a novel, cost-effective method for loudspeakers parameters measurement

7/31/2019 A Novel, Cost-Effective Method for Loudspeakers Parameters Measurement

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A Novel, Cost-Effective Method for Loudspeakers Parameters Measurement

Under Non-Linear Conditions

M. Faifer, R. Ottoboni, S. ToscaniDipartimento di Elettrotecnica, Politecnico di Milano

Piazza Leonardo da Vinci 32, 20133 Milano (MI), Italy

Phone: +39-02-23993727, Fax: +39-02-23993703, Email: [email protected]

Abstract The linear model of the dynamic low-frequency

loudspeaker is widely used nowadays, and it is well known how toassess its parameters from simple measurements. However, this is a

small-signal model which well represents the behavior of aloudspeaker only when slight diaphragm displacements are

considered. In many cases it is important to have a more accuratemodel which takes into account the nonlinear phenomena as well,

for example during the development of a feedforward loudspeakerlinearization system. It is quite simple to obtain such a model from

the small signal one by introducing the dependence of the force

factor Bl, the suspension stiffness Kms and the voice coil inductanceLe from the diaphragm position x. More difficult is to assess theseposition-dependent parameters. In this paper, novel and low cost

methods for the Bl(x) and Kms(x) identification are proposed. Someexperimental result will be presented, and in particular acomparison between the large-signal, low-frequency behavior of theloudspeaker and its model (whose parameters has been estimated

with the proposed methods) will be performed. Finally, this modelhas been employed in order to implement a simple model-basedmethod for the loudspeaker linearization. Other experimental resultswill be reported and the capability of the method to reduce the

second and third harmonic distortion at very low frequencies will bediscussed.

Keywords loudspeaker nonlinearities, loudspeaker linearization,parameters estimation, force factor, suspension stiffness.

I. INTRODUCTIONIn the field of electroacoustics, one of the most discussed

research activities is represented by the analysis of the

dynamic low-frequency loudspeaker nonlinearities [1], [2],

[3], [4], [5] and [6]. In particular the study of its large-signal

behavior is very important in order to improve the reliability

and the acoustic performances [7]. Basically, a significant

reduction of the harmonic distortion can be obtained through

two different ways. First of all, this could be achieved by

performing a careful optimization of the loudspeaker designand materials. Obviously, the manufacturing cost could

noticeably increase. The second way consists in the

development of an active linearization system, which is a sort

of algorithm which controls the diaphragm motion law. This

system can be either closed-loop or open-loop. A closed-loop

approach requires a very accurate sensor in order to estimate

the voice coil position. Such a sensor could be quite

expensive. In an open-loop solution there is no need for a

sensor, and because of the decreases of the electronic

components cost, this way is becoming more and more

interesting. In particular, if the large-signal behavior of the

loudspeaker were known, it should be possible to synthesize a

mirror filter [8] used to pre-distort the signal and thus

compensating its nonlinearities [9]. Thus, a loudspeaker

model which considers the nonlinear phenomena is required

in order to design the mirror filter. Previous studies have

shown that at low frequencies the main causes of the dynamic

loudspeaker nonlinearities are substantially three:

the suspensions stiffness Kms depends on the diaphragmexcursionx;

the force factorBlis a function of the voice coil positionwith respect to the air gap;

the value of the voice coil self-inductance Le is relatedwith its position.

So, the dynamic loudspeaker nonlinearities are related with

the dependence of these three parameters on the voice coil

position [10]. The classical small-signal model can be easily

adjusted in order to take into account these phenomena. It has

been shown that the obtained model well represents the large

signal behavior of the physical device assuming that itsparameters have been properly estimated. The evaluation of

the x-dependent parameters is quite tricky. This assessment

could be performed by feeding the loudspeaker with

appropriate signals, measuring the current draw and

employing least-square optimization techniques [11]. The

parameters could be also estimated with some spot

measurements on the physical device [12]. The drawback of

this approach is that often complex and expensive equipment

is needed.

