Guy Lemarquand

New structure of loudspeaker

We present a new structure of loudspeaker: the motor is ironless, the suspension is ferrofluidic, the moving part is piston like, with a concave dome. The absence of iron guarantees a small and constant inductance of the moving coil, as well as the absence of Eddy currents. The motor includes two circular joints, one on each side of the moving coil. These joints are ferrofluidic. They fulfil the guidance and centring function and the air tightness function. This structure is quite rigid. As there is no traditional suspension in this structure, the related non-linearities and hysteresis disappear.

I. THE NON LINEARITIES OF THE CLASSICAL STRUCTURE

Classical loudspeaker motors present well-known non linearities, which are drawbacks of the transducer. Numerous authors, such as Dobrucki

. These two effects are caused by the iron in the motor.

II. NEW STRUCTURES

A. The loudspeaker

) to get rid of these drawbacks. The motor is ironless and constituted by stacked rings of permanent magnets. These structures are studied with the use of the Coulomb's model of permanent magnets, that allows analytical calculations. The aim is to achieve a highly constant force factor along the Ox axis. The design can thus be optimized to create a uniform and high

level induction in the space where the coil moves. It also permits to work on the induction map so as to fix the ferrofluidic joints where they have to be.

The first consequence of the absence of iron in the structure is a very low and constant inductance of the coil. The second consequence is the absence of the reluctance effect and a great decrease of the Eddy currents, depending on the electrical conductivity of the permanent magnets' material. The third consequence is the increase of the magnetic induction and the decrease of the magnetic leakage, that leads to a good efficiency of the loudspeaker. The major goal is to create a very uniform induction in the space of the moving coil to cancel the harmonic distortion. The maximum excursion of the coil is limited in the presented structures by the inversion of the magnetic field in the areas outside the nominal displacement, which constitutes a good protection to limit the travel of the moving set when a peak of power is applied to the voice coil motor.

These structures are not very difficult to realize with the use of plastic bonded magnets. The motor includes two circular joints, one on each side of the moving coil. These joints are ferrofluidic. They therefore play an interesting part. They centre the moving set as a radial bearing. Indeed, the radial stiffness is high, while the axial one is low. The stiffnesses are independent from each other. As a consequence, the centring is good, and the axial resonance frequency is very low, with regard to a classical structure. These joints fulfil the air tightness function between the front acoustical pressure and the rear one. They also are an easy way for the thermal flow coming from the moving coil. The classical suspension has disappeared as well as its related drawbacks. Another peculiarity of the structure is the shape of the moving set. It is cylindrical, piston-like, and the transduction is done by a concave dome. This mechanical structure has high frequency vibration modes.

B. The loudspeaker cabinet

The cabinet associated to this loudspeaker is non conventional. As there is no motor in the center of the diaphragm, the whole rear surface is active and it generates a plane wave. The cabinet is in fact a cylindrical pipe that has the same diameter as the piston and behaves like a waveguide [START_REF] Suzuki | Sound radiation from a concave or a convex dome in a semi-infinite tube[END_REF]. The pipe is filled with an absorbing material.

III. PROTOTYPE

A prototype has been built, with the means of a lab -and not of an industrial group-. Of course, this first prototype shows some imperfections.

The piston diameter is 50 mm wide and it weights 5.1 g. The coil section is 2 mm x 2 mm. The force factor, Bl, is 5.85 N/A. The coil is shorter than the area of uniform induction along the Ox axis. The volume, V cab , of the "pipe cabinet" is 0.82 l.

A. Electrical properties

We use the equivalent electrical circuit proposed by Thiele and Small [START_REF] Thiele | Loudspeakers in vented boxes: Part 1 and 2[END_REF] and the corresponding notations for the parameters. The electrical resistance, r e , of the coil is 4.20 Ω. Its inductance, l e , is 210 µH. As there is no iron in the motor, this inductance is constant whatever the position of the moving set is. When the coil is wound on a capton stay, no Eddy currents appear, neither on the stator, nor on the moving set: the inductance and the resistance remain constant even at high frequencies. The value of the R ms parameter is 5.6 kg/s. This value is important because it results of the damping effect of the ferrofluidic joints. The mobile mass, M ms , is 5.1 10 -3 kg. The usual parameter C ms does not appear because the suspension stiffness is zero and so, the compliance is infinite. As the moving part is exactly a flat circular piston, the impedance Z ar is given by the Rayleigh formula in Eq.1.

Z ar = ρc S (kr) 2 2 + j 8 3π (kr) (1)
The impedance of the "pipe cabinet" is given by Eq.2.

