domain: spi sdv

text:
Davide Viovy

S Ferraro

Descroix

M Serra

T Duc Mai

A L Serra

M.-C Nguyen

A Eisele

L Perié

J.-L Viovy

Davide Ferraro
email: davide.ferraro@unipd.it

S Descroix

Integrated droplet microfluidic device for magnetic particles handling: application

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

Introduction

Droplet microfluidics offers competitive advantages as compared to conventional methods or monophasic microfluidic systems; notably, the possibility to aliquot samples and reagents into independent compartments, preventing undesired cross-contaminations, evaporation and non-specific adsorption onto channel surfaces [START_REF] Teh | Droplet microfluidics[END_REF]. Droplet microfluidics is thus experiencing a tremendous growth, especially in the bioanalytical field [START_REF] Shang | Emerging Droplet Microfluidics[END_REF][START_REF] Jebrail | Digital microfluidics: a versatile tool for applications in chemistry, biology and medicine[END_REF], as attested by both the increasing number of existing labscale protocols implemented [START_REF] Kaminski | Droplet microfluidics for microbiology: techniques, applications and challenges[END_REF][START_REF] Mashaghi | Droplet microfluidics: A tool for biology, chemistry and nanotechnology[END_REF] and the completely new directions in diagnostic and research opened [START_REF] Shembekar | Droplet-based microfluidics in drug discovery, transcriptomics and high-throughput molecular genetics[END_REF][START_REF] Murphy | Recent advances in the use of microfluidic technologies for single cell analysis[END_REF]. These advances are mainly due to the wide set of functionalities available in the droplet toolbox [START_REF] Seemann | Droplet based microfluidics[END_REF], such as droplet generation, merging, splitting, sorting and cell encapsulation. However, a robust method allowing the extraction and purification of target analytes from a complex matrix, which represents a fundamental step in many biochemical protocols has still not been established. In conventional lab-scale methods, these steps are typically achieved by exploiting the interaction with a solid phase, as in chromatography [START_REF] Tetala | Bioaffinity chromatography on monolithic supports[END_REF] or immunofiltration [START_REF] Morais | Immunofiltration: A Methodology for Preconcentration and Determination of Organic Pollutants[END_REF]. An alternative and expanding approach relies on functionalized magnetic particles, which presents high versatility in terms of surface chemistry functionalization and improved capture rate guaranteed by their micro/nanometric sizes. In a typical workflow, magnetic particles are dispersed in the sample solution in order to bind the molecules of interest; they are then trapped by a magnet placed next to the liquid holder while the supernatant is removed and finally released in a different buffer (e.g., for washing, elution, etc.).

Although this is an extensively diffuse strategy, these protocols still require manual labour-intensive work, leading to operator dependence, and suffer from contamination risks and limitations regarding the manipulation of small volumes. Some commercial automated systems exist, but they require devoted expensive ?robotic equipment and large consumption of disposable products, thus remaining expensive to implement and operate.

To overcome some of these limitations, droplet microfluidic systems have been developed in the last decade for the manipulation of functionalized magnetic particles [START_REF] Serra | The power of solid supports in multiphase and droplet-based microfluidics: towards clinical applications[END_REF]. In detail, the extraction process of magnetic beads confined in a droplet mainly involves two forces: the magnetic force (Fmag) acting on the beads and the interfacial force (Fint) associated to the droplet interface. The former can be approximated as: Fmag ≈(Q/r)•M•∇B [START_REF] Serra | The power of solid supports in multiphase and droplet-based microfluidics: towards clinical applications[END_REF], where Q, r and M are the quantity, density, magnetization of the beads, respectively, and B is the magnetic field. Differently, the latter can be written as Fint ≈ γ•L, where γ is the interfacial tension between the two fluids and 𝐿 is a representative length of the system, such as the radius of the droplet meniscus [START_REF] Serra | The power of solid supports in multiphase and droplet-based microfluidics: towards clinical applications[END_REF]. Therefore, to achieve magnetic particles extraction, the condition Fmag>Fint must hold. Early approaches rely on stationary [START_REF] Shikida | Using wettability and interfacial tension to handle droplets of magnetic beads in a micro-chemical-analysis system[END_REF][START_REF] Cui | Isolating Influenza RNA from Clinical Samples Using Microfluidic Oil-Water Interfaces[END_REF][START_REF] Park | Droplet-based magnetic bead immunoassay using microchannel-connected multiwell plates (µCHAMPs) for the detection of amyloid beta oligomers[END_REF][START_REF] Zhang | A surface topography assisted droplet manipulation platform for biomarker detection and pathogen identification[END_REF] or electrowetting-based microfluidic architectures [START_REF] Fouillet | Digital microfluidic design and optimization of classic and new fluidic functions for lab on a chip systems[END_REF][START_REF] Sista | Development of a digital microfluidic platform for point of care testing[END_REF][START_REF] Ng | Digital microfluidic magnetic separation for particle-based immunoassays[END_REF]. However, they are limited by the complexity of the fabrication, the low throughput and the relatively high droplet volume (typically several µl) required for efficient droplet manipulation. More recently, other strategies involving droplets continuously flowing in a microchannel have been proposed. Unfortunately, at the microscale, the extraction condition cannot be easily accomplished and, in fact, the magnetic beads separation is typically mediated by a hydrodynamic splitting: a parent droplet containing magnetic beads flows towards a channel junction, where it breaks into two daughter droplets while an asymmetric magnetic field allows the enrichment of beads in one of the daughter droplets [START_REF] Lombardi | Droplet microfluidics with magnetic beads: a new tool to investigate drug-protein interactions[END_REF][START_REF] Lee | Droplet-based microfluidic washing module for magnetic particlebased assays[END_REF]. Despite this approach allows high throughput handling, it suffers from low purification rate due to the inefficient removal of the supernatant. Moreover, the redispersion of the extracted magnetic cluster in another droplet, fundamental step in many protocols requiring buffer exchange, cannot be easily integrated.

Consequently, they show important limitations in the implementation of multi-steps protocols, requiring for example sequential extraction and redispersion steps. We proposed an alternative droplet microfluidic approach more adapted to these needs, named the magnetic tweezers technology [START_REF] Ali-Cherif | Programmable Magnetic Tweezers and Droplet Microfluidic Device for High-Throughput Nanoliter Multi-Step Assays[END_REF].

This system allows an efficient magnetic particles exchange between confined droplets flowing in a tube through a high and local magnetic force generated by a ferromagnetic tip. Despite this technology guarantees the efficient implementation of multi-steps protocols along with high programmability [START_REF] Mai | Single-step immunoassays and microfluidic droplet operation: Towards a versatile approach for detection of amyloid-β peptide-based biomarkers of Alzheimer's disease[END_REF][START_REF] Ferraro | Microfluidic platform combining droplets and magnetic tweezers: application to HER2 expression in cancer diagnosis[END_REF][START_REF] Ferraro | Continuous chemical operations and modifications on magnetic γ-Fe 2 O 3 nanoparticles confined in nanoliter droplets for the assembly of fluorescent and magnetic SiO 2 @γ-Fe 2 O 3[END_REF][START_REF] Teste | Chromatin immunoprecipitation in microfluidic droplets: towards fast and cheap analyses[END_REF], it suffers from low throughput (about 0.5 Hz) and, due to the capillary format, it cannot be used in protocols involving the recovery of the supernatant solution while the beads are discarded. The latter represents a crucial functionality in different biochemical processes requiring a clean-up step, as for beads-based DNA size selection that represents a key step in sample (called libraries) preparation for Next Generation Sequencing (NGS). These workflows typically rely on solid-phase reversible immobilization (SPRI) method [START_REF] Paithankar | Precipitation of DNA by polyethylene glycol and ethanol[END_REF][START_REF] Lundin | Increased throughput by parallelization of library preparation for massive sequencing[END_REF], in which carboxyl-coated magnetic microparticles can reversibly bind to target DNA under specific concentrations of polyethylene glycol (PEG) and NaCl salt. These solutes reduce the affinity of DNA for the beads, thus increase the minimal size that a DNA fragment must have to undergo stable adsorption. Consequently, the size of the target DNA can be tuned by controlling the volume ratio of SPRI kit (a commercial buffer solution containing magnetic beads, PEG and salt) to DNA solution [START_REF] Borgström | Large Scale Library Generation for High Throughput Sequencing[END_REF]. Exploiting magnetic solid phase extraction either an upper (right-side) or a lower (left-side) DNA size limit can be defined by discarding the beads or the supernatant solution, respectively. In the latter case, the beads are conventionally transferred in washing and elution buffers for the DNA recovery.

