Pierre Muret
email: pierre.muret@grenoble.cnrs.fr

Julien Pernot
email: julien.pernot@grenoble.cnrs.fr

Tokuyuki Teraji

Toshimichi Ito

Deep levels in homoepitaxial borondoped diamond films studied by capacitance and current transient spectroscopies

Deep level transient spectroscopies (DLTS) applied to

Schottky junctions made on homoepitaxial boron-doped diamond films show the existence of two traps. A deep acceptor, negatively charged and strongly attractive for holes, 1.57 eV above the valence band edge displays the characteristic features of a complex defect due to interacting centers and impurities, also displaying some evolutions after thermal cycles, possibly due to hydrogen effusion or diffusion. It is tentatively ascribed to associa-tion of a boron atom, a vacancy and several hydrogen atoms. A deep donor, 1.13 eV above the valence band edge, able to compensate the boron acceptors, is attributed to a defect correlated with dislocations. It could be due to the positively charged carbon vacancy. These conclusions are drawn from the Fourier transform-DLTS results coupled with isothermal time domain algorithms allowing the discrimination of multiple emission rates with high resolution.

Copyright line will be provided by the published 1 Introduction. A growing number of devices is based on boron doped diamond, a large gap semiconductor [1,2]. Like in other semiconductors, residual deep centers may hamper the improvement of device performances. But the electrical activity of defects and unwanted impurities are very scarcely documented [3][START_REF] Muret | Diamond Relat[END_REF][5], contrary to photoluminescence and cathodoluminescence due to radiative transitions of carriers. Non-radiative transitions are however an essential mechanism which controls the electrical characteristics of electronic devices through capture and emission of carriers trapped on the deep centres. Then, experimental evidence comes naturally from the transient response of junctions, which are sensitive to the trapping and de-trapping events. Such a study is presented here, addressing more samples and bringing new details than already published in [5]. Section 2 contains the experimental techniques and data while section 3 is devoted to the discussion of results.

Experimental techniques and results

Diamond growth, contacts fabrication and electrical measurements

Boron-doped homoepitaxial layers were deposited on Ib-type (100) diamond substrates, by microwave plasma assisted chemical vapor deposition (MPCVD), with conditions detailed in references [6,7]. Several Al Schottky contacts were fabricated on the oxidized surface using an electron-beam evaporator. Ohmic contacts were also made on the top surface of the diamond film with a gap of some tens of µm between the Schottky diode periphery and the ohmic contact. The resulting series resistance of each diode was low enough for allowing capacitance measurements at 1 MHz, which indicated a net acceptor concentration of (8±1)×10 16 cm -3 in the sample under study, which will be labelled A. Other homoepitaxial samples were prepared with different conditions in an other reactor, with a very light boron concentration of 7.5×10 14 B/cm -3 [START_REF] Wade | Diamond Relat[END_REF]9]. Ohmic contacts were deposited on the sample B of ref. [9], allow-ing current measurements in the bulk, because the sample has been treated in an oxygen plasma in order to remove the conductive surface layer. Fourier-transform deep level transient spectroscopy (DLTS) has been performed on the various samples, either with the capacitance detected mode in Schottky diodes of the sample A, or with photo-induced current transient spectroscopy (PICTS) in the sample B, provided only with ohmic contacts.

Fourier transform deep level transient spectroscopy

The capacitance due to the depletion zone of a junction is sensitive to the redistribution of charge which takes place when deep centres are loosing carriers by reemission either to the valence or conduction band. A transient current also exist because a corresponding flow of carriers is driven by the electric field into the external circuit. This capacitance or current transient can be analysed by Fourier transform, either as a function of temperature which controls the emission rates (DLTS), at some time windows t w , or varying the time window (Isothermal Transient Spectroscopy abbreviated in ITS) at several temperatures [10].

Figure 1 One of the Fourier coefficients, here a 1 , of the experimental capacitance transients in the diode 1 of the sample A, measured in time windows of 2.05 s and 10 s respectively for the peaks at the higher and lower temperatures, in full lines. Broken lines simulate the same DLTS spectra, assuming only one emission rate at each temperature, deduced from the Arrhenius diagram in Fig. 3. They deviate markedly from experimental data below 550 K, except for the peak maxima.

The Fourier coefficient a 1 of the capacitance transients measured in the diode number 1 and 2 of the sample A are displayed respectively in Fig. 1 and 2 for two time windows. They clearly show two peaks, occurring at the same temperatures in the two diodes, but with different amplitudes. This fact indicates that traps are not homogenously scattered in the sample A. The peaks appearing near 500 K and 630 K for the two time windows shown in this study will be labelled respectively "trap I" and "trap II". As it can be seen in figures, an other important difference between the two diodes is the width of the peaks related to trap I: in diode 2, the simulated DLTS spectra are very close to the experimental ones, whereas the peak occurring at the lower temperature is much broader in the diode 1. Such a discrepancy between the experimental and simulated spectra indicates that multi-emission rates probably occur in diode 1 for this trap or alternatively, that several traps superimpose. From the knowledge of the Fourier coefficients amplitude and net acceptor concentration, the trap concentrations are about 3×10 15 cm -3 in diode 1 for the two traps while in diode 2, they are close to 10 16 cm -3 and 10 15 cm -3 respectively for trap I and II. Because the product e p t w has a unique and known value when a particular Fourier coefficient is maximum, numerous data of e p (T) can be collected for each trap. Then, from an Arrhenius plot of e p /T 2 , the activation energy E a and the apparent capture cross section σ app , contained in the α constant, can be deduced respectively from the slope of the straight line and ordinate at infinite temperature. Such a diagram is recognised as the signature of the deep level as long as the experimental points are aligned along a straight line. In fact, the activation energy E a is the sum of ΔE B and the ionisation enthalpy ΔH T . For the two traps detected in the samples studied here, the Arrhenius diagram is plotted in Fig. 3, with data coming from DLTS spectra obtained in sample A (full symbols) and PICTS data [9] coming from sample B (open symbols). From the straight lines plotted in Fig. 3, using 40% of the data for trap I and 80% for trap II, one deduces E a1 = 1.57 ± 0.1eV, σ p1 = (2-8) ×10 -12 cm 2 for trap I and E a2 = 1.13 ± 0.06eV, σ p2 = (1-4) ×10 -18 cm 2 for trap II.

Although the points located outside the straight lines indicate that the model relying on a unique emission rate at each temperature ceases to be valid above some temperature, the activation energies can be derived from the straight portion with reasonable error bars, typically less than 7% for the relative uncertainty. As soon as these activation energies are known, the simulated DLTS spectra can be superimposed in Fig. 1 and 2 on the experimental data and the capture cross sections can be adjusted so that maxima match simultaneously those of the experimental curves for both time windows. The capture cross sections derived in such a way are more accurate than those obtained at the intercept of the vertical axis of the Arrhenius diagrams but can still be compared and justify the error bars mentioned above for σ p1 and σ p2 . It is worthwhile to notice that firstly this procedure is not commonly used by DLTS practitioners, and secondly each capture cross section thus obtained is extrapolated at infinite temperature because the capture barrier factor is incorporated into the activation energy. Other possible deviations of the extracted values from the real ones are discussed separately for trap I and II in the following.

Properties of the trap I

The DLTS spectra related to trap I show at least two anomalous behaviours: (i) a large broadening in some diodes ; (ii) a peak height which depends on the time window. These features are not expected if the emission rate distribution is discrete, with a value given by equation ( 1), or equivalently if the transient does not follow the simple exponential law exp(-e p t) at each temperature, t being the time. Since the e p derivation is based on this law, such anomalies induce deviations from the straight line in the Arrhenius diagram, as displayed in Fig. 3 for the highest temperatures. More sophisticated methods become necessary in the case of multiple emission rates or continuous distribution. Equation ( 1) can be considered as still valid, because it comes from the detailed balance principle, but a sum of terms or an integral form must be taken into account for the transient. The discrete distribution g

) exp( t e g pi i i - ∑
i or the continuous one g(e p ) has to be extracted by methods which both rely on time domain algorithms [11,12] and measurements done in a very large time range, because of the dispersion of e pi or e p over several decades. Such a constraint requires transients recorded on a logarithmic time scale, typically over 6 decades, as shown in Fig. 4 (a). According to the algorithm which is used, either DISCRETE [11] or FTIKREG [12], the distribution which is deduced appears respectively as discrete or continuous in Fig. 4 (b). As shown in Fig. 4 (a), the transients simulated with either the discrete or the continuous distribution cannot be distinguished. The isothermal spectra would be useful for discriminating between the two models if both the experimental and simulated ones were available. Such a work, which needs large handling of data, is in progress. Meanwhile, from the extent of the emission rate distribution over approximately six decades in Fig. 4 (b), it is easy to derive a broadening of 6×2.3kT close to 0.6 eV at the measurement temperature of 500 K, if the dispersion is supposed to affect only the activation energy. Alternatively, a distribution of the capture cross sections over up to six decades could comply with the experimental results, but is less probable because capture cross section is strongly dependent of the initial charge of the centre, which is not expected to experience large fluctuations. For these reasons, the values deduced for the parameters E a1 and σ p1 must be considered with caution and at least as average values with larger uncertainties than initially quoted.

