Sensorless Measurement of the Acoustic Impedance of a LoudspeakerRomain BOULANDET(1)(1) HEPIA, HES-SO University of Applied Sciences and Arts Western Switzerland, romain.boulandet@hesge.ch

Abstract
This paper presents an experimental approach for estimating the front-side radiation impedance of a loudspeaker.
This acoustic load impedance is the reaction of the sound field on the radiating diaphragm which may affect theloudspeaker response, particularly in the low frequencies due to the coupling with the room modes. In general,
this impedance can not be solved analytically because it depends both on the physical characteristics of the
speaker and the complexity of the acoustic field in which it radiates. Conventional measurement techniques
require sensors such as microphones and accelerometers. The approach proposed here relies mainly on thereciprocal mechanism of electrodynamic transduction, making it possible to use the loudspeaker as a sensor
simultaneously. This work shows that the front-side acoustic impedance can be derived from the speaker input
impedance and the knowledge of its electromechanical parameters. An analytical model is given and datameasured in actual rooms provides proof of concept. Potential applications are also discussed.
Keywords: Loudspeaker radiation impedance, Acoustic impedance meter, Sound field monitoring

1

INTRODUCTION

The complex interaction between a room and a loudspeaker is reflected by the air load on the radiating diaphragm. The real part of this acoustic impedance represents the sound power radiated into the air by thediaphragm and the imaginary part is the stored (reactive) power that is returned to the source. In general,
the front-side radiation impedance depends on both the loudspeaker configuration and the characteristics ofthe room. The radiation impedance of a vibrating surface can be calculated in some simple configurations
and analytical models can be found in [1, 2, 3]. State-of-the-art methods require sensors to measure acoustic
impedance. A general literature review of measurement methods for characterizing passive acoustic components
such as absorbing materials, mufflers or wind instruments can be found in [4]. An impedance sensor using anelectrodynamic transducer and two microphones to produce a controlled volume velocity source is described inRef. [5]. As shown in [5], this measurement device can be used to estimate the characteristics of the loudspeaker enclosure (eg volume size and thermal losses for a sealed enclosure, vent characteristics and Helmholtz
resonance for a bass-reflex speaker). In the case of a direct radiator loudspeaker system, the front-side acoustic
impedance can be measured very close to the diaphragm using a microphone and a nearby vibration sensor.
Doutres et al. (2010) developed a methodology for determining intrinsic properties of porous samples from themeasurement of the input impedance of an electrodynamic loudspeaker used to apply static and dynamic stresson it [6]. Recently, Ahadi and Bakhtiar (2014) proposed a computer-based measurement system using a similar
approach to assess the acoustic impedance of a closed duct [7]. Some difficulties in applying this method arereported in [8], where suggestions for improving the range of application are made based on FEM simulation.
Moreover, Boulandet et al. (2016) showed that the mobility characteristics of a mechanical structure can also
be detected from the variations of the input impedance of an electrodynamic actuator fixed thereto [9].
In this paper, we propose a straightforward method for evaluating the front acoustic load seen by a voice coilloudspeaker operating at low frequencies. Rather than using external sensors, we propose to estimate the acoustic load impedance directly through the reciprocal electrodynamic coupling, thus using the loudspeaker as anacoustic impedance meter. The remaining is organized as follow. In Section 2, we describe how to obtain
the acoustic load impedance from the loudspeaker input impedance. Section 3 describes the methodology andillustrates the performance of the concept using measured data and results computed using the finite element
method. Concluding remarks are provided, discussing potential applications in actual listening rooms.

