A Multiphysics Approach to the Design of Loudspeaker

A Multiphysics Approach to the Design of Loudspeaker
A Multiphysics Approach to the Design of Loudspeaker Drivers
Roberto Magalotti1
1
B&C Speakers S.p.A., Bagno a Ripoli(FI) - ITALY
*Corresponding author: rmagalotti@bcspeakers.com
Abstract: Loudspeaker drivers are energy
transducers: their main goal is to efficiently
convert electrical energy to acoustic energy
(sound) through the movement of mechanical
parts. As such, they are prime candidates for the
application of multiphysics methods and tools.
This paper will outline the growing set of tools
that COMSOL puts in the hands of the
loudspeaker designer; how they can be put to
practical use in everyday work; how they can be
applied to different kinds of electroacoustic
devices and systems (cone loudspeakers,
compression drivers, horns, waveguides) and
how the measured performance compares to the
simulations.
Keywords:
loudspeaker
electroacoustics, transducers.
drivers,
1. Introduction
A loudspeaker driver is a transducer, a device
designed to convert energy across different
physical domains. The customary definition of
loudspeaker
driver
is
“electroacoustic
transducer”, meaning that its role is to transform
an electrical signal into an acoustic one. The
definition neglects an intermediate step: except
for some esoteric types, all loudspeakers convert
the electrical signal first to mechanical
movement, then to sound pressure.
audio signal is fed to the voice coil, a conductor
wire winding around a cylindrical former. The
voice coil is immersed in the air gap, a small
cylindrical space where a strong magnetic field is
generated by a permanent magnet. There, the
electrical current interacts with the B field to
generate a Lorentz force acting along the coil
axis.
This part of the device is called a
“loudspeaker
motor”
and
effects
the
electromechanical transduction, the first half of
the driver's job.
The voice coil is attached to an extended
membrane, typically in the form of a paper cone,
that moves back and forth by effect of the
mechanical force, thus pushing and pulling on
the air and generating pressure waves, that can
be heard as sound. This is the mechanoacoustic
transduction, the second half of the driver's job.
The audible result depends on many boundary
conditions: whether the loudspeaker cone is in a
box, at the throat of a horn, inside a room or
outside in free-field, near a wall, a reflecting
floor, a corner and so on.
So the loudspeaker driver is a device
working across three different physical domains
– electromagnetism, mechanics, acoustics –
making it a prime candidate for the application
of multiphysics simulation techniques.
This paper will explore the set of tools that
COMSOL puts in the hands of the loudspeaker
designer, by looking at some simulations and
results.
2. Electromagnetism
2.1 Magnetic assembly and voice coil
Fig. 1 Structure of a dynamic loudspeaker
Fig. 1 shows the typical structure of an
electrodynamic loudspeaker, a device that has
been using the same basic principles since its
invention in the early 20th century. The electrical
A natural starting point is the loudspeaker
motor. Fig. 2 shows a simulation of the whole
assembly, including the voice coil. The
simulation is based on the AC/DC Module and
uses a Small Signal Analysis study. It’s a twostep study, solving first for the fields generated
by the permanent magnet, then adding the
current flowing in the voice coil as a frequencydomain perturbation. All interactions between
Excerpt from the Proceedings of the 2015 COMSOL Conference in Grenoble
magnetic and electric fields are taken into
account, therefore not only the Lorentz force is
computed, but also the additional magnetic fields
generated by the electrical current and the eddy
currents induced by the current variations (see
2.2).
Fig. 2 Simulation of magnetic assembly and voice coil
The magnetostatic simulation helps the
loudspeaker designer in sizing correctly the
permanent magnet and the polar expansions.
Once the voice coil is added, and its
displacement changed through a parametric
sweep, it is possible to compute the magnitude of
the force acting on the moving assembly for any
operating condition. The force vs. displacement
graph shown in Fig. 3 contains a wealth of
information about the excursion capabilities of
the loudspeaker motor, its reliability and its
nonlinear behavior in the large signal domain.
distribution of eddy currents at any frequency
can also be investigated through simulation.
In the example shown in Fig. 4, the voice
coil, not pictured, is centered in the gap and fed
with a sinusoidal signal at different frequencies.
The eddy currents induced in the steel structure
of the loudspeaker motor vary in density and
depth depending on the frequency of the signal
and the distance between the voice coil and the
metal parts, as shown by the images on the left.
A common method for counteracting the
eddy currents is to put a ring of conductive
material, aluminium in this case, inside the
structure. The simulation helps assess the
effectiveness of that short-circuiting ring.
Looking at the right side of Fig. 4, the ring
appears to do a good job in shielding from eddy
currents the bottom part of the structure,
especially at low frequency, but is much less
effective at high frequencies and in the air gap
area.