In this work, we present an original continuous droplet microfluidic device for magnetic particles handling based on a pair of integrated microscale soft magnetic structures, placed in correspondence of a T-shape junction. Notably, due to the optimized magnetic properties, these structures allowed the deflection and trapping of the particles in a specific region of the secondary channel, defined as capture region (or trap), as shown in Fig. 1a,b. Furthermore, this strategy was coupled to a hydrodynamic-assisted droplet deformation (or fingering, see inset Fig. 1a): a tunable hydraulic pressure (P) was applied to the secondary channel. This applied pressure allowed to finely control the parental droplet penetration in the capture region (fingering) and, consequently, to minimize the carryover of the initial matrix in the particle-enriched daughter droplet resulting from splitting, stored in the capture region (inset, Fig. 1b). A supernatant droplet was simultaneously generated from this process, which was pushed away by the continuous oil flow (Fig. 1b). Importantly, the low residual of supernatant in the extracted particle cluster is due to the integration of the magnetic microstructures which, differently than the previously presented works, allows Fmag>Fint, thus driving the droplet splitting. The device is capable to perform high-throughput extraction, redispersion and depletion of magnetic particles, assuring high purification efficiencies in a broad range of working conditions (continuous phase flow-rates, liquids viscosity and particles loads). Therefore, considering its capabilities, the device has been exploited for DNA size selection devoted to NGS library preparation.

In detail, this validation was performed at two levels of complexity. Initially, we successfully demonstrated left, right and double-side size selection on a DNA ladder sample; then, its compatibility with NGS workflows was validated by integrating the in-droplet size selection process in a DNA library preparation protocol including downstream sequencing.

Materials and methods

Device microfabrication

The microfluidic device comprises two parts produced by soft-lithographic techniques: the top one integrates the magnetic components, obtained by "doctor blade" strategy [START_REF] Teste | Selective handling of droplets in a microfluidic device using magnetic rails[END_REF][START_REF] Sayah | Elastomer mask for powder blasting microfabrication[END_REF], while the bottom hosts the microfluidic channels (main channel cross-section: 300 µm x 200 µm, secondary channel cross-section: 200 µm x 200 µm) and a pocket where the soft magnets will be inserted. In detail, a brass mold was prepared by micromilling (Minitech Machinery). A CycloOlefinCopolymer (COC, 8007, Topas) negative replica was then produced by hot embossing (5 bar, 130 °C, 20 min) of the brass mold using a hydraulic press (Scamex). For the fabrication of the top part, the COC replica included two pocket shape structures which imposed the design of the magnetic components. These pockets were filled with a mixture of Polydimethylsiloxane prepolymer (PDMS, Sylgard 184, Dow Corning, 1:10 base to curing agent ratio) and ferromagnetic iron powder (1-6 µm, GoodFellow), using a 70% (w/w) doping ratio. After degassing, the excess of magnetic PDMS was removed using a soft silicon blade in order to keep the pocket region of the mold filled with the magnetic polymer.

Conventional pre-polymer PDMS was then poured on the master and the whole system was cured in an oven at 70 °C for 48 h. The resulting structure was then carefully peeled-off from the mold. It consisted in a PDMS slab with extruded micrometric magnetic structures. The bottom part of the device was fabricated in PDMS by standard replica molding of the COC intermediate template.

Following an oxygen plasma treatment of both surfaces, a film of ethanol was dispersed on the activated bottom surface to facilitate positioning and alignment of magnetic components in the dedicated pocket. The final device was then cured on a heating plate at 50 °C for 20 min to allow complete ethanol evaporation and to support the bonding of the surfaces. PTFE tubings (0.6 mm OD, 0.3 mm, Sigma Aldrich) were installed in the two inlets of the main channel and in the secondary channel outlet of the device. Finally, the internal channel surfaces were treated with a solution of (Tridecafluoro-1,1,2,2-tetrahydrooctyl)-trichlorosilane in FC-40 oil (5% v/v) for 20 min to improve oil wetting and droplet stability. Fig. 1a,b reports a scheme of the final device, which presents a T-shaped geometry: the main longitudinal channel (channel A) is dedicated to the flow-driven transport of confined droplets, while a secondary channel (channel B) is used either for particles extraction/release or depletion. Fig. 1c shows a picture of the final PDMS device.

Experimental setup

The microfluidic device was connected with a droplet generation module (see Fig. 1a), composed by a pipetting robot (Rotaxys, by Cetoni), coupled with two syringe pumps (Nemesys, by Cetoni), guaranteeing continuous droplet generation and analysis, as presented in [START_REF] Mai | Single-step immunoassays and microfluidic droplet operation: Towards a versatile approach for detection of amyloid-β peptide-based biomarkers of Alzheimer's disease[END_REF][START_REF] Ferraro | Microfluidic platform combining droplets and magnetic tweezers: application to HER2 expression in cancer diagnosis[END_REF][START_REF] Ferraro | Microfluidic valve with zero dead volume and negligible back-flow for droplets handling[END_REF]. The channel B was connected to a pressure controller (MFCS, by Fluigent) and a tunable pressure (P) was applied in order to minimize the carryover of the initial liquid in the particle-enriched daughter droplet, stored in the capture region.

The activation of the magnetic structures was performed through an external permanent NdFeB magnet (N52, size 20 mm x 20 mm x 30 mm, by ChenYang technologies), placed with its magnetization direction parallel to the channel A (Fig. 1a) and installed on a linear translation stage (DDSM50, Thorlabs). Since the iron powder used for the soft magnets fabrication was characterized by a negligible residual magnetization, the resulting structures could be activated/deactivated by varying their mutual distance D from the magnet (Fig. 1a). In the ON state, the permanent magnet was fixed at 3 mm from channel A, being in contact with the device side (D= 0 mm). Then, in the OFF state, D was increased up to 18 mm. For droplets observation, a Macro Objective (MLH-10x) mounted on a CMOS camera (acA800-510um, by Basler) and white LED back light illumination (Schott Lighting and Imaging) were used.

Device characterization

Fifty droplets pairs were generated in a continuous oil phase (FC-40 oil + 2% perfluoro-1-decanol) for device characterization. The droplets, separated by a 300 nL oil gap, contained a solution of magnetic particles (1 µm diameter, Dynabeads MyOne) and water, respectively. For the determination of the extraction and purification rates of the device, 100 nL droplets loaded with 1 µg magnetic particles were transported by constant oil flow-rates ranging from 0.2 µL/s to 1.6 µL/s. To evaluate the extraction rate as a function of the dispersed phase viscosity and particle load, water phase droplets, pure and in solution with glycerol at different concentrations (viscosity between 1 mPa•s and 23 mPa•s), were loaded with various magnetic particle amounts (from 0.05 µg to 1.5 µg) and transported at a constant flow-rate (0.25 µL/s). A total number of 20 droplets for each investigated combination of particles load and viscosity were used. Viscosities were measured by rheometric characterization (MCR 302, by Anton Paar).

Data analysis was performed by ImageJ software and LabVIEW (National Instrument) home-made scripts. The extraction and purification rates were estimated according to the following expressions:

𝐸𝑥𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝐸𝑓𝑓(%) = 2 345678539 :;<96=> 2 >763?5 96=> • 100 𝑃𝑢𝑟𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝐸𝑓𝑓(%) = (1 - 𝜑 • 𝑉 HIJKLMJHN OPQNKRS 𝑉 SLKHTJ NKRS ) • 100
where Q and V are the loaded particles and the droplet volume. In particular, sub-droplets volumes V extracted subdrop ? were estimated using the analytical expression described in [START_REF] Musterd | Calculating the volume of elongated bubbles and droplets in microchannels from a top view image[END_REF], requiring as an input:

i) the channel height and width; ii) the droplet length, evaluated by image processing of a top view frame after the extraction. Q values were also evaluated by image-processing, by defining a region of interest (ROI) delimiting the particle cluster and estimating the integrated pixel intensity within the ROI, as described in [START_REF] Brouzes | Rapid and continuous magnetic separation in droplet microfluidic devices[END_REF]. Furthermore, the extracted cluster of particles was considered to be in a closepacked configuration [START_REF] Onoda | Random loose packings of uniform spheres and the dilatancy onset[END_REF], characterized by a constant compactness 𝜑 = 0.55.

Nucleic acids size selection

The DNA size selection was performed using the commercial Ampure XP kit (by Beckman Coulter).

The in-tube size selection was performed following the protocol indicated by the supplier. For the indroplet implementation, at first, a DNA ladder sample (0.1 µg/µL concentration, by ThermoFisher), characterized by sizes ranging between 25 bp and 700 bp, was used to demonstrate the left and rightside size selection. For the left-side size selection, 15µL DNA sample and the SPRI solution are mixed with a ratio 1.0x in order to capture DNA fragments larger than 100 bp on the particles.

Alternately, using the same sample volume, an SPRI/DNA ratio of 0.75x was chosen to define a right cut-off limit of 300 bp. In both cases, an aliquot (1 µg) of inert magnetic particles (by Sigma Aldrich) was added to the solution. This strategy allowed to benefit from in-droplet specific features (automation, higher purification), while demonstrating no interference neither with the active surface nor with the final DNA size selection spectrum and yield, as verified in-tube (see Supplementary material, Fig. S2).

A train of 40 droplets (100nL) with uniform content was generated for the right-side size selection protocol. Differently, the left-side size selection required the production of 40 pairs of droplets, each pair containing the sample (100nL volume) and an elution buffer (10 mM Tris-HCl, 1 mM disodium EDTA, pH 8.0, 200 nL) respectively, separated by a 300 nL oil gap.