Z cab = R cab + jωL cab + 1 jωC cab ( 2
)
where the compliance, C cab , is :

C cab = V cab ρc 2 = 5.4 10 -9 m 4 kg -1 s 2 (3)
The equivalent electrical circuit is shown in Fig. 2. Fig. 3 and Fig. 4 show the motion impedance versus the frequency. The first resonance, observed at 55 Hz, is caused by the compliance of the cabinet, C cab . This resonance is highly damped by the effect of the ferrofluidic joint, R ms . The second resonance, at 162 Hz, corresponds to the reflection at the end of the pipe. The third resonance, at 7700 Hz, is the dome proper mode. The phase of the impedance tends towards 90 o : the impedance is a pure inductance at high frequencies, which demonstrates the absence of Eddy currents.

B. Near-field acoustic radiation

Fig. 5 shows the near field response. This 50 mm diameter loudspeaker is a medium range one. The major imperfection appears clearly to be the resonance of the dome (7700 Hz). Three other structural resonances appear at 4470 Hz, 10 kHz and 14 kHz.

C. Advantages

The great advantage of this structure is its high linearity with regard to the displacement amplitude. All the responses are proportionnal to the excitation level. As a consequence, the sound has a high definition and seems very clear and crisp.

IV. CONCLUSION

The structure of this loudspeaker is a new concept that has to be studied further. The major non linarities have been suppressed. Thanks to the original magnetic circuit of the motor, the force factor is perfectly constant for large displacements of the moving part. As the motor is ironless, the electrical inductance is low and independent of the coil's position as well as of the frequency, because no Eddy currents appear. The absence of a traditional suspension makes the compliance depend only on the closed-box. So, the compliance is constant. The built prototype is quite promissive, in spite of its large mechanical resonance.

Fig. 1 .

 1 Fig. 1. Full structure geometry.

Fig. 2 .

 2 Fig. 2. Equivalent electrical circuit.

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Modulus of the motion impedance.


Guy Lemarquand

Romain Ravaud

Valerie Lemarquand

Claude Depollier

Mechanical Properties of Ferrofluids in Loudspeakers

This paper describes the mechanical properties of ferrofluid seals in ironless electrodynamic loudspeakers. The motor is constituted of several outer stacked ring permanent magnets. The inner moving part is a piston. In addition, two ferrofluid seals are used which replace the classical suspension. Indeed, these seals fulfill several functions. First, they ensure the airtightness between the loudspeaker faces. Second, they act as bearings and center the moving part. Finally, the ferrofluid seals also exert a pull back force on the moving piston. Both radial and axial forces exerted on the piston are calculated thanks to analytical formulations. Furthermore, the shape of the seal is discussed as well as the optimal quantity of ferrofluid. The seal capacity is also calculated.

I. INTRODUCTION

Electrodynamic loudspeakers have a well-known structure: a coil moves in front of the iron pole pieces of a magnetic circuit polarized by a permanent magnet. Moreover, it is also well-known that classical electrodynamic loudspeakers have drawbacks whose consequences are nonlinearities. Many authors highlighted these nonlinearities and proposed solutions to reduce or suppress them [START_REF] Dobrucki | Nontypical effects in an electrodynamic loudspeaker with a nonhomogeneous magnetic field in the air gap and nonlinear suspension[END_REF][2] [START_REF] Gander | Dynamic linearity and power compression in moving-coil loudspeaker[END_REF].The three major drawbacks are the following. First, the magnetic field in the air gap is non-uniform. Second, the inductance of the coil, which exert a non linear pull back force on the moving part [START_REF] Lemarquand | New structure of loudspeaker[END_REF], varies with the coil position. Both effects are sources of a distortion which increases with the coil displacement and the current intensity, so with the sound level. Third, Eddy currents in the pole pieces create a force which ejects the moving part and the coil out of the air gap, thus lessening the stability of the loudspeaker. The analysis of these drawbacks lead us to a peculiar design of loudspeaker motors, namely to ironless motors with ferrofluid seals. Furthermore, as these structures suppress some of the nonlinearities sources, such as the iron and the classical suspension, they have an improved behavior. Besides, the absence of iron enables analytical formulations to calculate their properties and authorizes multi-criteria optimization as well. This paper first reminds the drawbacks related to the iron. Then, it presents the proposed ironless structures with ferrofluid seals. Finally, it shows how the mechanical properties can be calculated.