Properties of the trap II

A peak similar to that related to trap II in diode A is found for the sample B [9] in PICTS response and the values of e p (T) are aligned on the same straight line as for sample A in the Arrhenius diagram (see Fig. 3). This is the firm indication of a common origin of the trap II in both samples. Moreover, DLTS spectra of trap II display full width at half maximum (FWMH) very close to those of simulated ones in Fig. 1 and2, indicating a single emission rate at each temperature. Then, the standard method used above for the derivation of E a2 and σ p2 can be considered as valid.

Within the uncertainty previously quoted, a capture cross section in the 10 -18 cm 2 range suggests that the centre responsible for the trap II is repulsive for holes, and hence positively charged, involving a donor character. This prop-erty is confirmed by the following experiment performed in sample B.

In sample B, the PICTS signal has been measured when the sample was both heated and submitted to a temperature gradient from the back to its top surface, due to thermal losses. The subsequent inhomogeneous thermal expansion has triggered a plastic relaxation of the diamond layer, because of the additional stress. A strong increase of the dislocations density resulted as it can be seen in the cathodoluminescence spectra taken before and after thermal cycle (Fig. 5), since the A band due to dislocations at 2.9 eV increased by a factor of 40. Moreover, the conductivity of the sample underwent a large decrease due to the compensation of boron acceptors, as already indicated in [9] and in a forthcoming publication. Compensation of acceptors must result necessarily from the presence of deep donors and therefore, trap II turns out to be a good candidate since it was detected after the decrease of the conductivity in sample B, displays a weak capture cross section in sample A and a common Arrhenius diagram in both samples. Moreover, the origin of this defect appears as connected to the presence of dislocations in the boron doped homoepitaxial layer.

Discussion and conclusions

Study of defects in diamond has been performed essentially by optical methods [START_REF] Zaitsev | Optical Properties of Diamond[END_REF], which probe radiative transition between a deep level and either the conduction band or valence band states, or even between two discrete levels like in donor-acceptor pair transitions. Hence, the final state is not necessarily known, contrary to the case of nonradiative transitions detected in DLTS. However, compari-son to data available in literature can be attempted. Several optical or photo-electrical studies [START_REF] Nesládek | [END_REF][15][16][17] and one based on electrical measurements [18] show a transition close to 1.5 eV which could be compared to the case of trap I. But these results have all been obtained in samples where either impurities such as Li, Na, S or N have been deliberately introduced or the sample has been bombarded by ions. Neither the sample A nor B have undergone such treatments. Hence, one is forced to conclude that the origin of the trap I is not related to the impurities mentioned above. More generally, with a broad distribution of emission rates at each temperature as demonstrated in the second section, an impurity center is not likely. A complex defect, with several species and/or distorted neighbouring site, and also a fluctuating environment, is much more probable. Such variable structural changes near the defect site could be made easier with the presence of vacancies. It must also be noticed that some metastability of this defect has been detected since its concentration profile is both non monotonic and changes according to the subsequent thermal cycles [5]. As only hydrogen is able to move inside the diamond lattice, this set of conjectures suggest that this defect could be related to one of the complex involving a substitutional boron, a carbon vacancy and zero to four hydrogen atoms, as discussed by Goss et al. [19]. The deep donor, 1.13 eV above the valence band edge, identified as the trap II, correlated with dislocations, may be tentatively ascribed to the positively charged vacancy, which is the stable state of the vacancy in boron doped diamond [20]. Detailed investigations, specially about the hydrogen content and profile will be helpful for a better understanding of these defects. The effect of non radiative transitions upon the deep levels population as probed by DLTS will remain the preferred method for studying this kind of problem, since it has already proven its interest in the present article.

Figure 2 2 α

 22 Figure2Fourier coefficients a 1 in diode 2 of sample A for two time windows in full lines: 2.05 s (the smaller peak) and 10 s (the taller peak). Although simulated spectra, in broken lines, are very close to experimental curves, the height difference of the peaks near 500 K (trap I) is not expected for a discrete emission rate.

Figure 3 :

 3 Figure 3 : Arrhenius diagram for the two traps. Full circles and squares come from DLTS spectra in sample A, while open squares from PICTS in sample B. Deviation from straight lines occur at highest temperature (and highest emission rates), specially for trap I.

Figure 4 :

 4 Figure 4 : (a) Experimental transient (open circles) measured at 500 K in the diode 3 of the sample A, which displays DLTS spectra (not shown) with intermediate FWMH between those of the diodes 1 and 2. (b) discrete and continuous distributions of emission rates deduced from the experimental transient (see text). In (a), solid lines are recalculated from these distributions, the upper from the discrete distribution, the lower from the continuous one, and shifted for clarity.

Figure 5 :

 5 Figure 5 : Cathodoluminescence spectra taken at 5 K and normalised on the FETO line: (a) before heating from ambient to circa 650 K; (b) after thermal cycle down to ambient temperature.

Copyright line will be provided by the publisher


Etienne Gaviot

Mehran Erza

Frédéric Polet

Lionel Camberlein

Bruno Bêche

Versatile Analytical Approach for Assessing Harmonic Distortion in Current-Driven Electrodynamic Loudspeakers

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Sound processing with current driven loudspeakers depends critically on high quality transducers together with their relevant and adequate circuitry. Although nowadays this approach is far from being spread in today's audio industry, the development of high-quality miniaturized devices based on new materials is bound to start a significant change towards current drive policy. Moreover, besides cogent technical developments, strong simplifications regarding classical analytical approaches are most welcome to foster the mandatory engineering work enabling to reach production stage.

The main purpose of this paper is to present a compact and articulate analytical model of current driven loudspeakers so as to assess their remaining non-linear shortcomings. Displacement, speed and acceleration (quantity directly associated with the resulting sound pressure) of the voice coil can be easily characterized up to their first fifth-order harmonic distortion components along the whole spectrum. To cover the latter, numerical simulations require numerous calculations to deal with a specific isolated flaw. On the other hand, any i th -order harmonic distortion line (HD i ) is classically referred to by way of plotting its respective acceleration level as a logarithmic ratio with that of the fundamental level. Conversely, we highlight in this paper the interest to depict each component together with the fundamental as a logarithmic ratio with the gravity acceleration level, allowing the observer to assess at a glance the spectral pattern of any i th -order harmonic distortion.

Straightforward models for coupling loudspeakers with electronics software are quite simple since based on RLC circuits [1]. Yet, despite picturing the impedance versus frequency relationship, such models are obviously unable to allow for the numerous shortcomings of loudspeakers, especially their nonlinear limitations. Indeed, nonlinear behaviours are out of reach of such models since neither electromechanical properties nor physical quantities, such as displacement, velocity, acceleration and force may be accurately accounted for. Since a long time, current drive of loudspeakers proved a well established way to reduce non linear distortion. However, despite numerous works and a significantly probing literature [2,[START_REF] Mills | Distortion Reduction in Moving-Coil Loudspeaker Systems Using Current-Drive technology[END_REF], quite few devices are nowadays available on the market. p-2/23 -As a matter of fact the following fundamental mechanical and electrical equations describing a generic transducer can be written as [START_REF] Mills | Distortion Reduction in Moving-Coil Loudspeaker Systems Using Current-Drive technology[END_REF]4,[START_REF] Klippel | Dynamic measurement and interpretation of nonlinear parameters of electrodynamic loudspeakers[END_REF]:

x k dt dx f dt x d M dx dL i i B m m m e l ⋅ + ⋅ + ⋅ = ⋅ ⋅ + ⋅ ⋅ 2 2 2 2 1 [N] (1) dt dx B dt di L i R e l e t e t ⋅ + + ⋅ = . ) ( ) ( [V]
(2)

As regards Eq.1, x is the displacement of the voice coil (m), B l the force factor (T.m) of the motor system, i the driving current, M m , the equivalent mass of the moving voice coil [kg], f m the mechanical damping parameter and drag force [N.s.m -1 ], k m the suspension stiffness [N.m -1 ], and L e the voice coil inductance. The left member includes the useful drive force due to the Lorentz interaction, acting together with an extraneous component, the solenoid force (also referred to as the reluctance force), stemming from the stocked energy within the moving coil [START_REF] Mills | Distortion Reduction in Moving-Coil Loudspeaker Systems Using Current-Drive technology[END_REF]4,[START_REF] Kaiser | Modelling of the Nonlinear Response of an Electrodynamic Loudspeaker by a Volterra Series Expansion[END_REF][START_REF] Klippel | Dynamic measurement and interpretation of nonlinear parameters of electrodynamic loudspeakers[END_REF].