Furthermore, a double-side size selection workflow was implemented. In particular, following the operations described above, at first a right side-selection protocol was performed. Then, the processed DNA solution was collected and used as starting sample for the following left-side size selection step.

The purified DNA were characterized by BioAnalyzer (Agilent) and by Qubit fluorometer using the QuantiT dsDNA HS assay kit. All experiments were performed in triplicate.

A left-side size selection step on a pooled DNA samples library was also performed. Starting from a set of synthetic DNA fragments (50 bp size), a specific library preparation protocol for cell lineage studies was performed in-tube, as described in [START_REF] Perié | The Branching Point in Erythro-Myeloid Differentiation[END_REF]. The library preparation consists in the amplification of the desired fragment in the genomic DNA of murine cells with specific primers by a nested PCR and a subsequent PCR to add the Illumina adapters for sequencing. A left-side size selection process was performed in tube and in-droplet (according to the protocol described above), considering a SPRI kit/sample volume ratio of 0.9x and, following the quality control check, both samples were sequenced using an Illumina MiSeq platform. Bioinformatics post-processing was performed using MultiQC tool [START_REF] Ewels | MultiQC: summarize analysis results for multiple tools and samples in a single report[END_REF].

Results and discussion

In the following sections, we will present the numerical investigation devoted to the microfluidic device conception and its experimental characterization in terms of capture and purification efficiencies, investigating different conditions (flow-rate, applied pressure, viscosity, particle load).

Finally, we will present the results for the DNA size selection and a comparable study with a conventional lab-scale protocol.

Numerically-assisted device conception for magnetic particles handling in droplets

Systematic finite-element simulations have been used both for the device conception and the soft magnets design (size, shape, mutual distance) in order to optimize the intensity of the resulting local magnetic force (Fmag) acting on the particles contained in the droplet, and to define a specific traplike energetic landscape for their deflection and trapping.

Numerical simulations were performed using the Comsol Multiphysics software. A spherical surface, with a radius 10-fold the largest feature of the device geometry, identified the simulation domain defined as air, characterized by non-magnetic properties. The behavior of each magnetic element in the geometry was described by its B-H curve: i) the magnetization Mmag of the NdFeB permanent magnet is 1•10 6 A/m (nominal value provided by the supplier); ii) the soft magnetic microstructures were assumed to be in the unsaturated linear portion of the experimentally derived curves [START_REF] Li | Design and fabrication of microfluidic mixer from carbonyl iron-PDMS composite membrane[END_REF] (taking into account the investigated magnetic field range), thus described by the equation: B=µ0µrH, where µr and µ0 are the relative magnetic permeability of the material (µr=1.883) [START_REF] Keyes | Computational simulation of a magnetic microactuator for tissue engineering applications[END_REF] and of the free space, respectively; iii) the cluster of 1 µg magnetic particles is defined by a saturated magnetization Mparticles=1•10 4 A/m [START_REF] Grob | Magnetic susceptibility characterisation of superparamagnetic microspheres[END_REF].

At first, three specific geometries were investigated, having a triangular (a), square (b) or trapezoidallike (g) shape (Fig 2). For each geometry, the spatial distribution of the magnetic flux density (B) was evaluated near the structures (Fig. 2a). The results show that B can be maximized in the capture region with the square or trapezoidal structures. Notably, for all the investigated geometries, B is higher than 0.25 T in the main channel, leading the particles magnetization (Mparticles) at saturation and justifying the definition of a constant Mparticles in the model (see Material and methods), which guarantees the maximization of its contribution to the magnetic force. As shown in Fig. 2b, the resulting Fmag, plotted along the secondary channel axis, presents similar trends for the three designs: it is positive in the main channel and decreases in intensity approaching the capture region, reaching a zero value inside the trap; below this zone, a symmetrical behavior is observed. Therefore, the magnetic microstructures generate a Fmag converging to the center of the capture region, which deflects the particles towards this area (Fig. 2c). Moreover, the trapezoidal-like shape allows to achieve the highest maximum force in the main channel (about 10.6 µN, corresponding to a maximum gradient of 430 T/m), compared with the square (about 8.5 µN) and the triangular (about 4.6 µN) designs.

Consequently, the trapezoidal-like structures were chosen as the final shape for the implementation of the microfluidic device.

Furthermore, a specific energetic landscape was achieved, characterized by the presence of energy minima in the capture region, as shown in Fig. 2d. In detail, two absolute minima are located in proximity of the center of the soft magnets' walls, while a third relative minimum, corresponding to a saddle point, can be identified in the middle of the trap. These minima represent equilibrium points for the deflected particles, allowing their spatial confinement in the trap and the retention against the continuous oil drag force. Importantly, as it will be shown in the next section, these numerical results are also experimentally confirmed.

Finally, the influence of the gap between the soft magnetic structures was investigated, showing comparable Fmag conditions for gap below 200 µm, while a strong decrease in the generated magnetic force is observed for gap larger than 300 µm (see Supplementary material, Note 1). Following the device conception and numerical investigation, a specific microfabrication process based on replica-molding [START_REF] Rogers | Recent progress in soft lithography[END_REF] with PDMS was developed (See Materials and methods). This overcomes the limitation of microsolidics [START_REF] Siegel | Microsolidics: Fabrication of Three-Dimensional Metallic Microstructures in Poly(dimethylsiloxane)[END_REF] approaches, based on the injection of a magnetizable material in a prefabricated microchannel, in terms of: i) maximum concentration of doping microparticles in PDMS [START_REF] Zhou | Microfluidic separation of magnetic particles with soft magnetic microstructures[END_REF]; ii) minimum distance required between the integrated structures and the channel transporting the droplets (tens of microns) [START_REF] Mcdonald | Fabrication of microfluidic systems in poly(dimethylsiloxane)[END_REF]. Therefore, for the first time in a droplet microfluidic device, engineered soft magnetic structures, composed of iron microparticles-doped PDMS (70% w/w), were integrated in direct contact with the channel transporting the droplets.

Notably, this strategy allowed to maximize the magnetic force contribution in the system, without requiring any extra-layer deposition to achieve surface compatibility with biological samples [START_REF] Tseng | Magnetic nanoparticle-mediated massively parallel mechanical modulation of single-cell behavior[END_REF].

Extraction, redispersion and depletion processes: implementation and performances characterization

The integration of the soft magnets in correspondence of the T junction and in direct contact with the droplet allowed the implementation of different magnetic beads handling functionalities, which represent crucial operations for magnetic solid phase extraction workflows, as reported in Fig. 3 and in Supplementary material, Video 1. A confined droplet containing superparamagnetic particles is transported in the main channel. Under the effect of the external magnetic field, the particles sediment in the bottom rear of the droplet, forming a compact cluster (Fig. 3a) before reaching the capture region. Approaching the soft magnetic structures, the droplet dynamically experiences a fingering in the capture region (Fig. 3b) while, simultaneously, the cluster migrates by magnetophoresis towards the trap (Fig. 3c). As the rear meniscus of the droplet moves beyond the capture region, the bridge linking the main droplet body to the finger thins out while the finger itself retreats (Fig. 3d). In addition, during this process, the finger meniscus applies an interfacial force (Fint) on the cluster, which tends to be pushed out from the capture region, in opposition to Fmag. Therefore, since the latter is intense enough to keep the interface in a deformed state (Fmag>Fint), it prevents the complete finger retreat [START_REF] Ménétrier-Deremble | Droplet breakup in microfluidic junctions of arbitrary angles[END_REF] and allows the particles cluster extraction from the droplet (Fig. 3e). In this case, the cluster is surrounded by a layer of aqueous solution preventing the particles to be in contact with oil or with the microchannel and soft magnets walls. Due to the T-shaped device design, two different operations can be performed on the extracted particle cluster by switching off the integrated structures. First, setting the pressure P to zero, it can be evacuated through the secondary channel (Fig. 3f-g) while the supernatant droplet keeps flowing in the main channel, allowing the depletion of unwanted molecules. In a second configuration, the cluster can be released (Fig. 3h), upon merging [START_REF] Ferraro | Controlling the distance of highly confined droplets in a capillary by interfacial tension for merging on-demand[END_REF] (Fig. 3i), in a different droplet with a predefined content, allowing consequent redispersion of the molecules of interest in a different solution for further reactions (Fig. 3j). Importantly, the integration of soft magnets is crucial to achieve the magnetic extraction, as shown in Supplementary material, Video 2: the mere presence of the external magnet leads to particles sedimentation but neither particle deflection nor their trapping can be obtained. To validate the potential of these integrated functionalities for the implementation of complex workflows, the device performance was evaluated in different experimental conditions.

At first, the magnetic particle extraction and purification efficiencies were investigated for 100 nL droplets loaded with a constant Q=1 µg for flow rate (f) ranging between 0.2 µL/s and 1.6 µL/s (corresponding to average droplet speeds from 3.3 mm/s to 26.7 mm/s) and screening different applied P conditions. As shown in Fig. 4a, for a fixed flow-rate (f=0.8 µL/s), the purification rate (orange bars) increases with P until reaching a maximum at 35 kPa. Indeed, the pressure applied on the perpendicular channel counteracts the fingering; at low pressure, the generated finger retains a supernatant volume larger than the magnetic cluster volume, negatively affecting the purification rate.

