PAPERS

Mechanical Resonances and Geometrical Nonlinearities in Electrodynamic Loudspeakers* NICOLAS QUAEGEBEUR AND ANTOINE CHAIGNE ([email protected])

([email protected])

UME–ENSTA, 91761 Palaiseau Cedex, France

For high amplitudes of vibration loudspeakers are subject to nonlinear phenomena that are responsible for audible distortions such as intermodulation or harmonic distortion. The lowfrequency model uses nonlinear lumped parameters and is only valid around the first mechanical resonance since it assumes a rigid body movement. In the present study a model of the electromechanical problem including the effects of diaphragm modal resonances using the state–space formalism is proposed. No approximation of the nonlinear behavior of the electrical parameters is necessary for the direct calculation of the vibration pattern. The model is developed for one degree of freedom (plane piston approximation) and then expanded to an n-degree-of-freedom system, allowing one to model mechanical resonances and geometrical nonlinearities of the system.

0 INTRODUCTION The aim of an electrodynamic loudspeaker is to transform an electrical signal into a sound pressure signal. Such a transduction should be linear. For high amplitudes of vibration, however, nonlinear phenomena occur and are responsible for audible distortions, such as intermodulation or harmonic distortion. For larger input power the distortions increase rapidly and can reach the order of the fundamental, which cannot be explained by early models based on single-valued small-signal parameters. The classical model for electrical and mechanical nonlinearities consists of a lumped-parameter model [1]–[3], which can be solved numerically or with the use of harmonic balance [4], [5], Voltera series [6], or fuzzy networks [7]. Those methods utilize the simplifying assumption that the diaphragm vibration pattern can be reduced to a single rigid-body mode so that good agreement with measurements can only be achieved in low frequencies (around the first resonance of the diaphragm). Previous experimental [8] and psychoacoustic [9] studies reported in the literature have shown that the highfrequency behavior influences the response of the loudspeaker and its global quality significantly, so that the mechanical behavior of the loudspeaker membrane can no

*Manuscript received 2007 May 31; revised 2007 December 13 and 2008 April 15. 462

longer be considered as a single-degree-of-freedom system accross the whole audible range and the mechanical resonances must be taken into account. Several experimental [10] and numerical studies have been performed to include those mechanical resonances using lumped parameters [11], finite and boundary elements [12], and analytic formulation [13], but all those techniques are limited to the linear range. In the present study a method of calculation for the electromechanical problem based on a state–space formalism is proposed. The idea is to use the nonlinear lumpedparameter model and include diaphragm vibration modes in the global formulation. Geometrical nonlinearities are then included using the works on nonlinear vibrations of thin structures [14], [15]. The results obtained by the present method are compared to mechanical measurements on a classical loudspeaker (165-mm Monacor-Stageline woofer, model SP-165 PA). The results show general agreement and allow one to model the loudspeaker over the audible range in the large-signal domain, including the effects of the geometrical nonlinearities. 1 NONLINEAR MODEL OF LOUDSPEAKER INCLUDING RESONANCE MODES AND GEOMETRICAL NONLINEARITIES 1.1 Nonlinear Lumped-Parameter Model For low frequencies the electrical, mechanical, and acoustical components of an electrodynamic loudspeaker J. Audio Eng. Soc., Vol. 56, No. 6, 2008 June

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RESONANCES AND NONLINEARITIES IN ELECTRODYNAMIC LOUDSPEAKERS

behave as concentrated elements whose properties can be expressed by lumped parameters [1], [2]. The assumptions of this model are a rigid-body displacement (singledegree-of-freedom model) and a linear behavior of the system. With these assumptions and by using electrical analogies, the electrodynamic loudspeaker can be modeled by an equivalent electrical circuit by introducing electrical (electric resistance of voice coil Re, voice-coil inductance L, and force factor Bl) and mechanical parameters (mass of the moving part Mm, stiffness of the suspension Km, and mechanical resistance of the suspension Rm). Good agreement can be achieved in the far field for small amplitudes of vibrations and in the low-frequency domain, but in the large-signal domain a variation of the lumped parameters can be observed with respect to the excursion of the voice coil x and the input current i. Those effects have been studied widely in past publications [3], [4], [6], [16]–[18], and dominant effects are assigned to the following nonlinear parameters: • Bl(x) Nonconstant force factor (electrodynamic driving) due to a variation of the magnetic field in the gap • L(x) Voice-coil inductance varying with displacement. • Km(x) Mechanical stiffness of the suspension depending on the excursion of the moving part (geometrical nonlinearities and viscoelastic effects). In previous studies [4], [7], [16] the dependency of those lumped parameters with respect to i and x is expressed with the use of polynomials as, for example, for the force factor Bl in the equation Bl共x兲 = Bl共1 + Bl1x + Bl 2 x 2兲

