AES - Voice Coil Impedance as a Function of Frequency and Displacement

Audio Engineering Society

Convention PaperPresented at the 117th Convention

2004 October 28–31 San Francisco, CA, USA

This convention paper has been reproduced from the author's advance manuscript, without editing, corrections, or considerationby the Review Board. The AES takes no responsibility for the contents. Additional papers may be obtained by sending requestand remittance to Audio Engineering Society, 60 East 42nd Street, New York, New York 10165-2520, USA; also see www.aes.org.All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from theJournal of the Audio Engineering Society.

Voice Coil Impedance as a Function of Frequency andDisplacement

Mark Dodd1, Wolfgang Klippel2, and Jack Oclee-Brown.3

1 KEF Audio UK (Ltd),Maidstone,Kent,ME15 6QP United [email protected]

2 Klippel GmbH, Dresden, 02177, [email protected]

3 KEF Audio UK (Ltd),Maidstone,Kent,ME15 6QP United [email protected]

ABSTRACT

Recent work by Klippel [1] and Voishvillo et al.[2] has shown the significance of voice coil inductance in respect tothe non-linear behaviour of loudspeakers. In such work the methods used to derive distortion require the inductanceto be represented by an equivalent circuit rather than the frequency domain models of Wright and Leach. A newtechnique for measurement of displacement and frequency dependant impedance has been introduced. The complexrelationship between coil impedance, frequency and displacement has been both measured and modelled, using FE,with exceptional agreement. Results show that the impedance model requires that its parameters vary independentlywith x to satisfactorily describe all cases. Distortion induced by the variation of impedance with coil displacement ispredicted using a lumped parameter method, this prediction is compared to measurements of the actual distortion.The possibility of using an FE method to predict distortion is demonstrated. The nature of the distortion is discussed.

Dodd et al. Voice coil impedance

AES 117th Convention, San Francisco, CA, USA, 2004 October 28–31 of 19

1. INTRODUCTION

In order to describe the performance of a loudspeakerprecisely, one must consider the electrical inputimpedance at higher frequencies. The voice coil doesnot operate in free air but close to conducting andmagnetically permeable structures: the pole tips, themagnet, the voice coil former, copper rings etc. Theimpedance can be only roughly modelled as a resistorRe and an ideal inductance Le, the occurrence of eddycurrents in these structures usually decreases theinductance of the coil and increases losses at higherfrequencies [3].

The nature of the voice coil impedance is dependentupon the geometry and physical properties of themagnet and surrounding assemblies. There are threeprominent methods of deducing the coil (blocked)impedance for a particular loudspeaker.

- The blocked impedance may be measured byclamping the voice coil stationary within themagnet assembly to remove the motional part of theimpedance. Blocked impedance may then bemeasured as with conventional impedancemeasurements.

- Various impedance models have been developed inorder to describe the electrical behaviour of thecoil, these models may be fitted to conventionalunblocked impedance measurements.

- Recently magnetic transient FEM has beensuccessfully applied to deduce the blockedimpedance [4].

At large amplitudes the impedance also variessignificantly with voice coil displacement. Whereas thefrequency dependence of the impedance is a linearphenomenon effecting the amplitude response, but notgenerating distortion, variation of impedance withdisplacement generates significant harmonic and inter-modulation distortion.

The nature of the impedance generated distortion is adependent upon the coil impedance, or more preciselyhow the coil impedance varies with frequency &displacement of the coil.

This paper investigates the voice coil impedance as afunction of frequency and of displacement.Measurements of frequency & displacement dependentimpedance have been made using a quasi-static methodusing the LPM module of the Klippel analyser [5].Frequency & displacement dependent impedance has

also been predicted using FEM analysis. A newimpedance model with displacement varying parametersis discussed and fitted to the measured data. Finally,distortion induced by the variation of impedance withcoil displacement is predicted using both lumpedparameter and FE methods, these predictions arecompared to measurements of the actual distortion. Thenature of this distortion is discussed.

1.1. Glossary

Bl(x) force factor (Bl product) Cms(x) = 1 / Kms(x) mechanical

compliance of loudspeaker suspension Exm exponent of imaginary part in WRIGHT

modelErm exponent of real part in WRIGHT

modelFm(x,i) reluctance forceKrm factor of real part in WRIGHT modelKxm factor of imaginary part in WRIGHT

modelK factor in LEACH modelKms(x) mechanical stiffness of loudspeaker

suspension Le(x) inductance used (cascaded model)L2(x) inductance (cascaded model)Leff(f,x) effective inductance depending onMms mechanical mass of loudspeaker

diaphragm assembly including air loadand voice coil

n exponent in LEACH modelRms mechanical resistance of total-

loudspeakerRe electrical voice coil DC-resistanceR2(x) resistance (cascaded model)Reff(f,x) effective losses due to eddy currents

depending on frequency anddisplacement

ZL(jω,x) complex excess impedance representingthe effect of the lossy inductance(motional impedance and Re isremoved)

� angular frequency �= 2�f



2. MODELLING

2.1. Lumped Parameter Model

MmsCms(x) Rms

Bl(x)

Re

v

Fm(x,I)

I

Bl(x)v Bl(x)IU

ZL(jω,x)

Figure 1 Electro-mechanical equivalent circuit of the loudspeaker

Figure 1 shows a simple equivalent circuit of aloudspeaker system. The dominant nonlinearities are thevariation of the force factor Bl(x), the mechanicalcompliance Cms(x) and the electrical impedanceZL(jω,x) with voice coil displacement x. The DCresistance Re and the motional impedance are notconsidered in the electrical impedance ZL(jω,x). Thusthe impedance ZL(jω,x) may be measured at theelectrical terminals by blocking the movement of thecoil and subtracting the resistance measured at a verylow frequency.

Different linear models have been developed to describethe frequency dependency of ZL(jω,x) with a minimalnumber of free parameters:

2.1.1. LEACH model

M. Leach [6] proposed a weighted power function of thecomplex frequency as an approximation for ZL

ZL(jω)= K·(jω)n ; ω= 2�f (1)

Although using only two free parameters this functioncan sometimes give a very good fit over a widefrequency range. Unfortunately, this function can not berepresented by an electrical equivalent circuit nor asimple digital system.