Considering Eq.2, R e is the voice coil electrical resistance [Ω], and e the fem [V] observed at the transducer terminals during operation. Although valuable to assess the electrical behaviour of the system, Eq.2 becomes factually useless in the advantageous case of a current-drive policy since the designer has only to optimize the left member of Eq.1 so as to drive flawlessly the moving coil. Then, the physical behaviour of a generic electrodynamic loudspeaker is exclusively imposed by Eq.1 and naturally devoid of the numerous non linear distortions stemming from the unfortunate compound of both Eq.1 and 2 addressing the voltage drive policy [START_REF] Erza | Contribution à l'étude des non-linéarités au sein d'une association amplificateur-transducteur -Atténuation des distorsions par un mode de contrôle en courant[END_REF][START_REF] Harwood | Loudspeaker Distortion Associated with Low-Frequency Signals[END_REF][START_REF] Dobrucki | Nonlinear Distortion of Woofers in the Fundamental Resonance Region[END_REF]. Although the only advantage under voltage drive is a natural damping in the resonance region of the spectrum (owing to the motional impedance), effective filtering solutions can be nowadays considered to promote current-drive [START_REF] Mills | Distortion Reduction in Moving-Coil Loudspeaker Systems Using Current-Drive technology[END_REF][START_REF] Klippel | Dynamic measurement and interpretation of nonlinear parameters of electrodynamic loudspeakers[END_REF], as is shown in the appendix illustrating a mere generic scheme. Then, one could acknowledge that the widespread voltage drive policy stems from commonly established practice and classical electronic convenience only.

HIGHLIGHTING THE RATIONALE OF THE APPROACH

Predominance and self-governing property of Eq.1 are clearly highlighted with the straightforward electrical schemes depicted in Fig. 1: indeed Eq. 2 entails most untoward compounded electromotive forces (inductive and motional emf s ), whereas any ideal current-drive policy makes the behaviour of the transducer independent of such extraneous e.m.f s . Therefore, the sources of non linearity associated with the latter are naturally rejected with a functional current-drive. However, to be operational, an adequate filtering policy has to clear out the resonance regime together with prospective untoward high frequencies.

dt dx B dt di L i R e l e t e t ⋅ + + ⋅ = . ) ( ) (

Fig. 1: Natural rejection of nonlinear components due to Eq.2 with an ideal current-drive policy

p-3/23 -Considering the displacement of the diaphragm and its derived quantities (speed and acceleration), the natural behaviour of the transducer only depends on Eq.1, where the nonlinear harmonic components stem from relevant effects regarding both parameters in the left member (B l , L e ), and those included in the right one (k m , f m ). Since representative of the effective sound pressure the acceleration is the pivotal quantity.

For the purpose of comparison between numerical and analytical results, we refer to the special case of a commercialized midrange loudspeaker (Morel ® EM 428) as a reference model, considering its nominal parameters [START_REF] Greiner | Loudspeaker Distortion Reduction[END_REF] together with its measured nonlinear characteristics obtained by way of a standard calibrating Klippel ® apparatus [11].

Nonlinear parameters can be described by way of Taylor classical truncated power-series [12,[START_REF] Jeffreys | Methods of Mathematical Physics[END_REF] that may be conveniently limited to the fifth order. Here, the following compact formulation allows us to describe harmonic distortions up to the fifth spectral line. To this end, the displacement can be described according to the following series:

( ) ψ ψ ψ ψ ψ ψ ω ω ⋅ ⋅ ⋅ ⋅ = ⋅ = ⋅ ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ≈ ⋅ = ⋅ = ∑ ∑ 5 5 4 4 3 3 2 2 1 1 1 ) ( e X e X e X e X e X e X e X X n i i i n i t j i i j (3)
On the other hand, considering the dependence of a given nonlinear parameter noted λ λ λ λ against a static value referred to as x (displacement in steady state), λ λ λ λ can be approached with a power-series expansion such as:

( )

5 5 4 4 3 3 2 2 1 0 x x x x x x ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + = λ λ λ λ λ λ λ [USI] (4)
For a given regime fitted with a fundamental frequency, the first fifth powers of the displacement 1<k<5 can be expressed with:

ε ψ ψ ψ ψ ψ ω + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ = = ⋅ ⋅ ⋅ ⋅ ⋅ 5 5 4 4 3 3 2 2 1 1 ) ( e x e x e x e x e x X X k k k k k j k k k k k k k k (5)
As each coefficient noted x i k related with Eq.5 can be easily determined by way of a classical truncated product, the set of relevant values is summed up in Table I. X picturing the sound pressure) are developed so as to conform with respective left and right members (LM and RM) of Eq.1 rewritten as:

X 1 x 1 1 = X 1 x 2 1 = X 2 x 3 1 = X 3 x 4 1 = X 4 x 5 1 = X 5 Ordrer II X 2 x 1 2 =0 x 2 2 = X 1 2 x 3 2 =2X 1 X 2 x 4 2 = X 2 2 +2X 1 X 3 x 5 2 =X 1 X 4 +2X 2 X 3 Order III X 3 x 1 3 =0 x 2 3 =0 x 3 3 = X 1 3 x 4 3 =3X 1 2 X 2 x 5 3 =2X 1 2 X 3 +3X 1 X 2 2 Order IV X 4 x 1 4 =0 x 2 4 =0 x 3 4 =0 x 4 4 = X 1 4 x 5 4 =4X 1 3 X 2 Order V X 5 x 1 5 =0 x 2 5 =0 x 3 5 =0 x 4 5 =0 x 5 5 = X 1 5
( ) [ ] ( ) ( ) ( ) Linear RM j m j m m linear non LM j l X k X f X M I B j _ _       ⋅ + ⋅ + ⋅ = ⋅ • • • - ω ω ω ω (6)
Here, RM exclusively fitted with linear quantities, is classically processed as:

ε ψ ψ ψ ψ ψ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ 5 5 4 4 3 3 2 2 1 1 e X e X e X e X e X X (x with k m ) (7a) ' 5 4 3 2 5 5 4 4 3 3 2 2 1 1 ε ω ω ω ω ω ψ ψ ψ ψ ψ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ • e X j e X j e X j e X j e X j X (x with f m ) (7b) ' ' 25 16 9 4 5 5 2 4 4 2 3 3 2 2 2 2 1 1 2 ε ω ω ω ω ω ψ ψ ψ ψ ψ + ⋅ - ⋅ - ⋅ - ⋅ - ⋅ - = ⋅ ⋅ ⋅ ⋅ ⋅ • • e X e X e X e X e X X (x with M m ) (7c)
Gathering the relevant terms yields:

[ ] ε ψ ψ ψ ψ ψ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 P P P P P e X e X e X e X e X M m RM , (8)
where specific polynomial quantities may be defined with:

      ⋅ + ⋅ ⋅ + ⋅ =       ⋅ + ⋅ ⋅ + ⋅ = 2 2 0 2 2 2 2 2 n n 1 n 1 2 n n 1 n 1 n ω τ p p M k p M f p m m m m P . [s -2 ] (9)
Then, considering a current excitation signal such as; i = i 0 .e jω ω ω ωt = i 0 .e ψ ψ ψ ψ , Eq.6 may be written as:

[ ] ∑ = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + n i i i m e X M e i X b X b X b X b X b b 1 i 0 5 5 4 4 3 3 2 2 1 0 ' ' ' ' ' P ψ ψ ε (10)
In this notation coefficients noted b' relative to X (jω) are given for each power with b' i = b i /2 i , considering so coefficients b i (0≤i≤5) measured in steady state [START_REF] Klippel | Dynamic measurement and interpretation of nonlinear parameters of electrodynamic loudspeakers[END_REF]. Then, assuming as relevant the identification of each component to a given harmonic frequency, the quantities summed up in Table I allow us to point out :

