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Rausch M.1, Kaltenbacher M.1, Kreitmeier
L.2, Krump G.2, Landes H.1, Lerch
R.1
1 Dep. of Electrical Measurement
Technology, University of Linz, Altenbergerstr.69, A-4040-Linz,
Austria
2 Harman Audio Electronic Systems GmbH,
Schlesische Straße 135,
D-94315 Straubing, Germany
Summary: In this paper the applicability of an
efficient numerical calculation scheme in the computer-aided-design
of electrodynamic loudspeakers is demonstrated. This numerical
technique is based on a Finite-Element-Method (FEM) and allows the
precise calculation of the magnetic, mechanical and acoustic fields
involved in acoustic radiation and considers all coupling terms
between the different physical quantities. Therewith, the complex
dynamic behavior of electrodynamic loudspeakers was studied.
The
validity of the computer simulations is verified with the help of
appropriate measurements. The calculated and measured values for
eigenfrequencies, axial pressure responses and electrical input
impedances are in excellent agreement. The applicability of the
presented calculation scheme with respect to the
computer-aided-design of moving coil drivers is demonstrated by
reporting two practical applications, the elimination of response
dips at intermediate frequencies and the examination of polar
responses.
1 INTRODUCTION
Fig. 1 shows the design of an electrodynamic loudspeaker.
Figure: Schematic of an electrodynamic loudspeaker
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A cylindrical, small light voice coil is suspended freely in a
strong radial magnetic field, generated by a permanent magnet. When
the coil is loaded by an electric voltage signal, the interaction
between the magnetic field of the permanent magnet and the current in
the voice coil results in an axial Lorentz force, acting on the coil.
Therewith, the whole structure, consisting of the voice coil, former,
suspension, surround and cone diaphragm, starts to move and generates
the acoustic sound.
Since in the case of a loudspeaker the
interaction with the ambient fluid must not be neglected, the
electrodynamic loudspeaker represents a typical coupled
magnetomechanical system immersed in an acoustic fluid. That is the
reason, why for the detailed finite element modeling of these moving
coil drivers the magnetic, the mechanical as well as the acoustic
fields including their couplings have to be considered as one system,
which cannot be separated. Due to the complexity of these multi-field
problems, the straight forward application of standard simulation
tools like commercially available finite element or boundary element
codes has shown only limited success.
2 FINITE ELEMENT MODEL OF A LOUDSPEAKER
In this paper, the governing equations
are solved using a Finite-Element-Method (FEM). The theory of the
underlying equations and finite element scheme has already been
reported in (1,2) and will not be repeated here.
In finite
element methods, the region of the electromechanical device with air
surroundings is subdivided into small discrete elements, the
so-called finite elements, as shown in Fig. 2.
Figure: 2D finite element model of an electrodynamic loudspeaker
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In case of an electrodynamic loudspeaker the voice coil and
aluminum former are discretized by so-called magnetomechanical finite
elements, which solve the equations governing the magnetic and
mechanical field quantities and take account of the full coupling
between these fields. Due to the concentration of the magnetic flux
within the magnet assembly, the magnet structure and only a small
ambient region have to be discretized by magnetic finite elements.
Furtheron, the surround, suspension and cone diaphragm are modeled by
standard mechanical finite elements. Finally, the fluid region in
front of the loudspeaker is discretized by acoustic finite and
infinite elements. A small part around the magnet assembly has to be
modeled by so-called magnetic-acoustic finite elements.
In order
to function properly, the infinite elements have to be located in the
far field of the moving coil driver. Consequently, a large number of
acoustic finite elements is necessary in the modeling of
electrodynamic loudspeakers. Therefore, a modified finite element
model has been applied recently, in which acoustic elements were
eliminated completely. The influence of the surrounding air, which
consists of mass-loading effects and damping due to the sound
emission, is now realized by so-called spring-elements, which have
been located on the outside boundary of the cone diaphragm and
surround. Therewith, the required finite elements as well as the
CPU-time can be reduced tremendously.
3 VERIFICATION OF THE COMPUTER MODEL
The verification of both computer models described above has been performed by comparing simulation results with corresponding measured data. In a first step, the frequency dependency of the electrical input impedance has been calculated. As can be seen in Fig. 3-a, good agreement between simulation results and measured data was achieved.
Figure: a) Comparison of computed and measured frequency response of the electrical input impedance and b) the axial pressure response at 1 m (voltage clamping)
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After this validation of the computational models, the axial
pressure response of the electrodynamic loudspeaker was measured and
compared with simulations. The input source is a voltage with nominal
1 W referred to 4
(2.0
V r.m.s.). The microphone is placed on the mid axis at a distance of
1 m from the loudspeaker. Again good agreement between simulation and
measurement was observed (see Fig. 3-b).
In the computer simulations of this electrodynamic loudspeaker
with acoustic finite and infinite elements, approximately 58.000
second order finite elements have been used, resulting in a total
number of about 190.000 unknowns. On a SGI, Octane 195 MHz the
calculation of a transient analysis with 5.000 time steps required
hours
of CPU-time and 860 MB of physical memory. In the modified finite
element model with additional spring-elements the total number of
unknowns is reduced to about 30.000, resulting in a decreased
CPU-time of
hours
and a required physical memory of 90 MB.
4 STUDIES IN LOUDSPEAKER-DESIGN PARAMETERS
As a first application of our
calculation scheme in the computer-aided-design of electrodynamic
loudspeakers, the elimination of response dips at intermediate
frequencies was considered. Measurements as well as simulation
results reveal two dips in the sound pressure response occuring at
approx. 400 Hz and 900 Hz (see Fig. 3-b).
The elimination of these dips is of great interest for loudspeaker
manufacturers, since a flat axial pressure response over a wide
frequency range is desired (3). In computer simulations two design
modifications could be established, both leading to an improvement of
these response dips. In the first modification, a new surround
material with an increased loss-factor was modeled. The increased
loss-factor results in a more effective absorption and termination of
the outward travelling energy and in reduced response dips. In the
second case, a surround with a flat section added was used. Due to
the modified geometry of the surround, the change of surround mass
and compliance results in a modified equivalent circuit and causes a
frequency shift of these response dips. With both modifications the
deviation can be held within
dB
over a wide frequency range and, therefore, an improvement in respect
to response flatness can be achieved.
In the second application,
design improvements in respect to directivity patterns of the
electrodynamic loudspeaker can be achieved. Fig. 4
shows the calculated polar response in dependence on the modulus of
elasticity E of the diaphragm for two frequencies.
Figure: Design Improvement: Polar response
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As can be seen, the decreased
modulus of elasticity results in a modified vibration pattern and in
a improved directivity pattern.
5 CONCLUSION
In this paper we have applied a new numerical scheme for the computer modeling of electrodynamic loudspeakers. Therewith, the optimization of loudspeaker-design parameters and limits by using minimal number of prototypes can be established.
REFERENCES
1. M. Kaltenbacher, H. Landes, R. Lerch, An
Efficient Calculation Scheme for the Numerical
Simulation of
Coupled Magnetomechanical Systems, IEEE Trans. on Magnetics, Vol.
33, No. 2, 1997
2. R. Lerch, M.
Kaltenbacher, H. Landes, F. Lindinger, Computerunterstützte
Entwicklung elektromech-
anischer Transducer, e&i
ÖVE-Verbandszeitschrift, Vol. 7/8, pp. 532-546, 1996
3.
M. Colloms, High Performance Loudspeakers, John Wiley &
Sons, Chichester, New York, 1997