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2

THEORYThis section provides the basic equations describing how to obtain the front-side acoustic load impedance ofa loudspeaker without using an additional sensor. An impedance-based modelling approach is used to derive
input/output relationships between variables in different parts of the loudspeaker. The lumped model discussed
below is appropriate in small signal operation when the transducer behaves sufficiently linearly [10].
The governing equations of the voice coil loudspeaker can be written in the complex frequency domain as [3]Sd p1 − Sd p2 = Zm (ω) v + Bl iu = Ze (ω) i − Bl v

(1)

where p1 and p2 are the acoustic pressures at the front and back of the diaphragm, respectively, v is thediaphragm velocity, i is the current flowing through the coil, Bl is the transduction coefficient, Sd is the effective
piston area representing the diaphragm radiating surface, and u is the voltage at the input terminals. In Eq. (1),Zm (ω) = jωMms + Rms + (jωCms )−1 is the mechanical impedance and Ze (ω) = jωLe + Re + jωL2 R2 (jωL2 + R2 )−1is the electrical impedance of the driver unit including eddy current [11]. Figure 1 gives the analogous electricalcircuit representation of the loudspeaker driver mounted in a baffle, where Zar1 and Zar2 denote the front andthe back-side acoustic radiation impedances.

Figure 1. Equivalent circuit representation of the voice coil loudspeaker mounted in a baffle; ε = −Blv is theback electromotive force, Fmag = Bli is the magnetic force on the current-carrying wires, and q is the volume
velocity.
In vacuum, i.e. p1 = p2 = 0 in Eq. (1), the speaker input impedance seen by the audio amplifier can be written
as:
u
(Bl)2
Zinv (ω) == Ze (ω) +(2)
i p1 =p2 =0
Zm (ω)In open space, the vibration of the diaphragm is influenced by the presence of air on both sides, which modifies
the speaker input impedance as
Zin (ω) =u(Bl)2= Ze (ω) +i
Zm (ω) + Zmr1 (ω) + Zmr2 (ω)(3)

where Zmr1 = Sd2 Zar1 and Zmr2 = Sd2 Zar2 denote the front and the back-side radiation impedances, respectively. As
shown in Eq. (3), the acoustic load exerted by the air on both sides of the diaphragm may affect the frequency
response of the loudspeaker.
From Eqs. (1-3), the transfer function between the input current and the output diaphragm velocity can be
expressed asv
1G(ω) = = − (Zin (ω) − Ze (ω))(4)
i
Bl
From Eqs. (3-4), and after some further manipulations, the front radiation impedance can be expressed as
Zmr1 (ω) = −

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Bl
− Zm (ω) − Zmr2 (ω)
G(ω)

(5)

As can be seen in Eq. (5), the front radiation impedance can be readily obtained from variations in the speaker
electrical input impedance and prior knowledge of the speaker fundamental parameters. Note that Zmr2 is substituted with


ρc2
Sd2
(6)
Zmb (ω) = jωMab + Rab +jωVb
in Eqs. (5) and (3) for a closed-box loudspeaker system, where Vb is the net enclosure volume, ρ is the densityof air, c is the speed of sound in air, Mab is an equivalent acoustic mass and Rab an acoustic resistance [3].

3

RESULTSThis section describes the methodology used to develop the concept of sensorless acoustic load impedance meterand illustrates the overall performance under free-field conditions and in a lightly damped room. As discussed
above, the approach proposed here for obtaining the front-side acoustic load at the loudspeaker diaphragm
requires prior knowledge of the electromechanical parameters of the loudspeaker driver. The equivalent electricalcircuit models derived in Section 2 are used in the following to find the Thiele-Small (T/S) parameters using a
curve fitting method.
3.1 Evaluation of Thiel/Small parameters
The values of the T/S parameters are determined by measuring the loudspeaker input impedance, near the
resonance frequency, at small signal levels for which the mechanical behavior of the driver is effectively linear(i.e., proportional to its input). Figure 2 shows the experimental setup used to evaluate the T/S parameters.

Figure 2. Experimental setup for measurement of the electrical impedance of the loudspeaker driver (a) when
the coil is blocked, (b) in vacuum, and (c) mounted in a baffle in open space. In (a), a specific locking system isused to prevent the diaphragm from vibrating, and thus the coil attached thereto; in (b), the loudspeaker driverunit is placed in a vacuum chamber to eliminate the effect of air on the diaphragm; in (c), the loudspeaker
driver is mounted in a baffle in an anechoic chamber.

