Measurement of loudspeaker parameters: A pedagogical approach
Antonin NOVAK1
Laboratoire d’Acoustique de l’Université du Mans, UMR CNRS 6613, Le Mans, France, antonin.novak@univ-lemans.fr(1) antonin.novak@univ-lemans.fr, https://ant-novak.com

Abstract
In this paper, we discuss a pedagogical approach to measuring an electrodynamic loudspeaker and understanding
the importance of the development of more precise models. The traditional way of measuring an electrodynamic
loudspeaker consists of adjusting the loudspeaker model parameters to get the best fit with the measured signals.
The method proposed in this paper is based on a straightforward idea that permits first to estimate the force
factor without any model assumption and then to separate the electrical and mechanical impedance intuitively.
Consequently, both impedances can be studied separately to highlight the importance of using models incorporating eddy currents on the electrical side and creep effect on the mechanical side. Python code to follow all thesteps is available online on https://ant-novak.com/pages/speaker_params/.
Keywords: Loudspeaker measurement, Thielle-Small parameters

1

INTRODUCTION

The basic principle of an electrodynamic loudspeaker, invented almost 100 years ago [9], is well-known thanks
to electrical analogies. These analogies have been next used in works of Thiele [13, 14] and Small [12] tomodel the low-frequency behavior of a loudspeaker thanks to parameters that are measurable. A general loudspeaker model is composed of three parts, the electrical one with electrical impedance Ze , the mechanical partwith mechanical impedance Zms and the acoustical part with load impedance Zal . These three parts are joined by
the force factor Bl (electro-mechanical) and by an equivalent surface area S (mechano-acoustical) (see Fig. 1a).
The acoustical part depends on the radiation (load) impedance of both sides of the membrane and is sometimes
included in the mechanical part (Fig. 1b).
In the traditional Thiele-Small model, the electrical part is represented by an R-L circuit in which Re and Le
represent the voice-coil losses and inductance respectively. An ideal mass–spring–damper system with mass
Mms , stiffness Kms , and mechanical resistance Rms representing the mechanical losses describes the mechanicalpart. If the acoustical impedance Zal is joined with the mechanical impedance Zms they form a mechanical
impedance Zma . Some works, trying to improve the original lumped-element linear Thiele and Small (TS)model, are presented below.
Ze

Zms

Zma = Zms + S 2 Zal

Ze

S

Bl

Bl

Zal

(a)

(b)

Figure 1. General model of a loudspeaker.

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1.1 Models of Voice-CoilThe classical loudspeaker motors include iron pieces whose electrical conductivity is relatively high leading to
the presence of eddy currents [11, 18]. There have been many attempts to include the effects of eddy currents
into the loudspeaker models by modifying the electrical part. Some of them remain simple, keeping the total
number of parameters low, others are more precise but for the cost of higher complexity. The model proposedby Thiele [14] put a resistance Rs in parallel with the inductance Le . This idea has been then extended by
Murray [6] and later by Klippel [3]. Another approach consist in applying frequency dependent elements, e.g.
the models of Vanderkooy [18], Wright [20], Leach [5], Thorborg and Unruh [17], or Brunet and Shafai [1]which uses fractional derivatives.
Electrical part models (Ze (ω))

Mechanical part models (Zma (ω))

jωLe

C1

Re

Mms

R1
R2

Rs
Thiele (1975)

Knudsen, Jensen (1993) [SLS]√√jωLe ||(K ω + jK ω)

Re

Mms

Kr ω Xr + jKi ω Xi

Mms

Wright (1989)

Rms

Cms (ω)

jωL1

Mms
Rs

R1
ωAms

jωL2

Murray (1994)

Thorborg et al. (2010)

jωL2
Re

C1 [1 − λ log(jω)]

Knudsen, Jensen (1993) [LOG]

jωLe
Re

Ce (jω)−βRms

Knudsen, Jensen (1993) [EXP]

Vanderkooy (1988)

Re

C2

jωLe

Mms

Rms

Cms (ω)λms (ω)R2
Klippel (2000)

Thorborg, Futtrup (2013)

K(jω)nRe

Mms

Leach (2002)

Re

Cα (jω)αRms

King, Agerkvist (2015)
√

√

Rs

K ω + jK ω

jωL0

jωLe

jωLeB

Mms

Thorborg, Unruh (2008)

Rv

C0

η, β

Novak (2015)

Figure 2. Historical evolution of electrical and mechanical part of loudspeaker models, taking into account eddy
currents and viscoelastic effects.