Fig. 4 Density of eddy currents induced by the voice
coil in the loudspeaker motor at 50 Hz (upper row)
and 500 Hz (lower row), with (left) and without (right)
short-circuiting aluminium ring.
3. Mechanics
Fig. 3 Force on the voice coil vs. coil displacement,
for a fixed current of 1 A
2.2 Eddy currents
The variation of current flowing in the voice
coil induces eddy currents in the metal parts of
the motor structure. The magnitude and spatial
3.1 Structure of a compression driver
In order to illustrate mechanical simulations,
it’s better to analyze a close relative of the cone
loudspeaker: the compression driver. In
professional audio, the compression driver is by
far the most commonly used high frequency
device. The typical range of compression drivers
Excerpt from the Proceedings of the 2015 COMSOL Conference in Grenoble
is from around 1 kHz to the upper limits of
human hearing, around 20 kHz.
The motor structure is the same as the cone
loudspeaker, while the membrane shape is
typically a dome, whose outer edge is flattened
and clamped to the structure and acts as a
suspension. The voice coil is attached between
the dome base and the suspension.
Fig. 5 Internal structure of a compression driver
Unlike the cone loudspeaker, though, the
compression driver membrane does not radiate
sound directly in open air, but through a system
of thin channels called a phase plug (black in the
figure, inside the motor on the right), whose role
is to bring the sound radiated from different parts
of the dome in the correct phase relationship and
to enhance the acoustic load, and therefore the
transduction efficiency. A loudspeaker horn is
customarily connected at the phase plug outlet.
Fig. 6 Rocking mode of a compression driver
membrane, with flat suspension (above) and
suspension ribs (below)
3.2 Eigenfrequency analysis
Since the compression driver acts on high
frequencies, where the wavelengths are smaller
and the displacements are in the sub-millimeter
range, the simulation must be much more
detailed than in the cone loudspeaker. Fig. 6
portrays an example of the eigenfrequency
analysis of a compression driver moving
assembly, used to optimize the materials,
thickness, weights and shapes of all the parts.
The simulation uses both the Solid
Mechanics and Shell physics settings from the
Structural Mechanics Module.
The eigenmode shown in the pictures is the
first one usually found after the fundamental,
pistonic mode and is called a “rocking mode”. In
this case, the simulation is aimed at designing
the small ribs in the suspension (seen in the
bottom part of the figure), in order to heighten
the eigenfrequency and reduce the amplitude of
the rocking motion.
4. Acoustics
4.1 Phase plug design
The phase plug is a device put between the driver
dome and the driver outlet. It is composed of
thin, concentric channels and its goal is to yield a
simple plane wavefront at the throat of the horn
at all frequencies.
The simulation can show the resonant
frequencies and mode shapes of the compression
chamber, through an Eigenfrequency study, as
well as help the designer refine the path and
expansion of the channels, to maximize the
efficiency and minimize internal resonances,
through Frequency Response and Optimization
studies.
Fig. 7 shows the results of such an optimization,
in the case of a phase plug with straight
channels. Notice how the resonances in the
Excerpt from the Proceedings of the 2015 COMSOL Conference in Grenoble
compression chamber (the dome-shaped cavity at
the top of simulation domain) have no effect on
the plane-wave propagation at the driver outlet.
Fig. 8 Acoustic pressure in a loudspeaker horn, at
1.25 and 10 kHz
Fig. 7 Sound pressure propagation through a
compression driver phase plug
4.2 Loudspeaker horn design
An especially effective application of FEA is in
the design of loudspeaker horns. Fig. 8
represents the sound pressure field propagation
inside a loudspeaker horn, from the circular
throat where the driver is connected up to the
sound radiating end, in this case an elliptical
mouth.
Other critical aspects of horn performance can be
evaluated through simulation. The graphs in Fig.
9 show the radiation pattern of the horn, i.e. the
size and shape of the area covered by the sound
emission on the horizontal and vertical plane of
the horn, and how it changes with frequency.
In this example, the graphs show that the
horizontal coverage of this horn is generally
broader and smoother that the vertical one, that
has many sidelobes and tends to decay faster at
high angles from the axis.
Fig. 9 Horizontal and vertical polar plots for the horn
in Fig. 8
Excerpt from the Proceedings of the 2015 COMSOL Conference in Grenoble
Another important evaluation criterion is the
acoustic load, or acoustic impedance at the
throat, that affects the overall efficiency of the
horn-driver system by interacting with the
mechanical impedance of the moving assembly.
The acoustic impedance in fig. 10 has a
good, uniform acoustic resistance from 3 kHz
up, and the reactive part is never larger than the
resistance, despite some irregularities in the
lowest range.
