Sabine Bottin-Rousseau

Victor T Witusiewicz

Ulrike Hecht

Jose Fernandez

Ana Laveron-Simavilla

Silvère Akamatsu
email: akamatsu@insp.jussieu.fr

Coexistence of rod-like and lamellar eutectic growth patterns

Keywords: Eutectic growth, directional solidification, microgravity, in situ experimentation

We present the first observations of a large-scale coexistence between rod-like and lamellar eutectic growth patterns during directional solidification of a eutectic alloy. In situ experiments with real-time optical monitoring were carried out under microgravity onboard the International Space Station (ISS). We used the transparent succinonitrile-d,camphor eutectic alloy that ordinarily forms rod-like patterns. At low growth velocity, short lamellae stabilized at the contact line with a sample glass wall. In the presence of a controlled transverse temperature gradient, the coupled-growth pattern experienced a global drift along an inclined isotherm, and a stable lamellar domain spread over the solidification front. The propagative nature of the lamellar-to-rod transition was evidenced. The advance of the lamellar/rod domain boundary was determined by a slowly amplifying varicose instability of the lamellae. On a large scale, the domain boundary underwent a dynamic faceting.

Binary eutectic alloys directly solidify into self-organized composites [START_REF] Dantzig | Solidification, 2nd Edition[END_REF]. Their growth is of interest in materials science [START_REF] Hecht | [END_REF][3][4][5][6] and the physics of nonequilibrium phenomena [7][START_REF] Fleury | Branching in Nature[END_REF]. During directional solidification at velocity V in a fixed temperature gradient G, two different solids grow simultaneously, and form planar two-phase patterns at the solid-liquid interface (Fig. 1).

The solidification dynamics is primarily governed by solute diffusion in the liquid (coupled growth). The interphase spacing λ falls close to a scaling length λm~V -1/2 , for which solute diffusion and capillary effects at the solid-liquid interface contribute equally to the average undercooling [START_REF] Jackson | [END_REF]. Coupled-growth patterns with band or hexagonal arrangements result in lamellar or rod-like microstructures in the bulk solid, respectively (Fig. 1a). Whether lamellae or rods prevail depends on alloy characteristics -rod-like patterns are favored when one solid phase has a much smaller volume fraction than the other. Complexity arises from the fact that λ can vary within a finite interval at given V. Outside their respective stability intervals, lamellae and rods undergo secondary, symmetry breaking instabilities, which lead to spatiotemporal behaviors that are sensitive to boundary and initial conditions. This phenomenology has been extensively studied [10], with key success gained from in situ experimentation [11] and numerical simulations [12]. However, the so-called lamellar-to-rod transition, that is, the relative stability of lamellar and rod-like patterns in a given system, remains an open question. Previous numerical studies considering a model eutectic with simplified characteristics established that, at given alloy composition, the lamellar and rodlike stability intervals generically overlap in the parameter space [13,14]. This makes likely a dynamic "metastability" between lamellae and rods -the question then being under which conditions the transition occurs, and in what form. Some mechanisms were proposed [13][14][15][16],

taking inspiration from experiments [15,16] and numerical simulations [13,14] that revealed an unsteady spatio-temporal dynamics involving mixed, disordered patterns during transients.

However, a coexistence between large lamellar and rod-like domains, giving full evidence of a genuine dynamic transition, has not been demonstrated so far. We present real-time observations of a lamellar-to-rod transition during directional solidification of a model transparent eutectic alloy, namely the succinonitrile-d,camphor (SCN-DC) eutectic alloy, which ordinarily forms rod-like eutectic microstructures [17,18]. In situ experiments -the first of their kind in a microgravity environment-were carried out with the TRANSPARENT ALLOYS (TA) apparatus of the European Space Agency (ESA) onboard the International Space Station (ISS) [19,20]. Microgravity in an orbiting facility provides a unique tool for achieving diffusion controlled crystal growth from the melt without convection, irrespective of thermal and/or solutal density gradients in the liquid [21,22].

During in situ eutectic solidification of SCN-DC on ground, convective fluid flow was observed with flow velocities of about 10 μms -1 close to the growth front [23]. Convection was absent in microgravity. Two distinct directional-solidification (DS) experiments were carried out. First, for reference, we observed rod-like patterns during regular directional solidification with the temperature-gradient axis well aligned with the pulling direction (regular-DS) [24,25]. Second, we employed a tilted-isotherm strategy using an intentional inclination of the temperature gradient from the pulling axis z (Fig. 1c). The isotherms, thus the growth front, were tilted by several degrees about the x axis (tilted-DS). This entailed a global drift of the pattern in the transverse direction y. In a previous work, tilted-DS was used to form a stable lamellar pattern from an unstable labyrinth pattern [26]. Here -our main finding-it permitted a coexistence of large rod-like and lamellar domains separated by a sharp, propagative boundary, at which the morphological transition was actually occurring (Fig. 1d). Fundamental questions concern the instability mode, the role of the global drift, and a possible pattern selection process at the domain boundary.

