modeling, fem analysis and dynamic simulation of a moving...
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Modeling, FEM Analysis and Dynamic Simulation
of a Moving Coil Loudspeaker
Ezio Santini, Sabrina Teodori
DIAEE, Department of Astronautic, Electrical and Energetic Engineering
SAPIENZA University of Rome, Via Eudossiana 18, 00184 Rome, Italy
[email protected], [email protected]
22nd International Symposium on
Power Electronics,
Electrical Drives,
Automation and Motion
Ischia (Italy) 18-20 June 2014
The goal: TO PROVIDE A SIMPLE AND EASY-TO-USE ALGORITHM FOR THE DETERMINATION OF THE MECHANICAL FORCE ACTING ON A LOUDSPEAKER MOVING COIL
USE OF THE DETERMINED QUANTITY: INPUT TO AN ACOUSTIC ANALYSIS SOFTWARE
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METHODOLOGY:
FEM ANALYSIS FOR PARAMETER EVALUATION
MECHANICAL EQUATIONS
FORCE FACTOR VS. DISPLACEMENT
IN COOPERATION WITH SICA ALTOPARLANTI S.R.L. (ITALY)
A loudspeaker is:
• a linear motor with a small displacement range • an electroacoustic transducer that produces sound in
response to an electrical audio signal input (voltage / current)
http://www.youtube.com/watch?v=3ZQqCyRQFB4
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How it is made? Mainly, it consists of:
• an annular permanent magnet
• a coil which is free to move into an airgap
• an iron structure as the pathway of magnetic circuit
• a plastic or paper cone
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How does it work?
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Outline:
• Calculation of the force factor (Bl) by studying the distribution of the flux density along the coil depth (FEM)
• Calculation of value of the coil self-inductance L (FEM)
• Simulation of the system by means of the Matlab tool Simulink
• Parametric analysis (what-if)
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CASE STUDY
WOOFER
Power 120 W
Range frequency 150/6000 Hz
Coil material Aluminum
Coil diameter 38 mm
Coil turns 63
Permanent magnet external diameter 124 mm
Permanent magnet internal diameter 44 mm
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Other dimensions
ANALITICAL AND ELECTRICAL MODELS DYNAMIC EQUATION OF THE SPEAKER MOBILE MASS
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),(1
txFxC
xRxMms
msms
axial displacement of the mobile coil
inverse of the spring force constant
damping coefficient
speaker mobile mass Lorentz force
ANALITICAL AND ELECTRICAL MODELS
EQUIVALENT CIRCUIT OF THE COIL
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• Voltage generator: represents the
input signal
• Series resistance: represents the
losses in the electrical conductors
• Variable inductance: the magnetic
energy stored into the winding
• Voltage generator: represents the
back emf.
ANALITICAL AND ELECTRICAL MODELS
ELECTRICAL CIRCUIT EQUATION
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dt
dxxBlixL
dt
dRivi )()(
electromotive force
resistive voltage drop
inductive voltage drops
ANALITICAL AND ELECTRICAL MODELS
CAD GEOMETRICAL MODEL
Notice that:
geometric and magnetic symmetry axis is present
Advantage in terms of:
• computing times
• improvement of the solution accuracy
Coil 2D geometry suitable for FEM
analysis - transverse section
ANALITICAL AND ELECTRICAL MODELS
MATHEMATICAL MODEL OF PERMANENT MAGNETS
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BB
HH Hr
cc ( )
Demagnetization B - H curve for a PM material for the easy axis (second quadrant).
Notice that:
large air gaps allows to consider the iron as operating in the linear part of magnetic characteristic.
FEM ANALYSIS
• Custom-made FEM analysis software has been used (2D FEM Software Amadeus®)
• The reference equation is the standard Poisson formulation of axisymmetric static magnetic fields
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1 ( ) 1rA AJ
r r r z z
Current density Permeability
Vector potential
FEM ANALYSIS BOUNDARY CONDITIONS
• No magnetic barrier.
But,notice that:
• Currents inputs exhibit intrinsically zero-divergence
then:
• B is practically zero at a given distance from the sources
then:
• Semi-circular boundary is a Dirichlet-type
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FEM ANALYSIS INPUTS FOR FEM SOLUTION
• Total current flowing into the conductor
• Magnetic properties of the materials:
Air
Aluminum
Iron
Permanent magnets
• Geometry of transducer in terms of nodes and edges.
