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IEEE AUTOTESTCON 2008 Salt Lake City, UT, 8-11 September 2008
A Novel, Cost-Eff Cost-Effective ective Method for Loudspeak Loudspeakers ers Parameters Measurement Under Non-Linear Conditions M. Faifer, R. Ottoboni, S. Toscani Dipartimento 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 to assess its parameters from simple measurements. However, this is a small-signal model which well represents the behavior of a loudspeaker only when slight diaphragm displacements are considered. In many cases it is important to have a more accurate model which takes into account the nonlinear phenomena as well, for example during the development of o f a feedforward loudspeaker linearization 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 K m mss and the voice coil inductance Le from the diaphragm position x. More difficult is to assess these position-dependent parameters. In this paper, novel and low cost methods for the Bl(x) and K m mss(x) identification are proposed. Some
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:
experimental result will be presented, and in particular a comparison between the large-signal, low-frequency behavior of the loudspeaker and its model (whose parameters has been estimated with the proposed methods) will be performed. Finally, this model has been employed in order to implement a simple model-based method for the loudspeaker linearization. Other experimental results will be reported and the capability of the method to reduce the second and third harmonic distortion at very low frequencie frequenciess will be discussed.
the factor factor Bl Bl is air is agap; function of the voice coil position withforce respect to the the value of the voice coil self-inductance Le is related with its position.
Keywords – loudspeaker nonlinearities, loudspeaker linearization, parameters estimation, force force factor, suspension stiffness. stiffness.
I.
INTRODUCTION
In 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 iimportant mportant in order to improve the t he rel reliability iability 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 design and 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
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the suspensions stiffness K ms ms depends on the diaphragm excursion x excursion x ;
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 t hat the obtained model well represents the lar large ge signal behavior of the physical device assuming that its parameters have been properly estimated. The evaluation of the x-dependent 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 ESTIMATION
Let’s suppose that x= x=0 0 is the rest position of the diaphragm. A simple and fast way to estimate Bl ( x) 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:
·
(2)
Knowing the instantaneous position and speed of the voice coil, and the instantaneous value of the induced emf e( e(t ) it is easy to calculate Bl ( x). 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 in order toopen impress thevoltage speed at to the theloudspeaker diaphragm, andrequired the instantaneous circuit 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. resolutio n. Supposing that the force factor has to be measured in the range x 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 x1−δ 1 and x2+δ 2, with δ 1 and δ 2 greater than zero. The position and the open circuit voltage of the tested loudspeaker should be acquired with adequate sampling rate 1/ 1/T T s. 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 to compensate the different delays introduced by the measurements devices. If these delays are unknown, it is recommended to shift the emf signal signal so that the zero crossings of the emf and and speed are almost aligned. The Bl The Bl ( x x((nT s)) signal should be calculated as the ratio (sample to sample) between the emf and diaphragm speed. Using the x x((nT s) signal, Bl ( x x)) can be estimated. Finally, an analytic expression of Bl ( x x)) could be obtained by performing a polynomial fitting. The proposed method has been employed in order to estimate Bl ( x) 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 results confirm the goodness of the approach showing a remarkable repeatability and a very low sensitivity to the excitation frequency. In Fig. 1 the Bl ( x) x) curve obtained using the proposed method is shown.
Fig. 1 – Fifth grade polynomial fitting of the force factor obtained with the proposed method
III.
SUSPENSION STIFFNESS ESTIMATION
The stiffness K ms x) is quite tricky to estimate; the main ms( x) 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. x) can be A possible method for the evaluation of K ms ms( x) obtained from the analysis of the second-order differential equation which describes the diaphragm motion. Indicating with F eel l ( x) x) the elastic force due to the suspension system, introducing the mechanical resistance Rms and the total moving mass M mass M ms ms this equation becomes:
12
(3)
Let’s suppose that the diaphragm is located in the equilibrium position x=X 0 because of the force due to the current i=I 0 flowing through the coil. The relationship between X between X 0 and and I I 0 is given by the following expression:
12 ·
(4)
where:
(5)
When small current perturbations δ i are considered, the previous differential equation can be linearized around the equilibrium point:
1 2
Having supposed that the perturbation is sinusoidal and substituting (8):
Usually, the first derivative of the force factor and the second derivative of the voice coil self-inductance with respect to x to x are are small, and can be neglected. This leads to:
(7)
The derivative of the elastic force with respect to x x calculated in x=X 0 is the differential stiffness K’ ms ms( X 0). Let’s suppose that the current perturbation is a sine wave at the angular frequency ω so that it is possible to employ the phasors analysis (phasors are written writte n in bold). Introducing the resonance angular frequency ω s( X 0) and the mechanical quality factor Qms( X 0) it can be written:
1 1
,, 1 1
(6)
(8)
where:
where Z e( ω, X ω, X 0) is the electrical impedance when the coil is vibrating around the equilibrium position X 0 with angular frequency ω . For a given X given X 0, when ω= ω s( X 0) it can be shown that the electrical impedance becomes purely resistive and reaches a maximum. From these observations, K ms ms( X 0) can be assessed using a method based on the following steps: •
feed the loudspeaker with a DC current I 0 so that the position of the diaphragm is X is X 0; • superimpose an AC current at the angular frequency ω measure the AC voltage at the terminals; • trace Z e( ω, X 0) magnitude plot by sweeping the current • frequency, and find ω s( X 0);
These operations have to be repeated with different values of of X X 0 according to the excursion range where the stiffness has to be estimated. It is possible to calculate the total moving mass M mass M ms ms using the added-mass technique; knowing ω s( X 0) and supposing that M that M ms X 0). However, ms is constant, it is easy to estimate K’ ms ms( X in the electro-mechanical model of the loudspeaker appears the stiffness, not the differential stiffness. Notice that:
(9)
(10)
The equation which gives the voltage v at the terminals is:
1 0 0 1
(16)
reminding the previous expression (15):
(12)
(13)
(15)
It is recommended to perform a grade n polynomial fitting in order to obtain an analytic expression of the differential stiffness K’ stiffness K’ ms x). At this point: ms( x).