(1)

where Bl denotes the force factor parameter evaluated in the small-signal domain, and Bl1 and Bl2 express the nonlinear tendency of the force factor Bl with respect to x. The formulation of Eq. (1) offers the advantage of allowing direct calculations with the harmonic balance and other methods but does not account for other typical effects such as hysteresis or a noninteger power dependency. These limitations of the polynomial formulation provide motivation for finding a more general formulation for solving the nonlinear problem. The nonlinear system, including nonlinear terms in the transduction equations [19], is then written as u共t兲 = Re i共t兲 + L 共x兲

di dx dL dx + Bl共x兲 + i dt dt dx dt

d2x

i2 dL dx . Bl共x兲i共t兲 = Mm 2 + Rm共x兲 + Km共x兲 x共t兲 − dt 2 dx dt

(2)

The low signal values of the nonlinear parameters are then Bl共x兲. Bl共x兲 = Bl +˜

(3)

[Only the force factor Bl(x) case is described here, but the same formulation is adopted for the inductance L(x), the mechanical stiffness Km(x), and the mechanical damping Rm(x).] J. Audio Eng. Soc., Vol. 56, No. 6, 2008 June

One can separate the linear and nonlinear parts of the problem and rewrite it using the state–space formalism, Y˙ = AY + B共Y 兲u共t兲 + N共Y 兲

(4)

where Y is the state vector and A, B(Y), and N(Y) are the state matrix, input vector, and nonlinear vector, respectively,

冢

冣

Bl Re 0 i 1 − − L L x L共x兲 0 1 , B共Y 兲 = , Y = dx , A = 0 0 Km Rm Bl dt 0 − − Mm Mm Mm

冢冣

冢

−

N共Y 兲 =

冉

冊

冢冣

冣

˜ Re BL共x兲 dx dL˜ dx Bl共x兲 Bl ˜ i− + + −i L共x兲 L共x兲 L共x兲 L dt dx dt 0 . ˜ Km共x兲 ˜ Rm共x兲dx i2 dL˜ Bl共x兲 ˜ + i− x− Mm Mm Mm dx 2 dx

(5)

Eq. (4) is then rewritten in its general form;

冢冣 i

dY = f 共Y, t兲, dt

Y=

x

dx dt

.

(6)

In the linear range, when Bl, L, Km, Mm, and Rm are constant and N(Y) is negligible, Eq. (6) is solved directly in both time and Laplace domains using state–space properties. (See Appendix 1 for details.) In the general case Eq. (6) is solved numerically in the time domain using a commercial solver with the use of fixed-point and Runge– Kutta methods. In the present problem i, x, and x˙ are part of the state vector Y, and all dependencies with respect to those variables can be included in the nonlinear part. We could also, for example, add eddy currents [20] in the electrical equation. The advantage of this formulation is that no assumption is made with regard to the law of the nonlinearities. The nonlinear description is not limited to polynomial approximation and, for example, incremental measurements can be implemented directly in Eq. (6). The direct calculations can be performed from nonlinear parameter measurements, and the effect of each nonlinear element can be calculated separately. This formulation can be used to explain typical nonlinearities in low frequencies but does not take into account the high-frequency behavior. The following section proposes a formulation that includes mechanical resonances, followed with another section to add geometrical nonlinearities. 1.2 Inclusion of Mechanical Resonances For higher frequencies of excitation “break-up” modes appear and modify the vibration pattern of loudspeakers [8], [10]. In that case the plane piston vibration assumption is no longer valid, and one needs to model the loudspeaker 463

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as a system with multiple degrees of freedom. The main assumption is that the diaphragm axisymmetric displacement field w(r, t) depending on the position r and time t can be expanded into separated variables [14], [15],

and A is a square matrix of dimension (2n + 1) representing the linear dynamics of the problem,

⬁

w共r, t兲 =

兺⌽ 共r兲q 共t兲. a

(7)

a

a=0

The functions ⌽a(r) in Eq. (7) represent the modal shapes, which can be either measured or calculated, and the functions qa(t) represent the time function associated with mode a. As a first approximation we assume that the dynamics of the system are linear, and that the functions qa(t) are solutions of a linear system of resonating filters, ␻2aqa + 2␮aq˙ a + q¨ a = Fa共i兲

(8)

where ␻a and ␮a represent respectively, the modal angular frequency and the modal damping of mode a. Those parameters have to be determined from modal analysis of the structure under consideration. The forcing term Fa associated with mode a is expressed as Fa = Ta

Bl i Mm

(9)

where Bli represents the global forcing term and Ta the fraction of energy distributed on mode a. In that case it can be shown that the global counter electromechanical force term Fcem is expressed as n

Fcem = Bl

兺 T q˙ .