Self-organizing band and hexagon patterns are frequent in spatially extended nonequilibrium systems [7]. They also are quite labile, and subject to secondary, symmetry breaking instabilities in response to slight changes in control parameters and boundary conditions [START_REF] Manneville | Dissipative Structures and Weak Turbulence[END_REF].

Such morphological-selection mechanisms are of great interest as regards the control of microstructures in materials [START_REF] Hecht | [END_REF][START_REF] Asta | [END_REF], as well as the imprint they can leave in natural environments [29,30]. In this context, situations leading to a long-lived coexistence of patterns separated by a domain boundary deserve special attention [31][32][33][34]. The band-hexagon transition belongs to this class [34][35][36]. It has been observed in various systems upon finite constitutional changes [37,38] or via active forcing [39][40][41], but it most often takes place via complex transients [42,43]. Controlling the coexistence of large band and hexagon domains via a non-intrusive means still remains challenging. In this respect, eutectic growth is a particularly instructive model system.

The eutectic SCN-DC alloy (13.8 mol%DC) solidifies into nearly pure SCN and DC crystal phases [17]. The small (≈ 24%) DC-phase volume fraction in the solid favors DC rods forming in a continuous SCN matrix (Fig. 1) [18]. A cartridge with a rectangular cross-section (6 × 1 mm 2 ; length: 70 mm) and optically flat fused-silica walls (Hellma) was filled (QinetiQ Space) with the molten alloy prepared with purified compounds under a protective atmosphere, and sealed. A crystal selector at the cold end was used to grow a single eutectic grain, except for a few subboundaries. The TA apparatus (Qinetiq Space) was designed from previous laboratory instruments [44], and adapted to the Microgravity Science Glovebox (MSG) facility onboard the ISS -technical features related to safety and automating constraints need not be mentioned here. The temperature gradient (G ≈ 50 ± 10 Kcm -1 ) was established between two metallic blocks separated by a 7-mm gap, each of them made of two pieces with independent thermal regulations, and good contact with the cartridge walls. A transverse gradient (Fig. 1c) could be generated by imposing slightly different temperatures at the two walls [26]. The tilt angle f of the isotherms was set to -6.2±0.3°. For solidification, the sample was pulled along z with a step motor toward the cold part. Observation from the exterior with a long-distance optics (and a camera) at an oblique incidence (θ = 43.6°; Fig. 1b) delivered dark-field images of the solidification front (spatial resolution: 3 μm) with sharp contrast between the bright DC phase and the dark SCN matrix [44]. Astigmatism due to plane diopters was partly corrected (Lambda-X). Good-quality images were reconstructed from focus series over about 2/3 of the growth front width along y, and rescaled numerically with a 1:1 aspect ratio. TA operations were essentially automatic, with telescience (parameters, sampled image series) from the operation center (E-USOC, Madrid, Spain) permitting near real-time feedback.

During regular-DS, a steady-state pattern was created at constant velocity (V = 0.04 μms -1 ), and its response to stepwise downward V changes was observed. The system was eventually left running for about 150 hours at V = 0.007 μms -1 (Fig. 2a). Rod-like patterns with local hexagonal order and topological defects were observed over the explored V range. A continual transverse stretching of the pattern (spatiotemporal diagram in Fig. 2a) was due to a slight forward curvature of the isotherms, thus of the envelope of the growth front (mean radius of curvature: 9-10 mm). This confirmed a previous analysis on-ground [24], which attributed this effect to the different thermal conductivities of the glass cartridge material and the organic alloy. At long times, the stretching was balanced by rod splitting events. This resulted in a broad λ distribution that roughly spanned between the lower (rod-elimination) and upper (rod-splitting) stability limits of hexagon patterns [25,45]. The average spacing λav was close to λm. The curvature effect also entailed the alignment of the subboundaries -but no crystal-orientation effects were detected [18,24], in contrast to observations reported in, e.g., Refs. [46,47]. At low velocity, a few short lamellae were observed along subboundaries, and in contact with the sample wall (Fig. 2a). Elongated rods in the core of the pattern were unstable against splitting. During tilted-DS, a global drift of the eutectic growth pattern along the transverse axis was observed. The drift velocity vd was approximately equal to Vtanf (the rods grow essentially perpendicular to the front envelope). At V = 0.04 μms -1 , hexagons gradually aligned with dense rows normal to the sample walls (Fig. 3a,inset). The foremost contact line (Fig. 1c) served as a source of new rods via splitting. No lamellae formed. Lamellae were observed at low velocity (V = 0.007 μms -1 ) after a maturation stage (Fig. 3). In the core of the pattern, short, isolated lamellae in rough alignment with a nearest-neighbor direction of surrounding hexagons (Figs. 2 and3c) were unstable, and broke up into rods, giving rise to a dynamics similar to that reported in Refs. [14,15]. Straight, long-lived lamellae appeared at the foremost contact line, perpendicular to it. Such a wall effect is usual in directionally solidified eutectics [48], and was reproduced numerically with a no-flux boundary condition [13] -it is well explained by symmetry arguments [34]. It is short-ranged -see a locally zigzagging pattern [49] on the left side of Fig. 3d. Globally, the lamellae elongated stable along the drift direction.