M m .X 1 .P 1 = b 0 .i 0 [fundamental frequency] (11)
M m .X 2 .P 2 = 0+ b' 1 .i 0 .X 1 M m .X 3 .P 3 = 0+ b' 1 .i 0 .X 2 + b' 2 .i 0 .X 1 2 M m .X 4 .P 4 = 0+ b' 1 .i 0 X 3 + b' 2 .i 0 .2.X 1 .X 2 + b' 3 .i 0 .X 1 3 M m .X 5 .P 5 = 0+ b' 1 .i 0 .X 4 + b' 2 .i 0 .(X 2 2 +2.X 1 .X 3 ) + b' 3 .i 0 .3.X 1 2 .X 2 + b' 4 .i 0 .X 1 4 M m .X 6 .P 6 = 0+ b' 1 .i 0 .X 5 + b' 2 .i 0 .(X 1 .X 4 +2.X 2 .X 3 ) + b' 3 .i 0 .(2X 1 2 .X 3 +3.X 1 .X 2 2 ) + b' 4 .i 0 .4X 1 3 .X 2 + b' 5 .i 0 .X 1 5
Such a writing is noticeably simplified when considering auxiliary coefficients c k =b' k .i 0 /M m together with inverted polynomial quantities such as:

( )

2 0 n 0 2 2 2 n n 1 Lim n, with n 1 n 1 n 1 1 ω ω = ∀       ⋅ + ⋅ ⋅ + ⋅ = = → F P F m m m m M k p M f p [s 2 ] ( 12
)
Then the set of relations Eqs.11 allows each component to be successively determined from its former ones: Such components may be also determined on their own, with for instance as regard the first fourth lines:

X 1 = F 1 .[c 0 ] ( 13
) X 2 = F 2 .[c 1 .X 1 ] X 3 = F 3 .[c 1 .X 2 + c 2 .X 1 2 ] X 4 = F 4 .[c 1 .X 3 + c 2 .2.X 1 .X 2 + c 3 .X 1 3 ] X 5 = F 5 .[c 1 .X 4 + c 2 .(X 2 2 +2.X 1 .X 3 ) + c 3 .
X 1 = [c 0 ].F 1 (14) X 2 = [c 0 .c 1 ].F 1 .F 2 X 3 = [c 0 2 .c 2 ].F 1 2 .F 3 + [c 0 .c 1 ].F 1 .F 2 .F 3 X 4 = [c 0 3 .c 3 ].F 1 3 .F 4 + 2.[c 0 2 .c 1 .c 2 ].F 1 2 .F 2 .F 4 + [c 0 2 .c 1 .c 2 ].F 1 2 .F 3 .F 4 + [c 0 .c 1 3 ].F 1 .F 2 .F 3 .F 4
X 5 and X 6 are not described in Eqs.14 due to their useless intricate shape besides being accessible with Eqs.13 for the purpose of illustration.

Application case and numerical specification to assess the results

As the sound pressure hinges on the acceleration of the diaphragm [1, [START_REF] Mills | Distortion Reduction in Moving-Coil Loudspeaker Systems Using Current-Drive technology[END_REF][START_REF] Klippel | Dynamic measurement and interpretation of nonlinear parameters of electrodynamic loudspeakers[END_REF] any relevant use of the latter results must consider the following transformations (for a given component regarding the n th order).

n 2 2 n n - X A ⋅ ⋅ = ω [m/s²] (15)
At first we consider the reference transducer whose measured B l static coefficients and nominal parameters are respectively summed up in Table II and Table III. Then, a moderate current regime is considered with p-5/23 -a 100 mA amplitude for the purpose of comparison of calculated acceleration values. Besides, results are also investigated with a numerical simulation operated with the Simulink ® software [START_REF] Klippel | Dynamic measurement and interpretation of nonlinear parameters of electrodynamic loudspeakers[END_REF].

Table III: Theoretical and measured nominal parameters of the EM 428 reference transducer

According to Table II, the pattern associated to the force factor against displacement is depicted in Fig. 2, where both even and odd parts are also illustrated. The operational range is voluntarily exceedingly spanned so as to highlight the relevant parts to be considered. running displacement x(t) as a feedback data for computing sequentially each other parameter. Indeed, the numerical scheme illustrated in Fig. 3 is fitted for the current supply mode [START_REF] Klippel | Dynamic measurement and interpretation of nonlinear parameters of electrodynamic loudspeakers[END_REF]. As enabled with this Simulink ® model, the current excitation signal can be processed, yielding both displacement, speed, and acceleration values in sampled time. Such behaviours are simulated before processing with a classical Fast Fourier Transform. As in the literature, the 0dB reference line level is that of the fundamental. Then, fundamental spectral lines can be investigated together with their respective sets of harmonics due to nonlinear distortions. Here, aside from B l all the other lumped parameters are considered at their rest position. The physical meaning regarding the lines pattern shown in Fig. 4 will be discussed hereafter since classical representations referred to the fundamental have clearly to be argued.

INVESTIGATING ON THE HARMONIC DISTORTION DUE TO THE SOLENOID FORCE

Considering the extraneous driving force within the left member of Eq.1, the mechanism can be assessed on its own so as to compare its untoward effects with the other sources of distortion. An exclusive solenoid drive would yield:

x k dt dx f dt x d M dx dL i m m m e ⋅ + ⋅ + ⋅ = ⋅ ⋅ ⋅ 2 2 2 2 1 [N] (16)
A very simple way to deal with such an interaction is to regard the quadratic character of the driving current in Eq. 16, with i(t)= i 0 .cos(ω ω ω ωt) and cos 2 (ω ω ω ωt) = ½.[1+cos(2ω ω ω ωt)]. The static component acts in continuously taking back the voice coil, and is superimposed with the fluctuation at the double-frequency. On the other hand, Table V illustrates the measurements values relative to the inductance power series coefficients.

Table V : Inductance nonlinear coefficient values

Then, assuming a significant current value with i 0 = 1 A, at the first order, the entailed static displacement noted X st would be close to: p-8/23 -

1 2 0 2 0 st l 4 1 4 1 ⋅ ⋅ ≈ ⋅ ⋅ = ⋅ ⋅ m e m k i dx dL k i X [m] (17)
According to Table III, the double-frequency fluctuation is superposed with a static displacement value given with: X st = -5,2.10 -7 m. Now, for the purpose of comparison the forces involved with both Lorentz useful mechanism and the solenoid extraneous one can also be evaluated. A quasi-static regime with a unity current yields a Lorentz force F L = B l .i ≈ ≈ ≈ ≈ + 5,93 N moving forward the diaphragm. On the other hand, with

[dL e /dx = -0,016 H/m], the solenoid force noted F S features a peak to peak absolute value given with:

F S = N 008 , 0 l 1 2 1 2 1 1 2 2 = ⋅ ⋅ ≈ ⋅ ⋅ ⋅ dx dL i e (18)
Such a tiny value, about a 0,35% magnitude of the Lorentz force allows us to posit a quasi linear behaviour as regards the double-frequency component superimposed with the static backward displacement. Then considering the double angular frequency ϖ ϖ ϖ ϖ =2.ω ω ω ω, the solenoid distortion component can be written:

      + ⋅ ⋅ + - = ⋅ ⋅ ⋅ = m m m m m M k j M f M X ϖ ϖ ϖ ϖ ϖ 2 ) 1( ) 1( 1 with 1 1 l 4 1 P P [m] (19)
As a result, the relevant acceleration (quantity to be considered for the audio signal), is given with:

[ ]

1 l 4 1 ) 1( 2 1 2 ϖ ω ϖ ϖ ϖ P - ⋅ ⋅ ⋅ = ⋅ = m M A [m/s²] (20)
Results of both Eq. 19 and 20 are compared with the Simulink ® numerical simulation as depicted in Fig. 5 for two different values of the driving current.

Fig. 5: Assessment of the solenoid effect

As the manifestation of the effect occurs at twice the shown frequency, it is clear from Fig. 5 that the extrema appearing at 44,45 Hz corresponds quite closely (88,90 Hz) but not exactly with the mechanical resonance frequency of the transducer (86,7 Hz). As a most important point that markedly differs from the other distortion sources, the solenoid effect doesn't abate as the frequency is increased.

INVESTIGATING ON THE RIGIDITY

The stiffness or rigidity parameter k m may be investigated either while considering its own power-series or that of its opposite quantity the so-called compliance C m =1/k m . Measured coefficient values for the stiffness are summed up in

Table VI: Nonlinear coefficient values for the suspension stiffness

Symmetry properties can be analysed the same way than that of the force factor, leading then to specific harmonic components as will be discussed hereafter when dealing again with the force factor analysis.

