1. Electrical Equivalent Circuit Model for
Dynamic Moving Coil Transducers
incorporating a semi-inductor
Knud Thorborg
1
, Andrew D. Unruh
2
1
Tymphany A/S, DK-2630 Taastrup – DENMARK
knud.thorborg@tymphany.com
ABSTRACT
A serial combination of inductor and resistor is traditionally used to model the blocked electrical impedance of a
dynamic moving coil loudspeaker. In practice, semi-inductive behaviour due to eddy currents and ‘skin effect’ in
the pole structure, as well as transformer coupling between the voice coil and conducting rings in the magnet system
can be observed, but are not well represented by this simple model. An improved model using a few additional
elements is introduced to overcome these limitations. This improved model agrees well with physical realities and is
easily incorporated into existing equivalent circuit models. The development of the model is explained and its use is
demonstrated. An example yielding more accurate box response simulation is also provided.
1. INTRODUCTION
An improved model of the electrical impedance of the dynamic loudspeaker was first developed in a convention
paper [1] and presented at the 122nd
AES conference in May 2007 in Vienna. The results are still valid, but already
by the time of the presentation in Vienna, the theoretical understanding of the model had been revised due to work

2. 2
that was done after the deadline for the submission of the paper. This makes it relevant to present here a more
complete work, augmented with more recent results.
The well-known equivalent circuit for a dynamic loudspeaker is shown in Fig.1. Here the admittance analogy is
used. This has the advantage that the electrical side of the system is “coupled” to the mechanical side through an
ideal transformer with a turns ratio of Bl:1, where Bl is the force factor of the speaker’s motor. The input to the
system is voltage and the output of the system is cone velocity u, as seen in Fig.1. Traditionally, when performing
loudspeaker calculations the radiation resistance of the cone operating into air is ignored since it has little influence
on the motion of the cone. The air load is regarded as part of the equivalent mass of the moving system MMS. RE and
LE are the resistance and the inductance of the voice coil. CMS is the compliance of the total suspension and RMS
represents the viscous damping in the system [2, 3].
The focus here is improving the electrical side of the equivalent circuit; to make the model agree more closely with
measurement, and to have the elements in the model represent the understood behaviour of the system (physical
modelling). The improved model - in connection with the “curve fitting” technique - makes it possible to more
correctly, and precisely derive the so-called Thiele Small parameters.
Fig.1 Traditional equivalent circuit diagram for a loudspeaker using the “admittance analogy”.

3. 3
2. IMPEDANCE
2.1. Blocked and Motional Impedance
The transformer coupling in Fig.1 makes it simple to transform the circuit to the primary side, and by doing so we
get the circuit in Fig.2. The electrical impedance of a loudspeaker ZS(f) can be considered to consist of two parts as
shown by the dashed boxes in Fig. 2. The components in the first box constitute of the “blocked” impedance ZE,
which for small signals is independent of the motion of the speaker cone. The components shown in the second box
constitute of the motional impedance ZM. Motional impedance occurs when the voice coil moves through the
magnetic field of the motor. This in turn sets up an electromotive force (i.e. e.m.f. or voltage) equal to Bl.
u, where u
is the cone velocity. The resulting current will oppose the velocity of the voice coil.
Fig. 2 Equivalent circuit diagram of Fig. 1 for a loudspeaker converted to the electrical side.
The mechanical parameters are converted to virtual electrical components by the equations shown below:
2
)Bl(
M
C MS
p  (F) (1)
MS
p
R
)Bl(
R
2
 (Ω) (2)

4. 4
2
)Bl(CL MSp  (H) (3)
uBlV  (V) (4)
The subscript “P” is used to designate the virtual electrical components in the parallel resonance circuit constituting
the motional impedance.
Fig. 2 shows a four-pole (two-port) electrical circuit model of a loudspeaker. The circuit will have a resonance
frequency, fS, and at this frequency the electrical impedance will attain a local maximum Z0 when 1/jωCP and jωLP
cancel each other. However, in practice a very small residual reactive impedance due to LE will move the measured
resonance peak a little upwards in frequency. Since at resonance the motional component of the electrical impedance
is equal to RP it follows that:
Ep RZR  0
(Ω) (5)
If the circuit shown in Fig. 2 is probed with a current i, then a voltage Bl.
u will appear at the output terminals and
we get:
u
Zi
Bl M
 (N/A) (6)
u(f) might be measured directly by a laser transducer. If V=i .
ZM(f) and u(f) are measured simultaneously in dB
scale the two curves should show congruence but with constant distance 20logBl.
2.2. Differences between Measured and Modeled Impedance
It is not a problem to measure Z0, but as will be shown later, the real part of the blocked impedance REAL(ZE) at fS
(in the following just called RES) is not exactly the same as RDC. It is always at least slightly greater but in some