Fig. 10 Acoustic impedance at the throat of the horn in
Fig. 8
4.3 Line array acoustical waveguide
A different example of sound propagation
analysis comes from the design of a highfrequency waveguide for line-array professional
sound systems, presented at the 2009 COMSOL
Conference in Milan [1]. The project goal was to
modify a waveguide so that the outgoing
wavefronts were as flat as possible in the vertical
plane. In other words, the shape of the outgoing
waves had to be cylindrical, diverging in the
horizontal plane, but staying straight in the
vertical plane up to the highest audio
frequencies.
The solution we found was to create an
acoustic lens inside the waveguide, by adding the
rigid obstacles seen in the bottom part of Fig. 11,
in a triangular shape. It’s easy to appreciate the
effect of the lens in flattening the curved
wavefronts to a straight segment.
Fig. 11 Design of an acoustical waveguide for line
array systems. The comparison shows the effect of the
acoustic lens in flattening the wavefronts at the outlet.
5. Thermodynamics
To make a good professional loudspeaker
driver, there's more than electromagnetism,
mechanics and acoustics. At the very least, one
has to consider thermodynamics, that plays a
vital role in the reliability and durability of the
device. Thermal failure, in the form of voice coil
burnout, is the most common cause of failure in
demanding applications.
The images in Fig. 12 are taken from a study
commissioned by B&C Speakers to the
University of Firenze in 2010 [2] with the goal
of analyzing the thermodynamics and heat paths
inside the motor of a large low-frequency
loudspeaker driver. The issue is especially
important when the material of the permanent
magnet is neodymium-iron-boron, that is known
to demagnetize easily under excessive heat.
Excerpt from the Proceedings of the 2015 COMSOL Conference in Grenoble
Fig. 13 Complete simulation of a compression driver
and horn.
7. Results
Fig. 12 Temperature map (above) and heat flux
(below) of a loudspeaker structure in operating
conditions.
The top image is a map of the temperature
distribution in working conditions, and shows
that the tips of the voice coil are hotter that the
rest, because they are away from the heat
dissipating effect of the steel structure around the
air gap. The bottom image shows the heat flow
out of the coil, through the steel and the
permanent magnet and eventually to the basket.
This kind of simulation is very useful to
assess the effect of heat-dissipating features, like
the fins located between the basket spokes.
The complete simulation also allows a
comparison of predicted and measured results on
the same device.
The graph in Fig. 14 compares the simulated
and measured electrical impedance of a
compression driver on a horn. Since the
loudspeaker is a transducer where the different
physical domains are strictly tied to each other,
the electrical impedance contains a wealth of
information not only on the electromagnetic
characteristics of the device, but also on the
mechanical and acoustical features. The
simulation tracks surprisingly well not only the
main features of the impedance curve, but also
many of the finer ones.
6. Multiphysics
6.1 Full model of a compression driver
Putting all of the tools together in a single
simulation yield results like the one in Fig. 13.
The products used include the Structural
Mechanics Module, the Thermoacoustics Study
from the Acoustics Module and the AC/DC
Module for the electric equivalent circuit of the
loudspeaker motor.
Fig. 14 Electrical impedance of a compression driver
on a horn, simulation vs. measurements.
The final test of performance for a good
loudspeaker is the transfer function between
input signal and sound output, that is, the
Excerpt from the Proceedings of the 2015 COMSOL Conference in Grenoble
frequency response. Fig. 15 compares the
simulated and measured frequency response of
the same compression driver. The fit between the
two curves and the amount of features
successfully replicated are excellent.
9. References
1. M. Cobianchi, R. Magalotti, “Optimization of
an Acoustic Waveguide for Professional Audio
Applications”, Proceeding of the COMSOL
Conference 2009, Milan
2. G. Grazzini et al, “Studio del raffreddamento
di un altoparlante”, Dipartimento di Energetica,
Università degli studi di Firenze (2010)
3. https://www.comsol.it/model/lumpedloudspeaker-driver-12295
Fig. 15 Frequency response of a compression driver
on a horn, simulation vs. measurements.
8. Conclusions
The loudspeaker driver is a multiphysics
device, where the physical phenomena in each
domain are strictly interwoven with each other.
The examples shown in this paper prove that any
attempt to model the behavior of the driver needs
a multiphysics approach to be successful.
COMSOL features a rich set of tools that
help the loudspeaker driver designer in applying
such an approach, and is constantly striving to
overcome the limitations of Finite Element
Analysis. For example, the combination of
Perfectly Matched Layers and Far-Field
Calculation make COMSOL a good solution for
open boundary acoustic problems. The
integration of lumped parameter modeling
through the use of equivalent electrical circuits is
also a handy tool for loudspeaker modelling [3].
Furthermore, the number of different tools in
the Acoustics module is constantly increasing.
The addition of the thermoviscous losses in
small passages has been instrumental in
improving the simulations of compression
drivers. Some of the new Physics interfaces, like
Ray Tracing and Acoustic Diffusion, further
extend the range of problems that COMSOL can
tackle.
Excerpt from the Proceedings of the 2015 COMSOL Conference in Grenoble
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