II. FORCE FACTOR ESTIMATIONLets suppose that x=0 is the rest position of the

diaphragm. A simple and fast way to estimate Bl(x) of a

loudspeaker comes directly from the induction law. An

electromotive force e(t) arises because of the voice coil

motion. Its expression is given by:

(1)

where u(t) is the moving assembly speed:

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(2)

Knowing the instantaneous position and speed of the voice

coil, and the instantaneous value of the induced emf e(t) it is

easy to calculate Bl(x). The suggested test setup is someway

similar to that used for the determination of the magnetization

characteristic of synchronous or DC machine: a prime motor

is required in order to impress the speed to the diaphragm,

and the instantaneous open circuit voltage at the loudspeaker

terminals should be measured. As prime motor, another

loudspeaker mounted in a face-to-face push-pull

configuration can be used. In order to reduce the effect of air

compression, it is recommended to make the chamber

between the two diaphragms as small as possible. This

annulus should also be sealed with particular attention. In this

way, at low frequencies the volume displaced by the two

loudspeakers is almost the same. The instantaneous position

of the tested loudspeaker diaphragm can be measured using a

laser distance sensor with proper requirements in terms of

bandwidth and resolution. Supposing that the force factor hasto be measured in the range x1 x x2, the test is performed

by feeding the motor loudspeaker with a low-frequency sine

wave so that the diaphragm of the loudspeaker under test

moves between x11 and x2+2, with 1 and 2 greater than

zero. The position and the open circuit voltage of the tested

loudspeaker should be acquired with adequate sampling rate

1/Ts. A DFT of the two signals has to be computed, and the

high frequency components (mainly due to disturbances and

quantization noise) should be removed from the spectra.

Then, the diaphragm speed is calculated performing a

frequency-domain derivation of the previously filtered

position signal. After returning in the time domain, emf,

position and speed signals must be time-aligned in order tocompensate the different delays introduced by the

measurements devices. If these delays are unknown, it is

recommended to shift the emfsignal so that the zero crossings

of the emfand speed are almost aligned. TheBl(x(nTs)) signal

should be calculated as the ratio (sample to sample) between

the emfand diaphragm speed. Using the x(nTs) signal, Bl(x)

can be estimated. Finally, an analytic expression of Bl(x)

could be obtained by performing a polynomial fitting.

The proposed method has been employed in order to

estimate Bl(x) of a commercial very low frequency

loudspeaker. The device has been tested at different

frequencies (12.5 Hz, 18.75 Hz and 25 Hz) in order to

validate the proposed method. The experimental resultsconfirm the goodness of the approach showing a remarkable

repeatability and a very low sensitivity to the excitation

frequency. In Fig. 1 the Bl(x) curve obtained using the

proposed method is shown.

Fig. 1 Fifth grade polynomial fitting of the force factorobtained with the proposed method

III. SUSPENSION STIFFNESS ESTIMATIONThe stiffness Kms(x) is quite tricky to estimate; the main

reason is that the behavior of the suspension system is fairly

complex even if the employed loudspeaker model describes it

in a very simple manner. It is possible to employ a more

complete and accurate model [1], but the assessment of the

parameters becomes even more difficult.

A possible method for the evaluation ofKms(x) can be

obtained from the analysis of the second-order differential

equation which describes the diaphragm motion. Indicating

with Fel(x) the elastic force due to the suspension system,

introducing the mechanical resistance Rms and the total

moving massMms this equation becomes:

12 (3)

Lets suppose that the diaphragm is located in the

equilibrium position x=X0 because of the force due to the

current i=I0 flowing through the coil. The relationship

betweenX0 andI0 is given by the following expression:

12 (4)

where:

(5)

When small current perturbations i are considered, the

previous differential equation can be linearized around

the equilibrium point:


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12

(6)

Usually, the first derivative of the force factor and the

second derivative of the voice coil self-inductance with

respect tox are small, and can be neglected. This leads to:

(7)

The derivative of the elastic force with respect to x

calculated in x=X0 is the differential stiffness Kms(X0). Lets

suppose that the current perturbation is a sine wave at the

angular frequency so that it is possible to employ the

phasors analysis (phasors are written in bold). Introducing the

resonance angular frequency s(X0) and the mechanical

quality factorQms(X0) it can be written:

1 1 (8)

where:

(9) (10)

The equation which gives the voltage v at the terminals is:

(11)

At low frequencies, the voltage drop across Le(x) isnegligible:

(12)

Linearizing the expression around the equilibrium point:

(13)

Having supposed that the perturbation is sinusoidal and

substituting (8):

,

1 1 (14)

where Ze(,X0) is the electrical impedance when the coil is

vibrating around the equilibrium position X0 with angular

frequency . For a givenX0, when = s(X0) it can be shown

that the electrical impedance becomes purely resistive and

reaches a maximum.