In contrast, the extraction efficiency (green bars) is not affected by the increase in P below 35 kPa, but it strongly decreases above this value due to the shorter length of the generated finger. As a matter of fact, this prevents part of the particles to enter the capture region, being directly dragged away in the parent droplet. These experimental results allowed the definition of a linear relationship (see Fig. 4b) between the optimal pressures P to be applied (investigated range between 0 and 60 kPa) and the associated flow rates values, permitting to conveniently determine the optimal beads trapping conditions. Importantly, by using the optimal P values, spontaneous droplet break-up at the junction [START_REF] Ménétrier-Deremble | Droplet breakup in microfluidic junctions of arbitrary angles[END_REF] cannot occur, thus the particles extraction is exclusively triggered by the magnetic force, as shown in Video S3. This unique skill of the device guarantees low carryover of supernatant, thus excellent performances even in the higher range of f considered, guaranteeing an unprecedented throughput (about 10 droplets/s) for extraction and redispersion processes in a continuous droplet microfluidic device (Fig. 4c). The volume of the extracted cluster containing the magnetic particles in a closed-packed configuration is lower than 3.5 nL, providing a purification efficiency higher than 96.5% from the supernatant. Furthermore, the achieved capture efficiency is higher than 90% for all the flow-rates considered, and higher than 95% up to 1.2 µL/s. To further characterize the device in a wide range of operational conditions, the extraction performances were then investigated as a function of the bead quantity (Q) and of the droplet viscosity (h) for a fixed flow-rate condition. In the operating diagram reported in Fig. 5a, four regimes are observed, depending on the force-interplay in the system, involving: the magnetic force experienced by the particles (Fmag), the interfacial force (Fint) which prevents the droplet break-up, the viscous a) b) c)

force (Fvis) acting against the particles mobility in the droplet. For low dispersed phase viscosities (below dashed line) and low particle loads (Q<0.1 µg) conditions, Fmag is smaller than Fint and the particles form a cluster but no extraction occurs (empty green circles), the cluster being dragged away from the trap during the meniscus finger retreat process. Differently, in case of higher viscosities (above dashed line), the extraction performances strongly depend on Q: for low particle loads (Q≲0.25 µg, empty red triangles), Fmag is smaller than Fvis, preventing the initial step of cluster formation and consequently the particles extraction; while, for high loads (Q≳0.25 µg, full blue triangles), Fmag is slightly higher than Fvis, therefore the particles will occupy the lower longitudinal section of the droplet and only a portion of the particles is extracted. Finally, in most of the investigated cases (h≲18 mPa•s, Q≤1.5 µg) an extraction efficiency higher than 95% is achieved (full black circles) demonstrating that our device guarantees a large set of conditions in which the extraction process is successfully achieved with high extraction and purification efficiencies, notably all across the viscosity range of interest for common biofluids (like whole blood [START_REF] Dormandy | Measurement of Whole-blood Viscosity[END_REF], cerebrospinal fluid [START_REF] Yetkin | Cerebrospinal Fluid Viscosity: A Novel Diagnostic Measure for Acute Meningitis[END_REF], urine, bile [START_REF] Reinhart | Viscosity of human bile sampled from the common bile duct[END_REF], etc.), typically from 1 to 6 mPa•s. characterized by a particle retention efficiency higher than 95%. For low viscosities and magnetic particles load (empty green circles), the particles aggregate in a compact cluster but no extraction occurs because Fmag is overcome by Fint. The dashed curve separating the circles and triangles domains represents the viscosity threshold above which viscous effects appear in the system. Above this line, viscous forces prevent the formation of a compact cluster, in the case of low particles load in the droplet (empty red triangles) or lead to a partial cluster formation and incomplete extraction of the particles (full blue triangles). b) The position of the extracted cluster in the capture region varies for increasing particle loads, thus for increasing cluster volumes (i. Q=0.1 µg, ii. Q=0.5 µg, iii. Q=1.00 µg). Scale bar=200 µm.

In the following proof-of-concept application, i.e. SPRI-based in-droplet DNA size selection, processed solutions are typically characterized by load and viscosity of Q≃0.04-0.07 µg and h≃3-10 mPa•s, respectively (corresponding to SPRI kit to DNA sample volume ratios between 0.6x and 2.5x), thus situating in a boundary region of the extraction domain in which interfacial and viscous effects may influence the device functionalities. In order to guarantee consistent operations over the entire range of conditions, the dispersed phase was supplemented [START_REF] Gu | Multifunctional Picoliter Droplet Manipulation Platform and Its Application in Single Cell Analysis[END_REF] by 1 µg surface inert superparamagnetic particles (see Materials and methods).

Finally, the reliability of the developed numerical model was confirmed by the experimental observations. In particular, the particles magnetophoretic trajectories to reach the trap (Fig. 2c) matched the experimental results reported in Fig. 3c and supplementary Video 1, and the position of the trapped cluster as a function of Q (Fig. 5b) corresponded to the energetic landscape numerical expectations (Fig. 2d). Indeed, small clusters (Q≈0.1 µg) preferentially arrange in either of the two absolute energy minima (biased by the flow direction, Fig. 5b,i), while increasing the particle load, the cluster shape and size varied accordingly: for Q≈0.5 µg, it was transversally oriented in the trap (Fig. 5b,ii), showing an elongated shape bridging the two regions corresponding to the minima, while for Q≈1.00 µg it filled the entire capture region (Fig. 5b,iii).

Size selection on a DNA ladder sample

Taking advantage of the different functionalities and the excellent performances of this in-droplet approach, tailored DNA size selections were then demonstrated. Left-side, right-side and doubleside size selection protocols were implemented, investigating different SPRI kit to DNA sample volume ratios and considering, at first, a synthetic DNA ladder as model sample.

In the integrated right-side size selection workflow (upper limit at 300 bp), droplets of constant volume (100 nL), containing a SPRI/DNA sample ratio of 0.75x, were generated and continuously processed by the device, following the operations described in Fig. 6a: particles were deflected towards the capture region and trapped by the soft magnets, allowing the extraction of the undesired DNA fragments, while DNAs with target sizes remained in the parent droplet that continued flowing in the main channel and was collected for downstream analysis. Meanwhile, the soft magnets were deactivated and the extracted sub-droplet was pushed towards a waste outlet via the secondary channel. The integrated structures were then activated again for processing the next droplet. This result is further reinforced considering the recovery rate (Fig. 6b, inset), evaluated as the intensity ratio (percentage) of corresponding peaks before and after the selection step, as a function of the fragments size. A significant reduction in the concentration of longer fragments is reported, with no significant loss of target DNAs during microfluidic handling. Furthermore, compatible sizeselected DNA concentrations resulted from in-tube (9.45±0.44 ng/µL) and in-droplet (8.77±0.39 ng/µL) processes (see Supplementary material, Fig. S3). (Tris-EDTA). Importantly, the low carryover of supernatant, representing a key feature of our concept, led to an important workflow simplification: the removal of the ethanol washing step. In most of the library preparation protocols, this represents a necessary process to remove the carryover of supernatant that might disrupt downstream enzymatic reactions; however, it still remains a critical stage, being the main source of low reproducibility [START_REF]SPRIselect:DNA Ratios Affect the Size Range of Library Fragments-v2 Reagents[END_REF] and reduced elution yields.

In order to implement this multi-step protocol, sequential extraction and release processes were performed. In detail, at first, the magnetic particles binding the target DNA (larger than 100 bp) were magnetically extracted from the initial droplet, stored in the capture region (Fig. 7a,i right panel) and released in the elution buffer droplet (Fig. 7a,ii left panel) while the initial droplet (containing DNA shorter than 100 bp) was transported to the main outlet. After a short beads incubation time (about 60 s), allowing the elution of target DNA molecules from particles while the droplet kept flowing in the capillary, a second magnetic extraction was performed to separate the particles from the supernatant solution containing the targeted fragments (Fig. 7a,ii right panel). The magnetic particles were trapped in the capture region and pushed along the secondary channel, while the droplet containing the target fragments kept flowing towards the main outlet, where it was collected for downstream analysis. the proper enrichment of fragments longer than 300 bp 100 bp ??? (Fig. 7b), as expected from the SPRI/sample volume ratio considered. Furthermore, evaluating the recovery rate as a function of the DNA size (see Fig. 7b, inset), no loss of target DNA is reported. Nevertheless, a slight divergence with respect to the theoretical profile is observed around the cut-off value (as also reported in Fig. 6, inset for the right-side case). Importantly, this trend is not related to the microfluidic handling but it is due to the non-ideal physical-chemical processes underlying SPRI method (see Supporting Material Fig. S4). In-tube and in-droplet workflows yielded comparable size-selected DNA concentrations of 7.53±0.41 ng/µL and 7.07±0.48 ng/µL, respectively (see Supplementary material, Fig. S3).

