Daniel Roger
email: daniel.roger@univ-artois.fr

Krzysztof Komeza
email: krzysztof.komeza@p.lodz.pl

Piotr Napieralski

Solid-state transformers of smart high-power battery charger for electric vehicles

Keywords: Solid-State Transformer (SST), medium frequency transformer, grain oriented electrical steel (GOES), wound core

There is no doubt that society will not turn back: most thermal vehicles will switch to electric ones, not only for cars but also for bigger vehicles. Fast chargers will become more and more essential parts of the global system, they will provide a charge control of batteries and galvanic insulation from the grid. At the present times, compact realizations are based on Solid-State Transformers (SSTs) operating at high frequencies with their well-known advantages and disadvantages, including power limits and reasonably high costs. The authors propose a low-cost structure using Grain Oriented Electrical Steel (GOES) wound cores and aluminum foil windings to achieve this goal. GOES has losses that increase with frequency but decrease with temperature; therefore, the authors propose a transformer structure allowing a hot core inside coils at usual temperatures. This idea is applied to a 400-kW compact, low-cost battery charger connected to a DC smart grid.

I. INTRODUCTION

Nowadays, reducing greenhouse gas emissions is a major challenge for limiting the planet's global warming. Electricity will play a more and more critical role. However, electricity is a secondary source; its production system must include much more renewable sources, which are intermittent for most of them. This intermittency may cause grid stability problems, becoming a major issue [START_REF] Dudurych | The Impact of Renewables on Operational Security: Operating Power Systems That Have Extremely High Penetrations of Nonsynchronous Renewable Sources[END_REF]. The electricity enduser can participate in this stability; incentives based on tariffs modulated according to demand are beginning to exist [START_REF] Ostergaard | Energy Security Through Demand-Side Flexibility: The Case of Denmark[END_REF].

It is estimated that there are currently over a billion classic cars in the world that must be replaced by electric ones. A similar change begins with medium-and heavy-duty vehicles [START_REF] Smith | Medium-and heavy-duty vehicle electrification: An assessment of technology and knowledge gaps[END_REF]. Many of these vehicles will operate on regular schedules (buses, garbage trucks, delivery vans, etc.) will be in their garages and connected to the grid when they will not be used. Thanks to advances in battery lifetimes [START_REF] Fichtner | Recent Research and Progress in Batteries for Electric Vehicles[END_REF][START_REF] Hu | Battery Lifetime Prognostics[END_REF], their big batteries will participate in the grid stability if the connection is made by smart chargers guaranteeing a full charge at the vehicle starting time. However, this contribution to the limitation of greenhouse gas emissions will demand many high-power bidirectional smart chargers designed with low environmental footprints and reduced construction costs. Technically, this function can be performed by high-power vehicle chargers. Such devices are compacts when the power circuits are Solid-State Transformers (SSTs), which provide bidirectional power transfers and voltage adaptations with the galvanic insulation compulsory for safety [START_REF] Leibl | Design and experimental analysis of a medium-frequency transformer for solid-state transformer applications[END_REF][START_REF] Liu | Compound Control System of Hybrid Distribution Transformer[END_REF][START_REF] Saleh | Solid-State Transformers for Distribution Systems-Part II: Deployment Challenges[END_REF][START_REF] Bahmani | Design Methodology and Optimization of a Medium-Frequency Transformer for High-Power DC-DC Applications[END_REF]. Currently available SSTs operate at high frequencies with transformers based on ferrite, nanocrystalline or amorphous magnetic cores [START_REF] Rylko | Magnetic material selection for high power high frequency inductors in dc-dc converters[END_REF]. Ferrites can operate at very high frequencies, over 20kHz, but the power is limited by the switching losses [START_REF] Nakmahachalasint | A frequencydependent hysteresis model for power ferrites up to curie temperatures[END_REF]. High powers can be reached using several elementary SST cells connected together. Amorphous and nanocrystalline materials can operate up to 1.2 T at lower frequencies with higher powers per SST cells [START_REF] Azuma | Core Loss in Toroidal Cores Based on Fe-Based Amorphous Metglas 2605HB1 Alloy[END_REF][START_REF] Kurita | 500 kVA Medium-frequency core-type amorphous transformers with alternately wound sheet winding for offshore DC grid[END_REF]. These materials can work up to 120°C and, if the maximum temperature is exceeded during an accidental overload, the core can permanently have lower performances [START_REF] Datta | Accelerated aging behavior of an iron base amorphous alloy for 60 Hz application[END_REF][START_REF] Mouhamad | Physicochemical and Accelerated Aging Tests of Metglas 2605SA1 and Metglas 2605HB1 Amorphous Ribbons for Power Applications[END_REF]. Grain Oriented Electrical Steel (GOES) is another option at a competitive cost because nanocrystalline material is several dozen times more expensive than GOES [START_REF] Huang | Optimal Design and Implementation of High-Voltage High-Power Silicon Steel Core Medium-Frequency Transformer[END_REF][START_REF] Kauder | Performance Factor Comparison of Nanocrystalline, Amorphous, and Crystalline Soft Magnetic Materials for Medium-Frequency Applications[END_REF][START_REF] Tumanski | Handbook of Magnetic Measurements[END_REF]. However, with GOES, the increase in core losses with frequency limits its use in a transformer made with a standard design.

When SSTs operate at high frequencies, coils must have a specific design for weakening skin and proximity effects in conductors that increase the AC resistances [START_REF] Baek | Design considerations of high voltage and high frequency transformer for solid state transformer application[END_REF]. The switching losses in power electronics, proportional to the frequency, are also relatively high.