From these observations, Kms(X0) can be assessed using a

method based on the following steps:

feed the loudspeaker with a DC current I0 so that theposition of the diaphragm isX0;

superimpose an AC current at the angular frequency measure the AC voltage at the terminals; trace Ze(,X0) magnitude plot by sweeping the current

frequency, and find s(X0);

These operations have to be repeated with different values

ofX0 according to the excursion range where the stiffness has

to be estimated.

It is possible to calculate the total moving massMms using

the added-mass technique; knowing s(X0) and supposing

thatMms is constant, it is easy to estimate Kms(X0). However,

in the electro-mechanical model of the loudspeaker appears

the stiffness, not the differential stiffness. Notice that:

1

0 0 (15)

It is recommended to perform a grade n polynomial fitting

in order to obtain an analytic expression of the differential

stiffnessKms(x). At this point:

(16)

reminding the previous expression (15):

1

(17)

The proposed method has been used for the Kms(x)

identification of the 250 mm very low frequency loudspeaker

whose force factor Bl(x) has been previously evaluated.


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Adopting the added mass method a moving mass of 58.6 g

has been measured. The experimental results are shown in

Fig. 2 and Fig. 3. The curves have been obtained analyzing

the loudspeaker over the excursion range from -9 mm to 9

mm with a 1 mm step. Notice that the fitting quality of the

differential stiffness obtained with a fifth grade polynomial is

very high.

Fig. 2 Fifth grade polynomial fitting of the differential suspension stiffnessobtained employing the proposed method. The marks show the experimental

points calculated from the resonance frequency measurements.

Fig. 3 - Polynomial fitting of the suspension stiffnessobtained with the proposed method

IV. LOUDSPEAKER LINEARIZATIONIn the last section of this paper, a simple loudspeaker

linearization method based on the mirror filter will be

proposed. The main assumption is that the nonlinear

phenomena are entirely located in the loudspeaker and in

particular that the acoustic load is almost linear. When a low

frequency loudspeaker is mounted in a properly designed

cabinet and the pressure levels are not very high, this

hypothesis can be considered true.

If an accurate model of the loudspeaker system were

available, it would be possible to predict the instantaneous

diaphragm position for a given input voltage v. But it would

be even possible to calculate what is the voltage signal to be

applied at the terminals so that the instantaneous diaphragm

motion law is x. This can be used to design a pre-distortion

filter inserted between the source and the power amplifier

connected to the loudspeaker.

In an ideal, perfectly linear transducer, the inductance, the

force factor and the suspension stiffness are constant. The

following expression can be written:

(18)

where Mmd is the moving assembly mass, Zma(s) is the

mechanical impedance due to the acoustic load, X(s)

represents the Laplace transform of the diaphragm positionx,

whileL-1 denotes the inverse Laplace transform operator.

Now, lets consider the loudspeaker nonlinearities andimpose that the instantaneous diaphragm position x is the

same as that calculated employing the linear model with the

input voltage v. In this case, the input voltage vf to be applied

at the terminals results:

2

(19)

where if is the current circulating in the coil of the nonlinearloudspeaker. Subtracting (18) from (19) yields:

1 1 1 1 1

1

2

(20)

The previous expression shows how the filter distorts the

voltage v so that the nonlinear speaker behaves like a linear

one. It contains the instantaneous diaphragm position and the

current of the ideal transducer when an input voltage v is

applied at its terminals. These quantities can be easily

computed through simple linear filtering operations

employing the small signal loudspeaker model. The

estimation of the current if is much more onerous since the


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integration of the nonlinear model is required. However,

nowadays in many speaker is installed a shorting ring over

the pole piece in order to reduce the voice coil self-

inductance (and its dependence on x) dramatically. So, if the

voice coil inductance is very low, it is possible to neglect the

difference between the inductive voltage drops in the

nonlinear and in the linear model:

0 (21)

In addition, it is possible to neglect the effect of the

reluctance force. This assumption is true with good accuracy

at the low frequencies: the excursion are large, thus the

nonlinearities due to the suspension stiffness and the force

factor are relatively strong. So, the relationship between v and

vfbecomes much simpler:

1 1

1 1 1 1 (22)

The estimation of the current if is no longer needed, so the

computational load is greatly reduced. It is possible to

implement the pre-distortion filter on a moderate powered

DSP. However, it should be noticed that because of the

loudspeaker asymmetry the linearization method could

require the injection of a DC voltage component at theterminals in order to center the diaphragm motion around

x=0. Since the audio amplifiers are AC-coupled, the

employment of the pre-distortion filter requires a special

power amplifier. It is important to stress that with this method

the possible presence of a DC voltage component is crucial.

The presented linearization system has been tested on the

commercial 250 mm very low frequency loudspeaker whose

Kms(x) and Bl(x) have been previously assessed. The

loudspeaker has been installed in a 40 l closed box, lined with

a 25 mm thick layer of wood wool. For a closed enclosure,

the mechanical impedanceZma(s) can be written as follows:

(23)Rmb represents the box losses, Kmb the spring effect of theenclosure air volume, Mmb the mechanical mass due to the

diaphragm radiation. The voice coil self inductance is pretty

low because of a copper shorting ring near the pole piece. For

the reason reported above, the effect of the inductance can be

neglected. Under this hypothesis, the model of the complete

loudspeaker system becomes:

(24)

having defined the parameters:

(25) (26) (27)

The values Mms=59.1 g, Rmt=2.35 kg/s and

Kmt(0)=3.78N/mm have been easily measured employing the

well known small signal techniques.

As said before, the linearization system requires an

accurate model of the loudspeaker system. In order to verify

this, the loudspeaker system has been fed with sinusoidal

signals at different frequencies and amplitudes. The voicecoil current, the diaphragm position and the near-field sound

pressure signals have been recorded with a daq USB board

connected to a PC. For the current measurement a shunt and

an isolation amplifier have been employed. A laser distance

sensor has been used as position transducer, while an high

quality microphone provides the sound pressure

measurement. The voltage has been applied for a properly

short time, in order to minimize the power compression

effects. The obtained results have been compared to that

achieved by a simulation performed on the loudspeaker

system model. As an example, some results are shown in Fig.

4and Fig. 5.

Fig. 4 Measured (dotted line) and predicted (continuous line)diaphragm position with 9V RMS 15 Hz sinusoidal input voltage

The results shows that the loudspeaker system model is

able to predict with remarkable accuracy the behavior of the

physical device even when the voice coil displacement is

high. This means that the estimation of the parameters (in

particularKms(x) andBl(x)) is quite good.


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Fig. 5 - Measured (square markers) and predicted (circular markers)on axis sound pressure spectrum at 1 m

with 9V RMS 15 Hz sinusoidal input voltage

Since a consistent model of the loudspeaker system is

available, it is interesting to test the performance of the

linearization system. In particular, its capability to reduce the

second and third harmonic distortion for a given fundamental

sound pressure level has been investigated. For each test, asinusoidal input voltage at the frequency f and RMS

amplitude V has been directly applied to the loudspeaker

(without pre-distortion). The on axis sound pressure at 1 m

has been recorded and in particular its fundamental amplitude

has been measured. Using the linear model of the

loudspeaker, it is possible to predict what is the sinusoidal

input voltage which gives the same fundamental SPL (sound

pressure level) assuming a perfectly linear loudspeaker. This

input signal has been processed with a MatLab program

which implements the nonlinear filter and calculates the pre-

distorted voltage. Then, this voltage has been applied to the

loudspeaker system using a DC-coupled amplifier and the

sound pressure has been acquired. Finally, to evaluate theperformance of the linearization system the sound pressure

spectrums obtained with and without the pre-distortion filter

have been compared. The tests have been performed

considering different fundamental sound pressure levels and

frequencies. Some results are shown in Fig. 6 and

summarized in TABLE I.