Left-side selection on pooled DNA samples libraries

To further evaluate the potential of our approach and to demonstrate the compatibility of the adapted in-droplet protocol with conventional NGS pipelines, a complex sample was processed and downstream sequenced on the Illumina MiSeq platform. A blind comparison test between in-droplet and conventional in-tube protocol was performed using a solution of pooled libraries of target size 220 bp prepared from 50 bp synthetic DNA samples. Following the in-tube library preparation (see Methods), a final left-side selection step is performed according to the in-droplet workflow described above (SPRI/DNA sample ratio: 0.9x) in order to deplete primer dimers and PCR byproduct and enrich the target sample. In Fig. 9a, the electropherogram describing the initial sample size distribution (red dashed curve) is compared with those of purified solution in-tube (red) and in-droplet (blue) mode. These electropherograms demonstrate that the in-droplet process allows an effective size selection, properly enriching the target DNA sizes and strongly reducing the concentration of undesired DNA fragments with respect to in tube-processing (Fig. 9a, inset). Notably, although the target fragments concentration (calculated as the ratio between the target peak and the total spectrum area) represented only 11.5% of the initial sample, a single in-droplet size selection allowed no loss in the target DNA molecules. Furthermore, a slightly higher enrichment efficiency was reported for the in-droplet implementation (64.8%) with respect to conventional in-tube methods (56.3%).

Following sample sequencing and data post-processing, a blind comparison between the hit counts associated to each sample processed in-tube and in-droplet is reported in Fig. 9b, showing a linear trend with high correlation (Spearman correlation sr=0.989). This demonstrates that the microfluidic approach simplification, associated with the removal of the ethanol washing step, does not affect downstream processes and leads to base calls comparable with conventional methods. This was further reinforced by the comparison of the Phred factor, the typical index used to quantify the quality of the overall sequencing workflow (including sample preparation, base calling and bioinformatic post-processing), which showed comparable values for both approaches (between 32.92 and 38.32), corresponding to a probability of correct base call between 99.95% and 99.99% for both methods (see Fig. 9b, inset and Supplementary material, Fig. S5). between the conventional approach (red line) and the microfluidic processing (blue line).

b) a) 4 Conclusions and perspectives

We presented here an integrated approach allowing magnetic solid supports handling between droplets continuously flowing in a microchannel. Using an innovative pair of integrated soft magnetic structures, coupled with a pressure-controlled droplet fingering, we demonstrated the implementation of all the fundamental functionalities required for magnetic solid-phase handling protocols. As demonstrated by the numerical simulation, the presence of the integrated magnetic PDMS structures allows the generation of a Fmag higher than Fint and converging towards a designed trap region. Then, as experimentally demonstrated, since the extraction process is driven by the soft magnetic structures, high purification rate (>96.5%) can be achieved. Furthermore, reliable and efficient extraction of magnetic particles in a broad range of dispersed phase viscosities (between 1 and 15 mPa•s) and particle loads (between 0.1 µg and 1.5 µg) was proven. Importantly, we demonstrated, for the first time in a continuous droplet microfluidic architecture, the capability to: i) discard the supernatant solution while the extracted cluster is transferred to a different droplet of deterministic content; ii) discard and/or redirect the cluster of particles to a distinct section of the microfluidic network to allow further downstream processes. Importantly, these operations were possible at an unprecedented throughput: an improvement by more than one order of magnitude in processing velocity was achieved with respect to the state-of-the-art technology [START_REF] Ali-Cherif | Programmable Magnetic Tweezers and Droplet Microfluidic Device for High-Throughput Nanoliter Multi-Step Assays[END_REF] thanks to the developed microfabrication strategy, allowing to maximize the magnetic force by introducing soft magnetic structures in direct contact with the droplet transporting the particles.

Thanks to above device's unique functionalities, we implemented a critical step of DNA sample preparation for NGS, namely the DNA size sorting. In particular, different multi-step DNA size selection protocols were successfully performed in the continuous droplet device on both a model DNA sample as well as on an NGS library, showing significant improvement with respect to conventional approaches, especially in terms of throughput, automation and workflow simplification, while presenting final yields and size distributions comparable with conventional methods and demonstrating compatibility with downstream sequencing processes.

Finally, we envision that the implemented protocols can be integrated with merging and thermal amplification stages [START_REF] Zhang | A review on continuous-flow microfluidic PCR in droplets: Advances, challenges and future[END_REF] in order to develop a complete sample-in/libraries-out droplet microfluidic platform, suitable for different NGS libraries preparation workflows. In addition, the versatility of our device allows the implementation of magnetic solid support-based protocols using a wide-variety of surface chemistries and targeting various applications (sample preparation, immunoassays, etc.).

Figure 1 :

 1 Figure 1: (a) Experimental setup overview and integrated soft magnets concept. (b) The particles confined in the parent droplet experience a magnetic force that induces their deflection, extraction and trapping in the capture region (inset). Subsequently, they can be either released in a second flowing droplets or discarded through the secondary channel, according with the desired protocol. c) Picture of the microfluidic device integrating the micrometric magnetic structures. Red dye is used to underline the channel design.

Figure 2 :

 2 Figure 2: Numerical simulation-assisted soft magnets design. a) Spatial distribution of the magnetic flux density for the three tested designs of soft magnets: triangular (a), square (b) and trapezoidal (g), separated by a 200 µm gap (scale bar: 200 µm). (b) Magnetic forces resulting from the investigated structures evaluated along the symmetry axis of the soft magnets (x-axis). The reference frame's origin coincides with the middle of the trap (see inset). (c) Magnetic force vector field direction (scale bar: 200 µm) and (d) energetic landscape around the capture region obtained through the integration of soft magnet.

Figure 3 :

 3 Figure 3: Image sequence describing the in-droplet magnetic particles handling functionalities. Scale bar: 500 µm

Figure 4 :

 4 Figure 4: Characterization of the device performances as a function of the pressure applied to the secondary channel (P) and of the carrier oil flow-rate (f ) for a fixed dispersed phase viscosity (h =1 mPa•s) and particle load (Q=1 µg). a)Considering a constant f=0.8 µL/s, an optimal applied pressure (P=35 kPa) can be defined, which maximizes both the extraction and purification efficiencies of the device. b) An optimal P is associated to each f under investigation, characterized by a linear relation (red line). Error bars are not visible when smaller than the corresponding data point. c) Using the optimized P associated to each f investigated, extraction (green) and purification (orange) efficiencies result higher than 90% and 95%, respectively, between 0.2 µL/s and 1.6 µL/s.

Figure 5 :

 5 Figure 5: (a) Characterization of the device performances as a function of the magnetic particle load (Q) and of the dispersed phase viscosity (η) for a fixed f =0.25 µL/s and P=11 kPa. a) In the operating diagram, four regimes can be identified and related to the interplay between the magnetic force on the particles (Fmag), the interfacial force of the droplet (Fint) and the viscous force of the dispersed phase solution (Fvisc), as indicated in the right panel. The full black circles represent the condition for an extraction regime,

Fig. 6b compares

  Fig. 6b compares the DNA size distribution of the initial (red electropherogram) and processed (black dashed electropherogram) sample, demonstrating that the in-droplet workflow allowed the proper depletion of unwanted fragments and the efficient enrichment of target DNA shorter than 300 bp.

Figure 6 :

 6 Figure 6: a) In-droplet right-side size selection workflow. The undesired fragments, longer than 300 bp (in blue), are extracted from the original solution through the soft magnets while the target DNAs (in red) remain in the supernatant solution (right panel). By switching off the soft magnets, the captured cluster is discarded.b) Results for the in-droplet right-side size selection step on a DNA ladder sample. The values on top of the peaks correspond to the DNA fragments length (bp). The first and last peaks are the BioAnalyzer internal standards. In the inset, the recovery rate as a function of the fragment length is shown. A slight divergencebetween the theoretical curve (in green) and the obtained profile (in blue) is observed. Note that the blue curve does not represent a data interpolation but a simple tracing to guide the reader.

Figure 7 :

 7 Figure 7: In-droplet left-side size selection workflow. (a) Grey structures indicate an OFF state of the device, while the ON state is represented in black. The magnetic particles bind fragments larger than 100 bp (in orange), while shorter fragments (in grey) remain in solution. The soft magnets allow the extraction of the target DNA through the separation of the particles from the supernatant solution (i, right panel). The extracted cluster is then released, upon merging (ii, left panel), in an incoming droplet containing the elution buffer (green droplet), by switching off the soft magnets. b) Results of the in-droplet left-side size selection performed on a DNA ladder sample.

Finally, following the

  protocols described above, right and left-side selection were sequentially performed to define a tight DNA range between 100 bp and 300 bp. This complex workflow integrating both beads depletion and extraction/redispersion steps allowed to correctly select DNA fragments of interest (Fig.8, black electropherogram) from the initial DNA ladder sample (Fig.8, red a) b) dashed electropherogram), as confirmed by evaluating the recovery rate (Fig.8, inset). This process demonstrated also compatible concentration resulting from in-droplet (4.05 ± 0.53 ng/µL) and in-tube (3.43 ± 0.66 ng/µL) processing.