Linearizing the expression around the equilibrium point:
(11)
At low frequencies, the voltage drop across Le( x x)) is negligible:
(14)
(17)
The proposed used for loudspeaker the K ms( x) x) identification of themethod 250 mmhas verybeen low frequency whose force factor Bl ( x) x) has been previously evaluated.
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.
diaphragm position for a given input voltage v. But it would be even possible to calculate what is the t he volt voltage age signal to be applied at the terminals so that the instantaneous diaphragm motion law is x. 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:
Fig. 2 – Fifth grade polynomial fitting of the differential suspension stiffness obtained employing the proposed method. The marks show the experimental points calculated from the resonance frequency frequency measurements.
(18)
where M md s) is the md is the moving assembly mass, Z m ma a( s) mechanical impedance due to the acoustic load, X ( s) s) represents the Laplace transform of the diaphragm position x position x,, while L while L-1 denotes the inverse Laplace transform operator. Now, let’s consider the loudspeaker nonlinearities and impose that the instantaneous diaphragm position x x is the same as that calculated employing the linear model with the input voltage v. In this case, the input voltage v f to be applied at the terminals results:
2 1 1 1 1 1 1 2
(19)
where i f is the current circulating in the coil of the nonlinear loudspeaker. Subtracting (18) from (19) yields:
Fig. 3 - Polynomial fitting of the suspension stiffness obtained with the proposed method
IV.
LOUDSPEAKER LINEARIZATION
(20)
In 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 tthe he acoustic load iiss almost linear. When a low frequency loudspeaker is mounted in a properly designed cabinet and the pressure levels are not very high, this
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
hypothesis can be considered true. If an accurate model of the loudspeaker system were available, it would be possible to predict the instantaneous
computed through simple linear filtering operations employing the small signal loudspeaker model. The estimation of the current i f is much more onerous since the
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 selfinductance (and its dependence on x 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:
(24)
having defined the parameters:
(25)
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 v f becomes much simpler:
1 1 1 1 1 1
(22)
The estimation of the current i f 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 the terminals in order to center the diaphragm motion around x=0. x= 0. Since the audio amplifiers are AC-coupled, the employment of the pre-distortion filter requires a special power amplifier. It is i s 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 x) and Bl ( x) x) have been previously assessed. The K m mss( x) 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 impedance Z m s) can be written as follows: ma a( s)
(26) (27)
The values M’ ms g, Rmt =2.35 kg/s and ms=59.1 K mt (0) (0)= = 3.78N/mm have been easily measured employing the mt 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 voice coil 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.
(23) Rmb represents the box losses, K m mb b the spring effect of the enclosure air volume, M mb the mechanical mass due to the mb 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:
Fig. 4 – Measured (dotted li ne) and predicted (continuous line) diaphragm position with 9V RMS 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 and Bl ( x)) x) and Bl x)) is quite good. particular K particular K ms ms( x)
TABLE I - second and t hird harmonic reduction achieved with the pre-distortion filter Harmonic reduction f [Hz]
Fig. 5 - Measured (square markers) and predict ed (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, a sinusoidal 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) l evel) assuming a perfectly linear loudspeaker. This T his input signal has been processed with a MatLab program which implements the nonlinear filter and calculates the predistorted 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 the performance 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.
Fundamental [dB SPL]
2nd harmonic
3rd harmonic
[dB SPL]
[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 less effective 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.
CONCLUSION
Two methods for the assessment of Bl ( x) x) and K ms x) has ms( x) 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 the proposed 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
Fig. 6 – Measured sound pressure spectrum with (circular markers) and without (square markers) pre-distortion filter for a 84.4 dB 15 Hz fundamental near-field SPL
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[9] W. Klippel, “Compensation for Nonlinear Distortion of Horn Loudspeakers by Digital Signal Processing”, Journal of the Audio Engineering Society, Volume 44, pp. 964-972, November 1996. [10] D. Clark, R.J. Mihelich, “Modeling and Controlling Excursion-Related Distortion in Loudspeakers”, presented at the 106 th Convention of the Audio Engineering Society, May 1999, Preprint 4862. [11] W. Klippel, “Measurement of Large-Signal Parameters of Electrodynamic Transducer”, presented at the 107th Convention of the Audio 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.