II. THE ELECTRODYNAMIC LOUDSPEAKER

A. Drawbacks of the Classical Iron Structures

It has to be noted that equivalent electric circuits are used to describe the electrodynamic loudspeakers. Thereby, the voice coil is modeled by an impedance which is both resistive and inductive. Moreover, the coil gives rise to a back electromotive force (EMF) which is proportional to the cone velocity. So, both acoustic and mechanic behaviors of the cone are to be found in the impedance variations. As a consequence, the loudspeaker output signal can be predicted from the electric input of the equivalent circuit. As long as the moving part can be considered as a rigid assembly, the model is valid in the corresponding frequency range.

1) The Inductance: As a matter of fact, the iron pole pieces increase the inductance of the moving coil. Consequently, the impedance increases with the frequency. On one hand, the loudspeaker is generally fed by an open loop voltage amplifier. On the other hand, the force exerted by the coil on the moving part is proportional to the current. So, the current decreases when the frequency increases and the force decreases too. Furthermore, the voltage and the force are out of phase. Consequently, the phase shiftings of the electric signal harmonics are modified. It can be noted that a short circuiting coil is sometimes added on the stator of the classical electrodynamic loudspeaker, in order to reduce the moving coil inductance and thus the distortion of the output signal. It is to be noted that another effect is related to the coil. Indeed, the latter behaves like a moving iron yoke coil for which the yoke is around the coil. So, when the coil moves, the yoke position changes and the reluctance of the magnetic circuit varies. Moreover, when the coil moves from the centered position of the yoke with regard to the coil the reluctance increases. Consequently, a force is created on the coil in such a direction that makes the coil move to decrease the reluctance. This results in an axial centering effect. Furthermore, the exerted force is proportional to the square of the current in the coil.So, this pull back force is non linear.

2) The Eddy Currents: It has been proved that the electrical impedance is modified by Eddy currents flowing in the solid iron poles. As a consequence, voice coil motors have a normal inductive behavior at low frequencies whereas the Eddy currents hinder the magnetic flux from penetrating the iron at high frequencies. Thereby, the result is the decrease of the effective permeability of the iron. This implies a decrease of the coil inductance, thus proving the existence of the Eddy currents. It can be noted that measuring the inductance is the easiest way to determine the frequency at which Eddy currents appear. Moreover, Eddy currents always appear to counteract their own causes. In the present case, they create a magnetic moment in the iron pole pieces, whose direction is the same as the one of the magnetic moment created by the current in the coil. The resulting system behaves like an Eddy current magnetic bearing. Unfortunately, the moving coil becomes axially unstable (but remains radially stable according to Earnshaw theorem). Indeed, the interaction of both magnetic moments creates axial forces that tend to eject the coil out of the air gap. These axial forces are another cause of distortion. Laminated pole pieces are a solution to avoid Eddy currents flowing circularly in the iron pole pieces around the coil axis.

III. THE IRONLESS STRUCTURE

A. Description

We propose loudspeaker motors totally made out of permanent magnets. Indeed, figure.1 presents a motor structure which consists of three outer stacked ring permanent magnets which are radially magnetized. It has to be noted that the middle ring magnetization has a direction opposed to the one of both extremity rings. Moreover, the moving part is an inner piston out of non magnetic material which is radially centered with the rings. One of its extremities is the emissive surface which is convex here, but could also be either plane or concave depending on the intended acoustic properties. Furthermore, a short coil is wound around the piston so that it is entirely in the region of uniform magnetic field, even when it moves. Besides, ferrofluid seals are located in the air gap between the inner piston and the outer ring magnets. As a consequence, the classical suspension disappears. The air in the rear cavity constitutes the loudspeaker pneumatic suspension. Therefore the suspension behavior is linear.

B. Properties

1) The absence of iron: As a matter of fact, the absence of iron suppresses the reluctance effect and its related non linearities. Moreover, the coil self inductance is dramatically decreased which is interesting for the loudspeaker impedance at high frequencies. Furthermore, as the permanent magnets have a small electric conductivity the Eddy currents in the pole pieces disappear too. It is emphazised that plastomagnets are well adapted for the realization of such motors and are electrically insulating. The iron suppression solves the saturation problem too, which was becoming a technical limit with the new rare earth permanent magnets. Consequently, some of the sources of distortion are removed. Furthermore, the design of the magnet assembly allows the use of a short coil entirely located in a region of uniform magnetic field. So, the force factor is significantly increased. For example, the device middle magnet of Fig. 1 is twice as long as the extremity magnets. Thus, the region of uniform field corresponds to the middle ring magnet length. Besides, ironless motors enable to control the magnetic leakage and even to design devices with no leakage at all [START_REF] Berkouk | Analytical calculation of ironless loudspeaker motors[END_REF]. Indeed, the ironless concept offers various design possibilities. In fact, the devices can be simple and use only two ring magnets (or even a single one) or they can be more complicated, with three or more ring magnets of various section shape. Furthermore, they have the great interest of being analytically tractable, a fact that enables their optimization. Besides, their optimization can be made with several criteria, such as the magnetic field uniformity, its intensity level but also the mechanical properties of the ferrofluid seals.