Figure 8 :

 8 Figure 8: Electropherogram describing the results of the double-size selection step performed in droplet on a DNA ladder sample. The dashed red curve represents the trace of the original sample, while the black trace is the results following the in-droplet processing. Et the inset ???

Figure 9 :

 9 Figure 9: a) In-droplet left-side size selection of a pooled library of DNA fragments. Black dashed line represents the starting sample size distribution while the red and blue curves correspond to the purified sample spectra obtained in-tube and in-droplet, respectively. The arrow indicates the peak corresponding to the target DNA size (220 bp). For each method, the recovery rate is plotted as a function of the sample size (inset). b) Blind comparison of hits number obtained from DNA sequencing following an in droplet or in-tube size selection. Each point represents a sample. The two methods present a high correlation coefficient. In the inset, the Phred quality score is plotted as a function of the position in the fragment, showing comparable results

Acknowledgements

The authors acknowledge Laurent Malaquin, Jérôme Champ and Dr. Patricia de Cremoux for fruitful discussions. This work was supported by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (reference: ANR10-NANO0207 and Labex and Equipex IPGG) and ERC CellO (FP7-IDEAS-ERC-321107). MS and AE were supported by a Curie Institute International PhD fellowship. High-throughput sequencing has been performed by the ICGex NGS platform of the Institut Curie supported by the grants ANR-10-EQPX-03 (Equipex) and ANR-10-INBS-09-08 (France Génomique Consortium) from the Agence Nationale de la Recherche ("Investissements d'Avenir" program), by the Canceropole Ile-de-France and by the SiRIC-Curie program -SiRIC Grant "INCa-DGOS-4654".


Ngac Duc Tan Vu

Ky Nguyen

Eric Semail
email: eric.semail@ensam.eu

Sensitivity of Torque Control for Seven-phase BLDC Machine with One Opened Phase under Constraints on Voltage and Current

Keywords: 7-phase BLDC machine, open-phase fault, faulttolerant control, parameter sensitivity, torque control, constraints on non-sinusoidal voltage and current, flux-weakening

This paper deals with control strategies when a sevenphase axial-flux brushless DC machine operates in one opencircuited phase fault by considering constraints on voltage and current. The constraints are related to the converter and machine design in terms of peak values of non-sinusoidal voltage and current. In addition, the sensitivity of the torque control on parameters of new imposed current references is analyzed under the base speed and in the flux-weakening region. The current references taking into account only first and third harmonics in healthy phases are proposed to ensure the torque optimization while phase currents and voltages are within their limits. The usefulness of the control strategies and the parameter analysis are verified by numerical results.

I. INTRODUCTION

Multiphase machines have been utilized in variable-speed applications for a half of century and a significant rise in this area has been recorded in recent years because of the developments in some specific uses [START_REF] Levi | Multiphase Electric Machines for Variable-Speed Applications[END_REF] and thanks to the advent of powerful processors. The design, modeling and control methods in healthy mode of operation are analyzed in [START_REF] Barrero | Recent Advances in the Design, Modeling, and Control of Multiphase Machines Part I[END_REF]. One main advantage of multiphase machines is to have better fault-tolerance from faults of the power converter in comparison with their 3-phase counterparts [START_REF] Huang | Multiphase machine theory and its applications[END_REF][START_REF] Toliyat | Analysis and simulation of five-phase variable-speed induction motor drives under asymmetrical connections[END_REF]. There is a variety of studies concerning opencircuited faults in the operation of multiphase machines. A constant torque can be obtained by a reconfiguration of control if the types of faults are detected and localized by a methodology as proposed in [START_REF] Trabelsi | Real-Time Switches Fault Diagnosis Based on Typical Operating Characteristics of Five-Phase Permanent-Magnetic Synchronous Machines[END_REF]. Many strategies have been proposed to determine optimal current references to maximize torque with a given copper loss level when one or more phases of the machines are open-circuited.

In [START_REF] Dwari | Fault-Tolerant Control of Five-Phase Permanent-Magnet Motors With Trapezoidal Back EMF[END_REF][START_REF] Mohammadpour | A Unified Fault-Tolerant Current Control Approach for Five-Phase PM Motors With Trapezoidal Back EMF Under Different Stator Winding Connections[END_REF][START_REF] Dwari | Open-circuit fault tolerant control of fivephase permanent magnet motors with third-harmonic back-EMF[END_REF], the fundamental and third harmonic current components for the excitation in healthy stator phases are introduced to control five-phase permanent-magnet motors with trapezoidal back-EMFs in open-phase faults. In these studies, there are eight unknown variables in current expressions that need to be determined (for example by Fmincon in MATLAB). By this way, machines with seven or more phases need more variables, which makes current solutions more complicated and less accurate because optimization algorithms are usually sensitive to the number of variables. In [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF], fault tolerant control strategies for a 7-phase machine using a vectorial multiphase description are verified with single and double open-circuit faults. However, these approaches have not considered the constraints on voltage and current of the drive system. These constraints play an important role in the sizing and cost of VSI as well as for the machine protection from thermal point of view. Besides, for wide speed-range applications such as electric vehicles or wind turbines, it is notable to take a flux-weakening strategy into account in the systems to meet the requirement of machine-converter voltage limits. In [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF], a variable speed control method including voltage and current limits is applied to a 5-phase permanent magnet synchronous generator but with sinusoidal back-EMFs only. In [START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF], the authors have proposed different control strategies for odd and even phase sinusoidal machines in one opened-phase faults. New current references for the degraded modes ensure the same magnetomotive force (MMF) as in healthy mode. However, these methods are only for sinusoidal back-EMF machines and the issue on maximum currents and voltages have not been considered. In [START_REF] Sun | Enhanced Optimal Torque Control of Fault-Tolerant PM Machine Under Flux-Weakening Operation[END_REF], an interesting approach is proposed, for 5-phase sinusoidal back-EMF machines, by defining a weighting factor which either optimizes copper losses in healthy operation or limits phase currents satisfying a required torque in case of one open-circuit faults, including the flux-weakening operation. However, the currents obtained by this approach contain high frequency components which limit the range of speed.

In this paper, based on [START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF], control strategies for one opened-phase fault is proposed to control the 7-phase brushless DC axial-flux machine with double identical rotors. The first and third harmonics of back-EMF are considered; therefore, the first and third harmonic currents are imposed to generate the torque. Simpler new current references are proposed with only three variables in the operation under base speed (with the control method of maximum torque per ampere (MTPA)) and four variables in the flux-weakening operation. Impacts of these variables are analyzed to optimize the mean torque amplitude and minimize the torque ripple while satisfying the constraints on peak values of non-sinusoidal current and voltage of the system, including flux-weakening region. The paper is organized as follows: Section 2 describes the model and parameters of the considered machine. The control strategy in a single open-circuit fault is demonstrated in Section 3. Section 4 analyzes the impacts of variations in the current parameters on the torque production.

II. MULTI-PHASE MACHINE MODELING AND PARAMETERS

The design of the 7-phase BLDC machine is proposed in [START_REF] Locment | Optimum use of DC bus by fitting the back-electromotive force of a 7-phase permanent magnet synchronous machine[END_REF][START_REF] Locment | Design and study of a multiphase axial-flux machine[END_REF] with axial flux and double identical rotors as shown in Fig. 1.

To identify the machine model, these following assumptions are considered: 7 phases are equally shifted and wye-connected; the machine has trapezoidal back-EMFs; the saturation of the magnetic circuits is not taken into account in the calculation of the back-EMFs and fluxes.

The voltage vector and torque can be written as follows:

𝑣 ⃗ = 𝑅𝑖 ⃗ + 𝛬 𝑑𝑖 ⃗ 𝑑𝑡 + 𝑒 ⃗ (1)
𝑇 𝑚 = 𝑒 ⃗.𝑖 ⃗ 𝛺 = 𝑒 ⃗ 𝑛 . 𝑖 ⃗ (2)
where 𝑣 ⃗, 𝑖 ⃗ and 𝑒 ⃗ are the 7-dimensional stator voltage, current and back-EMF vectors respectively; 𝑅 is the stator resistance of each phase; 𝛬 is a 7 by 7 stator inductance matrix; 𝑇 𝑚 is the torque and 𝛺 is the rotating speed of the machine; 𝑒 ⃗ 𝑛 = (𝑒 𝑎𝑛 , 𝑒 𝑏𝑛 , 𝑒 𝑐𝑛 , 𝑒 𝑑𝑛 , 𝑒 𝑒𝑛 , 𝑒 𝑓𝑛 , 𝑒 𝑔𝑛 ) is the speed-normalized back-EMF vector.

In fact, the speed-normalized back-EMFs of the machine contain many harmonics as expected by analyzing Fig. 2. The RMS values of the first and third harmonics account for the highest proportions while those of higher harmonics can be neglected [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF]. For simplicity of current and torque expressions, the back-EMFs are assumed to contain only the first and third harmonics as expressed in equation [START_REF] Huang | Multiphase machine theory and its applications[END_REF].