The longest ones were reaching 500 μm at the ending time of the experiment. Figure 4a illustrates the breakup process of the bulb-shaped free ends of the lamellae. As shown in Fig. 4b, the trijunction lines were affected by an undulating, varicose instability, with a characteristic length winst ≈ 32 μm close to λav ≈ 35.5 μm in the fresh rod pattern. The time τp between two successive breakup events, averaged over a long time -the instability being otherwise rather irregular-was of about 14 hours, that is about 10λ/V. The inset of Fig. 4b also makes it clear that the instability was spatially damped over a distance of about 3winst in the drifting reference frame (Y = y + vdt), and new rods were emitted one after another at the free lamella ends. The instability was "convected downstream" along the drift directionhere, the convective-instability phenomenology is disconnected from fluid mechanics [START_REF] Manneville | Dissipative Structures and Weak Turbulence[END_REF]. In contrast, the instability remained finite near the sample wall. In Fig. 5a, we represented a periodic lamellar domain in coexistence with a hexagonal rodlike pattern during an idealized tilted-DS experiment. In Fig. 5b, the breakup of a lamella is schematically represented in the reference frame of the temperature gradient -which makes the sample wall appear tilted. The free end of a lamella was actually recoiling at a mean rate of winst/τp in the drifting reference frame. The condition for a neat elongation of the lamellae in the solid is vd > winst/τp, hence the isotherm tilt angle must be larger than a minimum value, here of about 5°, which was indeed the case. It may be advanced that if the duration of the experiment had been long enough, the lamellae would eventually have connected the two walls of the sample. The tilted-isotherm configuration thus represents an efficient means for driving a global, directional motion of a domain boundary between self-organized patterns.

We measured the interphase spacing λ along the wall (i.e., at the foremost contact line) and the length L of the lamellae (or splitting rods). Two populations could be distinguished. Rods, or elongated rods (L < 2winst) were observed within small patches for λ close to λav, and lamellae over a much larger λ range, yet falling within the stable domain for rods. This is consistent with a wide overlap of the lamellar and rod stability domains. How the varicose instability of the eutectic lamellae scales with the control parameters still remains an open question. According to the present observations, the lamellar-to-rod transition does not follow a λ/λm ~ λV 1/2 scaling law, which may put into question the analysis proposed in Ref. [15]. In Ref. [13], a lamellar breakup arising from a homogeneous varicose instability (1-λ mode) was considered, but a definite scaling was not determined. Echoing a discussion in that work, the observed dynamics was reminiscent to that of a Rayleigh-Plateau instability of a liquid cylinder flowing from a nozzle [50]. This would suggest a dominating capillary effect, and a direct dependence on λ of the varicose instability. As compared to the stability analysis of a solid rod (e.g. a dendritic sidebranch) [51], the problem is complexified by the propagating dynamics and the tijunction lines. Further work would still be required for clarification. eutectic growth patterns do not arise from a uniform state via a bifurcation-like scheme, and the rod-emission mechanism involves a breakup of the trijunction line. A lamellar/rod DB with smoothly varying characteristics seems unrealistic [52]. Propagative band/hexagon DBs have been previously identified during transients in phase-field simulations of eutectic growth [14], and in Rayleigh-Bénard convection as well [42]. Here, the lamellar/rod DB propagated slowly at a constant, positive average rate (vd -winst/τp), and was observable over long times.

Interestingly, the DB gradually adopted a sawtooth-like shape on a typical scale of 10λ, with straight segments slanted at about ±30° from the x axis (see Figs. 1d, 3, and5a). This process could be observed thanks to the use of samples of large aspect ratio. It is analogous to a dynamic faceting, or Herring-like [53] instability of a DB between patterns of different symmetries with topological defects compensating for a difference of spacing of the two patterns [54]. In a phenomenological study [34], a straight DB between bands and hexagons with a dense row perpendicular to the bands was considered. Here, the DB superstructure resulted from the interplay between bands and hexagons on both sides of the DB, with dense hexagon rows parallel to the lamellae. A pattern selection mechanism associated with the propagating DB still remains to be evidenced -it has been predicted theoretically [32,33], and observed experimentally [34,55] in qualitatively different situations.