Harmonic components of displacement due to rigidity

Considering again Eq.6, the right member (RM) is now nonlinear, and Eq.7a has to be processed according to the power series describing the stiffness fitted with values noted k' and defined as:

( ) ε + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ = ⋅ = ∑ = 4 4 4 3 3 3 2 2 2 1 1 0 0 0 2 1 2 1 2 1 2 1 2 1 ' X k X k X k X k k X k K n i i i X (21)
Then Eq.6 can be reformulated as:

[ ] ε ω ψ ψ ψ ψ ψ ψ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⋅ ⋅ - = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 5 5 4 4 3 3 2 2 1 1 2 1 0 1 25 16 9 4 i e X e X e X e X e X M e B m [ ] ε ω ψ ψ ψ ψ ψ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ 5 5 4 4 3 3 2 2 1 1 5 4 3 2 e X e X e X e X e X f j m [ ] ε ψ ψ ψ ψ ψ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ 5 5 4 4 3 3 2 2 1 1 0 ' e X e X e X e X e X k ( ) ( ) [ ] ε ψ ψ ψ ψ + ⋅ ⋅ + ⋅ + ⋅ ⋅ + + ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ 5 3 2 4 1 4 3 1 2 2 3 2 1 2 2 1 1 2 2 2 ' e X X X X e X X X e X X e X k ( ) [ ] ε ψ ψ ψ + ⋅ ⋅ + ⋅ + ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ 5 2 2 1 3 2 1 4 2 2 1 3 3 1 2 3 2 3 ' e X X X X e X X e X k [ ] ε ψ ψ + ⋅ + ⋅ ⋅ + ⋅ ⋅ 5 2 3 1 4 4 1 3 4 ' e X X e X k [ ] ε ψ + ⋅ ⋅ + ⋅ 5 5 1 4 ' e X k
(22) Then, with regards to Table I, displacement values can be determined and each component identified according to the aforementioned way:

(23) M m .X 1 .P 1 = B l .i 0 [fundamental frequency] M m .X 2 .P 2 = -[k' 1 .X 1 2 ] M m .X 3 .P 3 = -[2.k' 1 .X 1 .X 2 + k' 2 .X 1 3 ] M m .X 4 .P 4 = -[k' 1 .(X 2 2 +2.X 1 .X 3 ) + k' 2 .(3.X 1 2 .X 2 ) + k' 3 .X 1 4 ] M m .X 5 .P 5 = -[k' 1 .(X 1 .X 4 +2X 2 .X 3 ) + k' 2 .(2.X 1 2 .X 3 +3.X 1 .X 2 2 ) + k' 3 .(4.X 1 3 .X 2 ) + k' 4 .X 1 5 ]
As before, for the purpose of simplification, auxiliary coefficients can be defined such as:

ϕ ϕ ϕ ϕ = B l .i 0 /M m Relative mass strength [N/kg] (24)
and:

m i i i M k 1 2 g - ⋅ = [N/(m i+1 .kg)] (25)
Then each component may be written as follows according to its former ones:

(26) X 1 = F 1 .[ ϕ ϕ ϕ ϕ ] X 2 = F 2 .[ g 1 . X 1 2 ] X 3 = F 3 .[ g 1 .2.X 1 .X 2 + g 2 .X 1 3 ] X 4 = F 4 .[ g 1 .(X 2 2 +2.X 1 .X 3 ) + g 2 .3.X 1 2 .X 2 + g 3 .X 1 4 ] X 5 = F 5 .[ g 1 .(X 1 .X 4 +2.X 2 .X 3 ) + g 2 .(2X 1 2 .X 3 +3.X 1 .X 2 2 ) + g 3 .4X 1 3 .X 2 + c 5 .X 1 5 ]
Those components may also be defined on their own, considering for instance the first four ones (the fifth one is not described as such, owing to a useless cumbersomeness):

X 1 = ϕ ϕ ϕ ϕ.F 1 X 2 = ϕ ϕ ϕ ϕ 2 . g 1 .F 1 2 .F 2 X 3 = ϕ ϕ ϕ ϕ 3 . [2. (27)

Application cases and comparison with numerical results

As before, acceleration values are determined with Eq.15, and relevant specifications regarding the considered driving current. In a first case, with all parameters aside k m being considered at rest (cf. Table III), the driving current is fixed with a 100 mA amplitude value. Then, with the classical way to refer all harmonic component levels to the fundamental values, Table VII makes clear the most significant level values regarding this special case.

Table VII: Level values of the fundamental line with a 100 mA current driving

Moreover, the behaviour of the first fifth components is depicted on Fig. 6. Here, the analytic model seems less fitting with the numerical approach whose values are ensured with another calculation developed with the Mathematica ® software. Indeed, at a frequency close to 25 Hz the fifth component values (then at 125 Hz) shows off a discrepancy between calculated levels reaching up to 7 dB. Aside from this noticeable local divergence the analytic model seems to behave properly. Considering now Fig. 7, as the driving current is significantly increased several noticeable discrepancies show up at low frequency values as regard both fourth and fifth harmonic components, with local inaccuracies reaching up to 10 dB. However, on the whole the analytic approach behaves properly and most especially as the frequency is increased.

IMPROVING THE HARMONIC LINES REPRESENTATION

In the first part of this paper, the classical way that hinges on taking the fundamental level as a reference has been used with the only purpose of comparison between numerical and analytic results. However, such a practice proves unsuitable for understanding the relative influence of the sources of distortion. Then, in the following we take as reference the acceleration level of the gravity constant (9.81 m/s²) as is classical in many a technical field of physics. As a first consequence one can assess the relative importance of the former cases that can be compared.

Investigating on the relative influence of the harmonic two

As an illustration, we consider the former distortion cases (with a 100 mA current driving) associated with the second harmonic components. Then, the effects respectively due to the force factor, the stiffness and the solenoid interaction, can be plotted while considering the gravity constant as the acceleration reference value. Results are depicted in Fig. 8, while taking account of the reference transducer data [START_REF] Greiner | Loudspeaker Distortion Reduction[END_REF].

p-12/23 -.

Fig. 8: Comparison between the relative effects of the second harmonic distortion lines

Straightforward acoustic considerations [1,6] allow to highlight the audible threshold with the following equivalence [0dB SL ⇔ -51dB g ]. It is clear from Fig. 8 that with such a low 100mA current value the solenoid effect is negligible while the force factor effect is predominant at low frequencies. Then, although unusual in the state of the art, taking the choice of a given acceleration value as a reference improves the understanding of the relative effects.

INVESTIGATING ON THE HARMONIC DISTORTION DUE TO THE DRAG FORCE

The mechanical resistance of driver suspension losses can be investigated with a view to designing specific metrological devices. Although the transducer considered for the illustration is not fitted for such a metrological purpose, it is of interest to assume the behaviour of its f m parameter as a nonlinear function of the speed of the diaphragm (v) and to determine the entailed distortion effect. To our knowledge such an investigation is in no way a classical approach described in the literature.

A simple law accounting for the Drag force

At first a general power-series function of speed can be considered to describe the mechanical damping parameter and drag force as follows:

( ) ε + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + = 5 4 3 2 v f v f v f v f v f f f 5 4 3 2 1 0 v m (28)
Although devoid of physical meaning, such a writing could be assessed with a speed observed at steady state. Then, considering back the physical meaning the coefficient f 1 is a rather insignificant quantity, just allowing for some sort of unsymmetrical interaction that we shall neglect in the following illustration. Conversely, f 2 is the prominent factor taking account of the drag force that is clearly a function of the volumetric mass (ρ ρ ρ ρ) of the fluid flowing around the transducer. Then with a clear physical meaning, Eq.28 could be written as:

( ) 2 2 2 v v f v f f f ⋅ ⋅ ⋅ ⋅ = ⋅ + ⋅ + = ρ ε 2 1 C ith w x 2 2 0 S v m [N/(m/s)] (29)
In such a relation, C x is a classical drag coefficient that we will take as the unity, and S the active surface of the diaphragm (57 cm² for the EM 428 transducer). Hence a general analytic rationale can be developed.