2) The moving part: The inner moving part is a piston with plane extremities. Besides, the classical suspension disappears. Instead, two ferrofluid seals are used which fulfill several functions. First, they ensure the airtightness between the loudspeaker faces. Second, they allow the heat transfer from the moving coil to the stator. Such a use is common in loudspeakers. But the ferrofluid seals also act as bearings and center the moving part by exerting a radial force on the moving part. Moreover, they exert a pull back force on the moving piston too. Consequently, they play the same part as the suspension and the spider. These functions are more innovative. It is to be noted that the suspension is due to the compression properties of the air in the rear cavity. In order to avoid the rest position offset when the atmospheric pressure varies, the latter is provided with a capillary tube whose extremity is at the external pressure. The ferrofluid seal exerts a light pull back force towards the stable equilibrium rest position. Moreover, this pull back force is ten thousand times smaller than the air compression force. Thus, the loudspeaker compliance is only related to the air compressibility which is perfectly linear. It is pointed out that the sliding of the piston on a ferrofluid allows very large displacements of the moving part. From this point of view, the piston length has to be chosen so that the ferrofluid seals remain confined between the piston and the stator over the whole displacement range. The structure of Fig. 1 is long because it is intended for large displacements. The only movement limitation is linked to the displacement speed. Indeed, it has been proven that intermodulation appears for high speeds (a few meters per second) and for a large bandwidth use of such ferrofluid suspended loudspeakers [START_REF] Lemarquand | Large bandwith loudspeaker emitting coherent acoustic waves: nonlinear inter-modulation effects[END_REF]. This phenomenon is due to the non linearity of the sound propagation in air.

C. Study and Calculations

An important point is the fact that the structure having no iron, we are able to calculate analytically the magnetic field, H, created by the magnets. Indeed, a coulombian model of the magnets, which represents each magnet by two charged surfaces, allows the calculation of all the components of the magnetic field they create [START_REF] Ravaud | Analytical calculation of permanent magnet rings[END_REF]. On one hand, the calculation of the radial component of the magnetic field is useful to characterize a structure with the field intensity as well as the field uniformity. The analytical formulation enables to optimize the design of the magnets according to criteria such as high field intensity and uniformity over a given displacement range of the coil, which will enhance the force factor. Moreover, the design and the optimization can take the magnetic leakage into account, depending if it is tolerated or not for the device. On the other hand, the same calculation is useful to characterize the field gradients too. Indeed, the ferrofluid seals form where the field gradients are high. Furthermore, the magnetic pressure in the ferrofluid, pm, can be calculated too, as pm(r, z) = µ 0 M H, where (µ 0 M ) is the magnetization volume density of the ferrofluid and µ 0 is the vacuum magnetic permeability. The ferrofluid seals shape can thus be determined as the seal contour is a magnetic isopressure. Of course, the shape also depends on the ferrofluid quantity used. Besides, the mechanical properties of the ferrofluid seal can be calculated. Indeed, they come from the ferrofluid seal potential energy which is defined by: E m = -(Ω) p m (r, z)dV where (Ω) is the ferrofluid seal volume. The radial centering force exerted on the moving part by the seal is expressed by: F r = ∆E m /∆r, where ∆r is the radial distance in length for which the piston is moved away and ∆E m the corresponding potential energy variation. Therefore, the radial stiffness can be calculated as well. Moreover, the same reasoning for axial displacements leads to the axial force and stiffness. The capacity of the seal can be evaluated too. It must be emphazised here that the analytical formulation enables a great flexibility in the optimization of the seals characteristics. Indeed, a multicriteria optimization gives the possibility of optimizing the whole device, from the force factor and the magnetic leakage to the seal shape and the ferrofluid quantity, as well as the radial and axial stiffnesses and the seal capacity.

IV. CONCLUSION

This paper shows that significant improvements are still possible for such well-known devices as electrodynamic loudspeakers. Indeed, ironless loudspeakers with ferofluid seals prove to have a very good linearity, as they suppress the source of many nonlinearities. Furthermore, their structure allows the analytical formulation of their magnetic and mechanic properties. In consequence, their optimization according to multiple criteria is not only possible but also very flexible. Eventually, it is to be noted that a first prototype was presented at the 120 th AES Convention in Paris and that the improvements made since are quite satisfying.

Fig. 1 .

 1 Fig. 1. Structure of an ironless loudspeaker.