(10)

a a

a=0

Eqs. (9) and (10) are introduced into the formulation of Eq. (6) to model the linear behavior of the structure including resonance modes. As the problem needs to be solved numerically, the modal expansion of Eq. (7) has to be truncated at a finite order N, and the mechanical problem is in this manner described as an N-degree-of-freedom system. The final linear state–space formulation for the N-degree-of-freedom problem is thus expressed in the equation (11)

Y˙ = AY + Bu共t兲

where a new state vector Y and a new input vector B of dimension (2N + 1) are introduced,

冢 冣 冢冣 i

Y=

464

dq0 dt ·· · dqn dt

,

B=

0 ·· · 0 0 ·· · 0

冢

··· − ··· ·· · ··· ···

0

··

· · ·

·

· · · −2␮N

冣

.

(13)

w 共r, t兲 = C共r兲Y

(14)

where the observation vector C(r) is introduced, C共r兲 = 共0 | ⌽0共r兲 · · · ⌽N共r兲 | 0 · · · 0兲.

(15)

As in the single-degree-of-freedom problem, the solution of the linear problem in both time and Laplace domains is given in Appendix 1 by substituting the right state and observation vectors. 1.3 Geometrical Nonlinearities The geometrical nonlinearities that occur for vibration amplitudes on the order of the diaphragm thickness are introduced using the dynamic analogs of the Von Ka`rma`n equations [14], [15], [21]. This formulation allows one to couple transverse and longitudinal displacements for amplitudes of vibration above the thickness of the vibrating structure. The main assumptions of this model are that the structure is thin, that the Kirchhoff–Love assumptions are verified, that only nonlinear terms of the lowest order are kept in the expression of strains as a function of the displacement, and that the in-plane and rotatory inertia terms are neglected. After projection on mode a, one obtains the modal equations,

兺兺兺⌫ + 兺 兺␤ q q

apur qpquqr

p

u

r

apu p u

p

(12)

Bl TN L 0 ·· · 1

The global displacement field is obtained from the state vector via Eq. (14),

␻2aqa + 2␮aq˙ a + q¨ a = Fa +

1 L

q0 ·· · qn

A=

Bl T0 Re 0 ··· 0 − L L 0 0 ··· 0 1 · · ·· ·· · · · · · · · · · 0 0 ··· 0 0 Bl −2␮0 −␻20 · · · 0 MmT0 · · ·· ·· · · ·· · ·· · · Bl 0 · · · −␻2N 0 T Mm N −

(16)

u

where coupling terms between oscillators are introduced using the cubic and quadratic terms ⌫apur and ␤apu. Since no modal coupling effects have been observed in practical cases (internal resonances, energy transfer between oscillators), and in order to measure the geometrical nonlinear terms in practical cases, all the coupling terms are neglected in Eq. (16). The dynamic behavior of each osJ. Audio Eng. Soc., Vol. 56, No. 6, 2008 June

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RESONANCES AND NONLINEARITIES IN ELECTRODYNAMIC LOUDSPEAKERS

cillator is then reduced to a Duffing equation as expressed by ␻2aqa + 2␮aq˙ a + q¨ a = Fa + ⌫aq3a + ␤aq2a.

(17)

This formulation explains typical nonlinear effects (harmonic distortion and intermodulation) that appear around modal resonance frequencies [15], [21]. In Eq. (17) cubic and quadratic nonlinearities are introduced through ⌫a and ␤a terms. For planar structures it has been shown that the quadratic terms ␤a vanish [21] and only cubic nonlinearities remain in Eq. (17). The quadratic term ␤a represents the asymmetry of the vibration pattern that might occur when the structure is stiffer for positive excursions than for negative excursions. As in the previous section, the problem is rewritten using state–space formalism, Y˙ = AY + B共Y兲u共t兲 + NLe共Y兲 + NLg共Y兲

