usability of deformed geometry solutions. The practical limit for
nonlinearity comes mostly from the ability to sufficiently deform the mesh geometry, since if thegeometry changes so much that a remeshing is needed, the discontinuity caused in the results by thechange in mesh geometry renders any distortion analysis results unusable.
2.3 Test configuration
The driver-enclosure combination used in the simulations represents the typical dimensions foundin miniature loudspeakers or headphone drivers. The diaphragm is 0.1 mm aluminum dome, and thesurround material parameters correspond to typical silicone rubber. The thickness of the surround isadjusted to 0.2 mm to yield the desired resonance frequency of 400 Hz with the cavity size used in thissimulation. The voice coil is modeled as a solid copper ring.
The electromagnetic force on the voice coil is approximated by an annular force acting on one edgeof the voice coil. In case of voltage drive, the force F is obtained from expression
𝐹 = 𝐵𝑙 ∙ (𝐸 − 𝐵𝑙 ∙ 𝑢)/𝑅(1)where Bl is the force factor, E in the input voltage, u the velocity of the edge, and R E the electricalresistance of the voice coil. In reality the force acts across the entire voice coil surface, and is place
dependent due to the inhomogeneous magnetic field B, but this approximation is accurate enough forthe purposes of this work, since the electromagnetic nonlinearity is left outside the scope of theanalysis.

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Figure 1 – The structure of driver and back cavity the model.
The simulations were run with two types of excitation: for acoustics models with deformablegeometry using an annular force acting on one corner of the voice coil, and for the CFD studies withstationary geometry a flow velocity source on the diaphragm, voice coil bottom surface, and thesurround, and with non-slip moving wall boundary condition for the side surfaces of the voice coil.

3. VARIABLE GEOMETRY SIMULATION
The first cases in this study describe the behavior of the loudspeaker when the geometry is
time-variant and the fluid is described using conventional acoustics equations without thermalconduction or viscous effects at the boundaries. The test signal used for the simulations is alogarithmic sweep, since it will allow the analysis of harmonic distortion in the post-processing. Threedifferent fluid models: linear fluid, adiabatic fluid with nonlinear compressibility, and ideal gas (withinternal thermal conductivity)
The cases studied include no electromagnetic damping (i.e. current drive), which here mostly
serves as a stability test for the method, as current drive is rarely used with microspeakers, moderate
electromagnetic damping (force factor Bl = 1), yielding the total Q factor of approximately 1 with thematerial parameters used here, and high damping (Bl = 5). The moderate damping corresponds topractical designs, and the high damping case is for computational testing, but unfeasible inmicropspeakers with the currently available magnetic materials. As and additional confirmation, thesmall-signal frequency responses were computed using a frequency domain formulation. The results
below are for the thermally conducting and viscous formulation. Lossless fluid models perform wellwhen electromagnetic damping is present, but they are unstable in the test case with current drive (i.e.
without electromechanical damping), since the driver surround is assumed to have no mechanicallosses. The computation time differences between various fluid models is small, but the computing
time for any time domain model is about two orders of magnitude higher than for frequency domain
models with the same number of frequency or time points.

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Figure 2 – Average outer surface acceleration of a logarithmic sweep signal with no electromagnetic damping(light green), moderate damping (dark green), and high damping (red).

Figure 3 – Frequency responses corresponding to the different values of electromagnetic damping. The riseof the responses near 10 kHz is due to the emerging diaphragm mechanical resonance.

4. STATIONARY GEOMETRY CFD SIMULATION
4.1 CFD simulation
As opposed to the wave equation used in conventional FEM solutions, the CFD equations do notsolve directly for particle velocity, but rather for energy variables of the field, which are in turn derived
from the velocity and pressure variables. From practical perspective the most critical difference
between the acoustic wave equation and CFD equations is that the CFD solutions allow the existenceof vortices in the field, while the classical derivation of the acoustic wave equation explicitly assumes
that the field is irrotational, and even viscothermal extensions to the acoustical wave equation do notassume a vortex component.
The computing time for stationary geometry time dependent CFD model is about one order ofmagnitude higher than that of a variable geometry acoustical model. This difference is due to both thecomplexity of the equations and the denser mesh needed. Practical experience indicates that the CFD
model should include turbulence. The amount of turbulent energy in the field is not large, but
apparently the turbulent energy transport helps to stabilize the model.
The behavior of the field in the loudspeaker assembly is illustrated by examining the sound field7335

generated by sinusoidal vibration at 450 Hz. The data is plotted at the velocity maximum, i.e.
displacement zero crossing.

Figure 4 – Flow velocity distribution at the peak of the diaphragm velocity.

Figure 5 – Pressure distribution at the peak of the diaphragm velocity. The actual pressure differences in thecavity are quite small, a few perecent, but the figure is scaled to illustrate the slight pressure inhomogeneity
near the moving boundaries.

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Figure 6 – Equal-velocity magnitude contours corresponding to Figure 4.

Figure 7 – Velocity field streamlines, line thickness scaled proportional to velocity to illustrate the dominantfeatures of the flow field.

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Figure 8 – Normalized streamline plot of the velocity field, all streamlines shown at the same level toillustrate the actual complexity of the flow field. It should be noted from Figure 7 that the flow velocity, andthus the energy in the vortices, is quite low in most of air space.
The important conclusion that can be drawn from the figures above is that the flow field intransducers is according to CFD results much more complex than the acoustical theory results would
lead to expect. The vortex formation is important especially in the air space between the diaphragmand the front plate and around the voice coil.
The acoustical models are excellent when estimating the resonance frequencies and pressure
distribution, but as the vortex formation cannot appear in the acoustical theory predictions, the local
flow velocities in the air space get underestimated, which in turn will result in underestimating theviscous losses.

5. CONCLUSIONSThe preliminary results in this work confirm that the behavior of miniature loudspeakers andrelated transducers is best understood using CFD models. However, as the models are computationallysignificantly heavier than wave equation based analysis, and the convergence of solutions is by nomeans guaranteed especially in more complex geometries, the use of lossy acoustics models is still
better suited for practical design work, especially when using time-dependent geometries, which areessential for proper analysis of transducer nonlinearities. Further work is needed on how to combine
the approaches needed for the various parts of the loudspeaker assembly into a computationallyefficient system model.

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