2.1.2. LR-2 Model

This model uses a series inductance Le connected to asecond inductance L2 shunted by resistance R2.

ZL(jω) = Le·jω + (R2·L2·jω ) / (R2 + L2·jω) (2)

Although this model uses three free parameters it oftenprovides a worse fit to measured ZL than the LEACHmodel. However, this model may be realised as an

electrical equivalent circuit (as shown in Figure 2 b) oras a digital IIR filter.

Le(x)Re

L2(x)

R2(x)

a)ZL(jω,x)

Reff(f, x)Leff(f, x) c)

b)

Figure 2 Representation of the electrical impedance

2.1.3. WRIGHT model

Wright [7] proposed a model using separate weightedpower functions in ω for both the real and imaginarypart of impedance.

ZL(jω)= Krm·ωErm + j·(Kxm·ωExm ) (3)

This model uses four free parameters and normally gives abetter fit than the other models with less parameters.Unfortunately, this function can not be directly realised asan analogue or digital system.

2.1.4. Effective inductance

ZL(jω) = Leff·(f)jω + Reff(f) (4)

M. Leach also proposed normalising the imaginary partof the electrical impedance ZL(jω) to the frequency jωand introducing an effective inductance Leff(f) whichvaries with frequency. The real part of ZL(jω) may beconsidered as a frequency depending resistance Reff(f)describing the losses due to eddy currents as shown inFigure 2c . Though the number of parameters is veryhigh , two parameters for each frequency point, bothparameters are easy to interpret and convenient forgraphical representation.

2.1.5. Large signal modelling

The linear models may be easily expanded to higheramplitudes by allowing each parameter to be dependentupon the displacement x.

For example considering the LR-2 model, the threeparameters Le(x), R2(x) and L2(x) are functions of thedisplacement x and may be approximated by a truncatedpower series expansion such as



0

( )N

ie i

i

L x l x=

=�(5)

20

( )N

ii

i

R x r x=

=�(6)

20

( )N

ii

i

L x xλ=

=�(7)

where the coefficients li, ri and �i are the free parametersof the model.

3. MEASUREMENT

The application of a model to a particular real objectusually requires an estimation of the free modelparameters in such a way that the model describes thereal object with maximal accuracy.

With the linear models straightforward techniques areavailable which may be applied for loudspeakers atsmall amplitudes.

Non-linear models require special techniques for theparameter identification. Static, quasi-static and fulldynamic techniques have been developed to measurethe force factor Bl(x), compliance Cms(x) andinductance parameter Le(x). The dynamic techniqueshave the advantage that an audio-like ac signal is usedfor excitation and the loudspeaker is operated underworking conditions.

The current version of the LSI module in the Klippelanalyser performs a dynamic measurement using a noisestimulus [8]. The free model parameters are optimisedto give the best fit between measured and modelledcurrent and displacement. The LR-2 model is currentlyconstrained so that the three lumped parameters varywith the same shape in x.

2 2

2 2

( ) ( ) ( )(0) (0) (0)

e

e

L x R x L xL R L

≈ ≈ .(8)

Although this assumption is a good approximation formost loudspeakers without shorting rings it is a purposeof the paper to investigate the validity of thisapproximation for more elaborate loudspeakers usingcopper cups and aluminium rings for reducing theinductance nonlinearity.

Clearly the measurement of suspension stiffness using adc offset gives significantly different results from adynamic measurement due to creep, relaxation and other

time dependent properties. Additionally, due to fluxmodulation, the force factor Bl(x) and the inductanceLe(x) are dependent on the current i. Thus using anaudio-like signal will produce far more meaningful datathan measurement with extremely small input current(coil offset generated by external force or pressure) orextremely large currents (coil offset generated by a dccurrent).

Despite these limitations a quasi-static technique is auseful method for investigating the variation of theimpedance with both frequency and displacement.

Measurements have been performed on two testloudspeakers. Both loudspeakers share the followingspecifications:

Voice coil diameter: 2” nominal (51.30mm ID)Voice coil DCR: 6.72 OhmsTurns in Voice coil: 126 TurnsFerrite ring magnetAnnular low carbon steel top-platePole/plate assembly type yoke.

The magnet assembly of loudspeaker 1 has analuminium shorting ring placed above the magnetic gap.The magnetic assembly of loudspeaker 2 has analuminium shorting ring placed below the magnetic gap.The geometry of the two loudspeakers is shown inFigure 26 & Figure 27. The pair serve to illustrate theinfluence of the effect of aluminium rings on thevariation of the voice coil inductance with displacement.

3.1. Mechanical Setup

A method for measurement of the displacementdependent impedance was developed at GP Acoustics(UK). The method is a simple modification to thestandard Linear Parameter Measurement (LPM) of theKLIPPEL Analyser system. The measurementspresented in this paper were performed by KlippelGmbH.

The loudspeaker is clamped in vertical position in theprofessional loudspeaker stand as shown in Figure 3.

An additional spider is attached to the diaphragm.

The spider holds an inner clamping part made ofaluminium which is secured to the lower rod (usuallyused for holding the microphone).

By shifting the lower rod a displacement may beimposed to the coil position. Clearly displacing the coilwill also change the other parameters such as Bl(x),Cms(x) and also loss factors.



The Distortion Analyser also provides a DisplacementMeter ([5] Hardware) this is used for measuring theoriginal rest position of the cone and to measure theimposed static displacement. Throughout this paper theconvention is that a positive displacement of the coilrefers to movement out of the magnet assembly.

Figure 3 Measurement Setup.

The loudspeaker under test is connected to DistortionAnalyser 2 allowing a simultaneous measurement ofvoltage, current and displacement signal.

Figure 4 Generating an additional DC offset by the lower rodconnected by an addition spider to the diaphragm.

3.2. Small signal measurements

The additional spider increases the stiffness of the totalsuspension and the resonance frequency. However, themodified loudspeaker is still a second-order system andcan be represented by the equivalent circuit in

Figure 1.

The module Linear Parameter Measurement (LPM) ofthe KLIPPEL Analyser system is used to measure thelinear parameter at each prescribed displacement.

The loudspeaker is excited by a multitone signal of 0.5V rms at the terminals. Since the voltage, current anddisplacement are measured simultaneously all of thelinear parameters can be identified instantaneously. Anadditional measurement with a mechanical perturbation(additional mass or measurement in a test enclosure) isnot required.