5. 5
cases the difference can be significant. This has the consequence that QES (the electrical Q-factor) and QTS (the total
Q-factor) are generally estimated lower than they really are, when calculated in the traditional way while assuming
RES=RDC [2, 3]. A further consequence is that the calculated values of Bl and SPLREF (the theoretical sensitivity at
fS if we had QTS = 1) are both higher than they really are. Consequently, in this paper, we will distinguish between
RES and RDC. This will be discussed in more detail. However, it should be noted that there are no simple means to
measure RES. A higher value RE>RDC at fS has previously been correctly attributed to eddy currents [4].
The shape of the loudspeaker impedance curve, ZS(f) is well known – see the measured curve (red) in Fig.3. It starts
near RDC at very low frequencies followed by a peak at the fundamental resonance (determined by the electrical
equivalents to MMS and CMS, - CP and LP respectively ). Above the resonance frequency there is an anti resonance at
fmin (mainly between LE and CP). Above this minimum the impedance according to the model should rise
proportionally with frequency (6dB/octave).
-10
0
10
20
30
10 100 1000 10000
Frequency [Hz]
dB/8ohm
Fig. 3 Impedance curve for a subwoofer with heavy 4 layer coil (red) compared to a curve calculated for the same
speaker using the simple equivalent circuit in fig.2 and RES=RDC. (blue)
To compare the measured curve with the one predicted by the simple equivalent circuit the impedance curve is
calculated and shown for the same speaker (the blue curve in Fig. 3).

6. 6
The curves shown are for a subwoofer with a heavy 4-layer coil representing a “worst case scenario” with respect to
the deviations between the measured and the calculated impedance curves, when the calculated curve is based on the
simple equivalent circuit and RES = RDC . These curves reveal very significant differences between traditional theory
and measured reality. At fmin the impedance magnitude predicted is very close to RDC. We see that the impedance
around fmin is significantly higher than predicted – and so we must expect a correspondingly significant deviation in
a simulated box response when compared to measured reality.
Another thing to note is that the measured slope of the impedance curve above fmin is typically closer to 3dB/octave
rather than the expected 6dB/octave if the only inductive element in the circuit were a conventional inductor.
2.3. Eddy Current Effects and Semi-Inductance
Vanderkooy [4] explains the 3dB slope of the impedance curve as the result of eddy currents in the iron core (mostly
in the speaker’s pole piece) and of the “skin effect”. As the skin depth decreases with the square root of frequency,
the electrical conductivity of the pole piece is correspondingly reduced. According to Vanderkooy, the skin depth is
given by:


2
 (m) (7)
Here  is the permeability and  is the conductivity of iron. If the coil was wrapped around an infinite long iron
core, Vanderkooy – using Maxwell’s equations – shows that the coil, above a very low transition frequency, is
acting as what he calls a semi-inductor, but below the transition frequency it behaves as a normal inductor. The
impedance of a semi-inductor is calculated as
jKZ  (Ω) (8)
This function will have a slope of 3dB/octave, just as we most often observe for speaker impedance at higher
frequencies. The unit for K is “semi-Henrys” [sH]. If the coil was wrapped around a core forming a closed magnetic
circuit we expect a similar result.