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1.2 Models of Suspensions
While the electrical part is quite well understood and modeled, the mechanical part of the loudspeaker remains
up to now the less understood part of the loudspeaker. Visco-elastic materials used for the surround and the
spider are very complex systems not only due to complex material structures but also due to their geometry.
Several studies [4],[16],[10] have shown that the traditional mass–spring–damper model fails to describe thecomplex viscoelastic behavior of these materials even at low amplitudes. One of the first attempts to take into
account the viscoelastic effects is presented in the study of Knudsen and Jensen [4] in which three models areproposed: (a) the Standard Linear Solid (SLS) model, (b) the EXP (exponential) model, and (c) the LOG (logarithmic) model. All three models have shown an improvement over the traditional model; the LOG performing
the best. The LOG model is then modified in the works of Thorgborg et al. [16, 15]. King and Agerkvist [2]use the EXP model applying fractional derivatives. Finally, in [7] a fractional element, often used in the fieldof rheology, is adapted for loudspeaker suspension modeling.
1.3 Which Model to Choose?
The number of existing models of voice-coil and surround is quite high and may lead to a source of confusion,
especially for students and young engineers. Each model has its pros and its cons, but none of the models describes the loudspeaker behavior perfectly. In general, if we want to have a better model describing loudspeaker
behavior more precisely, we need a model with more parameters. On the other hand, in many applications, thesimplicity (low number of parameters) wins over the model precision.
Side by side with the development of new models there are many methods for parameters estimation associatedwith the models. These methods are usually based on fitting algorithms and least square error (LSE) minimization that is applied to the measured data. Such an approach have two notable drawbacks. First, if a part of themodel does not fit the reality correctly, e.g., a loudspeaker with a huge creep effect modeled using a classical
TS model, the estimated parameters (including the electrical ones) are all in error. Second, fitting all the parameters at once does not provide any information from which part (i.e., voice-coil, magnetic circuit, suspension)
possible errors come from.
The following part of the paper proposes a measurement method based on an intuitive approach that also answers many questions one can ask regarding the choice of a model and the loudspeaker behavior.

2

PROPOSED METHOD

2.1 Prerequisites
In the following, we consider that the loudspeaker under test has been mounted in an "almost-infinite" baffle
and measured using any wide-band signal and that the voltage, current, and velocity signals have been acquired.
In the case of the example presented in this paper, we used a multi-tone signal consisting of 648 frequencies
f logarithmically spaced between 10 Hz and 20 kHz, forming an angular frequency vector ω = 2π f with 648
points. Next we suppose that the frequency-domain quantities, i.e. voltage U(ω), current I(ω), and velocity
V (ω) are calculated from the measured signals (see Table reftab:table1). For frequencies above 4 kHz, theTable 1. Measured and derived quantities used in the paper (Bl being the loudspeaker force factor).

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voltagecurrent

velocity

input impedance

electrical impedancemechanical impedancetime-domain

u(t)

i(t)

v(t)

—

—

—-

frequency-domain

U(ω)

I(ω)

V (ω)

Zin (ω) =

U(ω)
I(ω)

Ze (ω) = Zin (ω) − Bl V (ω)

Zm (ω) =Bl I(ω)V (ω)

vibrometer values V (ω) are set to zero to ignore high-frequency modal behavior. The data file and the Python
code with all the following steps are available online (https://ant-novak.com/pages/speaker_params/)[8].
2.2 Bl estimation
The input impedance Zin (ω) of the loudspeaker estimated as the ratio between the measured voltage U(ω) andcurrent I(ω) (see Table 1), depicted in Fig. 3 for the measured data, is given asZin (ω) = Ze (ω) + Bl

V (ω)
,
I(ω)