A sparse multitone signal used as excitation signalallows assessment of the distortion generated by theloudspeaker. During the small signal measurements themaximum distortion occurred 20 dB below thefundamental lines in the current spectrum. This showssufficiently linear operation of the loudspeaker [9].

3.3. Fitting of the inductance model

At first the linear parameters are measured at the restposition (x=0) and the different inductance models (LR-2, WRIGHT, LEACH) are used to describe theimpedance response, measured up to 18 kHz.

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

Impe

danc

e [O

hm]

Frequency [Hz]

10

20

30

40

50

60

70

80

90

100

110

Measured

LR-2

LEACH

WRIGHT

Figure 5 Magnitude of electrical impedance of loudspeaker 1measured and fitted by LR-2, Wright and LEACH model up to 18

kHz.



Figure 5 shows a measured curve and the fitted curvesusing the three models. The LEACH and WRIGHT areable to describe this particular impedance curve verywell. The LR-2 causes minor deviations about 500 Hzand 5 kHz. Although the WRIGHT usually gives thebest fit there are cases where the other models haveprovided a superior fit.

Since the test loudspeaker is based on a woofer intendedfor frequencies below 200 Hz the models are have alsobeen fitted using data only up to 2 kHz, Figure 6. In thisinstance all models are able to give a good fit to themeasured curve.

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

Impe

danc

e [O

hm]

Frequency [Hz]

10

20

30

40

50

60

70

80

90

100

110

Measured

WRIGHT

LEACH

LR-2

Figure 6 Magnitude of electrical impedance of loudspeaker 1measured and fitted by LR-2, WRIGHT and LEACH model up to 2

kHz.

3.4. Excess impedance ZL

The LPM module also calculates the amplitude andphase response of the excess impedance ZL(jω). Themeasured and the fitted curves are shown in Figure 7using the LEACH model. The measured curves arecalculated by subtracting the estimated dc resistance Re

and the motional impedance (calculated from theestimated parameters Bl, MMS, RMS, and CMS) from thetotal electrical input impedance. The magnitudeincreases with frequency, normally with a slope usuallyless than 6 dB per octave. The LEACH model uses aconstant slope corresponding with the exponent n inEquation (1). Close to loudspeaker resonance themagnitude of the calculated excess impedance variessignificantly. In this region the motional impedance isvery high (~100 Ohm), measurement and modellingerror (in the order of 1 %) will be assigned to the excessimpedance and makes accurate estimation of the excess

impedance below 100 Hz impossible. The calculatedphase of the excess impedance is about 68 degrees witha small decay to higher frequencies. The LEACH modelassumes a constant phase, which proves to be a goodapproximation for this particular loudspeaker.

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

hm]

[deg]

Frequency

-110

010

110

-150

-100

-50

0

50

Magnitude (measured)

Magnitude (fitted)

Phase (measured)

Phase (fitted)

Figure 7 Magnitude and phase of the electrical impedance ofloudspeaker 1 ZL(jω) measured (solid lines) and fitted by using the

LEACH model (dotted lines).

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

[Ohm

]

[deg]

Frequency

-210

-110

010

110

-150

-100

-50

0

50

100


Magnitude (fitted)

Phase (measured)

Phase (fitted)

Figure 8 Magnitude and phase of the electrical impedance ZL(jω) ofloudspeaker 1 measured (solid lines) and fitted by using the LR-2

model (dotted lines).

Figure 8 shows the excess impedance match using theLR-2 model. While the fitting above 1 kHz is good, atlower frequencies there are significant differences inboth phase and amplitude. The LR-2 model, and alsoother shunted models using more Li and Ri elements (i >2), behaves as an ideal inductance at very lowfrequencies giving a 6dB per octave slope and a phaseshift of 90 degrees. This property corresponds with theobservation that the eddy currents are frequency



dependent and will vanish at very low frequencies.Though a small increase in the measured phase at lowfrequencies supports this observation, the phase shift ofthe LR-2 model begins at a higher frequency than themeasured shift. Thus the LR-2 is usually limited to useover a frequency band of two decades. Using anadditional shunted section (R3 and L3) improves the fitsignificantly and results in a good description over thewhole audio band (three decades). The cascade ofshunted inductances is a minimum-phase system andcan be realised in the analogue or digital domain.

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

[Ohm

]

[deg]

Frequency [Hz]

-110

010

110

-150

-100

-50

0

50


Magnitude (fitted)

Phase (measured)

Phase (fitted)

Figure 9 Magnitude and phase of the electrical impedance ZL(jω) ofloudspeaker 1 measured (solid lines) and fitted by using the WRIGHT

model (dotted lines).

The excess impedance fitted by the WRIGHT model isshown in Figure 9. The magnitude response can beapproximated by smooth line having a different slope atlow and high frequencies (corresponding withexponents Erm and Exm). Contrary to the LEACHmodel the phase is not constant but depends on all fourparameters. The WRIGHT model also considers theresponse below 100 Hz, where measurement errors andnoise have corrupted the measurement, and generates adecrease of phase shift at very low frequencies. Sincethe WRIGHT model is not bounded to be minimumphase and not composed from a system lumpedelectrical elements it may be deceived by measurementartefacts when used to represent a measured curve. Thusa good match with the measured impedance curve doesnot guarantee that the parameters are meaningful.

3.5. Impedance versus displacement

Once measurements had been performed at rest position(x=0) a dc offset was imposed upon the coil using thelower rod in Figure 4. The Displacement Meter at the

hardware unit, Distortion Analyser 2, was used tomeasure the offset. The LPM module was then againused to measure the linear parameters. The excessimpedance ZL is displayed for loudspeaker 1 in Figure10 for a negative offset of -8mm (coil in) and a positiveoffset of 7.5mm (coil out), the impedance at the restposition is also shown. The variation of the impedancewith displacement may be clearly seen. It may beobserved that the inductance of the coil is effectivelyreduced in the region near to the aluminium shortingrings, above the gap in this case. Conversely as the coilmoved into the magnetic assembly, the effectiveinductance is increased as the coil moves away from thepole piece and top plate.