7. 7
According to Vanderkooy, we should expect a transition frequency when we have:
δ >

r
2
(9)
(r is the radius of the pole piece), and up to this frequency we should expect ordinary inductor behaviour. With a
pole piece radius of 10 mm, a relative permeability of 3000 and a conductivity 107
Ω-1
, he calculated the transition
frequency to be as low as 0,17 Hz However, Vanderkooy observed transition frequencies typically at 100-200 Hz.
This discrepancy, he suggested, was due to the openness of the magnetic circuit (air gap and high reluctance
magnet), resulting in a differential permeability of about 3 for the magnetic circuit instead of 3000 as would
normally be expected for iron. Inserting this value in (7) gives a 1000 times higher transition frequency – in
acceptable agreement with Vanderkooy’s observations. In the convention paper [1], this hypothesis was evaluated to
explain the transition from ordinary induction to semi-induction, but after the deadline for the paper, simulations of
the eddy currents in a magnet system revealed that the hypothesis was not correct. We have a “skin effect” far below
the observed transition frequency and we had to revise the theory seeking another explanation and re-considering the
real physical background for the high transition frequency.
Vanderkooy did not try to make an equivalent circuit incorporating the semi-inductor behaviour at higher
frequencies and the transition from inductor to semi-inductor. Developing such a circuit is the object of this paper.
The total loudspeaker impedance ZS as mentioned consists of two parts: the ‘blocked’ electrical impedance ZE and
the motional impedance ZM. The focus in the following will be on the “blocked impedance”.
2.4. Introducing air gap in the Vanderkooy model
Today we have tools, which were not available to Vanderkooy in 1989, such as finite element analysis programmes
able to calculate and visualize what happens in the iron when magnetized by current in a coil whether DC or AC.
Fig. 4 shows such a simulation. We see the right half (to the right of the symmetry line) of a cut through a speaker
magnet system probed with a 30 Hz sinusoidal current signal, having an amplitude of 1 Amp..

8. 8
We observe the coil – of course with maximum current density – as a red line. The colours around it indicate the
densities of eddy currents. Red is maximum density and the blue background shows very little or no current. The
red/yellow/green areas represent the scale of densities in the iron circuit.
The blue parts can be cut away without causing significant changes – except static field.
Fig. 4 Right half of a loudspeaker motor cut through, showing simulated eddy current densities in an iron circuit as
a result of 1 A at 30 Hz excitation signal in the voice coil (the red line – indicating maximum intensity).
An animation over some periods of a 30 Hz signal shows that the skin depth is not a conductive “egg shell” around
the iron. The current density spreads from the surface into the material as “rings in the water” (following Maxwell’s
equations). As the current changes direction we see a dark line of zero current moving into the iron and
disappearing. Inside this line the current has not yet changed direction – and “skin depth” is to be understood as an
equivalent value representing this picture with opposite currents inside and outside the zero line.
What do we learn from this simulation? We observe a “skin effect” as predicted by Vanderkooy, at very low
frequencies, here simulated by 30 Hz (in the simulated case formula (7) gives δ=0.53 mm) 30 Hz is much below the

9. 9
observed transition frequency and we see a “skin effect” as originally predicted and more than enough to give semi-
induction – so there must be another reason for the high transition frequency. The low resulting permeability for the
circuit does not affect the “skin effect”.
For the inductance of a blocked voice coil we have:
(H) (10)
(n is the number of turns in the voice coil, Riron is the reluctance in the iron circuit, and Rgap is the reluctance of the
resulting air gap. Due to the skin effect the induction L becomes a function of ω).
If we had no “skin effect” (non conducting iron core) this would lead to:
(H-1
) (11)
Liron is the induction we would have, if there were no air gap, LE would be the induction if we only had the
reluctance of the air gap.
This leads us to:
jωL=jωLE//jωLiron (Ω) (12)
If we re-introduce the conductive iron and the skin effect, from Vanderkooy model, and after substituting jωLiron
with EKj we then have :
ZE = RE + jωLEB + jωLE EKj (Ω) (13)
This formula leads directly to the equivalent circuit for the blocked impedance, as shown in Fig.5.
gapiron
n
)(L
RR 