Harmonic components of displacement due to Drag force

At first let's consider the auxiliary quantities Y and Y i defined with:

Y i = X 1 Y 2 = 2.X 2 Y 3 = 3.X 3 Y 4 = 4.X 4 Y 5 = 5.X 5 (30) p-13/23 - ε ψ ψ ψ ψ ψ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ 5 5 4 4 3 3 2 2 1 1 e Y e Y e Y e Y e Y Y
Then, as regards speed we have:

Y j X ω = • ,with: 2 2 2 Y X ⋅ - = • ω , 3 3 3 Y j X ⋅ - = • ω , 4 4 4 Y X ⋅ = • ω , 5 5 5 Y j X ⋅ = • ω (31)
On the other hand, the f m coefficient against displacement and successive derivatives is such as:

( ) ε + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ + ⋅ = • • • • • 5 5 5 4 4 4 3 3 3 2 2 2 1 1 0 0 2 1 2 1 2 1 2 1 2 1 2 1 X X X X X X m f f f f f f f (32)
With a driving current noted i = i 0 .e jω ω ω ωt = i 0 .e ψ ψ ψ ψ , the relevant part of Eq.1 can be formulated as:

(33) [ ] ε ω ψ ψ ψ ψ ψ ψ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⋅ ⋅ - = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 5 5 4 4 3 3 2 2 1 1 2 1 0 1 25 16 9 4 i e X e X e X e X e X M e B m [ ] ε ψ ψ ψ ψ ψ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ 5 5 4 4 3 3 2 2 1 1 e X e X e X e X e X k [ ] ε ω ψ ψ ψ ψ ψ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ 5 5 4 4 3 3 2 2 1 1 0 5 4 3 2 ' e X e X e X e X e X f j ( ) ( ) [ ] ε ω ψ ψ ψ ψ + ⋅ ⋅ + ⋅ + ⋅ ⋅ + + ⋅ + ⋅ ⋅ - ⋅ ⋅ ⋅ ⋅ 5 3 2 4 1 4 3 1 2 2 3 2 1 2 2 1 1 2 2 2 2 ' e Y X Y X e Y X Y e Y X e X f ( ) [ ] ε ω ψ ψ ψ + ⋅ ⋅ + ⋅ + ⋅ + ⋅ ⋅ - ⋅ ⋅ ⋅ 5 2 2 1 3 2 1 4 2 2 1 3 3 1 2 3 3 2 3 ' e Y X Y X e Y X e X f j [ ] ε ω ψ ψ + ⋅ + ⋅ ⋅ + ⋅ ⋅ 5 2 3 1 4 4 1 3 4 4 ' e Y X e X f [ ] ε ω ψ + ⋅ ⋅ + ⋅ 5 5 1 4 5 ' e X f j
In this writing the auxiliary parameters f' i = (1/2 i ).f i allow a simplified notation. Moreover, as before while considering the quantities arranged in Table I, we have the following identifications:

(34)

M m .X 1 .P 1 = B l .i 0 [fundamental frequency] M m .X 2 .P 2 = [f' 1 .ω 2 .X 1 2 ] M m .X 3 .P 3 = [f' 1 .ω 2 .2.X 1 .Y 2 + f' 2 .jω 3 .X 1 3 ] M m .X 4 .P 4 = [f' 1 .ω 2 .(Y 2 2 +2.X 1 .Y 3 ) + f' 2 .jω 3 .(3.X 1 2 .Y 2 ) -f' 3 .ω 4 .X 1 4 ] M m .X 5 .P 5 = [f' 1 .ω 2 .(X 1 .Y 4 +2Y 2 .Y 3 ) + f' 2 .jω 3 .(2.X 1 2 .Y 3 +3.X 1 .Y 2 2 ) -f' 3 .ω 4 . 4.X 1 3 .Y 2 -f' 4 .jω 5 .X 1 5 ]
As regards Eq.24. in the same way as the prior analysis (relative to k m ), such a writing may be reformulated as the following set of relations: (35)

X 1 = F 1 .[ ϕ ϕ ϕ ϕ ] [fundamental frequency] X 2 = F 2 .[ h 1 .ω ω ω ω 2 .X 1 2 ] X 3 = F 3 .[ h 1 .ω ω ω ω 2 .4.X 1 .X 2 + j.ω ω ω ω 3 .h 2 .X 1 3 ] X 4 = F 4 .[ h 1 .ω ω ω ω 2 .( 4
.X 2 2 + 6.X 1 .X 3 ) + j.ω ω ω ω 3 .h 2 .6.X 1 2 .X 2 -h 3 .ω ω ω ω 4 .X 1 4 ] X 5 = F 5 .[ h 1 .ω ω ω ω 2 .(4.X 1 .X 4 +12.X 2 .X 3 ) + j.ω ω ω ω 3 .h 2 .(6X 1 2 .X 3 +12.X 1 .X 2 2 ) -h 3 .ω ω ω ω 4 .8.X 1 3 .X 2 -h 4 .jω ω ω ω 5 .X 1 5 ],

with auxiliary parameters defined as:

m i i i M f 1 2 h ⋅ = (36)
Hence, here again each component can be expressed on its own, with for instance the first four components explicitly given with: (37)

X 1 = F 1 .[ ϕ ϕ ϕ ϕ ] [fundamental frequency] X 2 = F 1 2 .F 2 .ϕ ϕ ϕ ϕ 2 .ω ω ω ω 2 .[h 1 ] X 3 = F 1 3 .F 3 .ϕ ϕ ϕ ϕ 3 .ω ω ω ω 3 .

A prospective application

As a prospective application one could investigate on the volumetric mass that could be measured thanks to a transducer optimized to this end. The mass being a conservative quantity, the static pressure around the transducer could be the quantity under experiment. As an illustration with air, ρ ρ ρ ρ = 1,23 kg/m 3 at the atmospheric pressure, taking C x as the unity and S = 57 cm², as regard Eq.29 and Table III (f 0 =1,239 kg/s) p-14/23 -we have f 2 = (½.ρ ρ ρ ρ.C x .S) = 3,506.10 -3 N/(m/s) 3 . Then if the static pressure were to be increased the coefficient f m would behave according to Fig. 9.

Fig. 9: Influence of the static pressure on the drag force part of the f m coefficient

Harmonic components of acceleration due to the Drag force

With Eq.15 and taking account of the exclusive evenness associated with Eq.29, the set of Eqs.36 yields the three following harmonic components:

A 1 = -F 1 .ϕ ϕ ϕ ϕ.ω ω ω ω 2 (38)
A 3 = -9.F 1 3 .F 3 .ϕ ϕ ϕ ϕ 3 .j.ω ω ω ω 5 .h 2 (39)

A 5 = 150.F 1 6 .F 3 .F 5 .ϕ ϕ ϕ ϕ 5 .ω ω ω ω 8 .h 2 (40)

Then, special cases may be investigated involving both this analytic approach and relevant numerical Simulink ® computations.

Comparison between analytical and numerical results

Considering h 2 given with Eq.36, if P a is the atmospheric pressure and P the pressure experimented with, a given static pressure condition entails:

f 2 = (½.
[N/(m/s) 3 ] (41)
At first, it is clear that the fundamental is devoid of any useful information about the drag force, then about the static pressure. Conversely, according to Eq.39, the characterisation of the third line could be directly indicative of the pressure. Moreover, the sensitivity of a prospective device stemming from such an approach would be directly function of the third power of the current value, insofar as the other shortcomings (generating their part within this third component) would be duly rejected by way of any differential metrological approach in the state of the art [START_REF] Klippel | Tutorial :Loudspeaker nonlinearities -cause, parameters, symptoms[END_REF].

In a first case we take as before a quite low value for the driving current with 100 mA. Moreover the chosen relative static pressure value is also not that much noticeable with only twice the atmospheric pressure as is plotted in green in Fig. 9. Then, according to the aforementioned interest about the gravity reference level, analytical and numerical results can be compared and made clear with Fig. 10.

p-15/23 - Indeed, the gravity reference way to represent the specific pattern relative to each component is not flawed with the pattern of the fundamental, as is unfortunately the case with the classical way to refer to the level of the latter. Then, Fig. 10 allows a genuine observation about the real level featured by the harmonic components. Here the extremum associated with the third component (at the specific frequency 3x86,7Hz) is relevant with the maximum value characterizing the sensitivity of a device whose operation would be fostered by our rationale. Here, with the relatively moderate value of the driving current, the signal would be just close to the audible threshold.

More explicitly, considering the preceding notations together with Eq.39, the third acceleration component analysis may also be stated as the product of a specific metrological constant (K a ) by a relevant function of frequency (ϒ ϒ ϒ ϒ (ω ω ω ω) ) so as to highlight the parameter of sensitivity for a dedicated sensor (S )[(m/s²)/(kg/m 3 )].

[ ] ρ ω ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = 3 0 3 3 1 5 x 4 3 3 S C 8 9 - A i j M B m l F F , or: A 3 = K a .ϒ ϒ ϒ ϒ (ω ω ω ω) .i 0 3 .ρ ρ ρ ρ or: A 3 = S .ρ ρ ρ ρ (42)
Then aside from those technical prospective considerations, it is necessary to get back to the assessment of the analytical approach, with a second case to be investigated with both approaches considering a more significant current value: Here, the current is now set to 250 mA while maintaining the static pressure value at twice the atmospheric pressure.

Then, a pertinent comparison between analytical and numerical results is allowed while considering the patterns of the harmonic components depicted in Fig. 11. As expected, the relevant level of the acceleration signal are noticeably increased with the driving current, suggesting that some degree of freedom would be allowed as regards the design of a specific sensor. In this way it is clear that harmonic components of both third and fifth orders are substantially enhanced.