In conclusion, real-time observations of a lamellar-to-rod transition in a directionally solidified eutectic alloy were presented for the first time. At low growth rate, a long-lasting coexistence of extended rod-like and lamellar domains was realized, thus bringing a clear demonstration of a dynamic metastability between band and hexagon patterns in a nonequilibrium system. Straight lamellae were stabilized in contact with a wall, and a varicose instability on their free end was made convective-like due to a global drift along inclined isotherms. This also offers a laboratory exemplification of how a self-organized composite architecture with a two-domain superstructure in the solid can be produced. This could inspire new protocols involving a propagative domain boundary in materials of potential interest for micropatterning [6,[56][57][58]. Importantly, this work provides a long-awaited reference in eutectic solidification with real-time imaging in purely diffusive conditions. It adds to the success of recent in situ microgravity solidification studies in dilute alloys [11,22]. Much work remains to be done. Reproducing the reverse morphological transition -from rods to lamellae-as it has been observed in a confined system [16] would be highly informative.

Further numerical work aiming at a systematic investigation of the lamellar-to-rod transition would be of great value.

Figure 1 :

 1 Figure 1: a) Schematic rod-like and lamellar eutectic growth patterns (λ: interphase spacing). b) Regular directional solidification (DS). V: pulling velocity. θ: viewing angle. c) Tilted-DS. f: tilt angle of the isotherm. vd: drift velocity. f: foremost contact line. d) Image: rod-like and lamellar domains in coexistence during tilted-DS of a eutectic SCN-DC alloy [detail; red frame in c)]. The growth axis z points towards the reader. The DC crystals appear white, the continuous SCN matrix dark. V = 0.007 μms -1 . Bar: 50 μm. Color online.

Figure 2 :

 2 Figure 2: Directional solidification of a eutectic SCN-DC alloy in microgravity. Growth patterns at V = 0.007 μms -1 : a) regular-DS experiment; b) tilted-DS experiment. These "panoramas" were reconstructed from a lateral scan of the solidification front. Vertical dimension: 1 mm (thickness of the sample). Insets: large-magnification views of the framed details. Right panels: spatio-temporal diagrams recorded along a fixed line parallel to y [duration times (horizontal dimension): a) 23.6; b) 172.8 hours].

Figure 3 :

 3 Figure 3: Tilted-DS eutectic patterns: a) 31.3; b) 88.8; c) 146.3; d) 203.8 hours after a V change from 0.04 [inset in a)] down to 0.007 μms -1 . Panel c) and Fig. 2b were recorded closely in time.

Figure 4 :

 4 Figure 4: a) Detail of the tilted-DS pattern (V = 0.007 μms -1 ). Binarized images: three times (τ1 = 12 hours). The images on the left (τ = 0) and on the right (2τ1) were taken after two subsequent breakups of the central-lamella end. Right panel: raw 2τ1 image. Horizontal dimension of the images: 110 μm. b) Shape of one side of the central lamella at three times. y=0: sample wall. Inset: same curves in the drifting reference frame (arbitrary y origin).

Figure 5 :

 5 Figure 5: Schematic representations: a) Coexistence of lamellar and rod-like domains. b) Breakup of a lamella end in the reference frame of the temperature gradient

Acknowledgments

We thank Patricia Ott for technical help. We gratefully acknowledge Mathis Plapp for enlightening discussions, and our late colleague Gabriel Faivre for sharing his profound thinking with us. We thank the European Space Agency (ESA), QinetiQ Space (Antwerp, Belgium), and Lambda-X (Nivelles, Belgium) for the development of the TRANSPARENT ALLOYS instrument. We are grateful to the E-USOC Madrid for the preparation and the operating of the experiments, in smooth coordination with NASA. This research was realized thanks to the support from the German Space Agency (DLR) under grant number 50 WM 1743, and from the French Center for Spatial Research (CNES).


Hamid Ben Ahmed

Olivier Bethoux

Antoine Cizeron

Emmanuel Hoang

Anthony Juton

Eric Laboure

Adrien Mercier

Eric Monmasson

Javier Ojeda

Loïc Queval

Ghislain Remy

Electric Traction Chain with Segmented Power Supply In alphabetical order

Keywords: Non-standard electrical machine, Emerging topology, System integration, Converter machine interactions, Automotive application

We present an electric traction chain with segmented power supply for automotive applications. The concept is to split the windings of the electrical machine into sub-windings by bundle of turns. Each sub-winding is then fed by an independent low-voltage power electronics converter. We describe here the concept in details and discuss its advantages and its drawbacks. This allows us to identify the scientific and technical obstacles that will need to be solved.

Introduction

Electric vehicles are gradually being manufactured by every automaker, pushed by the regulations to reduce greenhouse gases. Among the various manufacturers, the permanent magnet synchronous motor seems to have achieved consensus. But the architecture of the complete traction chain is not yet standardized. For example, the voltage level often differs depending on the power requirements. A first approach is to increase the voltage when the power rating increases, just like in power grids. However, above 48 V safety standards impose additional constraints that can be difficult to manage and costly. This is probably why some manufacturers opted for "low voltage" drives. The different technological choices observed today is a motivation for prospective studies about non-standard electric traction chains, such as the one discussed in this contribution.