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Figure 3 illustrates the input impedance of the loudspeaker measured and calculated for different configurations.
The Matlab function lsqcurvefit was used to solve the data-fitting problems in the least squares sens. As can beseen in Fig. 3, there is a good agreement between the measured and fitted data, especially around the resonancefrequency. As the frequency increases, some discontinuities in the frequency response can be observed. They
are partly due to the break-up modes of the diaphragm, as shown in Fig. 3. Imperfect clamping of the coil
may also explain the differences shown between the measured and fitted data in Fig. 3 (b).

Figure 3. Measured and fitted data of the electrical impedance of the loudspeaker driver (a) when the coil isblocked, (b) in vacuum, (c) mounted in a baffle in anechoic chamber and (d) closed-box loudspeaker mounted
in a baffle in anechoic chamber.

The parameters Re , Le , R2 and L2 are estimated from the speaker input impedance with the coil blocked, as
shown in Fig. 3 (a), when the loudspeaker impedance only depends on the physical properties and the geometryof the coil. The parameters Mms , Rms , Cms and Bl are estimated from the speaker input impedance in vacuum,
as shown in Fig. 3 (b). As given in Eq. (2), the speaker input impedance in vacuum reveals the effect of themechanical part in addition to that of the coil, but does not include the acoustic load exerted on the diaphragm.
In free-field conditions, as given in Eq. (3), the speaker input impedance also depends on the acoustic radiation
on both sides of the diaphragm. It can be seen in Fig. 3 (c) that the resonant peak is a bit lower and slightly
shifted lower in frequency compared to that in vacuum. This is due to the coupling between the diaphragmand air which has the effect of adding an acoustic resistance and an acoustic mass to the loudspeaker. The
parameters Mab , Rab and Cab in Eq. (6) are deduced from the input impedance shown in Figs. 3 (c) and (d).
The estimated values of the T/S parameters are summarized in Tab. 1.

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Table 1. Thiele/Small parameters of the loudspeaker measured at small signal levels.
Parameter
dc resistance of the coilInductance of the coil
Resistance due to eddy currents
Para-inductance of the voice coilTransduction coefficientMoving mass
Mechanical resistanceMechanical compliance
Projected area of the diaphragm
In vacuum resonance frequency
Acoustic compliance of the cabinet

Notation
Re
Le
R2
L2
Bl
Mms
Rms
Cms
Sd
fs
Cab

Value
5.19
0.78
0.37
0.38
9.3
23.7
3.11
0.646
227
41
0.03097 10−6

Unit
Ω
mH
Ω
mH
N A−1
g
N s m−1
mm N−1cm2
Hz
m3 Pa−1

3.2 Experimental proof of concept
Once the T/S parameters of the loudspeaker system have been determined, equations (4) and (5) can be used
to estimate the output velocity to input current relationship and the front radiation impedance, respectively.
Figure 4 shows the experimental configurations used to provide a proof of concept. A Polytec ISV-500 Laser
Doppler Velocimeter is used to measure the normal velocity of the diaphragm. A Brüel and Kjaer 1/2 inch
free-field measurement microphone is used to measure the acoustic pressure. A Brüel and Kjaer LAN-XI TYPE
3160 multichannel analyzer is used to measurement. Note that the loudspeaker is driven by a voltage-controlledcurrent source (not shown here), which is designed to simultaneously measure the current and voltage at the
input terminals.

Figure 4. Experimental setup for estimating the front-side acoustic load impedance on the closed-box loudspeaker system (a) placed in an anechoic room and (b) located near a corner in a lightly damped room.