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Magnetic permeability
Differential magnetic permeability
Residual flux density Br
FEM ANALYSIS INPUT QUANTITIES FOR INDUCTIVE PARAMETERS EVALUATION :
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QUANTITIES VALUE
Total current 1 A
Magnetic potential boundary condition A = 0
Ceramic magnet (second quadrant)
µr=1,671 Br=0,42
Iron µr=10000
FEM ANALYSIS RESULTS
EQUIPOTENTIAL LINES IN:
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No-load case Load case
Notice that:
• influence of the currents
flowing into the coils on the
magnetic field is truly minimal
in fact, the aim is:
• B in the coil should not
vary in the speaker operation.
PARAMETER IDENTIFICATION BY FEM
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• Several configurations of the moving coil have to be analyzed, representing its displacement during the electromechanical energy conversion
FLUX LINES IN THE COIL:
in central position 2 mm displaced in vertical
direction
Notice that: • when the coil goes out of the air
gap, the flux lines are not
anymore perfectly orthogonal
to the coil displacement
direction
then
• the force produced is not
parallel to the coil axis
then • there is a sound distortion
The coil has been moved by 1 mm steps, in a range that goes from 4 mm over the central position to 4 mm lower.
B IN THE COIL
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coil position
[mm]
average flux
density [T]
-4 0.58
-3 0.68
-2 0.77
-1 0.84
0 0.85
1 0.83
2 0.77
3 0.67
4 0.57
average B on the coil
in different positions
B vs. coil depth for the central
position of the coil itself.
Notice that:
• B in the coil decreases
when the coil moves out of
the air gap
then
• Lorentz force on the coil
decreases
PARAMETER IDENTIFICATION BY FEM
FORCE FACTOR (Bl)
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ì
coil position
[mm] force factor [T m]
-4 4.40
-3 5.10
-2 5.82
-1 6.30
0 6.40
1 6.26
2 5.79
3 5.06
4 4.26
ì
force factor on the coil in different coil positions force factor vs coil position
PARAMETER IDENTIFICATION BY FEM
FEM ANALYSIS RESULTS
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SELF-INDUCTANCE L:
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Inductance trend vs. coil displacement
Notice that:
•the presence of the airgap has a
smoothing effect on the
inductance behavior
•This variation is a non-linearity for
the simulation model
•A functional relationship between x
and Bl(x) must be arrived at
PARAMETER IDENTIFICATION BY FEM
SIMULATOR
INTEGRATION OF DYNAMIC EQUATION
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) tF(x,xC
1xRxM
ms
msms
• Simulink has been used in order to build a magnetic motor simulator
• Dynamic equation has been integrated with Matlab
• The force factor function Bl(x) has been obtained by means OLS interpolation of the
FEM data.
• In the case under investigation, such relationship was found to be:
6.4x 0.1342- )Bl(x 2
SIMULATOR
SIMULATION MAIN SYSTEM
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Notice that:
As result of the simulation, it is
possible to obtain
• mechanical answer of the
loudspeaker mobile mass
to an audio signal
• audio output deriving by
the transduction.
SIMULATOR
SUBSYSTEM SIMULATION
REPRESENTING EQUATION:
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t)F(x,xC
1xRxM
ms
msms
Notice that:
Input
• voltage generated by an
audio signal
• normalized wave
SIMULATOR
INPUT AND OUTPUT WAVEFORMS
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Comparison between input (yellow) an output
(purple) waveforms
Notice that:
the instrument operates as a
low-pass filter:
• the inertia of the mobile
mass
• when high frequencies are
present, the inductance of
the coil causes a significant
cut to the output wave
amplitude.
CONCLUSIONS
• The analysis method, through a FEM software, of a common loudspeaker has been described.
• The study is based on the electromagnetic phenomena in the magnetic motor.
• Forces acting on the moving coil, magnetic energy stored and flux linkages have been studied in detail.
• Through the mechanical model it has been possible to study and observe the mechanical answer of the transducer to the input electromagnetic forces.
• An electromechanical simulator of the loudspeaker has been created linking these two analysis.
• Through the simulator it is possible to perform a first approximation study of the loudspeaker, that allows to design new devices or to improve existing models.
• This allows to limit the experimental tests and to verify the measurements on existing devices.
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THANKS FOR YOUR ATTENTION!
22nd International Symposium on
Power Electronics,
Electrical Drives,
Automation and Motion
Ischia (Italy) 18-20 June 2014