We propose here a traction chain with segmented power supply [START_REF] Hoang | Electric machine powered at low voltage and associated multicellular traction chain[END_REF]. The idea came from a discussion between an electrical machine engineer and a power electronics engineer that can be summarized by two questions: 1 -How to modify the coupling of the windings of a machine to optimize its convertible power? [START_REF] Hoang | Proposition pour accroître la puissance convertible par un ensemble onduleur de tension -machine synchrone à commutation de flux à double excitation dans une application motorisation de véhicule hybride ou électrique[END_REF] 2 -How to segment the power electronics converter to optimize the converter-machine integration? [START_REF] Forest | An isolated multicell intercell transformer converter for applications with a high step-up ratio[END_REF] The concept can contribute to the development of integrated traction chains. In this context, we review in section 2 the topics of machine windings reconfiguration, splitting of the phase windings, segmentation of power electronics converter and integrated motor drive. The proposed traction chain is detailed in section 3. Advantages and drawbacks of the architecture are discussed in sections 4 and 5. Then our first prototype is introduced in section 6. Finally, the scientific and technical obstacles that will need to be solved are described in section 7.

State of the art of integrated traction chains a) Machine windings reconfiguration

Electrical engineers are familiar with the "star-triangle" start-up technique for an asynchronous machine fed by a fixed-frequency fixed-voltage grid, which enables to reduce the magnitude of the inrush currents. Similarly, the Dahlander pole-changing circuit allows for varying the speed of the motor by altering the wiring connections inside the machine. In reference [START_REF] Kume | A wide constant power range vector-controlled ac motor drive using winding changeover technique[END_REF], the authors "proposed an ac spindle motor drive with a wide constant power range by switching the winding connections of the motor". They use the same technique as the one used to dynamically change the number of turns in the windings of a tap-transformer. In reference [START_REF] Gerrits | Electrical gearbox equivalent by means of dynamic machine operation[END_REF], power electronics solutions have been proposed to change the configuration of the coil. Finally, in [START_REF] Boxriker | Increasing the operating range of permanent magnet synchronous motors by switching the winding configurations[END_REF] a modeling of the dynamic reconfiguration of the coils is proposed.

b) Splitting of the phase windings of the machine

The phase windings of an electric machine can be split to add redundancy to the electric drive [START_REF] Quéval | A switched reluctance motor drive using photovoltaic transistors: principle, prototype, experimental and numerical results[END_REF] or to adapt the voltage ratings of the machine and of the power electronics [START_REF] Wang | Integrated modular motor drive design with GaN power FETs[END_REF]. Alternatively, multi threephase machines can be obtained by splitting the different sectors of a three-phase machine [START_REF] Sala | Power-sharing control in bearingless multi-sector and multi-three-phase permanent magnet machines[END_REF] or by grouping the phases of multi-phase drives [START_REF] Golovanov | 4 MW class high power density generator for future hybrid-electric aircraft[END_REF].

c) Segmentation of the power electronic converter

In reference [START_REF] Waide | Energy-efficiency policy opportunities for electric motor-driven systems[END_REF], the authors "present the findings of the first global analysis of energy consumption in electric motor driven systems and the options to reduce it". On page 56, they show that the segmented operation of the converter can bring important benefits. This is tightly related to losses in power electronics components during partial load operation [START_REF] Pai | A calorimetric method for measuring power losses in power semiconductor modules[END_REF][START_REF] Chang | Adopting MOSFET multilevel inverters to improve the partial load efficiency of electric vehicles[END_REF].

d) Integrated motor drive

In reference [START_REF] Abebe | Integrated motor drives: state of the art and future trends[END_REF], dating back to 2016, the authors present a comprehensive review of integrated motor drives based on 203 references. In addition, we should mention the projects of the Fraunhofer Institute for Integrated Systems and Device Technology [START_REF] März | Power electronics system integration for electric and hybrid Vehicles[END_REF] as well as the EMiLE project [START_REF]EMiLE Electric motor integrated power electronics with smart stator teeth[END_REF]. The Wisconsin Electric Machines and Power Electronics Consortium is no exception, demonstrating its expertise in this area in [START_REF] Brown | Power converter design for an integrated modular motor drive[END_REF]. Several integration projects are under development [START_REF] Wang | Integrated modular motor drive design with GaN power FETs[END_REF][START_REF]Overview of the VTO electric drive technologies program[END_REF]. Additional elements on this part of the state of the art can be found in [START_REF] Cizeron | Perspectives d'intégration pour la chaîne de conversion d'énergie des véhicules électrifiés[END_REF].

Presentation of the proposed traction chain with segmented power supply

To clarify the following discussion, let us define a few terms:

• A turn consists of one electrically conductive insulated wire wound in a certain shape (Fig. 1a).

• A coil is made of one or several turns connected in series and/or parallel (Fig. 1b).

• A winding is made of one or several coils connected in series and/or parallel. Each winding has its own two terminals by which it is supplied (Fig. 1c). For illustration purpose, we consider the traction chain shown in Figure 2a). The machine is generic: the topology of the rotor is not specified. The 2-pole 3-phase electrical machine has concentrated armature windings. There are three windings (6 terminals). The windings are star-connected. Each winding is made of two series-connected coils. Each coil has three turns and is wound around a different tooth. The machine is fed by a 3-phase converter having a DC side voltage . It is possible to segment the power supply by splitting the windings into sub-windings.