2

Eiron LLnnL
iron 111
22
 gapRR

10. 10
Fig. 5 Hypothetical circuit for the blocked impedance incorporating a “semi inductor”.
Note that we insert a smaller inductor LEB, representing mainly the part of the voice coil which is above the pole
piece (the upper overhung). This is a leakage inductor – to be explained in more detail later. This we assume to
behave more or less as an ordinary inductor.
KE is calculated by Vanderkooy, see [5], equation (27), the approximation is for higher frequencies (above
transition):



j
rnZ 22
 (Ω) (14)
(r is the pole radius, valid for δ<<r (see (7)), n is the number of turns per m).
From this we get:


l
r
nKE
22
 (sH) (15)
Here l is the active coil length (r and l are both in mm). The length l is thought to be a part of an infinite coil
wrapped on an infinite long pole. A short circuiting ring in the magnet system coupling tightly to the coil might
shorten the “active” coil length - more about this later. We cannot use this formula to calculate KE for a speaker, we
are too far from the ideal conditions under which this equation is valid, but it provides an indication of the factors
determining the value of KE.

11. 11
As we later make a more complete circuit, we call the hypothetical circuit in Fig.5 “The practical blocked
circuit”.“Practical” because it is a circuit usable for most practical applications.
3. VERIFICATION OF THE PRACTICAL MODEL
A PC-based data acquisition system (Listen, Inc. SoundCheck) was used to measure the impedance curve for a
subwoofer at 1/12-octave intervals from 10 Hz to 20 kHz. The data were fit to a model as shown in Fig.6,
combining the blocked impedance from Fig. 5 and the motional impedance from Fig. 2 and using the "Solver"
function in Microsoft Excel.
Fig.6 Equivalent circuit with the traditional blocked impedance substituted with the “practical blocked circuit”
from Fig. 5. This circuit is recommended for practical use .(e.g. for calculation of Thiele Small parameters and box
simulations)
The speaker tested was a 10” Peerless XXLS model 830843. This is a subwoofer featuring an aluminium spacer on
the pole piece and an aluminium short circuiting ring (see Fig.11).
Curve fitting can be regarded as a “short cut” to solving equations with several independent variables. Here we have
seven. The results of this fitting are shown in Fig. 6 (magnitude) and Fig. 7 (phase). The resulting parameters are
shown in Table 1. We obviously get a very good fit, and results look very plausible. In order to have a good fit, we
have to have good initial estimates for the model parameters, the curve fitting will then finish the job..

12. 12
Fig. 7 Measured magnitude data (red curve) and curve fit, Peerless 830843. The fig is made to the circuit of Fig. 6.
Unit was measured in free air
Fig.8 Measured phase data and curve fit, Peerless 830843. Blocked impedance fit to the circuit shown in Fig. 6.
Unit was measured in free air. The artefact at 700 Hz is due to rim resonance

13. 13
Table 1. Peerless 830843 parameters. Blocked
impedance fit to the circuit shown in Fig. 6. Unit was
measured in free air
To test the reliability of the results, we want to measure and fit the blocked impedance separately. For this purpose,
we introduce a high strength adhesive between the voice coil former and the pole piece of the driver to eliminate
voice coil motion. This eliminates the effects of the back electro-motive force, allowing the blocked impedance to
be examined without having to remove the motional component of the electrical impedance.
The results of these measurements are shown in Fig. 9 (magnitude) and Fig. 10 (phase), and the blocked electrical
parameters are shown in Table 2.
Parameter Value Units
RE 6,3 Ω
LEB 0.319 mH
KE 0.133 Semi-Henrys
LE 2.38 mH
CP 442 F
RP 114 Ω
LP 86.4 mH
fS 25,72 Hz
Z0 122,00 Ω
RES= Z0 -RP 6,34 Ω
QMS 8,28
QES 0,45
QTS 0,43

14. 14
Fig. 9 Measured magnitude data and curve fit according to the circuit in Fig. 5 for the blocked Peerless 830843
Fig. 10 Measured phase data and curve fit according to the circuit in Fig. 5 for the blocked Peerless 830843. The
artifact found around 4 kHz in the measured data is the result of a mechanical resonance in the blocked system.

