LUMPED IMPEDANCE MODELLING

Fig. 1 shows a simple equivalent circuit for a direct
radiating loudspeaker. The dominant nonlinearities are
the variation of the force factor Bl(x), the mechanical
compliance Cms(x) and the electrical impedance ZL(jw,x)
with voice coil displacement x. The DC resistance Re
and the motional impedance are not considered in the
electrical impedance ZL(jw,x). Thus the impedance
ZL(jw,x) may be measured at the electrical terminals by
blocking the movement of the coil and subtracting the
resistance.

Different linear models have been developed to describe
the frequency dependence of ZL(jw,x) with a minimal
number of free parameters:

Leach model

M. Leach [4] proposed a weighted power function of the
complex frequency as an approximation for ZL

ZL(jw)= K·(jw)n ; w=2pf (1)

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.

LR-2 Model

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

ZL(jw)= Le ·j.
+(R2 ·L2 ·j.
)/(R2 + L2 ·jw) (2)

Although this model uses three free parameters it often
provides a worse fit to measured ZL than the Leach
model. However, this model may be realised as an
electrical equivalent circuit (as shown in Fig. 2b) or as a
digital IIR filter.

Wright model

J. Wright [3] proposed a model using separate weighted
power functions in .
for both the real and imaginary
part of impedance.

ZL(jw)= Krm ·wErm + j·(Kxm ·wExm ) (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.

Effective inductance

ZL(jw) = Leff·(f)j.
+ Reff(f) (4)

M. Leach also proposed normalising the imaginary part
of the electrical impedance ZL(jw) to the frequency j.
and introducing an effective inductance Leff(f) which
varies with frequency. The real part of ZL(jw) may be
considered as a frequency dependent resistance Reff(f)
describing the losses due to eddy currents as shown in
Fig. 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.

Large signal modelling

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 Le(x), R2(x) and L2(x) are functions of the
displacement x and may be approximated by a truncated
power series expansion such as

where the coefficients li, ri and li are the free parameters
of the model.

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