The conventional way is to split the windings by coil [START_REF] Kume | A wide constant power range vector-controlled ac motor drive using winding changeover technique[END_REF][START_REF]EMiLE Electric motor integrated power electronics with smart stator teeth[END_REF]. In Figure 2b), each winding is split into two sub-windings (the winding a is split into a1 and a2, etc.). As a result, there are 6 sub-windings (12 terminals). Each sub-winding is now made of a single coil. Note that each sub-winding is wound around a different tooth. To obtain the same magnetic state, the machine is now fed by two independent 3-phase converters having a DC side voltage 2 ⁄ .

The original idea proposed here is to go further and to split the windings by bundle of turns [START_REF] Hoang | Electric machine powered at low voltage and associated multicellular traction chain[END_REF]. In Figure 2c), each coil is split into three sub-windings (the sub-winding a1 is split into a11, a12 and a13). As a result, there are 18 sub-windings (36 terminals). Each sub-winding is now made of a single turn. Note that several sub-windings are now wound around the same tooth. To obtain the same magnetic state, the machine is now fed by six independent 3-phase converters having a DC side voltage 6 ⁄ .

The proposed concept does not necessarily imply to split the windings up to the turn. This is only one option, among others. Similarly, the reduced converter DC side voltage does not imply to have a low voltage DC bus. Indeed, depending on the DC side connection of the converters, one could have a low voltage DC bus (parallel connection) or a high voltage DC bus (series connection) [START_REF] Wang | Integrated modular motor drive design with GaN power FETs[END_REF]. Finally, we underline that the concept equally applies to machines with distributed windings.

Main advantages of the concept

In this section, we discuss in detail several points that could be listed as advantages of the concept.

a) Low voltage traction chain

By concept, the converter DC side voltage is reduced in comparison to the equivalent standard solution. The DC bus can then be "low voltage" to limit electrical hazards, or "medium voltage" to stay under Paschen's minimum. In the previous example, with parallel connection of the converters, each winding (Fig. 2c) can now be fed from a low voltage DC source (e.g. 48 V). This is magnetically equivalent to powering the classical machine (Fig. 2a) from three times this voltage (e.g. 144 V) in order to apply the same ampere-turns. From the DC side of the converter to the electric machine, all the components operate at reduced voltage. As consequence, on the one hand it possible to use transistors that do not require to withstand high voltage and have therefore low on-state resistance, and on the other hand it facilitates the use of low-voltage battery packs.

b) Improved availability rate

The adaptation of the structure of the electric machine to the structure of the power electronics converter can allow for a better integration and an optimized response to the specifications (cost, safety of operation, conditions of use, etc.). This directly impacts the availability rate. In addition, the resilience is enhanced by the choice of a multi-converting architecture. Indeed, the management of the degraded mode seems to be facilitated by splitting the windings. Whether it is a subwinding or the associated converter that is failing, two strategies are possible: increase the current of the other sub-windings of the same phase to keep the same magnetic state or passivate the corresponding sub-windings of the other phases to restore a balanced magnetic state.

c) Reduced stress on the insulation The ⁄ are better distributed within each sub-winding powered by a converter because they have few turns. This is even more important as the switching frequency is high (SiC, GaN). This reduces the stress on winding insulation. Alternatively, the machine could be designed to maintain the same ageing, but as the voltages are lower, it becomes possible to reduce the "grade" of the insulation.

d) Multi-source power supply

In Figure 2c), the DC source is not shown. It is possible to consider the use of a DC multi-source power supply (battery, fuel cell, supercapacitors, etc.) with differentiated uses. It would also possible to manage the energy storage of each DC source by the associated power electronics converter (BMS).

e) Additional degrees of freedom of control

The additional degrees of freedoms offered by the structure could allow for a dynamic optimization of the number of active sub-windings depending on the point of operation (management of low torque/high speed operating points, etc.). Inactive sub-windings could be used as sensors to perform real-time diagnostic. In addition, certain sub-windings could be dedicated to the perform specific tasks, such as active centering to limit the wear of the bearings (active magnetic bearing).

f) Low power per switch

The multiplication of low power switches does not necessarily reduce reliability. Indeed, the splitting of the power per switch is suitable for integration and large-scale industrial realization (CMS technology). This could result in better reliability through perfectly mastered processes.