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

[Ohm

]

Frequency [Hz]

0

10

20

30

40

50

60

70

- 8 mm

0 mm7.5 mm

Figure 10 Magnitude of electrical impedance ZL(jω) of Loudspeaker 1measured at rest position (solid line), at –8 mm (dotted line) and 7.5

mm (dashed line)

The phase of the excess impedance ZL was alsocalculated and is shown in Figure 11 at the same coilpositions. Whereas the impedance magnitude varies byup to 40 % at higher frequencies the phase stays almostconstant at 70 degrees.

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

[deg]

Frequency [Hz]

-150

-100

-50

0

50

100

150

-8 mm

0 mm

7.5 mm

Figure 11 Phase of electrical impedance ZL(jω) of Loudspeaker 1measured at rest position (solid line), at –8 mm (dotted line) and 7.5

mm (dashed line)



3.6. Non-linear Parameters

Having identified the linear parameters for differentvalues of coil offset, the displacement dependency ofthe parameters were be calculated using the MathProcessing Software (MAT), a free programmable(SCILAB or MATLAB) module [10] for the KLIPPELAnalyser. This module imports all of the resultsmeasured by the LPM for all measured voice coiloffsets and calculates the coefficients li, ri and �i inequations (5) - (7).

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

-5.0 -2.5 0.0 2.5 5.0

[mH

]

Displacement X [mm]

Le(X)

L2(X)

Figure 12 Inductance Le(x) and L2(x) of loudspeaker 1 versusdisplacement x.

10.0

12.5

15.0

17.5

20.0

22.5

-5.0 -2.5 0.0 2.5 5.0

[Ohm

]

Displacement X [mm]

R2(X)

Figure 13 Resistance R2(x) of loudspeaker 1 versus displacement x.

Figure 12 and Figure 13 show the parameters Le(x),

L2(x) and R2(x) of the LR-2 model versus displacementx for loudspeaker 1 (ring above the gap). The LR-2model is used as the parameters have an analoguerepresentation and are easy to interpret. Corresponding

to our observations regarding the measured excessimpedance, it can be seen that the fitted model alsoidentifies the effective inductance as reducing when thecoil is close the aluminium shorting ring above themagnetic gap. It is interesting to observe that the shapeof the parameter functions; Le(x), R2(x) and L2(x); isvery similar in this instance. In this case the assumptionof (8) appears to be valid.

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

100 101 102 103 104

[mH

]

Frequency [Hz]

-7.0 mm0.0 mm

7.0 mm

Figure 14 Effective inductance Leff(f,x) versus frequency f ofloudspeaker 1 plotted for the rest position (solid line) and –7 and +

7mm displacement (dotted and dashed line, respectively).

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

100 101 102 103 104

[Ohm

]

Frequency [Hz]

-7.0 mm

-0.0 mm

7.0 mm

Figure 15 Effective resistance Reff(f,x) versus frequency f ofloudspeaker 1 plotted for the rest position (solid line) and –7 and +

7mm displacement (dotted and dashed line, respectively).

Use of the effective resistance Reff(f,x) and the effectiveinductance Leff(f,x), as defined in equation (4) andillustrated in Figure 2 c, simplifies interpretation of theexcess impedance. Figure 14 shows the effective



inductance Leff(f,x) for three different voice coildisplacements based on the LR-2 model. It is clearlyshown that the voice coil inductance decreases if thecoil moves outwards and increases as the coil moves in.

At low frequencies the effective inductance Leff(f,x) isequal to the sum of Le(x) and L2(x) and the effectiveresistance Reff(f,x), Figure 15, is close to zero. At highfrequencies the Leff(f,x) is equal to Le(x) only and theeffective resistance Reff(f,x) becomes equal to R2(x).

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

-5.0 -2.5 0.0 2.5 5.0

[mH

]

Displacement [mm]

133 Hz

1546 Hz

18000 Hz

Figure 16 Effective inductance Leff(f,x) versus displacement x ofloudspeaker 1 plotted for frequencies 133 Hz, 1545 Hz and 18 kHz.

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

-5.0 -2.5 0.0 2.5 5.0

[Ohm

]

Displacement [mm]

133 Hz

1546 Hz

18000 Hz

Figure 17 Effective resistance Reff(f,x) versus displacement x ofloudspeaker 1 plotted for frequencies 133 Hz, 1545 Hz and 18 kHz.

Figure 16 and Figure 17 shows the variation of effectiveinductance and resistance versus displacement x forselected frequencies 133Hz, 1545Hz and 18kHz.

Clearly all the curves have a distinct asymmetry anddecrease with positive displacement.

A second loudspeaker has been made using the samesuspension and motor structure but with the aluminiumring located below the magnetic gap.

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

[Ohm

]

Frequency [Hz]

0

10

20

30

40

50

60

70

80

-7.5 mm

0 mm

7 mm

Figure 18 Magnitude of electrical impedance ZL(jw) of loudspeaker 2(ring below) measured at rest position (solid line), at –7.5 mm (dotted

line) and 7 mm (dashed line)

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

[deg

]

Frequency [Hz]

-150

-100

-50

0

50

100

150

-7. 5 mm

0 mm 7 mm

Figure 19 Phase of electrical impedance ZL(jw) of Loudspeaker 2measured at rest position (solid line), at –7.5 mm (dotted line) and 7

mm (dashed line)

The magnitude and phase of the electrical impedanceZL(jω) of the second loudspeaker are shown in

Figure 18 and

Figure 19 for three coil displacements.



0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

100 101 102 103 104

[mH

]

Frequency [Hz]

-7.0 mm

-0.0 mm

7.0 mm

Figure 20 Effective inductance Leff(f,x) versus frequency f ofloudspeaker 2 (ring below) plotted for the rest position (solid line) and

–7 and + 7mm displacement (dotted and dashed line, respectively).

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

100 101 102 103 104

[Ohm

]

Frequency [Hz]

-7.0 mm

-0.0 mm

7.0 mm

Figure 21 Effective resistance Reff(f,x) versus frequency f ofloudspeaker 2 (ring below) plotted for the rest position (solid line) and

–7 and + 7mm displacement (dotted and dashed line, respectively).