15. 15
Table 2. Peerless 830843, blocked impedance
parameters. Result of fitting to the circuit shown in
Fig. 5
Again, the curve fit and the measured data are in excellent agreement. Note that the blocked impedance parameters
shown in Tables 1 and 2 show good agreement. This tell us that the voice coil does not need to be held motionless
to accurately measure the blocked impedance parameters.
An important thing to note is that the RE found by curve fitting is significantly higher than RDC (about 0,5 Ω higher).
4. EVALUATION OF THE COMPLETE MODEL
4.1 Applying Transformer Theory
Considering the rather complex geometry of the motor in a modern speaker (see Fig. 11), it is very difficult to pre-
calculate the electrical impedance of the blocked voice coil. The figure shows a cutaway of the sub woofer used for
the verification, in part 3. It will be observed that the voice coil does not, in its full length, enclose the iron pole
piece. Furthermore, the coil outside the air gap is close to concentric high conductive aluminium parts, below the air
gap to the short circuiting ring (D) and above the air gap to the pole extension (A). Such components, made of
conducting material and concentric to the voice coil are normally found in quality speakers, as they have good effect
on reducing distortion and extending frequency range in the upper end, by reducing coil inductance.
Parameter Value Units
RE 6,19 Ω
LEB 0.315 mH
KE 0.134 Semi-Henrys
LE 2.29 mH
RES 6,26 Ω

16. 16
To evaluate the influence of the coupling between the coil and the highly conductive elements introduced into the
magnet system (short circuiting ring, copper cap and high conductive extension to the pole piece) we will apply
some transformer theory.
Fig.11 Cross-section of a subwoofer (Peerless 830843). The part of the voice coil inside the air gap contributes to
the semi-inductance, while the remaining voice coil might be regarded primarily as a transformer with highly
conductive material (the aluminium short- circuiting ring (D) and pole extension (A)) that are concentric with the
voice coil as a one-turn secondary.
Generally it will be more practical to look at three parts of the voice coil separately (but of course they interact). The
part of the coil above the air gap is the “upper overhung” - in most cases this is in free air and must be expected
mainly to act as an ordinary inductor (contributing to LEB), but in the present case this contribution is reduced due to
coupling to the aluminium pole extension. The part of the coil inside the air gap encloses the pole piece, and is
enclosed by the iron of the pole plate. In both of these we observe high eddy current densities (see Fig. 4) with about
the same “skin depth”, thus working in parallel and reducing KE as compared to a situation with iron only on one

17. 17
side. Below the air gap the “lower overhung” part of the coil couples to the pole piece on the inside and to the
aluminium short circuiting ring on the outside. If there were no short circuiting ring, this part of the coil would have
a significant contribution not only to the magnitude of KE, but also to its dependency on coil position - a primary
cause to loudspeaker distortion. A momentary current in the coil will induce a current in the opposite direction in
the aluminium ring. This will reduce the magnetizing of the iron and the eddy currents in the pole piece, making this
part of the coil resistive, and reducing the resulting KE.
To evaluate the impact of the short circuiting rings (inclusive pole extension and copper cap if any) we will look at a
“transformer” with the part of the coil coupling directly to these parts as primary and all the concentric conducting
material as a “one turn” closed secondary, see Fig. 12. .
Fig.12 Equivalent circuit for a transformer with a turns ratio of n:1 shown with the secondary short-circuited.
Secondary winding resistance is converted to primary side as ΔR. LEP and LES’ are primary and secondary leakage
inductances (the secondary converted to primary side)
RDC is the primary winding resistance; ΔR is the secondary winding resistance converted to the primary side. L0 is
the primary inductance and LEP and LES’ are the primary and the secondary leakage inductions, LES’ is converted to
the primary side.
The primary leakage inductance in a transformer represents the part of the flux from the current in the primary
winding not enclosed by the secondary – and vice versa. The tighter the coupling between the windings, the less

