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Audio Engineering Society

Convention Paper

Presented at the 117th Convention

2004 October 28–31

This convention paper has been reproduced from the author's advance manuscript, without editing, corrections, or consideration

by the Review Board. The AES takes no responsibility for the contents. Additional papers may be obtained by sending request

and remittance to Audio Engineering Society, 60 East 42 nd Street, New York, New York 10165-2520, USA; also see www.aes.org.

Journal of the Audio Engineering Society.

Voice Coil Impedance as a Function of Frequency and

Displacement

Mark Dodd 1 , Wolfgang Klippel 2 , and Jack Oclee-Brown. 3

1 KEF Audio UK (Ltd),Maidstone,Kent,ME15 6QP United Kingdom.

Mark.dodd@KEF.com

2 Klippel GmbH, Dresden, 02177, Germany.

wklippel@klippel.de

3 KEF Audio UK (Ltd),Maidstone,Kent,ME15 6QP United Kingdom.

Jack.oclee-brown@KEF.com

ABSTRACT

Recent work by Klippel [1] and Voishvillo et al.[2] has shown the significance of voice coil inductance in respect to

the non-linear behaviour of loudspeakers. In such work the methods used to derive distortion require the inductance

to be represented by an equivalent circuit rather than the frequency domain models of Wright and Leach. A new

technique for measurement of displacement and frequency dependant impedance has been introduced. The complex

relationship 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 independently

with x to satisfactorily describe all cases. Distortion induced by the variation of impedance with coil displacement is

predicted 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.

San Francisco, CA, USA

Dodd et al.

Voice coil impedance

1. INTRODUCTION

also been predicted using FEM analysis. A new

impedance model with displacement varying parameters

is discussed and fitted to the measured data. Finally,

distortion induced by the variation of impedance with

coil displacement is predicted using both lumped

parameter and FE methods, these predictions are

compared to measurements of the actual distortion. The

nature of this distortion is discussed.

In order to describe the performance of a loudspeaker

precisely, one must consider the electrical input

impedance at higher frequencies. The voice coil does

not operate in free air but close to conducting and

magnetically permeable structures: the pole tips, the

magnet, the voice coil former, copper rings etc. The

impedance can be only roughly modelled as a resistor

Re and an ideal inductance Le, the occurrence of eddy

currents in these structures usually decreases the

inductance of the coil and increases losses at higher

frequencies [3].

1.1. Glossary

Bl(x) force factor ( Bl product)

C ms (x) = 1 / K ms (x) mechanical

compliance of loudspeaker suspension

Exm exponent of imaginary part in WRIGHT

model

Erm exponent of real part in WRIGHT

model

F m (x,i) reluctance force

Krm factor of real part in WRIGHT model

Kxm factor of imaginary part in WRIGHT

model

K factor in LEACH model

K ms (x) mechanical stiffness of loudspeaker

suspension

L e (x) inductance used (cascaded model)

L 2 (x) inductance (cascaded model)

L eff (f,x) effective inductance depending on

M ms mechanical mass of loudspeaker

diaphragm assembly including air load

and voice coil

n exponent in LEACH model

R ms mechanical resistance of total-

loudspeaker

R e electrical voice coil DC-resistance

R 2 (x) resistance (cascaded model)

R eff (f,x) effective losses due to eddy currents

depending

The nature of the voice coil impedance is dependent

upon the geometry and physical properties of the

magnet and surrounding assemblies. There are three

prominent methods of deducing the coil (blocked)

impedance for a particular loudspeaker.

- The blocked impedance may be measured by

clamping the voice coil stationary within the

magnet assembly to remove the motional part of the

impedance. Blocked impedance may then be

measured as with conventional impedance

measurements.

- Various impedance models have been developed in

order to describe the electrical behaviour of the

coil, these models may be fitted to conventional

unblocked impedance measurements.

- Recently magnetic transient FEM has been

successfully applied to deduce the blocked

impedance [4].

At large amplitudes the impedance also varies

significantly with voice coil displacement. Whereas the

frequency dependence of the impedance is a linear

phenomenon effecting the amplitude response, but not

generating distortion, variation of impedance with

displacement generates significant harmonic and inter-

modulation distortion.

The nature of the impedance generated distortion is a

dependent upon the coil impedance, or more precisely

how the coil impedance varies with frequency &

displacement of the coil.

This paper investigates the voice coil impedance as a

function of frequency and of displacement.

Measurements of frequency & displacement dependent

impedance have been made using a quasi-static method

using the LPM module of the Klippel analyser [5].