The effect of changing the location of the shorting ringcan be seen most clearly in Figure 20 & Figure 21.Loudspeaker 1 exhibited an effective inductance whichdecreased as the coil moved out of the magnetic gap.This trend is close to the reverse for loudspeaker 2, theeffective inductance of the coil remains almost constantas the coil moves out the gap and into free air. When thecoil moves inward toward the now internally locatedring, the effective inductance is seen to fall.Additionally, the phase varies significantly more withthe displacement.

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

-5.0 -2.5 0.0 2.5 5.0

[mH

]

Displacement [mm]

133 Hz

1546 Hz

18000 Hz

Figure 22 Effective inductance Leff(f,x) versus displacement x ofloudspeaker 2 (ring below) plotted for frequencies 133 Hz, 1545 Hz

and 18 kHz.

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

-5.0 -2.5 0.0 2.5 5.0

[Ohm

]

Displacement [mm]

133 Hz

1546 Hz

18000 Hz

Figure 23 Effective resistance Reff(f,x) versus displacement x ofloudspeaker 2 (ring below) plotted for frequencies 133 Hz, 1545 Hz

and 18 kHz.



0.35

0.40

0.45

0.50

0.55

0.60

0.65

-5.0 -2.5 0.0 2.5 5.0

[mH

]

Displacement X [mm]

Le(X)

L2(X)

Figure 24 Inductance Le(x) and L2(x) of the loudspeaker 2 (ringbelow) versus displacement x.

14

15

16

17

18

19

20

21

-5.0 -2.5 0.0 2.5 5.0

[Ohm

]

Displacement X [mm]

R2(X)

Figure 25 Inductance R2(x) of the loudspeaker 2 (ring below) versusdisplacement x.

Figure 24 and Figure 25 show the estimated parametersLe(x), R2(x) and L2(x) of the LR-2 model. Contrary tothe parameters of loudspeaker 1 both the inductance L2

and the shunt R2 decay symmetrically for positive andnegative displacement. The inductance Le(x) is stillasymmetric but increases in an unusual way if the coilmoves outwards. This is a very interesting result as itdemonstrates a case were the assumption of (8) wouldnot be valid.

4. FEM MODELING

4.1. The Use of FEM

The most obvious application of FEM is for literalmodelling in which the aim is to produce a resultequivalent to a measurement of some particular aspectof performance such as distortion spectra or frequencyresponse. FEM allows the geometry and materialproperties to determine the behaviour by applying theappropriate physical laws. However, while this gives awealth of information about how a loudspeaker behavesunder particular circumstances it does not explain thebehaviour or help to improve it.

A different approach is to represent the device with asystem of analytic equations having variablesrepresenting simplified physical aspects. Thisparameterised approach allows the engineer to workwith manageable data and to clearly identify howimprovements to a design may be realised in terms ofthe simplified parameters. However, while this gives aclear set of specifications & targets for design orredesign it does not tell the engineer whether or howthese may be achieved.

It is clearly most effective to use FEM and lumpedparameter modelling together along with measurementsto allow the reduction, as far as is possible, of theassumptions within the lumped parameter modelsthrough, for example, use of various FEM derived ormeasured transfer functions & parameters. Use of FE asa design tool has forced the development of this strategy[4]. The process essentially allows the engineer toconsider a particular FEM result not alone but withinthe context of other physical systems & parameters andin a way to which they are accustomed. Secondly theuse of the methods together allows the engineer a routefrom the parameters to the physical systems. Anyidentified parameter goals may be readily researchedusing the relevant FE model to investigate how orwhether improvement may be achieved, and this isperformed in terms of the actual physical geometry andmaterials of the design not comparatively arbitraryparameters.

Until FEM evolves to the extent that a model is able tofully represent all the physical systems of a loudspeakerand hardware allows physical changes to be computedin real time, linking various results & measurementsusing assumptions and relationships from lumpedparameter modelling is the most puissant methodcurrently available.



4.2. Modelling Method

A discretized model of the magnet assembly wasproduced using the Flux2D a programme supplied byCEDRAT. The model domain was axisymmetric withsecond order elements. The model boundary is definedwith an outer annular ‘infinite region’. In this region theelements have a modified co-ordinate system in whichthe outer nodes are at an infinite distance. Thistechnique avoids the errors due to modelling only asmall region of space [11][12].

Evaluating the impedance requires the current andvoltage through the voice coil. This has been achievedby means of Transient Magnetic FEM in which avoltage source has been coupled to the voice coil region[11]. The result we are seeking is the current at eachtime step.

In the coil region of the FEM model the current isrestricted to flow uniformly since the coil is a strandedconductor. In the other conductive regions the magneticforces produced by the current are allowed to force thecurrent flow into a skin on the surface of the conductor.The discretization is critical for this analysis and asuitable ‘skin’ of quadrilateral elements must be formedon the outer surfaces of conductive regions [11]. Aftersufficient time-steps for the starting transient to settle, asteady-state waveform of current versus time may beextracted from the solution files, along with the drivingvoltage. This analysis includes the effect of eddycurrents induced in the pole and the aluminium ring;these may be seen in Figure 27 and Figure 26. Theeffect of the eddy currents is to produce a field opposingthe voice coil flux. This has the effect of reducing thevoice coil impedance.

Figure 26 Loudspeaker 1. Lines of constant power density at 31.25Hz,x=4.

Figure 27 Loudspeaker 2. Lines of constant power density at 31.25Hz,x=-5.75mm

In both loudspeakers the lines of equal power densityare dense in the steel pole surface and more widelyspaced in the aluminium rings. This is largely due topermeability of the steel being very much higher. Withincreasing frequency both the extent and thickness ofthe skin reduce.

The current waveform flowing through the coil isextracted by evaluating the current through the coil ateach timestep. A typical result is shown in Figure 28.

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Figure 28 Current waveform for Loudspeaker 2 at 31.25Hz, x=-5.75mm.

Subsequently the voice coil’s ‘blocked’ electricalimpedance may be calculated by applying Ohm’s law tothe fundamental components of the waveforms. Eachloudspeaker was solved for seven frequencies in fivepositions. The positions were chosen to be coincidentwith the measurement positions.

4.3. FEM results

Figure 29 shows the FE calculated impedance comparedto the measured data for loudspeaker 1 with the coil inthe rest position. At low frequencies the small errors inthe modelled mechanical impedance cause somesignificant artefacts in the measured data. At higherfrequencies the agreement is very good.