Fig. 6 Measured sound pressure spectrum with (circular markers) andwithout (square markers) pre-distortion filter

for a 84.4 dB 15 Hz fundamental near-field SPL

TABLE I - second and third harmonic reductionachieved with the pre-distortion filter

f [Hz]Fundamental

[dB SPL]

Harmonic reduction

2nd harmonic

[dB SPL]

3rd harmonic

[dB SPL]

10 75.4 0.4 0.4

10 77.1 1.7 1.7

15 82.3 3.4 3.4

15 84.4 4.6 4.6

30 93.7 6.3 6.3

30 96.2 5.5 5.5

45 99.4 5.8 5.8

It can be noticed that at very low frequencies, where the

effect of the voice coil inductance can be neglected and the

voice coil excursions are large, the method can achieve a

significant reduction of the second and third harmonic

distortion. At the higher frequencies the method becomes lesseffective because the nonlinear effects of the inductance get

stronger while those related to the stiffness get weaker. The

assumptions on which the method is based on becomes not

completely true.

V. CONCLUSIONTwo methods for the assessment ofBl(x) and Kms(x) has

been proposed. These methods are very simple and do not

require expensive instrumentation; in addition, they can be

easily automated. The suspension stiffness and force factor of

a commercial loudspeaker have been evaluated using theproposed procedures. This has permitted to build up a

nonlinear model of this loudspeaker mounted into a closed

enclosure.

Tests shown how the model can accurately predict the non

linear low frequency behavior of the physical devices. The

obtained results prove the goodness of the model parameter

estimation.

Finally a model based sensorless loudspeaker linearization

system has been proposed. Its capability to reduce the second

and the third harmonic distortion has been proved.

REFERENCES

[1] M.H. Knudsen, J.G. Jensen, Low-Frequency Loudspeaker Models thatInclude Suspension Creep, Journal of the Audio Engineering Society,Volume 41, pp. 3-18, January/February 1993.

[2] M. Navarri, E. Bellati, F. Tordini, R. Toppi, A Novel SynthesisApproach to Loudspeaker Design, presented at the Italian AudioEngineering Society Seminar on Loudspeaker and Nonlinearities,Milan, September 20 2003.

[3] H. Schurer, A.G.J. Nijmeijer, M.A. Boer, C.H. Slump, O.E. Herrmann,Identification and Compensation of the Electrodynamic TransducerNonlinearities, IEEE International Conference on Acoustics, Speech,and Signal Processing, Volume 3, pp. 2381 2384, April 21-24 1997.


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[4] J. Christophorou, Low Frequency Loudspeaker Measurements with an

Accelerometer, Journal of the Audio Engineering Society, Volume28, pp. 809-816, 1980.

[5] E.S. Olsen, Measurement of Mechanical Parameter Nonlinearities ofElectrodynamic Loudspeakers, presented at the 98th Convention of theAudio Engineering Society, paper 4000, February 1995.

[6] R.H. Small, Direct-Radiator Loudspeaker System Analysis, Journalof the Audio Engineering Society, Volume 20, pp. 383-395, June 1972.[7] W. Klippel, Nonlinear Large-Signal Behavior of ElectrodynamicLoudspeakers at Low Frequencies, Journal of the Audio EngineeringSociety, Volume 40, p. 483, 1992.

[8] W. Klippel, The Mirror Filter - A New Basis for Reducing NonlinearDistortion and Equalizing Response in Woofer Systems, Journal of theAudio Engineering Society, Volume 40, pp. 675-691, September 1992.

[9] W. Klippel, Compensation for Nonlinear Distortion of HornLoudspeakers by Digital Signal Processing, Journal of the AudioEngineering Society, Volume 44, pp. 964-972, November 1996.

[10] D. Clark, R.J. Mihelich, Modeling and Controlling Excursion-RelatedDistortion in Loudspeakers, presented at the 106th Convention of theAudio Engineering Society, May 1999, Preprint 4862.

[11] W. Klippel, Measurement of Large-Signal Parameters ofElectrodynamic Transducer, presented at the 107th Convention of theAudio Engineering Society, New York, September 24-27 1999,Preprint 5008.

[12] D. Clark, Precision Measurement of Loudspeaker Parameters,Journal of the Audio Engineering Society, Volume 45, pp. 129-141,

March 1997.

a novel, cost-effective method for loudspeakers parameters measurement

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