(1)

where Bl is the force factor of the loudspeaker and Ze (ω) is the electrical impedance of the voice coil, which
can be expressed as
V (ω)
Ze (ω) = Zin (ω) − Bl
.
(2)
I(ω)
The main idea of the estimation of the force factor Bl independently of other model parameters (such as thosein Fig. 2) is that the voice-coil impedance Ze (ω) has no reason to exhibit any resonance at low frequencies.
V (ω)
The resonance behavior is due to the mechanical part, in other words, due to the term Bl. Consequently,
I(ω)
we can evaluate Eq. (2) for different estimated values of Bl. Such value Bl that minimizes the resonant behavior
of Ze (ω) is then selected as the Bl value of the loudspeaker. This process is graphically represented in Fig. 4.
In this particular example the Bl = 4.8 Tm.
2.3 Electrical ImpedanceOnce the value of Bl is estimated, we can separate the electrical impedance Ze (ω) using Eq. (2). The ThieleSmall model predicts the electrical impedance behavior being influenced by two constant parameters, resistance
Re and inductance Le . The resistance is the real part of Ze (ω), and the inductance is the imaginary part
Ze (ω) divided by the angular frequency ω. These two quantities are deduced from the real data from themeasurement and depicted in Fig. 5. It is clear from both figures that both quantities are highly dependent on
frequency contrary to the Thiele-Small model in which both quantities are constant. For this reason they arecalled "apparent resistance" Re (ω) and "apparent inductance" Le (ω).
The reason for such a variation with frequency is the presence of eddy currents in the iron pole pieces. They
increase the apparent resistance due to the thermal effects of eddy currents passing inside the iron and decrease
the apparent inductance by mutual inductance. The eddy currents depend on frequency due to the skin effect
[18].

25

|Zin| [ ]

20
15
10
5
0 1
10

102

103
Frequency [Hz]104

Figure 3. Input impedance of the loudspeaker estimated from the measured voltage and current.

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25

8

real(Ze) [ ]

20
15
10
5
0 1
10

102

103
Frequency [Hz]real(Ze) [ ]Bl = 0.0
Bl = 1.0
Bl = 2.0
Bl = 3.0
Bl = 4.0
Bl = 5.0
Bl = 6.0

7Bl = 4.4
Bl = 4.6
Bl = 4.8
Bl = 5.0
Bl = 5.2

6
5
4 1
10

104

102
Frequency [Hz]

103

25

1.0

20

0.8

apparent Le [mH]apparent Re [ ]Figure 4. Graphical representation of the force factor Bl estimation from the measured data. The value of Bl
that removes the resonance from the electric impedance Ze (ω) is selected as the best estimate of Bl.

15
10
5
0
0.0

2.5

5.0

7.5 10.0 12.5 15.0 17.5 20.0
Frequency [kHz]

0.6
0.4
0.2
0.0
0.0

2.5

5.0

7.5 10.0 12.5 15.0 17.5 20.0
Frequency [kHz]n Z (ω) oe(b) imagωno(a) real Ze (ω)

Figure 5. Apparent resistance and inductance of the loudspeaker voice-coil, both depending on frequency, estimated from the measured data.
The effect of eddy currents can now be included in the model of the loudspeaker by choosing any of themodels, e.g., one of those presented in Table 1. The model fitting can be done only on the isolated electricalpart Ze (ω) with no influence on other parameters of the loudspeaker.
2.4 Mechanical Impedance
In the next step, we can estimate the mechanical impedance Zm (ω) that includes also the acoustical radiation
impedance. The mechanical impedance Zm (ω) (or mechanical admittance Ym (ω) = Zm−1 (ω)) can be deduced as
Zm (ω) = Bl

F(ω)
,
V (ω)

(3)

In Fig. 6 both the mechanical admittance and mechanical impedance are depicted as a function of frequency.
By observing both impedance curves, we can note two distinct parts. At low frequencies (up to 300 Hz) the
behavior is driven by a damped mass-spring system with a resonance frequency near 55 Hz. At high frequencies(above 300 Hz) the behavior starts to become less predictable (see Fig. 6b) due to a non-piston motion of themechanical part (break-up, modal behavior). Since the modal behavior is out of the scope of this paper andsince we consider only the piston motion, we use only the data from low frequencies (between 20 Hz and
250 Hz) for further analysis of mechanical part.