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Figure 29 Example FE impedance calculation (dotted) measured data(solid). x=0 loudspeaker 1.

At around 1kHz the measured value is consistentlybelow the FE value. To determine the cause of thisdiscrepancy a conventional impedance measurement ofloudspeaker 2 was made. The loudspeaker voice coilwas then glued in position and a direct measurementmade of the blocked impedance. The result in Figure 30clearly shows that at 1kHz the blocked impedance ishigher than the free impedance. This curious differenceis thought to be an artefact resulting from the motionalimpedance due to non-pistonic diaphragm motion.

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Figure 30 Conventional impedance measurement (solid) directlymeasured blocked impedance (dashed) loudspeaker 2.

The impedance results are illustrated opposite togetherwith the equivalent data from the measured values usingthe method described in Impedance versus displacementsection. To allow the large number of results to becompared contour plots have been used.

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Figure 31 upper graph FEM impedance magnitude loudspeaker 1.Lower graph measured data of loudspeaker 1.

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Figure 32 upper graph FEM impedance magnitude of loudspeaker 2.Lower graph measured data of loudspeaker 2.

It is evident that in both cases the FEM yields the sametrend of impedance variation as the measurements. Themeasured data exhibits some noise around thefundamental resonance as mentioned in Excessimpedance. There is also evidence of non-pistonicmodal behaviour which has resulted in non-inductiveimpedance variations.



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Figure 33 Loudspeaker 2 upper graph real part of FEM impedance.lower graph imaginary part of FEM impedance.

Figure 33 shows the real and imaginary parts of theFEM derived impedance for loudspeaker 2. Thedifference in symmetry of the impedance mentioned inNon-linear Parameters can be clearly seen.

5. PREDICTION OF DISTORTION

5.1. Lumped parameter-based prediction

If non-linear parameters of the equivalent circuit inFigure 1 are measured then the state signal (current,displacement) and the sound pressure output can bepredicted for any excitation signal. This technique isuseful for several applications: diagnostic testing of aloudspeaker; assessment or improvement of a newdesign; auralization of large signal performance withmusic or other test signals [1]. The Simulation module(SIM) [13] has been used to predict the distortion basedupon the measured non-linear parameters Le(x), R2(x)and L2(x) above.

The results of these simulations may be directlycompared with the results of the 3D Distortionmeasurement module (DIS) [14] which provide a directmeasure of the actual distortion of the loudspeaker. Inthis way we are able to validate the measured non-linearparameters of the LR-2 model by comparing predictedand measured distortion generated in the input current.

Measurement of distortion in the electrical input currentreveals the distortion generated by the varying input

impedance almost directly (provided the loudspeaker isconnected to an amplifier with a suitably low outputimpedance). The distortion generated in the current willalso appear in the displacement, velocity and soundpressure output. However, the distortion generated bydisplacement varying force factor Bl(x) and complianceCMS(x) appear in the current only close to the resonancewhere the velocity and back EMF are high. Dopplerdistortion and any other radiation distortion will not bedetected in the input current.

A very important aspect is the selection of the teststimulus. A single tone reveals only harmonic distortionwhich is, for displacement varying input impedanceZL(jω,x), relatively low. This is because at lowfrequencies the variation of ZL(jω,x) with displacementx is relatively small, seen in Figure 10 and

Figure 18 for both loudspeakers. In addition at highfrequencies the voice coil displacement becomes verysmall and the variation of ZL(jω,x) with displacement isminimal.

However, a two-tone signal comprising a bass tone atfrequency f2 and a probe tone at higher frequency f1 is amuch more revealing signal [15]. The bass tone is set toa fixed frequency below resonance to produce a largedisplacement of ~5mm peak. The second ‘probe’ tone isvaried from 200Hz to 18 kHz to represent any audiosignal in the pass-band of the loudspeaker. The varyingimpedance ZL generates not only harmonics of bothtones but additionally difference and summed-toneintermodulation components, which may exceed theharmonics significantly.

There is a simple relationship between shape of a non-linear parameter and the order of the resulting distortioncomponent. A distinct asymmetry of the parameter,such as in Figure 16 and Figure 17, will generatedominant 2ndorder distortion, which will outweigh3rdorder and higher distortion. Conversely a symmetricalcurve, such as in L2(x) and R2(x) in Figure 24 andFigure 25, will generate strong 3rd-order and other odd-order distortion components.

5.2. FEM prediction

The FE provides an alternative means of predicting thedistortion. Using the FEM software a full kinematicsolution for an arbitrary input signal may be computed.This solution can again be used to predict the distortion,this time from FE models adapted from those developedin order to model the voice coil impedance. For thepurpose of this paper, this type of FEM solution appears



for only one frequency; the solution time is very longand use of this type of modelling is thus restrictive.The two-tone intermodulation stimulus must be solvedwith a small time step defined by the higher frequencyover two periods of the lower frequency, provided thatthe frequencies are integral multiples. It is estimatedthat solving with a two-tone intermodulation stimulusfor the seven octave spaced frequencies up to 2kHzwould take 180 hours of processor time on a 2GHz PC.

As we have seen, the results of the FE correlate wellwith measured ZL(jω,x). The use of the measured non-linear parameters to calculate the distortion is dependentupon the LR-2 fitting and the assumption that the modelis adequate to describe the behaviour of the system.Additionally, as previously discussed, the measurementmethod (quasi-static) may have a bearing upon thededuction of the LR-2 parameters and indeed themeasured ZL(jω,x). The full kinematic analysis does nothave these limitations as it returns directly to thefundamental physical relationships in order to calculatethe system output. It is also able to account for morecomplex phenomena such as the effect of currentmagnitude on the impedance response. The FE allowsapplication of specific laws of physics to model thebehaviour whereas the lumped parameter modelmatches specific effects in such a way that the resultingnon-linear system of equations behaves in a closelysimilar way to the loudspeakers measured behaviour.The FE method used here is also able to represent othernon-linear relationships using user prescribed powerseries. This facility could be used to model the Cms(x)non-linearity for example.

It would be possible to use the ZL(jω,x) results fromFEM to determine the LR2 parameters and predict thedistortion with the SIM module. This has not beendone here since the results would be derived fromalmost identical data. In practice the time saving of thismethod is substantial and it is anticipated that furtherwork will be done using this method. Where furtherunderstanding of the physics is an aim the detailedresults of a direct FEM approach are likely to outweighthe time cost.