18. 18
leakage inductance is observed. So, thicker insulation (distance) between the windings results in more leakage
inductance. The leakage inductances will depend on the tightness of the coupling between the voice coil and the
“one turn secondary” .
In the resulting “Complete Circuit” we have to insert ΔR//L0 in series with KE//LE as shown in Fig. 13. LEP and LES’
contribute to LEB . We could make a similar evaluation for the pole extension and for a possible copper cap, but for
practical reasons we will approximate this single complex entity to represent the full effect of all concentric
conducting material in the speaker.
It is easily observed that the circuit has the expected features. At very low frequencies the influence of ΔR vanishes
as it is in parallel with L0 – and at DC we find RDC as it should be. If we had no high conductive elements concentric
to the voice coil, we would find a much higher value for KE, ΔR//L0 would disappear, but KE would give a
significantly higher contribution to RES . If the one turn secondary is a copper cap extended in the whole length of
the voice coil (SD, symmetric drive [16]), the coupling between the voice coil and the “secondary” becomes so tight,
that we can ignore all leakage inductances and the blocked impedance will become totally flat, having the value of
RDC+R, for frequencies higher than the frequency at which R=ωL0.
(A calculation of the ΔR contribution (the example in part 3.) from the “lower overhung” part of the coil gives about
0,44 Ω (72 turns and aluminium ring ø63mm and thickness 5mm, active height 12,5mm and conductivity 0,027 Ω-1
).
To this we need to add a minor contribution from the upper part of the coil coupling to the massive pole extension
(less than 0,1Ω) – and we are close to the 0,5 Ω that we had to add to RDC to get the value that was found by curve
fitting the impedance data).
4.2 Complete Impedance Model
The total equivalent circuit diagram for the improved impedance model is shown in Fig. 13. Here we have added the
combination L0//ΔR.

19. 19
Fig. 13 “New Complete Model”, an improved electrical equivalent circuit model for a dynamic transducer. ΔR is
due to resistance in short circuiting rings. If no conductive material in the magnet system, ΔR//L0 is to be taken out,
but KE becomes significantly higher.
The impact of short circuit rings in the magnet system is to reduce the part of the voice coil that contributes to the
semi-inductor. L0 and ∆R define the lower frequency limit for this effect. The KE//L term, on the other hand,
represents the part of the voice coil that is solely coupling to the iron core of the pole piece, and LEB represents the
overhung part of the voice coil, and the leakage inductions in the “voice coil/short circuiting ring” transformer.
The data obtained by measuring the 830843 in free air was used to fit the ‘complete’ blocked impedance model
shown in Fig.13. We now have nine independent parameters. Fig.14 shows the measured impedance magnitude and
Fig.15 shows the phase.

20. 20
Fig. 14 Measured magnitude data and the corresponding curve fit, Peerless 830843. Blocked Impedance Fit to
Complete Circuit Shown in Fig.13. Unit was measured in Free Air
Fig. 15 Measured phase data and the corresponding curve fit, Peerless 830843. Blocked impedance fit to the
complete circuit shown in Fig.13. Unit was measured in free air

21. 21
Table 3 shows the resulting parameters. Compared to the data shown in Table 2, the data given in Table 3 produces
a slightly better curve fit, although the improvement is too small to be easily seen in the figures. The values for LEB,
KE, and LE are not strongly dependent upon which model is used – simplified or complete. In addition, the data
again show that it is not necessary to eliminate voice coil motion to measure the blocked parameters; they can be
derived from the free air impedance. Note that RDC+ΔR in table 3 is 6.37Ω as compared to RE=6.29 from table 1 - a
very good agreement.

22. 22
Table 3. Peerless 830843 Parameters. Blocked
Impedance Fit to Complete Circuit Shown in Fig. 8.
Unit was measured in Free Air
To find the transition frequency, for the example speaker, between R (0.5) and L0, we seek the frequency where
RL  0
(16)
Parameter Value Units
RDC 5.83 Ω
LEB 0.307 mH
KE 0.138 Semi-Henrys
LE 2.24 mH
L0 10.8 mH
∆R 0.540 Ω
CP 445 F
RP 116 Ω
LP 86.0 mH
fS 25,74 Hz
Z0 122,26 Ω
RES=Z0 -RP 6,36 Ω
QMs 8,33
QES 0,46
QTS 0,43

