The Transient Magnetic Behaviour of Loudspeaker Motors' Mark Dodd

1
The Transient Magnetic Behaviour of Loudspeaker
Motors.Mark Dodd
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Loudspeaker Basics

• Loudspeakers convert an electrical signal to a
  mechanical signal and then to an acoustic signal.

• Moving coil linear actuator voltage -gt motion.

• Diaphragm motion -gt sound pressure.

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Loudspeaker Basics.

• Rigid body motion assumed.
• Mechanically loudspeaker is equivalent to a mass,
  m mechanical resistance rmc and spring stiffness
  k.
• Equating forces gives Drive force F

• For a small current i

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Loudspeaker Basics.

• The acoustic pressure P produced, at distance r
  by a sinusoidal point source with volume velocity
  U at angular frequency w is given by

• Where volume velocity U is produced by a
  diaphragm with area S moving at velocity, dx/dt
  is given by

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Pass Band Response.

• Motion mass-limited, decreasing amplitude with
  frequency.

m
0.28m
0.21m
0.14m
70u
0
20
50
100
100
200
500
1k
1k
2k
5k
10k
10k
20k
20
20k
Frequency
Hz
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Pass Band Response.

• Velocity amplitude decreases above resonance
  frequency.

-40
dB
-54
-68
-82
-96
-110
20
50
100
100
200
500
1k
1k
2k
5k
10k
10k
20k
20
20k
Frequency
Hz
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Pass Band Response.

• Radiation efficiency increases with increasing
  frequency.

ohm
4.3k
3.3k
2.2k
1.2k
100
20
50
100
100
200
500
1k
1k
2k
5k
10k
10k
20
20k
Frequency
Hz
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Pass Band Response.

• Motional impedance is reflected in electrical
  domain.

• Net result is an SPL response with a flat pass
  band.

dB
110
80
dB
Uin2.83Vrms, Distance1m
100
64
90
48
80
32
70
16
60
0
20
50
100
100
200
500
1k
1k
2k
5k
10k
10k
20k
20
20k
Frequency
Hz
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Previous Approach

• Consider the static flux from permanent magnet
  separately from flux produced by current flowing
  through the coil.

• Use FEM to calculate static field calculate
  Bl(x) for different coil displacements x.

• Calculate Flux through coil for an arbitrary
  current.

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Polynomial method

• Assume that Lvc is a function of x but not i.

• Express force variation with current in terms of
  coil inductance

FBl(x)I-Lvc(x)i2.
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Polynomial method

• Eddy Currents.
• AC signal through coil results in alternating
  magnetic field.
• EMF induced in any conductive loops round field.

• Current flow in skin depth.

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Polynomial method

• Eddy currents
• assumed to be confined to a cylindrical region
  equal to skin depth adjacent to coil.
• Permeability fixed at average static value.
• Constant current density in conductive region.
• Total Flux in gap now given by

• Flux from eddy currents used to derive Lvc as a
  function of frequency and coil position Lvc(x,f).

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Polynomial method

• Equating forces.

• Equating voltages.

• Runge Kutta method used to numerically solve
  equations.
• Spectrum of distorted sinusoidal waveform
  produced is analysed with FFT.

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New Magnet Topology
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Flux2D Magneto-static FEM.

• Magneto-static solver uses static version of
  Maxwell's equation.

• Force on a coil is calculated using

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Flux2D Magneto-static FEM Results.

• The force versus displacement may be calculated
  for a range of coil displacements.

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Flux 2D Magneto-static FEM Results.

• Voice coil self Inductance may be calculated for
  different coil positions and currents.

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Flux2D Magneto-dynamic FEM

• Linear steady state sinusoidal solution.

• No static magnetic field.
• Sinusoidal input to coil induces eddy-currents in
  conductive regions.
• Use of permeability from static solution would
  give quick calculation of small signal dynamic
  fields.
• Could be used to generate Lvc(x,f) for parametric
  model.

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Flux2D Transient Magnetic FEM

• Allows solution of stationary voice coil
  permanent magnet with poles of non-linear steel
  for time-varying currents fields.
• Based on dynamic Maxwell's equation.

• Solution is made for 40-120 time-steps per period.

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Flux2D Transient Magnetic FEM

• Mesh uses quadrilateral elements to mesh skin
  and give a fast solution time.
• Mesh has frequency dependant spacing
  perpendicular to metal surface.

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Flux2D Transient Magnetic FEM

• Sinusoidal voltage source input allows
  calculation of blocked impedance of coil.
• Voice coil is stranded conductor constant
  current density.
• Iron NdFeB are solid conductors with eddy
  currents.
• Hysteresis is neglected.
• Calculation may be for different coil positions
  and voltage inputs.

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Flux2D Transient Magnetic FEM

• Flux at 80Hz.

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Flux2D Transient Magnetic FEM

• Permeability at 80Hz.

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Flux2D Transient Magnetic FEM

• Power density currents at 80Hz various coil
  displacements.

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Flux2D Transient Magnetic FEM

• Power density currents at 20Hz, 80Hz 240Hz.

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Flux2D Transient Magnetic FEM With Kinematics

• Mass, compliance and mechanical resistance
  included in the model.
• Acceleration is derived from driving force.
• Displacement mesh allows solution with coil
  displaced axially.
• Rigid body motion assumed.
• Compliance assumed to be linear.
• Iron BH curve and compliance force stiffness
  curves do not include hysteresis.

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Kinematic Validation.
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Flux2D Transient Magnetic FEM with Kinematics
Results.

• Power density at 20Hz 40vrms input with coil
  motion.

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Flux2D Transient Magnetic FEM with Kinematics
Results.

• magnetic field in gap plotted along contour, at
  20Hz 40vrms input as a function of time.

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Flux2D Transient Magnetic FEM With Kinematics
Results.

• waveform and spectra of eddy currents in pole

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Flux2D Transient Magnetic FEM with Kinematics
Results.

• Acceleration waveform and spectra at 20Hz for
  40vrms input.

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Flux2D Transient Magnetic FEM With Kinematics
Results.

• Relative Harmonic distortion of cone acceleration.

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Flux2D Transient Magnetic FEM With Kinematics
Results.

• Measured Versus FEM of SPL Magnitude of
  impedance.

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Conclusion.

• Flux2D is able to predict loudspeaker motor
  distortion from geometry and material properties.
• FEM results contain a wealth of data including
  some which cannot be readily measured.
• Possible improvements are
• Include FEM derived Force versus Displacement
  data for suspension.
• Include magnetic Hysteresis of iron.
• Include mechanical Hysteresis of suspension.
• Better material property data required!