Frequency & displacement dependent impedance has

frequency

and

displacement

Z L (j w ,x) complex excess impedance representing

the effect of the lossy inductance

(motional impedance and R e is

removed)

angular frequency = 2 f

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

Page 2 of 19

Dodd et al.

Voice coil impedance

2. MODELLING

electrical equivalent circuit (as shown in Figure 2 b) or

as a digital IIR filter.

2.1. Lumped Parameter Model

Z L (j w ,x)

R e

C ms (x)

M ms

R ms

F m (x,I)

Z L (j w ,x)

L 2 (x)

R e

L e (x)

Bl(x)v

Bl(x)

Bl(x)I

R 2 (x)

Figure 1 Electro-mechanical equivalent circuit of the loudspeaker

L eff (f, x)

R eff (f, x)

Figure 1 shows a simple equivalent circuit of a

loudspeaker system. The dominant nonlinearities are the

variation of the force factor Bl(x), the mechanical

compliance C ms (x) and the electrical impedance

Z L (j w ,x) with voice coil displacement x. The DC

resistance R e and the motional impedance are not

considered in the electrical impedance Z L (j w ,x) . Thus

the impedance Z L (j w ,x) may be measured at the

electrical terminals by blocking the movement of the

coil and subtracting the resistance measured at a very

low frequency.

Different linear models have been developed to describe

the frequency dependency of Z L (j w ,x) with a minimal

number of free parameters:

Figure 2 Representation of the electrical impedance

2.1.3. WRIGHT model

Wright [7] proposed a model using separate weighted

power functions in w for both the real and imaginary

part of impedance.

Z L ( j w)= K rm ·w Erm + j ·( K xm ·w Exm )

(3)

This model uses four free parameters and normally gives a

better fit than the other models with less parameters.

Unfortunately, this function can not be directly realised as

an analogue or digital system.

2.1.4. Effective inductance

2.1.1. LEACH model

Z L (j w ) = L eff ·(f)j w + R eff (f)

(4)

M. Leach [6] proposed a weighted power function of the

complex frequency as an approximation for Z L

Z L ( j w)= K ·( j w) n ; w = 2 f

(1)

M. Leach also proposed normalising the imaginary part

of the electrical impedance Z L ( j w) to the frequency j w

and introducing an effective inductance L eff (f) which

varies with frequency. The real part of Z L ( j w) may be

considered as a frequency depending resistance R eff (f)

describing the losses due to eddy currents as shown in

Figure 2c . Though the number of parameters is very

high , two parameters for each frequency point, both

parameters are easy to interpret and convenient for

graphical representation.

2.1.5. Large signal modelling

Although using only two free parameters this function

can sometimes give a very good fit over a wide

frequency range. Unfortunately, this function can not be

represented by an electrical equivalent circuit nor a

simple digital system.

2.1.2. LR-2 Model

This model uses a series inductance L e connected to a

second inductance L 2 shunted by resistance R 2.

The linear models may be easily expanded to higher

amplitudes by allowing each parameter to be dependent

upon the displacement x.

For example considering the LR-2 model, the three

parameters L e (x), R 2 (x) and L 2 (x) are functions of the

displacement x and may be approximated by a truncated

power series expansion such as

Z L ( j w) = L e · j w + ( R 2 · L 2 · j w ) / ( R 2 + L 2 · j w)

(2)

Although this model uses three free parameters it often

provides a worse fit to measured Z L than the LEACH

model. However, this model may be realised as an

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

Page 3 of 19

Dodd et al.

Voice coil impedance

(5)

time dependent properties. Additionally, due to flux

modulation, the force factor Bl(x) and the inductance

L e (x) are dependent on the current i. Thus using an

audio-like signal will produce far more meaningful data

than measurement with extremely small input current

(coil offset generated by external force or pressure) or

extremely large currents (coil offset generated by a dc

current).

L x

( )

l x

R x

( )

= Ã

r x

(6)

(7)

L x

( )

Despite these limitations a quasi-static technique is a

useful method for investigating the variation of the

impedance with both frequency and displacement.

Measurements have been performed on two test

loudspeakers. Both loudspeakers share the following

specifications:

where the coefficients l i , r i and i are the free parameters

of the model.

3. MEASUREMENT

The application of a model to a particular real object

usually requires an estimation of the free model

parameters in such a way that the model describes the

real object with maximal accuracy.

With the linear models straightforward techniques are

available which may be applied for loudspeakers at

small amplitudes.