(18)

where A denotes the linear part of the problem, equivalent to Eq. (13), NLe(Y) and NLg(Y) contain all the electrical, mechanical, and geometrical nonlinear terms of the electromechanical transduction, and B(Y) denotes the input vector, which depends on Y. Since the maximum excursion is reached around the first resonance frequency and decreases linearly with increasing frequency, the dependency of nonlinear electrical parameters with respect to the amplitude of vibration of higher modes is neglected. In that case the nonlinear electrical parameters (L(x) and Bl(x) are assumed to be only dependent on the excursion of the first mode of vibration. For those parameters, x(t) ⯝ q0(t). Under those considerations one obtains for the nonlinear terms NLe(Y) and NLe(Y) NLe共Y兲

冢

冉

冊

˜ Bl共q0兲 dq0 dL˜ dq0 Bl共q0兲 Bl ˜ Re i− + + −i − ˜ ˜ L dt dq0 dt L共q0兲 L共q0兲 ˜ L共q0兲

=

0 ·· · 0

T˜ 0Bl共q0 兲i +

i2 dL˜ 2 dq0

T˜ NBl共q0 兲 i +

i2 dL˜ 2 dq0

冣

This general formulation allows one to include geometrical nonlinearities that typically appear around mechanical resonance frequencies and for their submultiples (angular frequencies such as ␻ ⳱ ␻a/m for m an integer). As previously, the global system is solved using the fixed-point method and the Runge–Kutta algorithms. No assumption is made on the form of the nonlinearities, except for the geometrical nonlinearities, which are assumed to be polynomial expansions of order 3. 2 EXPERIMENTAL AND NUMERICAL APPLICATION 2.1 Case under Study In this section we propose a method for analyzing the response of a typical loudspeaker up to 6 kHz, while the low-frequency model is limited to 600 Hz for this driver. The loudspeaker of interest is a 165-mm-diameter medium woofer (Monacor-Stageline model SP-165 PA) represented in Fig. 1. The first step of the analysis is to determine the nonlinear lumped parameters as explained in [5] and then to determine the modal parameters associated with the radiating structure. The results obtained by our model are compared with measurements for high amplitudes of vibration, and the influence of geometrical nonlinearities is finally discussed. 2.2 Parameters of the Loudspeaker Considered 2.2.1 Nonlinear Lumped-Parameter Analysis The analysis of nonlinear lumped parameters is similar to the study of Scott et al. [22], who proposed a method for analyzing nonlinear parameters in low frequencies. The principle is to estimate (with the use of a least-squares method) the linear lumped parameters for different static excursions xi of the moving coil. This method is known as point-by-point dynamic method, and more details about the technique can be found in [22], [23]. The parameters of interest are the voice-coil inductance L(xi), the force factor Bl(xi), and the incremental stiffness Kinc(xi) (the tangent in the force deflection curve defined as the derivative of the normal stiffness) associated with the first mode. The regular stiffness Km(xi) used in Eq. (2) is then calculated by integrating the incremental stiffness Kinc(xi). The other lumped parameters (electrical resistance Re, global mass

(19)

冢 冣 0

NLg共Y兲 =

0 ·· · 0

.

+ ·· · ⌫Nq3n + ␤Nq2n ⌫0q30

(20)

␤0q20

J. Audio Eng. Soc., Vol. 56, No. 6, 2008 June

Fig. 1. Loudspeaker studied. (Monacor-Stageline SP-165 PA). 465

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Modal angular frequencies ␻p (or modal frequencies fp) Modal damping factor ␮p Modal excitation Tp, normalized to the first mode value T1 Modal shapes ⌽p.

Mm, and mechanical resistance Rm) are taken as being invariant with respect to xi. The results obtained for a maximum excursion of x ⳱ ±3 mm are presented in Fig. 2.

• • • •

2.2.2. Linear Mechanical Parameters In order to assess the mechanical properties of the diaphragm, a modal analysis is performed using the software IDEAS娀. The linear modal characteristics are expressed by four parameters:

The results obtained for the loudspeaker considered, below 6400 Hz, are represented in Table 1. In the present modal analysis only axisymmetric modes are retained. (Asymmetric modes have little influence on the sound pressure field.)

Fig. 2. Measurements of nonlinear lumped parameters depending on static position of coil xi. (a) Inductance L. (b) Force factor Bl. (c) Mechanical stiffness Km. 466

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As emphasized by Skrodzka and Sek [10], the first modes (below 3 kHz for the loudspeaker considered) correspond to resonances of the surround. Their influence in the vibratory and acoustical responses can be neglected due to high damping and low radiation efficiency, but those modes are nevertheless responsible for audible distortions generated by geometrical nonlinearities. The other modes above 3 kHz correspond to resonances of the inner cone that are less damped and more excited that the surround resonances. Their influence in the global response will thus be more significant than the surround resonances.