23. 23
In the example, the frequency is about 8 Hz.
For practical use, L0 may be neglected, and the “practical” circuit shown in Fig. 5 can be used with RE =RDC + R,
where RE is determined by curve fitting. The exact fitting is most critical in the frequency range up to fmin.
5. PRACTICAL IMPLEMANTATION OF THE NEW MODEL
With the computer techniques today, it is of little problem to make (or modify) simulation programmes that are able
to handle the new model. One such programme is already available, the LspCad version 6.32, now updated to handle
the “Improved Equivalent Circuit” (not the full version, but the “practical version” as shown in Fig. 6). It should be
noted that the above mentioned program includes a shunt resistor to LEB, which is not included in our circuit. In the
program, a very high value should be assigned to this shunt resistor to eliminate its effect. In case of a very flat
impedance curve, due to an extended copper cap, it might be possible to use LEB//REB as L0//ΔR to improve the curve
fit. More information about this programme can be found at the following link: http//www.jidata.com. Simulation
results using this program is shown in Fig. 15.
70
80
90
100
10 100 1000
New parameters Old parameters
Fig. 15 Box simulation for Peerless 830843 using old and new parameters
Tymphany intends to make the parameters based on the new model available for all speakers, as soon as the
necessary measurement facilities are established. This however cannot be done overnight. The idea is to use a

24. 24
measuring system based on SoundCheck (as used in the example) and a laser transducer to the measure velocity
curve (simultaneously with the impedance curve) and use curve fitting to determine fS and QMS. Bl is found using
(6). These parameters are exported to the final fitting programme to make sure the mechanical parameters are as
correct as possible. The laser transducer will also be used to determine SD. Measuring frequency response and
velocity simultaneously gives us the possibility to find SD by means of curve fitting. This technique gives better
results than to be found by just measuring (for a single roll surround) the wave top to wave top diameter. The whole
identification procedure should be programmed to be executed automatically using suitable “macros”. Wolfgang
Klippel has further showed interest in incorporating the new model into his LPM module in his “Klippel Analyzer”.
When such an integration is realized, it will make the new model parameters available for all using “The Klippel
Analyser” – and will improve the results obtained with this very useful equipment.
.
6. CONCLUSION
An improved blocked electrical impedance model for loudspeaker drivers was introduced that was found to be
highly predictive of the actual driver behavior over a wide bandwidth - and therefore we conclude that the model is
close to describing correctly the physical reality of the speaker’s blocked impedance. This model is applicable to
most, if not all, electro dynamic transducers. The model incorporates a semi-inductor (the Vanderkooy Model [5])
due to eddy currents and “skin effect” in the pole piece. A discrepancy in Vanderkooy’s theory was that he found a
surprisingly high transition frequency between ordinary induction behavior at low frequency to semi-induction
behavior at higher frequencies. This paper shows that we, in fact, have semi-induction down to an extreme low
transition frequency, and that the observed high transition frequency is a rather simple consequence of the air gap in
the magnetic circuit.
Further it is shown, how the effect of high conductive material concentric to the voice coil (short circuiting rings in
the magnet system, pole extension or copper cap on the pole piece) can be treated as a one turn secondary in a
transformer with the voice coil as primary. These effects are introduced in the new model with the addition of a few
new elements. A further advantage of the equivalent circuit proposed here, is that it also copes well with
loudspeakers having a very flat impedance curves.

25. 25
The New Model will allow us to make box simulations more correctly, will be useful for calculation of dividing
networks, and will provide a sound basis on which further work on linear and nonlinear behaviour of loudspeakers
can be conducted.
7. ACKNOWLEDGEMENTS
Tymphany Corporation supported this work. The authors would like to thank John Vanderkooy, Wolfgang Klippel,
Bob True, and Christopher J. Struck, and the team of colleagues at Tymphany Taastrup in Denmark, for all their
useful comments and suggestions, especially Ulrik Skov who has worked on the simulations.
8 APPENDIX
Refinement of the motional Impedance model
In this paper we have focused upon the “blocked impedance”. We have taken for granted that the “motional
impedance” could be described as a simple parallel resonance circuit as shown in Fig. 2. But is this the full truth?
What about “creep”?
Peerless for about 25 years has measured TS parameters by first measuring the velocity curve with laser transducer
around resonance and by finding the peak and the 3dB frequencies of the curve determining fS and QMS. These values
along with impedance measurements were then used to determine Bl (assuming RES same as RDC) and finally
calculating the remaining parameters .This has generally given very reliable and well-defined parameters, but it was
observed that the measured velocity curve over a broader frequency range showed a lack of symmetry inconsistent
with the behavior predicted by the simple circuit model.
Before introducing the laser measurements, Peerless – like most other speaker manufacturers - used the “added
mass” method to find MMS. Here another observation was made: The lower the measured resonance (due to extra
mass) the lower the value for Z0, where as the simple theory predicts a frequency independent value.
Similarly, in reflex boxes: Measurements of the impedance curve generally showed the upper resonance peak higher
than predicted by the simulation program – and the lower resonance not as high as predicted.
The “creep” theory [17] predicts frequency dependent compliance - increasing compliance the lower the frequency.
This might explain the first observation, but obviously does not explain the next two.