g) Improved filling factor

Pushing the concept, each winding could be split into as many sub-windings as there are turns. In that case, each single-turn sub-winding (a11, a12 and a13 in Figure 3) could be connected directly to a printed circuit board (PCB). The placement of the single-turn coils can then be optimized, making it possible to improve the filling factor and/or the thermal design. To illustrate this point, we performed thermal simulations for two types of windings. Considering the requirements for an electric vehicle motorization [START_REF] Mcfarland | Performance and efficiency comparisons for interior PM and flux-switching PM machines with ferrite magnets for automotive traction applications[END_REF] using a flux switching synchronous machine with hybrid excitation [START_REF] Hoang | A new structure of a switching flux synchronous polyphased machine with hybrid excitation[END_REF], we carried out an initial electromagnetic sizing. We determined that the armature windings should have 5 turns. The simulations presented in Figure 3 were performed for an operating point (65 N•m, 5400 rpm) on the rated power curve (30 kW). For this operating point, the current density of the field winding is 12 A/mm 2 and the one of the armature winding is 8.6 A/mm 2 . On the left of the figure, the armature windings are modeled as homogeneous domains [START_REF] Idoughi | Thermal model with winding homogenization and FIT discretization for Stator Slot[END_REF]. On the right of the figure, the armature windings are split into 5 single-turns sub-windings. Each sub-winding is placed in the iron core and isolated from it by a 0.15 mm thick sheet of mica. The temperature distribution obtained by controlling the placement of the windings is more homogeneous.

h) Improved compactness

A "system-level design" can result in higher power densities. Indeed, as [START_REF] El-Refaie | High specific power electrical machines: a system perspective[END_REF] points out: "Focusing solely on the machine specific power can lead to a sub-optimal solution at the system-level".

i) Suitable for wide-bandgap semiconductors

Power converters using conventional semiconductors like silicon, are not suitable for the integration within an electrical machine. Indeed, the maximal operating temperature of a silicon module is much lower than the temperatures that can reign in the housing of an electric machine. It makes it difficult to mutualize the cooling systems of the power electronics and of the motor. But wide-bandgap semiconductors, such as Gallium Nitride (GaN) and Silicon Carbide (SiC), can tolerate much higher operating temperatures. It makes it possible to reconsider a deep integration of power converters within electrical machines [START_REF] Cizeron | Perspectives d'intégration pour la chaîne de conversion d'énergie des véhicules électrifiés[END_REF]. The power converter could then be installed directly on the back side of the PCB of Fig. 3. Other advantages related to the use of wide-bandgap semiconductors (low commutation losses and high frequency) would further improve the compactness of the concept (smaller passive components and thermal sinks) [START_REF] Gurpinar | Single-phase T-type inverter performance benchmark using Si IGBTs, SiC MOSFETs, and GaN HEMTs[END_REF].

Main drawbacks of the concept

In this section, we discuss in detail several points that could be listed as drawbacks of the concept.

a) Large number of sensors

The proposed solution relies on many sensors, with their number increasing with the number of subwindings. This can cause problems in terms of cost, real-time information processing and resilience. It will probably have to be managed by a high-performance on-board controller having many inputs/outputs, therefore complex to develop and expensive.

b) Difficulty of synchronization

One may suspect a sensitivity of the concept to the quality of the synchronization of the switching events for the sub-windings located on the same teeth. This is related to the low leakage inductance between them (since they are almost perfectly coupled magnetically). The difficulty of synchronizing all the converters and balancing the currents in the sub-windings may require different duty cycles for each inverter. Because of the strong magnetic coupling between the sub-windings, if the inverters are out of sync or apply different duty cycles to counteract the disparities in each sub-winding, current ripples limited only by the leakage inductance appear. A trade-off must be found between the unbalanced "low-frequency" currents and the increase in "high-frequency" current ripples. This compromise will guide the choice of the structure to be adopted for the control system. A first study [START_REF] Cizeron | Prediction of PWM-induced current ripple in subdivided stator windings using admittance analysis[END_REF][START_REF] Cizeron | Modélisation d'enroulements statoriques subdivisés pour l'évaluation d'une commande équilibrante[END_REF] provides a better understanding of these phenomena.

Field coil

Armature coil c) Rethinking the control From the previous paragraph, it will be understood that the splitting creates difficulties from the point of view of the control. This requires establishing a "good" dynamic model of the machine for its control. Such model will have to consider the splitting and the additional magnetic couplings that it implies. This will most likely lead to the design of a hierarchical control with at least 3 levels:

• A low-level control that will manage the synchronization of the command sent to the sub-windings.

• A "classic" vector control-type control level that will control the currents of the active sub-windings and the torque of the machine.