Even if only the axisymmetric modes are retained, the modal shapes associated with modes 4 and 5 exhibit a nonradial dependency. This phenomenon can be attributed to misalignment of the structure or small nonradial imperfections in the membrane shape and has little influence on the results of the model. 2.2.3 Nonlinear Mechanical Parameters The evaluation of the parameters ⌫p (cubic coefficient) and ␤p (quadratic coefficient), which represent the geometrical nonlinearities that occur at medium and high fre-

Fig. 2. Continued Table 1. Axisymmetric modal analysis of loudspeaker below 6400 Hz*. Damping ␮p (%)

Mode p

Frequency fp (Hz)

1

118

2

1150

5.6

0.14

3

2060

5.4

0.136

4

3550

8.34

0.358

5

4300

6.50

−0.75

6

5280

6.74

−1.51

48

Excitation Tp

Modal Shape ⌽p

1

* The two first modes correspond to surround modes, the other modes correspond to resonances of the truncated cone. J. Audio Eng. Soc., Vol. 56, No. 6, 2008 June

467

QUAEGEBEUR AND CHAIGNE

quencies, is performed for each mode p by exciting the structure at the resonance of the mode considered. It can be shown (see Appendix 2) that when the structure is excited at f ⳱ fp, the magnitude of the nonlinearities is directly related to the nonlinear coefficients: • The magnitude of the second harmonic at f ⳱ fp is directly related to the quadratic coefficient ␤p. • The magnitude of the third harmonic f ⳱ fp is directly related to the quadratic and cubic coefficients ␤p and ⌫p. The idea is to evaluate the quadratic coefficients first (with measurements of the second harmonic level at each resonance) and then to determine the cubic coefficients. (Refer to Appendix 2 for more detail.) 2.3 Comparison Measurements / Simulation 2.3.1 Linear Range Neglecting the nonlinear terms NLe(Y) and NLg(Y) in Eq. (18), one obtains the linear system, which represents the vibratory behavior of the structure in the frequency range considered (depending on the number of modes retained). For the present study the results are presented in Fig. 3 and are compared to measurements of the velocity performed at the center of the structures for small amplitudes of vibration. In Fig. 3, the dashed line represents the solution of the Thiele and Small problem (one degree of freedom) and the solid line the solutions of the problem considering the six first axisymmetric modes shown in Table 1. Good agreement with measurement is thus achieved up to 6 kHz, whereas the lumped-parameter model is only valid below 600 Hz. In this range, considering six modes of vibration,

PAPERS

our model predicts the vibration pattern with a maximum error of 3 dB. 2.3.2 High Amplitudes of Vibration For high amplitudes of vibration the main goal is to determine the harmonic distortion generated by the system with respect to the frequency of excitation. The magnitudes of the fundamental and the second and third harmonics are evaluated numerically and experimentally for each frequency, and the results are presented in Fig. 4 for both measurements (points) and predictions (lines). In Fig. 4, it appears that around the first resonance (plane piston mode) the nonlinear behavior is explained correctly by the nonlinear single-degree-of-freedom model (dashed line). For frequencies above 600 Hz this rigidbody assumption fails to identify performance features related to the diaphragm bending modes: • When the excitation frequency fe is near the resonance frequency fp, that is, fe ⯝ fp, the geometrical nonlinearities give rise to harmonic distortion in the velocity (and pressure) signal. • When the excitation frequency fe is near a submultiple of the resonance frequency fp, that is, n fe ⯝ fp, with n being an integer, the frequency of harmonics n corresponds to the resonance frequency fp, and the geometrical nonlinearities give then rise to an increase in the harmonic n in the velocity signal. Those observations are grossly predicted by the present model using six modes of vibration with cubic and quadratic nonlinearities in their modal equation. The discrepancy between model and measurements can be reduced by

Fig. 3. Comparison of velocity measurements at center (䡩 䡩 䡩) and the simulation results. – – – Thiele–Small model (one degree of freedom); —— solution obtained with six modes of vibration. 468