26. 26
The explanation is to be found in a 25-year-old “Peerless Ring Binder Catalogue”. In the technical supplement to
this, called “PSAM” (Peerless Standard for Acoustic Measurements), there is a note stating the following:” It should
be mentioned that QMSB in an empty box becomes higher than predicted by the theory and so does Z0. The reason is
that RP --- should be split up into RP’ >RP and Rs to be placed in series with LP to represent loss in speaker
suspension. The impact of this -- diminishes as LP is shunted by LB - and Z0B might be 20% higher than Z0".
(Subscript B indicates data in box and LB is box compliance).
70
75
80
85
90
95
100
105
110
10 100 1000 10000 100000
Frequency [Hz] 80 dB~8 ohm
Calculated Impedance
Measured Impedance
Reflex Box Impedance - with Rs
Reflex Box Response - with Rs
Fig. 16 Calculated impedance (magnitude), measured impedance, calculated reflex box impedance and response.
Calculations are made with the extra resistance Rs in series with LP.
We introduce this resistor in our circuit and use the “solver” to optimize the parameters. The result is shown in Fig.
16. (We find Rs=0,66  and RP' =185.7 ). The figure also shows the impedance and response curve for a reflex
tuning (the classical tuning with box volume the same as VAS for the driver and port mass the same as MMS). This
should give two symmetric peaks at each side of the original free air resonance of the driver. It is however observed
that the upper peak is significantly higher than the lower. We can approximately substitute the impact of the series
resistance Rs by a frequency dependent parallel resistor, RP''= (L)2
/Rs - the lower the frequency the smaller this
value and the higher the damping – therefore, less damping is present at the high frequency resonance peak than at

27. 27
the low frequency resonance peak. It is worth noting that the ordinary RP when the voice coil former is made of
aluminum - as it often is - is dominated by eddy current losses in the former material.
Fig. 17 shows how the situation would have been if a series resistor had not been added to LP. Here we see the
opposite result, The lower peak higher than the upper. This is due to the influence of the real part of KE having less
impact on the lower frequency peak than on the upper frequency peak.
70
75
80
85
90
95
100
105
110
10 100 1000 10000 100000
Frequency [Hz] 80 dB~8 ohm
Calculated Impedance
Measured Impedance
Reflex Box Impedance w/o Rs
Reflex Box Response w/o Rs
Fig. 17 Same as Fig. 16, but without Rs. Note that now the lower resonance peak in the reflex impedance is higher
than the upper. This is due to the real part of KE having greater impact the higher the frequency
It is obvious that the curve fit is more exact with than without the series resistor.
To further justification for a series resistor, parameters were measured for a speaker, again Peerless 830843 (a newer
and a little improved sample), first in free air, and then loaded with an extra mass (21g). In the first case we found
fS=26,2Hz and RP=144,8Ω and in the second fS*=23,4 Hz and RP* =128,2 Ω. For this speaker we found CP=0,083 F
and a calculation based on these figures gave Rs=0,65 Ω and RP=291 Ω. (This figure is higher than for the first
830843 sample used for this evaluation. This is due to an improved spider, but the surround is the same).
9. REFERENCES

28. 28
[1] K. Thorborg, A.D. Unruh and C.J. Struck: “An Improved Electrical Equivalent Circuit Modelfor Dynamic
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[10]M. Dodd, W. Klippel, and J. Oclee-Brown, “Voice Coil Impedance as a Function of Frequency and
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29. 29
[14]A. Unruh, C. J. Struck, “Linear Array Transducer Technology”, presented at the AES 121st
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[17]Knudsen, Jensen and Grue: “Low Frequency Loudspeaker Models that include Suspension Creep”, J. Audio
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[18]Peerless "PSAM" technical supplement to "Ring Binder" catalogue (first published approximately 1980)