5.3. Results

5.3.1. Lumped Parameter Model results

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Figure 34 Second –order intermodulation distortion in the inputcurrent measured and predicted by using the lumped parameter

method (SIM) for loudspeaker 1

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Figure 35 Third –order intermodulation distortion in the input currentmeasured and predicted by using the lumped parameter method (SIM)

for loudspeaker 1

Figure 34 and Figure 35 show the measured andpredicted 2nd-order and 3rd-order distortion forloudspeaker 1. The 2nd-order distortion is dominant andis caused by asymmetry of the parameters Le(x), L2(x)and R2(x).



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Figure 36 Second–order intermodulation distortion in the input currentand predicted by using the lumped parameter method (SIM) for

loudspeaker 2 (ring below)

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Figure 37 Third–order intermodulation distortion in the input currentmeasured and predicted by using the lumped parameter method (SIM)

for loudspeaker 2 (ring below)

Loudspeaker 2, with shorting ring below the gap, givesa slightly better performance than loudspeaker 1 forfrequencies below 1 kHz. Here the 2nd-order distortioncomponents are smaller but still dominant as theasymmetry remains in the non-linear parameters. Athigh frequencies the loudspeaker 2 generates similardistortion levels to loudspeaker 1 which correspondswith the increasing asymmetry of the effectiveinductance Leff(x,jω) in Figure 22.

Note that the distortion increases with frequencysimilarly to the increase in the variation of theimpedance ZL with x shown in Figure 10.

5.3.2. FEM results

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Figure 38 Acceleration response spectra of kinematic model with two-tone excitation at 12.5Hz & 2kHz. Level of 2kHz 2.83v, 12.5Hz level

adjusted to give same excursion as lumped parameter model.Loudspeaker 2.

The discretized geometry of loudspeaker 2 was solvedusing a full kinematic solution with a two tone inputsignal in order to reveal intermodulation distortioncomponents as previously discussed. The model wasexcited with an input signal comprising a 12.5 Hz toneand a 2Khz tone.

Figure 38 shows, by means of an FFT, the accelerationresponse spectra of the model to this signal. The twoexcitation tones may be clearly seen on the spectra at12.5Hz and 2kHz. In addition, due to the non-linearitiesof the model, distortion artefacts have been produced.These artefacts are both harmonic products of the twotones, occurring at multiples of the two basefrequencies, and intermodulation products resultingfrom the interaction of the two excitation signals,occurring at frequencies of the sum and differencebetween the base frequencies and their multiples.

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Figure 39 Acceleration response spectra of kinematic model with two-tone excitation at 12.5Hz & 2kHz – detail of 2kHz region.

Loudspeaker 2.

An enlarged view of the spectra is shown in Figure 39.This view clearly shows that the product visible at 4kHz



is composed from three distinct peaks. These peakswould appear to be the results of additional distortionproducts as well as the intuitive second harmonic of the2kHz tone.

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Figure 40 Current flow in aluminium ring inside magnet, excited withtwo-tone signal at 12.5Hz & 2kHz. Loudspeaker 2.

The FEM solution allows generation of additional datasuch as the current flow in the aluminium ring as shownin Figure 40. It is difficult to think how else this type ofdetailed information may be generated. FEM isexcellent at providing data for situations that would beextremely difficult to measure or otherwise predict.

A number of observations may be made from Figure 40.Firstly the induced current in the aluminium ring ispredominantly at high frequencies, the eddy currents &skin depth increase in severity with frequency as theirinduction is proportional to current flux. Additionallythe modulation of the current with voice coil position isclearly shown.

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Figure 41 Current flow in poles, excited with two-tone signal at12.5Hz & 2kHz. Loudspeaker 2.

Figure 41 shows a similar result for the current in thepole. In this case a far greater proportion of lowfrequency current flows. The overall level of the currentis significantly higher than that of the aluminium ringeven though the resistivity of steel is higher. This is

thought to be a consequence of the close proximity ofcoil and poles as well as the high magnetic permeabilityin the poles.

A comparison of the FEM predicted levels of distortionand the measured and lumped parameter predictedlevels of distortion has not been performed. The type ofanalysis performed here is extremely time consumingboth in terms of model setup and solution. The datagenerated & shown in the previous figures is notsufficient in quantity to provide accurate & comparablevalues for the relative levels of the intemodulationproducts.

5.4. Discussion of modelling

Throughout this paper FE and lumped parameter modelshave been applied to model static voice coil impedanceat a constant temperature. This leaves only themagnetic and electrical domains to determine the coilimpedance. Helpfully FE considering both these twodomains is available. All that is required is theappropriate material properties and geometry.

The FE path followed to its logical conclusion wouldprovide results equivalent to measurements. A completeFEM approach to modelling the full behaviour of aloudspeaker would require a large number of physicaldomains to be considered; mechanical, acoustic,magnetic, electrical & thermal. Solving in the timedomain allows non-linear effects to occur. Fullyapplied, this approach could result in the ultimatesimulation of reality that we are currently able toconceive. However, coupling FEM models results in alarge increase in computation time compared to solvingthe models separately; this is further exacerbated by thelong thermal time-constants found in loudspeakers. Theresulting FE model would be impracticably slow tosolve even in 2D and would not give the engineer thenecessary data to improve the design.

Even with the relatively simple case considered in thispaper the disadvantages of an exclusively FE methodare clear. Calculation of the intermodulation distortionusing the SIM module can be performed in ~30 seconds,faster than an equivalent measurement, the kinematicFE solution for the seven frequencies used for theZL(jω,x) modelling is estimated to take in the region of100 hours solving time on an equivalent PC. Thelumped parameter method is faster by factor of around10,000. Applying Moores’ law [16], the FEMcomputation is likely to be able to match the currentlumped parameter computation speed in approximately35 years.



With lumped parameter models the question, 'whichphysical domains must we consider?', is replaced by arequirement to understand the physics involved so thatappropriate simplifications may be made. This allowsdevelopment of a simplified system with appropriateparameters for representation of an effect. The systemand parameters may be defined in numerous ways tosuit the users requirements in order to provide therequired accuracy, complexity, practicality etc.