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determining the optimum linear and nonlinear coefficients that fit the experimental data. 3 CONCLUSION A method of analysis of the nonlinear behavior of loudspeakers is proposed. The scheme is based on state–space formalism, which allows a model with any kind of nonlinearity. (No assumption on the nonlinearity law with respect to current or displacement is made.) The method can either be developed for a single-degree-of-freedom system (loudspeaker at very low frequency) or for an Ndegree-of-freedom system to include “break-up” modes of vibration. The results show very good agreement with measurements in both low- and high-frequency linear domains. The study points out that an N-degree-of-freedom formulation is necessary to model accurately the velocity behavior of transducers over the audible range. It is also shown that geometrical nonlinearities can be observed for excitation frequencies around mechanical resonance frequencies and for submultiples of resonance frequencies. 4 REFERENCES [1] R. H. Small, “Closed-Box Loudspeaker Systems, Part I : Analysis,” J. Audio Eng. Soc., vol. 20, pp. 798–808 (1972 Dec.). [2] R. H. Small, “Vented-Box Loudspeaker Systems, Part II : Large-Signal Analysis,” J. Audio Eng. Soc., vol. 21, pp. 438–444 (1973 July/Aug.).

[3] W. Klippel, “Diagnosis and Remedy of Nonlinearities in Electrodynamical Transducers,” presented at the 109th Convention of the Audio Engineering Society, J. Audio Eng. Soc. (Abstracts), vol. 48, p. 1115 (2000 Nov.), preprint 5161. [4] W. Klippel, “Nonlinear Large-Signal Behavior of Electrodynamic Loudspeakers at Low Frequencies,” J. Audio Eng. Soc., vol. 40, pp. 483–496 (1992 June). [5] S. T. Park and S. Y. Hong, “Development of the Two-Stage Harmonic Balance Method to Estimate Nonlinear Parameters of Electrodynamic Loudspeakers,” J. Audio Eng. Soc., vol. 49, pp. 99–116 (2001 Mar.). [6] P. G. L. Mills and M. O. J. Hawksford, “Distortion Reduction in Moving-Coil Loudspeaker Systems Using Current-Drive Technology,” J. Audio Eng. Soc., vol. 37, pp. 129–148 (1989 Mar.). [7] S. M. Potirakis, G. E. Alexakis, M. C. Tsilis, and P. J. Xenitidis, “Time-Domain Nonlinear Modeling of Practical Electroacoustic Transducers,” J. Audio Eng. Soc., vol. 47, pp. 447–468 (1999 June.). [8] N. W. McLachlan, “On Symetrical Modes of Vibration of Truncated Conical Shells with Application to Loudspeaker Diaphragms,” Proc. Phys. Soc., vol. 44, pp. 408–425 (1932). [9] F. E. Toole, “Loudspeaker Measurements and Their Relationship to Listener Preferences, Part 1,” J. Audio Eng. Soc., vol. 34, pp. 227–235 (1986 Apr.). [10] E. B. Skrodzka and A. P. Sek, “Comparison of Modal Parameters of Loudspeakers in Different Working Conditions,” Appl. Acoust., vol. 60, pp. 267–277 (2000 July). [11] D. J. Murphy, “Axisymmetric Model of a Moving

Fig. 4. Comparison of velocity measurements at center of loudspeaker for an electric power of 60 W (䡩 䡩 䡩) and simulation of velocity (——) for high amplitudes of vibration. Black data—fundamental; dark grey—second harmonic; light grey—third harmonic for each frequency. – – – results obtained considering only one degree of freedom; —— results obtained considering six modes of vibration. J. Audio Eng. Soc., Vol. 56, No. 6, 2008 June

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Coil Loudspeaker,” J. Audio Eng. Soc., vol. 41, pp. 679–690 (1993 Sept.). [12] O. Von Estorff, Boundary Elements in Acoustics (WIT Press, TU Hamburg-Harburg, Germany, 2000). [13] A. M. Bruneau and M. Bruneau, “Electrodynamic Loudspeaker with Baffle: Motional Impedance and Radiation,” J. Audio Eng. Soc., vol. 34, pp. 970–980 (1986 Dec.). [14] J. Hadian and A. H. Nayfeh, “Modal Interaction in Circular Plates,” J. Sound Vib., vol. 142, pp. 279–292 (1990 Oct.). [15] O. Thomas, “Analyse et mode´ lisation de vibrations non-line´ aires de milieux minces e´ lastiques,” Ph.D. dissertation, Universite´ Pierre et Marie Curie, Paris VI, France (2001). [16] A. Dobrucki, “Nontypical Effects in an Electrodynamic Loudspeaker with a Nonhomogeneous Magnetic Field in the Air Gap and Nonlinear Suspensions,” J. Audio Eng. Soc., vol. 42, pp. 565–576 (1994 July/Aug.). [17] G. Lemarquand, “Ironless Loudspeakers,” IEEE Trans. Mag., vol. 43, pp. 3371–3374 (2007). [18] M. Berkouk, V. Lemarquand, and G. Lemarquand, “Analytical Calculation of Ironless Loudspeaker Motors,” IEEE Trans. Mag., vol. 37, pp. 1011–1014 (2001).