𝑒 𝑗𝑛 = 𝐸 𝑚1 𝑠𝑖𝑛 (𝜔𝑡 -(𝑗 -1) 2𝜋 7 ) + 𝐸 𝑚3 𝑠𝑖𝑛 (3 (𝜔𝑡 -(𝑗 -1) 2𝜋 7 )) (3)
where 𝑒 𝑗𝑛 is the speed-normalized back-EMF of phase j; 𝐸 𝑚1 , 𝐸 𝑚3 and 𝑘 𝑒 = 𝐸 𝑚3 /𝐸 𝑚1 are the amplitudes and the ratio of the first and third harmonics of the back-EMF respectively; 𝜔 = 𝑝𝛺 is the electric speed of the machine.

The electrical parameters of the 7-phase machine and converter for validating the efficiency of the control strategies and analyzing the sensitivity of current parameters are shown in Table 1. In this machine, the amplitude of the third harmonic back-EMFs equals approximately 19% of the first harmonic ones (𝑘 𝑒 = 𝐸 𝑚3 /𝐸 𝑚1 = 0.19). The converter is supplied by a DC bus voltage of 200 V and the allowed peak phase current is 7.5 A.

III. CONTROL METHOD IN DEGRADED MODE

Without loss of generality, it can be assumed that the opencircuited fault happens in phase a. The control strategy is adapted for other open-circuited phases. When one phase is opencircuited, it is necessary to keep continuous operations of the drive with acceptable performances. However, the imposed current references in healthy mode are often unsuitable for the faulty cases in some applications such as offshore wind energy systems due to mechanical resonance created by torque ripples; therefore, these currents need to be modified [START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF]. In [START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF], the authors have proposed new sinusoidal current references for healthy phases of sinusoidal back-EMF seven-phase machines in a single opened-phase fault. These sinusoidal currents are assumed to have the same amplitude which is, for the same torque, 23% higher than in normal operation. Nevertheless, the constraints on current and voltage and methods for trapezoidal back-EMF machines have not been considered.

In this paper, new non-sinusoidal current references for the non-sinusoidal back-EMF seven-phase machine are proposed. The first and third harmonic currents are considered to contribute to the torque production. The new current references in the natural frame are denoted by 𝑖 𝑏 ′ , 𝑖 𝑐 ′ , 𝑖 𝑑 ′ , 𝑖 𝑒 ′ , 𝑖 𝑓 ′ and 𝑖 𝑔 ′ and expressed by equations [START_REF] Toliyat | Analysis and simulation of five-phase variable-speed induction motor drives under asymmetrical connections[END_REF][START_REF] Trabelsi | Real-Time Switches Fault Diagnosis Based on Typical Operating Characteristics of Five-Phase Permanent-Magnetic Synchronous Machines[END_REF][START_REF] Dwari | Fault-Tolerant Control of Five-Phase Permanent-Magnet Motors With Trapezoidal Back EMF[END_REF][START_REF] Mohammadpour | A Unified Fault-Tolerant Current Control Approach for Five-Phase PM Motors With Trapezoidal Back EMF Under Different Stator Winding Connections[END_REF][START_REF] Dwari | Open-circuit fault tolerant control of fivephase permanent magnet motors with third-harmonic back-EMF[END_REF][START_REF] Locment | Vectorial Approach-Based Control of a Seven-Phase Axial Flux Machine Designed for Fault Operation[END_REF]. ))) (𝑠𝑖𝑛(𝜔𝑡 + 𝜑 𝑑 + 𝜑 1 ) + 𝑘 𝑖 𝑠𝑖𝑛(3(𝜔𝑡 + 𝜑 𝑑 + 𝜑 3 ))) } [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF] where 𝐼 𝑚1 and 𝐼 𝑚3 are the amplitudes of the first and third harmonic currents respectively; 𝜑 𝑏 , 𝜑 𝑐 and 𝜑 𝑑 are constant initial phase angles of phase b, c and d respectively, determined by the method in [START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF]; 𝜑 1 and 𝜑 3 are adjustable phase angles of the first and third harmonics of currents.

𝑖 𝑏 ′ = 𝐼 𝑚1 𝑠𝑖𝑛(𝜔𝑡 + 𝜑 𝑏 + 𝜑 1 ) + 𝐼 𝑚3 𝑠𝑖𝑛(3(𝜔𝑡 + 𝜑 𝑏 + 𝜑 3 )) (4) 𝑖 𝑐 ′ = 𝐼 𝑚1 𝑠𝑖𝑛(𝜔𝑡 + 𝜑 𝑐 + 𝜑 1 ) + 𝐼 𝑚3 𝑠𝑖𝑛(3(𝜔𝑡 + 𝜑 𝑐 + 𝜑 3 )) (5)
𝑖 𝑑 ′ = 𝐼 𝑚1 𝑠𝑖𝑛(𝜔𝑡+𝜑 𝑑 + 𝜑 1 ) + 𝐼 𝑚3 𝑠𝑖𝑛(3(𝜔𝑡+𝜑 𝑑 + 𝜑 3 )) (6)
𝑖 𝑒 ′ = -𝑖 𝑏 ′ ( 7
)
𝑖 𝑓 ′ = -𝑖 𝑐 ′ ( 8
)
𝑖 𝑔 ′ = -𝑖 𝑑 ′ (9)
The proposed current references with the same amplitude enable an equal distribution of copper losses in all healthy phases of the machine. Hence, the requirements of the machine design are significantly simplified. The above references also satisfy the assumption of the wye connection because the sum of all instantaneous currents is zero. Besides, the waveform of current in each phase with MTPA control is like the corresponding experimental back-EMF as shown in Fig. 2. This similarity allows the machine to obtain the maximum torque.

From equations (2-9), the generated torque could be expressed in equation [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF] in which 𝑘 1 = 2sin (3𝜋/7) and 𝑘 3 = 2𝑘 𝑒 sin (9𝜋/7) are constants determined from the machine parameters; 𝑘 𝑖 = 𝐼 𝑚3 /𝐼 𝑚1 is the ratio between the third and first harmonics of the currents. The torque-time and torque-speed characteristics are functions of 𝐼 𝑚1 , 𝑘 𝑖 , 𝜑 1 and 𝜑 3 . Adjustments of these parameters result in changes in the average torque and torque pulsation of the machine. This study investigates the impacts of these parameters on: the torque (average and ripple values), the speed range, the maximal values of current and voltage.

Additionally, the limits on voltage and current of the machine are shown in equations [START_REF] Jen-Ren | Disturbance-free operation of a multiphase current-regulated motor drive with an opened phase[END_REF][START_REF] Sun | Enhanced Optimal Torque Control of Fault-Tolerant PM Machine Under Flux-Weakening Operation[END_REF] and the absolute values of phase voltages are calculated as in equations [START_REF] Locment | Optimum use of DC bus by fitting the back-electromotive force of a 7-phase permanent magnet synchronous machine[END_REF][START_REF] Locment | Design and study of a multiphase axial-flux machine[END_REF][START_REF] Vu | Torque optimization of seven-phase BLDC machines in normal and degraded modes with constraints on current and voltage[END_REF](16)(17)(18).

[
where 𝐼 𝑚𝑎𝑥 and 𝑉 𝑚𝑎𝑥 are current and voltage limits according to the converter and machine parameters; L is the self-inductance of each phase; 𝑀 1 , 𝑀 2 , and 𝑀 3 are mutual inductances between one phase and its adjacent and non-adjacent phases of the machine.

In the paper, neglecting the impact of a homopolar voltage component, the value of 𝑉 𝑚𝑎𝑥 is equal to a half of the DC bus voltage 𝑉 𝐷𝐶 in the Pulse Width Modulation technique. The value of 𝐼 𝑚𝑎𝑥 can be determined by the peak value of instantaneous currents related to VSI components or the maximum thermal current depending on temperature of windings. According to the machine parameters in Table 1, the verification of the control strategies is carried out with 𝑉 𝑚𝑎𝑥 = 100 𝑉 and 𝐼 𝑚𝑎𝑥 = 7.5 𝐴. This value of 𝐼 𝑚𝑎𝑥 is only for a short-term operation and will decrease in a long-term use according to the thermal constraint of the machine. Operations under and above the base speed (37 rad/s) are analyzed in next sections. Fmincon function in MATLAB is an optimizer that can estimate the parameters of the current references to maximize the torque and respect the constraints on peak values of non-sinusoidal current and voltage.

