The Transient Magnetic Behaviour of Loudspeaker Motors. Mark Dodd

1FLUX Users Club 2001 - Aix en Provence - 27, 28 September

2001

The Transient Magnetic Behaviour

of Loudspeaker Motors.

Mark Dodd


2001

Loudspeaker Basics

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

Permanent Magnet

Iron pole

Iron pole

diaphragm moving as rigid body mass m

voice coil in magnetic gap

suspensions with combined spring constant k

Moving coil linear actuator: voltage -> motion.

Diaphragm: motion -> sound pressure.


2001

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:

F mx r x kxmc

F Bli

For a small current i:


2001

Loudspeaker Basics.

Pr

jU

j t kr

e 1

4

( )

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

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

U x S


2001

Pass Band Response.

Motion mass-limited, decreasing amplitude with frequency.

20 50 100 100 200 500 1k 1k 2k 5k 10k 10k 20k 20 20k

0

70u

0.14m

0.21m

0.28m

Frequency Hz

m


2001

Pass Band Response.

Velocity amplitude decreases above resonance frequency.

20 50 100 100 200 500 1k 1k 2k 5k 10k 10k 20k 20 20k

-40

-54

-68

-82

-96

-110

Frequency Hz

dB


2001

Pass Band Response.

Radiation efficiency increases with increasing frequency.

20 50 100 100 200 500 1k 1k 2k 5k 10k 10k 20 20k

100

1.2k

2.2k

3.3k

4.3k

Frequency Hz

ohm


2001

Pass Band Response.

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

20 50 100 100 200 500 1k 1k 2k 5k 10k 10k 20k 20 20k

60 0

70 16

80 32

90 48

100 64

110 80

Frequency Hz

dB

dBUin=2.83Vrms, Distance=1m

Motional impedance is reflected in electrical domain.


2001

Previous Approach

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

Calculate Flux through coil for an arbitrary current.

Total static vc emf _

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


2001

Polynomial method

Express force variation with current in terms of coil inductance:

L x ivc vc emf( ) _

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

F=Bl(x)I-Lvc(x)i2.


2001

Polynomial method

Eddy Currents.

AC signal through coil results in alternating magnetic field.

EMF induced in any conductive loops round field.

2

e td t

dt( )

( )

Current flow in skin depth.


2001

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).

Total static vc emf eddycurrent _


2001

Polynomial method

Equating forces.

Bl x idL x f

dxi mx r x k x xvc

mc( )( , )

( ) 2

U R i Bl xdL x f

dxx L x f

di

dtevc

vc ( ( )( , )

) ( , )

Runge Kutta method used to numerically solve equations.

Spectrum of distorted sinusoidal waveform produced is analysed with FFT.

Equating voltages.


2001

New Magnet Topology

Infiniteregion

NdFeB

NdFeB

Top plate

Pot

Yoke

Axis of rotational symmetry


2001

Flux2D Magneto-static FEM.

Flux2D Magneto-static FEM.

( )1

0 0

rcA J H

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

• Force on a coil is calculated using


2001

Flux2D Magneto-static FEM Results.

Flux2D Magneto-static FEM Results.

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


2001

Flux 2D Magneto-static FEM Results.

Flux 2D Magneto-static FEM Results.

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

Inductance versus current for static coil.

0

12

34

5

67

89

10

-10.00 -5.00 0.00 5.00 10.00

Current A.

Inducta

nce m

H


2001

Flux2D Magneto-dynamic FEM

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.


2001

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.


2001



Mesh uses quadrilateral elements to mesh ‘skin’ and give a fast solution time.

Mesh has frequency dependant spacing perpendicular to metal surface.


2001


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.


2001



Flux at 80Hz.


2001



Permeability at 80Hz.


2001



Power density currents at 80Hz various coil displacements.


2001



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


2001

Flux2D Transient Magnetic FEM With

Kinematics

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.


2001

Kinematic Validation.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

-1

0

1

2Force (Acceleration) of Kinematic Test.

Time S

For

ce N

1 2 3 4 5 6 7 8 9 10 11-200

-150

-100

-50

0Spectrum of Kinematic Test.

Frequency Hz

dB (

re f

unda

men

tal)


2001

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

Flux2D Transient Magnetic FEM with Kinematics Results.



2001



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


2001

Flux2D Transient Magnetic FEM With Kinematics Results.


waveform and spectra of eddy currents in pole

0.15 0.2 0.25-400

-200

0

200Current Through iron.

Time s

curr

ent

A

20 40 60 80 100 120 140 160 180-50

-40

-30

-20

-10

0

Frequency Hz

dB I

re:

Fun

dem

enta

l

Spectrum of Current through iron,


2001

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



0.15 0.2 0.25

-50

0

50

Voice Coil acceleration

Time S

N

0 20 40 60 80 100 120 140 160 180 200-50

-40

-30

-20

-10

0

10

Frequency Hz

dB S

PL

re f

unde

men

tal

Spectrum of Voice Coil acceleration


2001



-50

-40

-30

-20

-10

0

10

10 100 1000 10000

Frequency Hz.

dB

(re

fun

dem

enta

l)

2F FEM

3F FEM

2F measured

3F measured

Relative Harmonic distortion of cone acceleration.


2001

10

30

50

70

90

110

130

10 100

Frequency Hz

dB

SP

L, d

B |z

| re

1oh

m.

SPL 200w FEA

SPL 200w measured

|Z| 200w FEA

|Z| 200w measured



Measured Versus FEM of SPL & Magnitude of impedance.


2001

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!

Conclusion.Conclusion.

Documents

The Transient Magnetic Behaviour of Loudspeaker Motors. Mark Dodd