The ability of FE to allow display of the fieldsconcerned may well prove enlightening to the engineer.However, while FE provides extensive information, theoverall principles by which a loudspeaker operates arebest illustrated using lumped parameter models. UsingFE to produce data for a lumped parameter modelcombines these viewpoints to give an engineer the mostenlightening information in a reasonable time. Anumber of FE models may be used to provide values forthe various model parameters. Furthermore these FEresults show how the fields concerned change for theparameter modelled. Even without modelling the resultof the non-linearities such as the intermodulationdistortion the FEM ZL(jω,x) is extremely useful to theengineer. Another very significant result if that if oneaspect of the design is altered only the appropriatemodel need be resolved.

As shown here for the voice coil impedance, withcareful consideration and investigation, the mostimportant information can be compressed into a fewmeaningful parameters and an appropriate lumpedmodel. The benefits of an integrated modellingapproach, such as that described in [4], cannot beemphasised too strongly. This approach becomes evenmore appropriate when nonlinearities are to bemodelled.

6. CONCLUSION

Detailed knowledge of the voice coil inductivebehaviour is essential for the design of loudspeakerswith minimal distortion.

In this paper, the variation of voice coil impedance withposition and frequency has been both measured, using anew quasi static technique, and modelled, using FE,with exceptional agreement. It is possible to use thisinformation to investigate the effect of the shorting ringsand to derive other indications for improvements.

The measured or modelled voice coil impedance may beexpressed in terms of a few meaningful parametersusing Wright, Leach or LR-2 models. The LR-2 modelallows the expression of the variation with displacement

in a form that may be incorporated in a non-linearlumped parameter model. An extended model with moreRL cascades (LR-3) can provide a better fit whenrequired. Loudspeaker 2 impedance results show thatall LR-2 parameters must vary independently withposition to satisfactorily describe the non-linearbehaviour of some loudspeakers over the full frequencyrange. A non-linear lumped parameter model using thenew LR-2 model has been successfully used to predictthe intermodulation distortion of a loudspeaker.

Using modern (full kinematic FE) numerical methods itis beginning to become possible to deduce the finalbehaviour of a loudspeaker (linear and non-linear) basedupon the geometry and the material properties. FEallows additional data and visualisations to be computedthat it are not possible to deduce using measurement orother simulations. Most usefully numerical models mayalso be applied to conceptual loudspeakers. However,describing the loudspeaker using one numerical modelrequires massive computation, furthermore the resultswill inevitably be complex and difficult to interpret.

This paper has demonstrated the equivalency ofmeasurement and appropriate careful numericalmodelling. The results of the two techniques can both beused to generate non-linear lumped parameter models.An alternate route is to simply use the kinematic FEmethod to model the full case and miss out theintermediate impedance results. This alternative is,surprisingly, a less powerful technique as it is greatlyless efficient and does not provide the highly insightfulZL(jω,x) results. After further consideration it isbelieved that similar findings will emerge with theincreasing application of numerical techniques toloudspeaker design. An approach for the use of FE andother numerical techniques is presented. Usingnumerical methods to provide data for lumpedparameter models the geometry & performance may belinked in an efficient way which conveys the maximumunderstanding.

6.1. Acknowledgements

Many thanks to the Klippel GmbH & GPAcousticsengineering teams for their help and support for thispaper. Thanks to Joerg Panzer for use of hisvisualisation program VACS.

7. REFERENCES

[1] W. Klippel, “Prediction of Speaker Performance atHigh Amplitudes”, Presented at the 111th



Convention of the Audio Engineering Society,preprint 5418, November 2001. J. Audio Eng. Soc.,Vol. 49, No. 12, p. 1216, December 2001.

[2] A. Voishvillo, V. Mazin, “Finite-Element Methodof Modeling of Eddy Currents and Their Influenceon Nonlinear Distortion in ElectrodynamicLoudspeakers”, Presented at the 99th Convention ofthe Audio Engineering Society, preprint 4085,September 1995.

[3] J.Vanderkooy, “A Model of Loudspeaker DriverImpedance Incorporating Eddy Currents in the PoleStructure” J. Audio Eng. Soc., Vol. 37 No 3 pp.119-128; March 1989.

[4] M. Dodd, “The Development of a ForwardRadiating Compression Driver by the Applicationof Acoustic, Magnetic and Thermal Finite ElementMethods,” Presented at the 115th Convention of theAudio Engineering Society, preprint 5886,September 2003.

[5] “Manual of the KLIPPEL Analyzer System”,Klippel GmbH, www.klippel.de, 2004.

[6] W.M. Leach, “Loudspeaker voice-coil inductancelosses: circuit models, parameter estimation, andeffect on frequency response”, J. Audio Eng. Soc.,Vol. 50, No. 6, 2002.

[7] J.R. Wright, “An empirical model for loudspeakermotor impedance”, J. Audio Eng. Soc., Vol. 38,No. 10, 1990.

[8] “Large Signal Identification (LSI)”, Specificationof the KLIPPEL Analyzer module, Klippel GmbH,www.klippel.de, 2003.

[9] “Maximizing LPM Accuracy”, Application noteAN 25 of the KLIPPEL Analyzer System, KlippelGmbH, www.klippel.de, 2004.

[10] “Processing Software (MAT)”, Specification of theKLIPPEL Analyzer module, Klippel GmbH,www.klippel.de, 2003.

[11] M.A. Dodd, “The Transient Magnetic Behaviour ofLoudspeaker Motors” Presented at the 111th

Convention of the Audio Engineering Society,preprint 5410, November 2001.

[12] FLUX2D 7.60 User Manual, CEDRAT, MeylanFrance, 2002.

[13] “Simulation (SIM)”, Specification of the KLIPPELAnalyzer module, Klippel GmbH, www.klippel.de,2003.

[14] “3D Distortion Measurement (DIS)”, Specificationof the KLIPPEL Analyzer module, Klippel GmbH,www.klippel.de, 2003.

[15] W. Klippel, “Assessment of voice-coil peakdisplacement Xmax”, J. Audio Eng. Soc., Vol. 51,No. 5, pp. 307-324, 2003.

[16] G. Moore “Cramming more components ontointegrated circuits,” Electronics, Vol. 38, No. 8,April 19, 1965.

AES - Voice Coil Impedance as a Function of Frequency and Displacement

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