IV. ANALYSIS OF EFFECTS OF CURRENT PARAMETERS

Operation below the base speed Under the base speed (for example at 20 rad/s), the phase currents are controlled to be maximal (corresponding to a maximum torque) while the phase voltages are under their limit. The optimal values of the amplitude 𝐼 𝑚1 , ratio 𝑘 𝑖 and initial phasors (𝜑 1 , 𝜑 3 ) of the current references are determined by Fmincon and remain unchanged during this operation (𝐼 𝑚1 = 8.6 𝐴, 𝑘 𝑖 = 0.19, 𝜑 1 = 𝜑 3 = 0 𝑟𝑎𝑑). An analysis using the torque expression [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF] to investigate the impacts of the current parameters on the machine torque control will be conducted in this section. For simplicity, only the initial phase angle of the third harmonics 𝜑 3 is considered and denoted by 𝜑 while that of the first harmonics is nullified (𝜑 1 = 0). In addition, when one of the above parameters of the current references is being considered, the others are constant and equal to their optimal values. Variations of each current parameter (𝐼 𝑚1 , 𝜑 or 𝑘 𝑖 ) within its range and consequent changes in the phase currents and the torque generation at 20 rad/s are derived as follows:

1) Effects of the first harmonic current amplitude (𝐼 𝑚1 ) In the current equations (4-9), 𝐼 𝑚1 is proportional to the peak value of phase currents of the machine. To guarantee the constraint on current, the value of 𝐼 𝑚1 must be less than or equal to the optimal value of 8.6 A which is determined by Fmincon. Besides, the relationships between the average torque, torque ripple and 𝐼 𝑚1 are easily analyzed by using the torque equation [START_REF] Fall | Variable speed control of a 5-phase permanent magnet synchronous generator including voltage and current limits in healthy and open-circuited modes[END_REF]. When 𝐼 𝑚1 increases from 0 to 10 A, the average torque also linearly rises from 0 to about 70 Nm while the torque ripple value does not change and remains equal to 7.2%. It means that 𝐼 𝑚1 has no effect on the torque ripple but it is proportional to the average torque.

2) Effects of the current phase (𝜑 3 = 𝜑; 𝜑 1 = 0) In Fig. 3, when 𝜑 varies within the interval [-𝜋/2; 𝜋/2], enabling sine functions of phase angles in equation ( 10) to take on their full range of values [-1; 1], the peak value of the nonsinusoidal healthy phase currents varies between 7.5 A and 10.2 A. The lowest peak current is 7.5 A when 𝜑 is zero, which means that the constraint on current is satisfied only at that point. Furthermore, the waveform of the currents is also modified. Fig. 4 shows that there are major changes in the torque ripple (from 7% to 63%) but minor variations in the average torque in terms of 𝜑. The optimal value of 𝜑 is zero because at this point, the average torque is maximum at 59.6 Nm while the torque pulsation is minimum at 7.2%.

3) Effects of the current ratio (𝑘 𝑖 )

The range of the current ratio 𝑘 𝑖 = 𝐼 𝑚3 /𝐼 𝑚1 is supposed to increase from 0 to 1. The variation of 𝑘 𝑖 affects the peak value of phase currents as shown in Fig. 5. The value of 𝑘 𝑖 must be equal to 𝑘 𝑒 = 𝐸 𝑚3 /𝐸 𝑚1 = 0.19 so that the peak value of all phase currents is equal to its limit 𝐼 𝑚𝑎𝑥 = 7.5 𝐴 and the control method MTPA is implemented. In addition, Fig. 6 shows that the torque ripple dramatically increases when 𝑘 𝑖 varies far away from the value of 𝑘 𝑒 . The torque average undergoes a moderate increase when 𝑘 𝑖 gains from 0 to 1.

4) Summary of the functioning under the base speed

From the analyses above, the average torque is sensitive to 𝐼 𝑚1 while the torque ripple is significantly affected by 𝜑 and 𝑘 i .

It can be seen that 𝑘 i = 𝑘 e = 0.19, 𝐼 𝑚1 = 8.6 𝐴 and 𝜑 = 0 𝑟𝑎𝑑 are the optimal solution of the proposed current references. At these values, the highest average torque equals 59.6 Nm (approximately 80% of the heathy torque as shown in Fig. 11) with a small torque ripple (7.2%) in one opened-phase fault. The flux-weakening operation When the speed exceeds the base speed (37 rad/s), the machine turns into the flux-weakening operation. To keep the phase voltages within the constraint value (Fig. 7) and obtain a maximal average torque, the absolute value of the d-axis currents must increase while that of the q-axis currents need to decrease, keeping the peak phase currents equal to 𝐼 𝑚𝑎𝑥 (Fig. 8). As a result, in natural frame, the first harmonic amplitude 𝐼 𝑚1 , current ratio 𝑘 𝑖 and initial phases 𝜑 1 and 𝜑 3 of phase currents need to be adapted accordingly at every speed over the base speed. In general, each of the current parameters can be a constant or variable. Because of 4 parameters keeping constant or changing, there are 16 possibilities for the variations of phase currents. However, it is necessary to have at least one adjustable variable among 4 parameters, hence 15 options are available to vary the current parameters to allow the machine to operate in the fluxweakening region. In these options, only cases with variations of 𝜑 1 and 𝜑 3 (for example, strategy 1 as defined in Table 2) can make the peak current value of the healthy phases equal to Imax in the flux-weakening operation. In the previous section, it has been shown that parameters φ and k 𝑖 affect seriously the peak phase current and the torque ripple. Therefore, strategy 1 produces unacceptable torque ripple increasing from 7.2% under base speed to 60% at 50 rad/s due to adjustments of 𝜑 1 and 𝜑 3 (Fig. 9). Therefore, because 𝐼 𝑚1 has no effect on the torque ripple, a simple solution in which only 𝐼 𝑚1 is varied to respect the voltage constraint and minimize the torque ripple is proposed and named strategy 2 and defined in Table 2. With the strategy 2, the peak value of healthy phase currents changes from Imax=7.5A to lower values in the flux-weakening region while their waveforms are identical, for example, the peak value decreases from 7.5 A at 20 rad/s to 2 A at 45 rad/s in Fig. 10. In Fig. 11, the average torques of strategies 1 and 2 are the same under the base speed and equal to 80% of the value in healthy mode. In the flux-weakening region, the torque of the strategy 2 is slightly lower than that of the strategy 1. Nevertheless, in Fig. 12, the torque ripple of strategy 2 at all speeds is tiny and unchanged at 7.2% much smaller than that of strategy 1. These results can be compared with a classical strategy producing no torque ripple: in Figs. 11 and12, the torque ripple is completely eliminated but the average torque is equal to only 58% of the healthy mode as discussed in [START_REF] Vu | Torque optimization of seven-phase BLDC machines in normal and degraded modes with constraints on current and voltage[END_REF]. In this strategy, current references in d-q frames are imposed as constant values. V. CONCLUSIONS For a 7-phase BLDC machine with non-sinusoidal back-EMFs, the paper has analyzed control strategies imposing two harmonics of phase currents to generate torque and guarantee the constraints on peak values of non-sinusoidal voltage and current. When the torque ripple is not imposed to zero, the proposed current references enable the machine to produce a torque in oneopened phase fault equal to 80% of the heathy torque with a low torque ripple of 7%. On the contrary, if the torque ripple is totally nullified, the average torque probably drops to 58% of the healthy mode. In addition, effects of parameters of the current references on torque control have been analyzed in the operation under the base speed and in the flux-weakening region. The system is simulated by MATLAB to verify the usefulness of the control strategies. In future, the methods will be implemented to the experimental system.

Fig. 1 .

 1 Fig. 1. The 7-phase axial-flux double rotor BLDC machine.

  amplitude of 1 st harmonic back-EMF 𝐸 𝑚1 (V/rad/s) 2.38 Speed-normalized amplitude of 3 rd harmonic back-EMF 𝐸 𝑚3 (V/rad/s) 0.45 DC bus voltage 𝑉 𝐷𝐶 (V) 200 Maximum peak current (A) 7.5

Fig. 2 .

 2 Fig. 2. Current reference with MTPA control and experimental back-EMF in phase e of the machine at 26 rad/s.

Fig. 3 .Fig. 4 .Fig. 5 . 6 Fig. 6 .

 34566 Fig. 3. Effect of 𝜑 = 𝜑 3 on the peak value of healthy phase currents when phase is open-circuited and phase current waveforms at 𝜑 = 0 rad and 𝜑 = 1 rad.

Fig. 7 .

 7 Fig. 7. Voltage-speed characteristics and instantaneous phase voltage at 45 rad/s over the base speed (37 rad/s) when phase a is open-circuited with strategy 1.

Fig. 8 .

 8 Fig. 8. The peak values of healthy phase currents are unchanged (7.5 A) at speeds over the base speed (37 rad/s) when phase a is open-circuited with strategy 1.

Fig. 9 .ϕ 1 ϕ 3 ϕFig. 10 .

 9310 Fig. 9. Initial phases (φ 1 ,φ 3 ) and torque ripple when phase a is open-circuited with strategy 1.

Fig. 11 .

 11 Fig. 11. Average torques in healthy mode and when phase a is open-circuited with different strategies.

Fig. 12 .

 12 Fig. 12. Torque ripples in healthy mode and when phase a is open-circuited with different strategies.

TABLE II .

 II TWO STRATEGIES IN FLUX-WEAKENING OPERATION

	Strategy	𝑰 𝒎𝟏 (A)	𝒌 𝐢	𝝋 𝟏 (rad)	𝝋 𝟑 (rad)
	1	variable	variable	variable	variable
	2	variable	0.19	0	0