where we introduce the state vector Y, the state matrix A, and the input vector B as in the following equation:

冢冣 冢冣 i

Y=

B=

x

dx dt

A=

,

冢

Re L

0

0

0

−

Bl Mm

−

−

Bl L

1

Km Rm − Mm Mm

冣

,

1 L 0

(22)

.

0

Then the knowledge of the displacement x(t) is simply obtained by the linear system x共t兲 = CY = 共0

1

(23)

0兲Y.

Eq. (21) can be solved analytically in both the time and the Laplace domain using state–space methods. Indeed it is shown that the transfer function G(s) between x(s) and u(s) in the Laplace domain (where s represents the Laplace variable) is simply obtained by the equation

x共s兲 = G共s兲u共s兲 = C共sI − A兲−1B =

Bl 2

[19] H. Schurer, “Linearization of Electroacoustic Transducers,” Ph.D. dissertation, University of Twente, Enschede, The Netherlands (1997 Nov.). [20] W. M. Leach, Jr., “Loudspeaker Voice-Coil Inductance Losses: Circuit Models, Parametric Estimation, and Effect on Frequency Response,” J. Audio Eng. Soc., vol. 50, pp. 442–450 (2002 June). [21] N. Quaegebeur and A. Chaigne, “Nonlinear Vibrations of Loudspeaker-like Structures,” J. Sound Vib., vol. 309, pp. 178–196 (2007 Oct.). [22] J. Scott, J. Kelly, and G. Leembruggen, “New Method for Characterizing Driver Linearity,” J. Audio Eng. Soc., vol. 44, pp. 258–265 (1996 Apr.). [23] IEC 62458, “Measurement of Large Signal Parameters,” International Electrotechnical Commission, Geneva, Switzerland (2007 June). APPENDIX 1 STATE-SPACE FORMULATION FOR SINGLE-DEGREE-OF-FREEDOM PROBLEM IN THE LINEAR RANGE The basic one-dimensional linear formulation of the Thiele–Small problem is expressed directly in the time domain using the state–space formalism, as presented in the equation (21) Y˙ = AY + Bu共t兲 470

(24)

LMms + S 共ReMm + LRm兲 + s共ReRm + Bl2 + LKm兲 + ReKm 3

which corresponds to the classical Thiele–Small solution [1]. In the time domain the solution is obtained directly, assuming that u(0) ⳱ 0, by x共t兲 = C

兰e t

0

A共t−␶兲

Bu共␶兲 d␶.

(25)

APPENDIX 2 HARMONIC BALANCE FOR NONLINEAR MECHANICAL PARAMETER ESTIMATION Considering that the excursion of the voice coil is small, the electrical parameters are taken to be constant. In that case the system, including the geometrical nonlinearities, is reduced to u共t兲 = Rei共t兲 + L

di + Bl dt

兺T q˙

a a

Bl T i = −␻2aqa − 2␮aq˙ a − q¨ a + ⌫aq3a + ␤aq2a . Mm a

(26)

Moreover, at high frequencies the electrical impedance is purely inductive (the induction of the voice coil is preponderant), and the counterelectromechanical force can then be neglected in the electrical equation of Eq. (26) so that the term Bl ⌺Taq˙ a vanishes. Now we look for a solution of this system of equations using the harmonic balJ. Audio Eng. Soc., Vol. 56, No. 6, 2008 June

PAPERS

RESONANCES AND NONLINEARITIES IN ELECTRODYNAMIC LOUDSPEAKERS

ance method in order to estimate the coefficients ⌫a and ␤a from distortion measurements. We assume that the structure is excited around a mechanical resonance, and that the displacement is reduced to a single mode, w共r, t兲 = ⌽共r兲q共t兲.

(27)

This assumption is not valid in general, but allows us to obtain a simple formulation of the displacement field and to determine an approximation of the nonlinear coefficients. We apply the input signal u(t) that is sinusoidal with angular frequency ⍀, so that u共t兲 = Ue j⍀t + U*e−j⍀t

(28)

and the current i(t), which is assumed to be a linear function of the input voltage, is expressed similarly, i共t兲 = Ie j⍀t + I*e−j⍀t

(29)

where * denotes complex conjugate. Inserting Eq. (29) into Eq. (26), one obtains U = I共Re + jL⍀兲.