Voice coil diameter: 2” nominal (51.30mm ID)

Voice coil DCR: 6.72 Ohms

Turns in Voice coil: 126 Turns

Ferrite ring magnet

Annular low carbon steel top-plate

Pole/plate assembly type yoke.

The magnet assembly of loudspeaker 1 has an

aluminium shorting ring placed above the magnetic gap.

The magnetic assembly of loudspeaker 2 has an

aluminium shorting ring placed below the magnetic gap.

The geometry of the two loudspeakers is shown in

Figure 26 & Figure 27. The pair serve to illustrate the

influence of the effect of aluminium rings on the

variation of the voice coil inductance with displacement.

Non-linear models require special techniques for the

parameter identification. Static, quasi-static and full

dynamic techniques have been developed to measure

the force factor Bl(x), compliance C ms (x) and

inductance parameter L e (x). The dynamic techniques

have the advantage that an audio-like ac signal is used

for excitation and the loudspeaker is operated under

working conditions.

3.1. Mechanical Setup

A method for measurement of the displacement

dependent impedance was developed at GP Acoustics

(UK). The method is a simple modification to the

standard Linear Parameter Measurement (LPM) of the

KLIPPEL Analyser system. The measurements

presented in this paper were performed by Klippel

GmbH.

The current version of the LSI module in the Klippel

analyser performs a dynamic measurement using a noise

stimulus [8]. The free model parameters are optimised

to give the best fit between measured and modelled

current and displacement. The LR-2 model is currently

constrained so that the three lumped parameters vary

with the same shape in x.

L x

( )

R x

( )

L x

( )

(8)

The loudspeaker is clamped in vertical position in the

professional loudspeaker stand as shown in Figure 3.

(0)

An additional spider is attached to the diaphragm.

The spider holds an inner clamping part made of

aluminium which is secured to the lower rod (usually

used for holding the microphone).

Although this assumption is a good approximation for

most loudspeakers without shorting rings it is a purpose

of the paper to investigate the validity of this

approximation for more elaborate loudspeakers using

copper cups and aluminium rings for reducing the

inductance nonlinearity.

Clearly the measurement of suspension stiffness using a

dc offset gives significantly different results from a

dynamic measurement due to creep, relaxation and other

By shifting the lower rod a displacement may be

imposed to the coil position. Clearly displacing the coil

will also change the other parameters such as Bl(x),

C ms (x) and also loss factors.

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

Page 4 of 19

Dodd et al.

Voice coil impedance

The Distortion Analyser also provides a Displacement

Meter ([5] Hardware) this is used for measuring the

original rest position of the cone and to measure the

imposed static displacement. Throughout this paper the

convention is that a positive displacement of the coil

refers to movement out of the magnet assembly.

3.2. Small signal measurements

The additional spider increases the stiffness of the total

suspension and the resonance frequency. However, the

modified loudspeaker is still a second-order system and

can be represented by the equivalent circuit in

Figure 1.

The module Linear Parameter Measurement (LPM) of

the KLIPPEL Analyser system is used to measure the

linear parameter at each prescribed displacement.

The loudspeaker is excited by a multitone signal of 0.5

V rms at the terminals. Since the voltage, current and

displacement are measured simultaneously all of the

linear parameters can be identified instantaneously. An

additional measurement with a mechanical perturbation

(additional mass or measurement in a test enclosure) is

not required.

A sparse multitone signal used as excitation signal

allows assessment of the distortion generated by the

loudspeaker. During the small signal measurements the

maximum distortion occurred 20 dB below the

fundamental lines in the current spectrum. This shows

sufficiently linear operation of the loudspeaker [9].

3.3. Fitting of the inductance model

Figure 3 Measurement Setup.

At first the linear parameters are measured at the rest

position (x=0) and the different inductance models (LR-

2, WRIGHT, LEACH) are used to describe the

impedance response, measured up to 18 kHz.

The loudspeaker under test is connected to Distortion

Analyser 2 allowing a simultaneous measurement of

voltage, current and displacement signal.

110

100

LR-2

Measured

WRIGHT

Figure 4 Generating an additional DC offset by the lower rod

connected by an addition spider to the diaphragm.

LEACH

5 10 20

50 100 200 500 1k 2k

5k 10k

Frequency [Hz]

Figure 5 Magnitude of electrical impedance of loudspeaker 1

measured and fitted by LR-2, Wright and LEACH model up to 18

kHz.

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

Page 5 of 19

AES - Voice Coil Impedance As A Function Of Frequency And Displacement.pdf

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