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140

0.4

120

0.3

|Zma| [Nsm 1]|Yma| [mN 1s 1]

0.5

0.2
0.1
0.0
101

100
80
60
40
20

102

103

Frequency [Hz]

0 1
10

102
Frequency [Hz](a) mechanical admittance103(b) mechanical impedance

Figure 6. The absolute value of the mechanical admittance (a) and the mechanical impedance (b) as a functionof frequency, estimated from the measured data.
Mechanical Resistance
The Thiele-Small model considers the mechanical impedance to behave like a mass-spring-damper system withmass Mma , stiffness Kma , and mechanical resistance Rma related to the mechanical impedance as
Zm (ω) = jωMma + Rma +

Kma
.
jω

(4)

The real part of the mechanical impedance Zm (ω) represents mechanical losses. According to Eq. (4), themechanical resistance is constant and equal to Rma . The real part of the mechanical impedance Zm (ω) depicted
in Fig. 7 for the measured data shows an important frequency dependency with frequency and is called "apparent
mechanical resistance". Indeed, this kind of behavior is related to a creep effect [4, 7] and can be taken into
account by applying a model that includes these effects (see Table 1).

apparent Rma [Nsm 1]

5
4
3
2
1
0

0

50

100
150
Frequency [Hz]

200

250

Figure 7. The real part of the mechanical impedance representing the mechanical losses, estimated from themeasured data.

Moving Mass and Stiffness
Finally, the imaginary part of the mechanical impedance Zm (ω) represents the mechanical energy storage. According to the Thiele-Small model, the stiffness Kma is related to the moving mass Mma as
Kma = ω 2 Mma − imag {Zm (ω)} ω.

(5)

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To find both, one can fit the imaginary part of the mechanical impedance to the measured data. However, it
is known that the stiffness is frequency dependent due to the creep effect and that the moving mass can befrequency dependent as well due to the inertial air load [19]. Nevertheless, at very low frequencies the moving
mass should not vary with frequency a lot [19]. We consider it to be constant in the selected frequency region
from 20 Hz to 250 Hz and we apply a similar technique as we used for the force factor estimation by plotting
the apparent stiffness Kma (ω) calculated using Eq. (5) for several estimated mass Mma .

Mma = 15.0
Mma = 16.0
Mma = 17.0
Mma = 18.0
Mma = 19.0

8
6
4

apparent Kma [Nmm 1]

apparent Kma [Nmm 1]

10

2
0

0

50

100
150
Frequency [Hz]
(a)

200

250

Mma = 15.8
Mma = 16.0
Mma = 16.2
Mma = 16.4
Mma = 16.6

3.5
3.0
2.5
2.0
1.5

0

50

100
150
Frequency [Hz]

200

250

(b)

Figure 8. Estimation of the apparent stiffness Kma (ω) and the mass Mma from the measured data.
Fig. 8 shows the apparent stiffness Kma (ω) for different values of Mma . The rough estimation (Fig. 8a) withMma varying from 15 to 19 grams show that there is not a single flat Kma (ω) curve. For low estimates of Mma(e.g., 15 g) the apparent stiffness drops quickly to negative values. On the other hand for high estimates of Mma(e.g., 19 g) the apparent stiffness raises with frequency to very high values. A detailed variation of the mass
Mma (Fig. 8b) shows that it is indeed very difficult to select a single value of Mma and one associated curveof Kma (ω). It is clear that the Kma (ω) is rising with frequency, which is a well-known phenomena due to thecreep effect [4, 7], but for more detailed analysis one would need to chose one of the models from Table 1 andfit the parameters to the data.

3

Conclusion

This paper shows that the measurement of linear loudspeaker parameters can be made more intuitively andpedagogically than just by blindly fitting the model to measured data. The estimation consists of several consecutive steps. First, we estimate the force factor Bl independently of the chosen model. Then, we separate
the electrical and the mechanical impedance, and we show that the classical Thiele-Small parameters that areconsidered constant vary with frequency. These steps may help to understand why there is a large amount ofloudspeaker models available in the literature and their relation to physics. We also show that the mechanicalpart can be estimated only in the low-frequency region to avoid break-up and modal behavior influence while
the electrical part can be estimated in a full frequency region.

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