GETTING AHEAD TO IMPROVE HARMONIC LINES REPRESENTATION

As a major improvement, taking the gravity as a reference proves to be of help for interpreting the behaviour of harmonic distortion. A second provision may be considered while plotting each component according to its own real (eventually audible) frequency value. Fig. 12 translates in this way the data depicted in Fig. 4. As real frequency values are directly displayed on such a representation, physical properties are made much clearer. At first, whatever the component a maximum shows up at the natural mechanical frequency of the transducer. Then, as a prominent property each harmonic component shows off a second maximum exactly arranged at its own specific place as the product of the fundamental resonance frequency by the considered order of said component. Now, considering the fifth component, as is also clear with Fig. 4., a quite peculiar singularity occurs around the special frequency value [42.6 x 5 = 213 Hz]. Such a "Moot Extremum" point was closely observed with both the analytic and numerical approaches, up to now without any obvious physical meaning nor mathematical justification as regards the set of Eqs.13 [START_REF] Klippel | Dynamic measurement and interpretation of nonlinear parameters of electrodynamic loudspeakers[END_REF][START_REF] Gaviot | Metrological prospects for the assessment of transition plateaus[END_REF].

On the other hand, the falling off regarding each harmonic component at high frequencies is easily quantified, at first while perusing the figure, but more precisely with the analytic approach applied on both sides of the decade As an a priori non obvious property only limited to the harmonic components investigation (since it would be false with a study on intermodulation generation), the set of Eq.13 clearly indicates that any given alteration regarding a component noted n, (due to a given change of relevant coefficient), should not interact with any former component (n-1 for instance).

Furthermore the way to display each component against its own frequency confirms its interest when considering the abovementioned Fig. 6 associated with the stiffness analysis. Expanding each frequency scale as before, Fig. 13 can be considered with a view to getting specific conclusions and understanding about the effective behaviour of the system. As in the former case, all components display a first maximum at the fundamental mechanical frequency of the transducer. Again, each harmonic component is fitted with a second maximum exactly arranged at its own specific place as aforementioned.

As a major difference when compared with the force factor investigation, here the level abatement while increasing frequency is quite more marked although with the same property (-40 dB g fall marking one apart from the former one). Hence we have: Such a behaviour discrepancy is quite naturally explained while considering the shift order stemming from the left member of Eq.10 (while multiplying by the current) as regards the analysis of the force factor.

SPECIAL BEHAVIOUR WHILE INVESTIGATING ON A PARAMETER EVENNESS

Considering now the constitutive coefficient involved within the power-series relative to the nonlinear parameters, even and odd parts can be isolated according to the straightforward following relation standing for Eq.4:

( )

) ( ) ( 2 / 1 1 2 1 2 2 / 1 2 2 0 1 0 x x x x x x odd even n i i i n i i i n i i i λ λ λ λ λ λ λ λ + =       ⋅ +       ⋅ + = ⋅ + = ∑ ∑ ∑ = - ⋅ - ⋅ = ⋅ ⋅ = (43)
Then, a most interesting point can be underlined if we artificially consider only the even part of such a parameter. As regards the force factor B l and the former representation given in Fig. 12, zeroing the odd coefficients with b 1 =b 3 =b 5 =0 in Table II, entails the results depicted in Fig. 14. Fig. 14: Calculated behaviour of the harmonic components displayed according to their own frequency in the case of an evened nonlinear force factor and a 100 mA driving current Dashed line are the result of the analytic approach while same symbols as those used before are representative of the numerical results. It is clear from Fig. 14 that the even components, 2, 4, and 6, (still represented in dotted lines for the purpose of comparison), are totally rejected.

p-19/23 -Another most noticeable point concerns the abovementioned "Moot Extremum" point associated with the second order discontinuity that was assigned to the fifth component. Then, zeroing the odd coefficients made this point disappear as a sharp discontinuity, being now replaced by a smooth minimum. So as to check upon it, other numerical computations (still operated with Simulink ® ) were carried out for this point together with an apt set of other ones closely arranged around it: As a result, both analytic and numeric approaches brought evidence about this major change in the pattern of this component. Moreover, after trying the respective effects of selectively zeroing coefficients b 1 and b 3 , the smoothing change proved to be clearly attributed only to b 1 . At this point, Fig. 14 displayed without such a discontinuity makes clear a trough-like set of minima highlighted with a falling off dashed line.

Since in no way experimental measurements (and especially thermal ones) are up to now available, we can only assume that the whole physical meaning could be probably explained by way of a minimum entropy generation as the driven system with its electromagnetic energy storage properties generates its harmonic lines [START_REF] Kreyszig | Advanced Engineering Mathematics[END_REF][START_REF] Prigogine | From Being to Becoming: Time and Complexity in the Physical Sciences[END_REF][START_REF] Gaviot | Towards a thermodynamic assessment of transition plateaus[END_REF]. Then, thanks to the thermal know-how of the laboratory this hypothesis will be investigated in the future of this work.

CONCLUSION

This paper has presented a simple analytic approach allowing to investigate on the harmonic distortion of electrodynamic loudspeakers. As a prominent advantage such an approach allows a quick representation of the components as regards analysis up to the fifth one. The approach is concomitantly supported with numerical computations carried out with a Simulink ® model. For the purpose of comparison the measured parameters of a midrange generic transducer have been chosen. Although slight discrepancies may appear, the analytic approach proved to be valuable. Moreover it allows us to analyse the evenness property of a given influence parameter, and an example has been discussed with the force factor. On the other hand, the approach proved to be able to account for nonlinear parameters function of the speed of the diaphragm. Then, as a prospective investigation a metrological application is given as regards the drag force measurement and any parameter relevant with it.

As a noticeable point we have also shown the interest to figure out the acceleration results linked to the entailed audio reproduction, as referred to the gravity constant. Moreover, by way of expanding the display of each component to its real perceived frequency, we have highlighted a specific characteristic as regard two observable distinct maxima respectively occurring: for the first, at the natural mechanical frequency of the fundamental, and for the second one, showing off at the same value times the order of the considered component. We expect to provide an articulate energetical interpretation of such component patterns in the next coming future.

p-21/23 -Fig. 15: Generic simple scheme allowing to reject the mechanical resonance For the purpose of basic experiments integrated power amplifiers can be tested (TI ® OPA 549 for instance). As regards Fig. 15, the transconductance ratio characterizing the converter that feeds the transducer is defined with:

( ) ( ) 3 B 2 B R / R Z 1 R 1 + + ⋅ = in V I (A/V) (44)
Now, the nominal parameters (Table III) of the Morel ® EM 428 reference transducer can be considered so as to adjust the components values. Then, resulting behaviours regarding impedances (network and transducer) together with the transconductance ratio as regard amplitude and phase shift curves are depicted in Fig. 16.

Fig. 16: Respective behaviours of the reference transducer and filtering network

Considering such a basic design, one has to adjust R 2 >>R B since the R 2 value acts on the whole spectral behaviour aside from the resonance regime. Moreover, as regards the stability of the amplifier, R B has to be kept with a quite low value with R B =1Ω Ω Ω Ω in the given example. As regards Fig. 16, a symmetric behaviour can be observed with regard to the transducer and the filtering network. Component values are chosen so as to get an electrical Q close to 0.45, allowing then to compensate precisely for the transducer behaviour.

As a result, while calibrating the input voltage with a 0.2V RMS level, the filtering efficiency can be compared with two other cases as depicted in Fig. 17 illustrating the acceleration of the diaphragm against p-22/23 -frequency. Indeed the behaviour of a simple unfiltered design supplying a unity current is plotted in black, while the natural filtering characterizing a voltage drive (1V RMS ) is plotted in brown. As is shown by the red curve representing the behaviour of the filtering design, the system operates properly. As a matter of fact any further filtering function (at high frequencies) can be easily compounded to such a system with a classical feedforward scheme. However, any industrial project should consider operational schemes out of such a mere theoretical stance. Indeed, due to its size (and cost) the abovementioned inductance L 2 (80mH) could not really be considered with an air-cored copper device, but would require either a ferrite coil structure or a virtual implantation. In this way, with the latter, temperature drift self-compensation solutions are under experiment in the laboratory together with more sophisticated filtering schemes.