Figure 5 illustrates the frequency response function between the input current and output velocity for different
configurations. As can be seen in Fig. 5, there is a good agreement between the measured data and the model
given in Eq. (4). For the loudspeaker used for this study, the useful frequency range extends over a decade
between about 25 Hz and 300 Hz. For higher frequencies, it can be seen that the measured and estimated data
are different due to unmodelled dynamics in the loudspeaker system and inaccuracies in the coil inductanceestimation. It can be noted in particular in Fig. 5 (d) that room modes can be clearly distinguished from theestimated data.

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Figure 5. Frequency response function between the input current and the output velocity as given in Eq. (4)of (a) the loudspeaker mounted in a baffle in anechoic chamber, (b) the closed-box loudspeaker mounted ina baffle in anechoic chamber, (c) the closed-box loudspeaker placed in an anechoic room and (d) in a lightly
damped room.

Figure 6 illustrates the frequency response function of the specific acoustic radiation impedance, i.e. the relationship between the front-side acoustic pressure and the normal velocity of the diaphragm, in different configurations. As can be seen in Fig. 6, there is a good agreement between the measured data and the model
given in Eq. (5). In Fig. 6 (a), the measured and estimated data are also compared to the model of a circular
piston of radius r = 0.085 m mounted in an infinite baffle [3]. As shown in Fig. 6 (a), the useful frequency
range is larger with the baffle configuration and ranges between about 60 Hz and 300 Hz for the loudspeaker
used for this study. For the closed-box configuration, the useful frequency range is from 70 Hz to 250 Hz,
as shown in Figs. 6 (c) and (d). It can be noted in particular in Fig. 6 (d) that room modes can be clearly
distinguished from the estimated data. As noted above, the measured and estimated data are different outside
the useful frequency range due to unmodelled dynamics and inaccuracies in the coil inductance estimation.

4

DISCUSSIONAs demonstrated above, the radiation characteristics of a direct-radiator loudspeaker system can be estimatedwith good accuracy directly through the reciprocal electrodynamic coupling. This requires monitoring both thecurrent and voltage at the loudspeaker terminals and prior knowledge of the fundamental parameters of theloudspeaker drive unit and enclosure. The useful frequency range is, however, limited around the mechanicalresonance frequency of the loudspeaker system. In particular, the break-up modes of the diaphragm, not considered in the model, can seriously affect the performance of the concept. The electronic circuit used to drive
the voice coil loudspeaker while simultaneously monitoring the voltage and current at the terminals can also
limit performance by disrupting the measurement of the input impedance. Compensation of the frequency response of the electronic circuit makes it possible to reduce the differences. Note that several of the Thiele/Small

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Figure 6. Frequency response function of the specific acoustic impedance (see Eq. Eq. (5)) of (a) the loudspeaker mounted in a baffle in anechoic chamber, (b) the closed-box loudspeaker mounted in a baffle in anechoic chamber, (c) the closed-box loudspeaker placed in an anechoic room and (d) in a lightly damped room.

parameters are a function of cone/voice coil excursion and therefore depend on the input voltage/current level.

5

CONCLUSIONSThis paper presents a convenient methodology to estimate the front-side radiation impedance of a voice coilloudspeaker. A lumped-parameter model is derived to obtain the acoustic load radiation characteristics of thesurrounding sound field, including room modes, from a measurement of the speaker input impedance. A proof
of concept is provided based on data measured in an anechoic room and in a lightly damped room. The usefulfrequency range is around the resonance frequency of the closed-box loudspeaker system, i.e. from about 70
Hz to 250 Hz for the loudspeaker considered in this study. The concept is able to evaluate the acoustic load
exerted by the surrounding sound field on the diaphragm, especially at very low frequencies. This allows an
accurate detection of the room modes that may impact the response of the speaker depending on the locationin the room.

6

ACKNOWLEDGMENTS

The author wishes to thank François Bugnon for his technical assistance to the experimental configuration andVirgile Favre for his help during the measurement. The author also wishes to thank PSI Audio for providing
the speakers used in this study.

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