• A supervision control level that will have to decide in real time which elementary sub-windings are active, and which are inactive. An inactive sub-winding is not necessarily at rest, in fact it could serve as a sensor. Thus, the supervisor requires a precise monitoring of the machine in order to offer an optimized real time "dispatching" of the current references to the sub-windings, both in healthy and degraded mode [START_REF] Monmasson | FPGA-based controllers[END_REF]. The hierarchical structure of the command and the diversity of the tasks to be carried out seems to point out to the use of a SoC FPGA (System-on-Chip Field Programmable Gate Array). Indeed, such component incorporates both high-performance ARM processor cores (1 GHz) that are suitable for complex treatments (we think here of the supervisor) and a FPGA perfectly suited to the low-level control carried out in parallel [START_REF] Monmasson | FPGA-based controllers[END_REF].

d) Complex connections

The multiplication of the sub-windings leads to an increased number of connections. Non-standard solutions could be explored to reduce cost and increase reliability.

e) Electromagnetic compatibility

A high winding filling factor improves thermal exchanges but leads to an increase of the common mode parasitic capacity: the electromagnetic compatibility is therefore to be monitored.

f) Isolation of a defective cell

If a power converter or its associated sub-winding fails, the continuity of service is provided by the disconnection of the converter. This is based on the ability to guarantee that the switch will fail in open circuit and not in short circuit.

g) Consumption of copper

By design, the proposed traction chain is low voltage high current. Therefore, the consumption of copper, a critical resource, is expected to be important. Nevertheless, one can hope for an easy recycling of the windings.

h) Difficulty of recycling

An integrated system is generally considered more difficult to recycle at the end of its life.

Prototype

A prototype has been built by modifying a 3-phase synchronous machine (Fig. 4). Each armature winding has been divided into three sub-windings (9 sub-windings in total, 18 terminals). Three independent 3-phase 2-level MOSFET voltage source converters whose DC buses can be decoupled have been built to feed the machine. With the goal of later integrating the converter and the machine, all the semiconductors have been assembled on the same round shaped PCB. The system is sized for a 48 V DC bus voltage and a 10 A peak phase current. For comparison, the classic solution with a single 3phase inverter would require a 144 V DC bus voltage for the same current. In Fig. 5, we show the measured steady-state currents of the 9 sub-windings. The three 3-phase systems have electrical disparities. As a result, we observe a current imbalance between the different subwindings. But it does not affect the total ampere-turns in the machine. And it could be regulated by implementing a suitable control strategy.

Scientific and technical obstacles

The above discussion allows us to list the scientific and technical obstacles on which it is necessary to work in order to develop the proposed concept.

a) Modeling the traction chain

First, it is necessary to establish numerical models aiming at estimating the performance of the traction chain and at controlling it. Each component of the chain must be modeled with enough accuracy to obtain a representative model of the entire chain, but it needs to be fast enough for control design purpose.

b) Implementation of the hierarchical control

Second, we need to develop the control software that would ideally be deployed on an embedded target, to ensure the best performance (torque, speed, efficiency, noise, electromagnetic compatibility...) both in healthy and degraded mode.

c) Analysis of the degrees of freedom offered for auxiliary functions

Third, the number of degrees of freedom resulting from the splitting is to be explored. Each sub-winding is not necessarily powered, especially when the full power is not required. It then becomes possible to use these sub-windings for auxiliary functions (active magnetic bearing) or as sensors to perform realtime diagnostics.

d) Study of the impact of the design and quality of the connections and sensors

Fourth, as the number of connections and sensors increases with splitting, a reflection is to be carried out on the impact of their design and quality. A sensitivity study would allow us to know which sensors are essential for proper functioning, and which ones can be removed in order to get as close as possible to a "sensorless" control. In parallel, non-standard connections could be explored to reduce cost and increase reliability. e) Co-optimization of the converter-machine Finally, the proposed converter-machine should be co-optimized including the architecture, the sizing and the control. The optimal system should be compared with the equivalent conventional one. The ecodesign of the traction chain would be a complementary design objective.

Conclusions

This article introduces a concept for a traction chain with segmented power supply that is deemed promising for electric vehicles integrated traction chains. The concept could be applied to aeronautics applications too. After describing the proposed architecture in detail, we listed the main pros and cons. This allowed us to identify a set of scientific and technical obstacles. To overcome them, competences spanning the full spectra of electrical engineering will be required, stimulating sustained collaboration between electrical machine engineers and power electronics engineers.

Figure 1 -

 1 Figure 1 -Definitions of a) one turn, b) one coil, and c) one winding with its two terminals.

  Fig. 2: Example of splitting: from a conventional traction chain to a traction chain with segmented power supply.

Fig. 3 :

 3 Fig. 3: (left) Splitting the windings up to the turn and attaching them to a PCB. Adaptation of the Figure 6b of the patent [1]. (right) Example of better thermal management thanks to optimized armature conductor placement.

Figure 4 :

 4 Figure 4: (left) Prototype 3-phase machine with 9 sub-windings fed by three independent 3-phase converters. (right) Prototype round-shaped PCB with three independent 3-phase converters (Ø20 cm).

Figure 5 :

 5 Figure 5: Measured steady-state currents of the 9 sub-windings of the prototype.

Acknowledgements

The power converter was designed and built by SIREPE (www.sirepe.fr) thanks to the financing of the laboratories GeePs (www.lgep.supelec.fr) and SATIE (http://satie.ens-paris-saclay.fr). The support in "collective intelligence" was provided by MCP factory (www.mcpfactory.fr), thanks to the financing of ENS Paris-Saclay (https://ens-paris-saclay.fr).