Figures captions

Fig. 2 :

 2 Fig. 2: Static changes measured for the force factor against displacement (Klippel ® testing bench)On the other hand, Simulink ® is a block diagram environment for multidomain simulation and Model-Based Design. It supports system-level design, simulation, automatic code. and continuous test. Then, as the representative equations of a transducer depend both on time and displacement, such a numerical solver can take into account all the involved parameters. All physical quantities, such as displacement, speed and acceleration, can be advantageously modelized considering any parameter described according to Eq.4. Then, nonlinear quantities can be coupled with articulate function blocks allowing to account for the evolutive parameters [B l (x), k m (x), L e (x), f m (v)]. As depicted in Fig.3, the model relies on processing the

Fig. 3 :Fig. 4 :

 34 Fig. 3: Simulink ® nonlinear numerical evolutive model for a current driven loudspeaker

Fig. 6 : 7 .Fig. 7 :

 677 Fig. 6: Behaviour of the first five harmonic components regarding nonlinear stiffness (with 100 mA)Hence, in order to fathom about the degree of prospective discrepancy to be expected, an other case is investigated with a significantly increased current value, now fixed at 250 mA. Then, relevant results are plotted and made clear in Fig.7. Since the purpose of this illustration is only to detect eventual flaws of the analytic model, fundamental level values are not given (being considered useless in this case).

  ρ ρ ρ ρ (P/Pa) .C x .S) = (P/P a ).3,506.10 -3

Fig. 10 :

 10 Fig. 10: Influence of the drag force through the third and fifth components (P = 2 bar , I = 0.1A)

Fig. 11 :

 11 Fig. 11: Influence of the drag force through components 3 and 5, for a 2 bar static pressure (0.25A)

Fig. 12 :

 12 Fig. 12: Real behaviour of each harmonic component displayed according to their own frequency in the case of a nonlinear force factor and a 100 mA driving current

  [10kHz -100kHz]. Then, asymptotic level lessening values, all with a -40 dB g fall apart from the former one, are characterized as follows: Fundamental behaviour : invariant level Asymptotic fall of harmonic component 2: -40 dB g /decade Asymptotic fall of harmonic component 3: -80 dB g /decade Asymptotic fall of harmonic component 4: -120 dB g /decade Asymptotic fall of harmonic component 5: -160 dB g /decade Asymptotic fall of harmonic component 6: -200 dB g /decade

Fig. 13 :

 13 Fig. 13: Behaviour of each harmonic component displayed according to its own frequency in the case of a nonlinear stiffness parameter and a 100 mA driving current

  Fundamental behaviour : invariant level Asymptotic fall of harmonic component 2: -80 dB g /decade Asymptotic fall of harmonic component 3: -120 dB g /decade Asymptotic fall of harmonic component 4: -160 dB g /decade Asymptotic fall of harmonic component 5: -200 dB g /decade

Fig. 17 :

 17 Fig. 17: Compared behaviours with fundamental schemes (respectively unfiltered I-drive and V-drive naturally filtered with the motional impedance of the Morel ® ® ® ® EM 428)

Fig. 1 :Fig. 3 : 5 : 6 :Fig. 7 :Fig. 8 : 9 : 10 :Fig. 11 :Fig. 12 : 13 : 14 : 15 : 16 : 17 :

 13567891011121314151617 Fig. 1: Natural rejection of nonlinear components due to Eq.2 with an ideal current-drive policy Fig. 2: Static changes measured for the force factor against displacement (Klippel ® testing bench) Fig. 3: Simulink ® nonlinear numerical evolutive model for a current driven loudspeaker Fig. 4: Behaviour of the first six harmonic lines according to both analytic and Simulink ® approaches Fig. 5: Assessment of the solenoid effect Fig. 6: Behaviour of the first five harmonic components regarding nonlinear stiffness (with 100 mA) Fig. 7: Behaviour of the first five harmonic lines regarding nonlinear stiffness (with 250 mA) Fig. 8: Comparison between the relative effects of the second harmonic distortion lines Fig. 9: Influence of the static pressure on the drag force part of the f m coefficient Fig. 10: Influence of the drag force through the third and fifth components (P = 2 bar , I = 0.1A) Fig. 11: Influence of the drag force through components 3 and 5, for a 2 bar static pressure (0.25A) Fig. 12: Real behaviour of each harmonic component displayed according to their own frequency in the case of a nonlinear force factor and a 100 mA driving current Fig. 13: Behaviour of each harmonic component displayed according to its own frequency in the case of a nonlinear stiffness parameter and a 100 mA driving current Fig. 14: Calculated behaviour of the harmonic components displayed according to their own frequency in the case of an evened nonlinear force factor and a 100 mA driving current Fig. 15: Generic simple scheme allowing to reject the mechanical resonance Fig. 16: Respective behaviours of the reference transducer and filtering network Fig. 17: Compared behaviours with fundamental schemes (respectively unfiltered I-drive and V-drive naturally filtered with the motional impedance of the Morel ® ® ® ® EM 428)

Table I : Coefficients values relative to the first fifth powers of the displacement X 2 HARMONIC COMPONENTS OF DISPLACEMENT DUE TO THE FORCE FACTOR

 I

Aside from B l expanded according to Eq.4, we consider Eq.1 with linear parameters. The successive derivatives of the displacement (the speed • X , and the acceleration

3.X 1 2 .X 2 + c 4 .X 1 4 ] X 6 = F 6 .[c 1 .X 5 + c 2 .(X 1 .X 4 +2.X 2 .X 3 ) + c 3 .(2X 1 2 .X 3 +3.X 1 .X 2 2 ) + c 4 .4X 1 3 .X 2 + c 5 .X 1 5 ]

Table II : Force factor coefficients entailing the investigated nonlinear behaviour of the transducer

 II

	Morel ® Midrange	EM 428-8Ω Ω Ω Ω	∅ ∅ ∅ ∅4", ∅ ∅ ∅ ∅2.1" voicecoil
	Nominal parameters	Theoretical values	Measured values
	Mass M m	6,55 g	6,529 g
	Mechanical damping f m	0,86 kg/s	1,239 kg/s
	Rigidity k m	1136 N/m	1938 N/m
	Resonance Freq. F 0	66,29 Hz	86,7 Hz
	Force factor B l	5,4 T.m	5,934 T.m
	Resistance DC R e	5,4 Ω	5,41 Ω
	Inductance (para) L e	0,36 mH	0,356 mH
	Mechanic Quality factor Q m	3,03	2,871

Table VI .

 VI

	k 0 [N/mm]	k 1 [N/mm 2 ]	k 2 [N/mm 3 ]	k 3 [N/mm 4 ]	k 4 [N/mm 5 ]	k 5 [N/mm 6 ]
	1,579	0,33332	0.058674	0.036590	0.025980	0

Table I : Coefficients values relative to the first fifth powers of the displacement X Table II: Force factor coefficients entailing the investigated nonlinear behaviour of the transducer Table III: Theoretical and measured nominal parameters of the EM 428 reference transducer Table IV: Acceleration reference values associated to the fundamental (current 100 mA) Table V : Inductance nonlinear coefficient values Table VI: Nonlinear coefficient values for the suspension stiffness Table VII: Level values of the fundamental line with a 100 mA current driving

 I

p-4/23 -

p-9/23 -

p-16/23 -

ACKNOWLEDGMENT

The authors would like to thank Michel Bruneau (Professor Emeritus) for all the most valuable and fruitful discussions we had all along this work. His full-fledged experience in the field of acoustics and genuine kindness have been most contributing to improve the quality of this paper. The authors wish to thank Romain Raveau (Whylot enterprise) for permission to publish the technical aspects of this study.

PATENT PROTECTION POLICY

The authors would like to point out that after legal registration several aspects dealt with in this work are the subject of a FR patent application.

APPENDIX

The transfer function from drive voltage to displacement shows off a natural electrical damping on the Q factor of the system under operation. Then, doubts cast upon the efficiency of the current-drive policy are most often based just on the lack of such natural damping usually entailing an oversized Q that results as a consequence [START_REF] Mills | Distortion Reduction in Moving-Coil Loudspeaker Systems Using Current-Drive technology[END_REF][START_REF] Klippel | Dynamic measurement and interpretation of nonlinear parameters of electrodynamic loudspeakers[END_REF]. In other words, low damping factors associated with current-drive are a crucial problem and are probably the major reason why current drive is still not adopted by the industry. However, effective solutions based on active and passive processing as well as driver and enclosure techniques may be considered as an alternative to the natural filtering of current brought about by the motional impedance. Moreover, operative filtering schemes can be properly operated (and even quite enhanced, insofar as the resonance drift may be compensated against temperature). As a fundamental requisite the Thévenin impedance seen by the driver (due to the amplifier and passive circuits) has to remain close to infinity (Norton source). Then, one has to discard any solution involving circuit arms connected directly in parallel with the driver, that could untowardly lower the impedance in a given part of the spectrum. A simple generic scheme is depicted in Fig. 15 with a view to rejecting the mechanical resonance while subjecting the voltage to current conditioner to an optimized transconductance function.