(30)

The displacement q(t) can be expressed with a sum of sinusoids, such as −j⍀t −2j⍀t q共t兲 = Q0 + 共Q1e j⍀t + Q* 兲 + 共Q2e2j⍀t + Q* 兲 1e 2e −3j⍀t + 共Q3e3j⍀t + Q* 兲 3e

(31)

where Q0 denotes the dc displacement and Q1, Q2, and Q3 the rates of the fundamental and the second and third harmonics, respectively, in the displacement signal. The aim of the present action is to show that the factors Q2 and Q3, which measure the rate of the second and third harmonics in the velocity signal, are directly dependent

on the quadratic coefficient ␤a and the cubic coefficient ⌫a. Assuming that the system is weakly nonlinear, we have

冦

|Q1| Ⰷ |Q0| |Q1| Ⰷ |Q2|

(32)

|Q1| Ⰷ |Q3|.

In a first approximation the rates of the second and third are directly related to the quadratic and cubic coefficients ␤a and ⌫a such as Q2 ⯝ ␤a

Q3 ⯝ ⌫a

冉 冉

Q21 ␻2 − 4⍀2 + 4j␮a⍀

冊

Q31 ␻2 − 9⍀2 + 6j␮a⍀ − 6⌫aQ21

冊

(33) .

Eqs. (33) show that in a first approximation the harmonic n increases in two distinct cases: • When the fundamental Q1 increases, then the harmonic n also increases. • When the excitation frequency ⍀ is around the submultiple of order n (that is ⍀ ⳱ ␻/n), then the denominator of Eq. (33) increases and gives rise to an increase in harmonic n. In this manner, by measuring the growth of harmonics Q2 and Q3 with respect to the level of the fundamental Q1, one obtains an estimate of the nonlinear mechanical coefficients. Fig. 5 represents the measured behavior of the second and third harmonics with respect to the fundamental level for an excitation frequency corresponding to the sixth axisymmetric mode, which confirms the formulation of Eq. (33).

Fig. 5. Evolution of (a) second harmonic and (b) third harmonic in velocity signal at center of loudspeaker SP-165-PA with respect to level of fundamental for an excitation frequency of 5280 Hz (corresponding to the sixth axisymmetric mode). Reference (0 dB) corresponds to 125 mm/s. Points represent measured levels, lines refer to polynomial interpolations. Second harmonic level Q2 varies as the square of the fundamental Q21 and third harmonic as Q31 for low values of fundamental and as Q1 for higher values [see Eq. (33)]. J. Audio Eng. Soc., Vol. 56, No. 6, 2008 June

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QUAEGEBEUR AND CHAIGNE

PAPERS

THE AUTHORS

N. Quaegebeur Nicolas Quaegebeur was born in Orsay, France, in 1981. He studied at the Ecole Normale Supe´ rieure de Techniques Avance´ es (ENSTA) in Paris and obtained a Master’s degree in acoustics from the University of Paris, Institut de Recherche et de Coordination Acoustique Musique (IRCAM), in 2004. He received a Ph.D. degree in mechanics and acoustics from the Ecole Polytechnique in 2007 for studies of nonlinear vibrations and sound radiation of thin loudspeaker-like structures. He is currently working at the Groupe d’Acoustique de l’Universite´ de Sherbrooke (GAUS), Sherbrooke, Quebec, Canada, on active control of sound transmission by plates. ●

Antoine Chaigne was born in Versailles, France, in

472

A. Chaigne 1953. He studied at the Ecole Normale Supe´ rieure in Cachan, where he obtained his Agre´ gation in applied physics in 1975. He obtained a Master’s degree in acoustics from the University of Le Mans in 1976 and a Ph.D. degree in acoustics from the University of Strasbourg in 1981. He started his career as a senior lecturer at the Ecole Nationale Supe´ rieure des Te´ le´ communications in Paris in 1985. He then took the lead of the acoustics laboratory at this institution, where he became a full professor in 1991. Since 2000 he has been the head of the Mechanical Engineering Department at the Ecole Nationale Supe´ rieure de Techniques Avance´ es (ENSTA), which is part of the Paris Institute of Technology. His research interests include musical acoustics and sound analysis.

J. Audio Eng. Soc., Vol. 56, No. 6, 2008 June