modeling, fem analysis and dynamic simulation of a moving...

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

ezio.santini@uniroma1.it, sabrina.teodori@uniroma1.it

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