The paper proposes an alternative solution based on transformers operating at medium frequencies (several kHz) and designed with GOES wound cores [START_REF] Hamdinou | Potentiel des aciers électriques à grains orientés pour la réalisation de transformateurs compacts à forte densité de puissance[END_REF][START_REF] Ichou | Conception de Solid-State Transformers (SST) de fortes puissances construits autour de transformateurs dotés de noyaux entoulés en acier électeique à grains orientés[END_REF]. Such cores can operate at high peak flux densities (1.8 T), with rectangular voltages corresponding to simple converter topologies and low switching losses [START_REF] Ichou | Assessments of High-Power Solid State Transformers based on Grain-Oriented magnetic cores[END_REF][START_REF] Hatakeyama | Core Loss Estimation of Various Materials Magnetized With the Symmetrical/Asymmetrical Rectangular Voltage[END_REF]. The GOES has a very stable crystalline structure and a high Curie temperature (747°C). This material has been available for a long time, and it now has a high technology readiness level (TRL9). Moreover, the losses decrease when the core temperature increases [START_REF] Chen | Influence of temperature on magnetic properties of silicon steel lamination[END_REF], because of the natural increase in the material resistivity, like every metal. The thin inorganic coating layers on both faces of the GOES strips appear during the metallurgic process; they withstand high temperatures. Therefore, a GOES wound core can operate up to very high temperatures (500°C) without significantly decreasing its magnetizing capability [START_REF] Ichou | Assessment of a grain oriented wound core transformer for solid state converter[END_REF][START_REF] Ababsa | High-Temperature Magnetic Characterization Using an Adapted Epstein Frame[END_REF].

The new concept involves designing a medium-frequency transformer with a hot core inside windings at standard temperatures. With this concept, it is possible to reduce core losses and transformer sizes significantly. Of course, this concept requires a slightly more complicated construction ensuring thermal insulation between the core and the winding with a simple air-cooling system.

First, the paper presents the charger structure chosen among existing SST ones. Next, the hot core transformer is presented. An experimental approach shows that the highfrequency core losses decrease with temperature. Then a lowcost foil winding design is considered. The analyses are based on Dowel's analytical model, a FEM simulation, and an experimental investigation. The final part presents the preliminary design of a high-power charger.

The first version of the paper was presented at the EEEIC2021 conference in Bari, Italy in September 2021 [START_REF] Roger | Solid-state transformers of smart high-power battery charger for electric vehicles[END_REF].

II. SMART CHARGER STRUCTURE

Several SST structures can be used for building smart chargers with high power densities.

Generally speaking, SSTs are often modular to split the power into several SST cells for high-power applications. Specific control systems guarantee the good repartition of the global energy flow between the cells.

Transformers made with ferrites cores can operate at very high frequencies but at flux densities under 0.4 T. The ferrite relative permeabilities are in the range of 2000, limiting the average field line length and the area of windows available to windings. High frequencies make it possible to design transformers and other passive components of small sizes. Still, the switching losses in power electronic active components limit the rated power of SSTs elementary cells.

Amorphous and nanocrystalline materials can operate up to more or less 1.2 T but at lower frequencies than ferrites [START_REF] Kurita | 500 kVA Medium-frequency core-type amorphous transformers with alternately wound sheet winding for offshore DC grid[END_REF], [START_REF] Huang | Optimal Design and Implementation of High-Voltage High-Power Silicon Steel Core Medium-Frequency Transformer[END_REF]. Their permeabilities are higher, which allows to design cores with larger windows for the coils. Like ferrites, these materials can operate permanently up to 120-140°C [START_REF] Azuma | Core Loss in Toroidal Cores Based on Fe-Based Amorphous Metglas 2605HB1 Alloy[END_REF]. Amorphous and nanocrystalline materials' relatively low Curie temperature explains their maximum operating temperature.

For both technologies, the maximum core temperatures are lower than the standard insulation classes of the coils, which are 155°C (class F), 180°C (class H), or 200°C (class C) [START_REF]IEEE Standard Test Procedure for Thermal Evaluation of Systems of Insulating Materials for Random-Wound AC Electric Machinery[END_REF]. Therefore, the maximum temperature of cores limits the performance of the windings. To reduce the cost and the complexity, the proposed structure is made of a single highpower SST cell made with a GOES wound core transformer operating in the kHz range.

Generally speaking, the apparent power S of a transformer depends on two main parameters of the core shape: the area for windings 𝐴 ! and that for the magnetic flux 𝐴 " . The four additional major parameters are the RMS current density in coils 𝐽 #$% , the maximum flux density in the core 𝐵 & , the frequency f, and the filling factor 𝐾 ' , which depends on the actual conducting section in the window area. A well-known handbook on transformer and inductor design [START_REF] Colonel | Transformer and Inductor Design[END_REF] applies the Ampere and Faraday laws with the previously defined parameters, to obtain (1):

𝑆 = 4.44 𝐾 ' 𝐴 ! 𝐴 " 𝐽 #$% 𝐵 & 𝑓 (1)
The author specifies that, for identical current density, induction, and frequency, the transformer volume is proportional to [𝐴 ! 𝐴 " ] ) * ⁄ . Changing the coefficient 4.44, this expression can be applied to rectangular voltages of any duty cycles [START_REF] Hamdinou | Potentiel des aciers électriques à grains orientés pour la réalisation de transformateurs compacts à forte densité de puissance[END_REF]. For the same apparent power S, the transformer size (𝐴 ! 𝐴 " ) is smaller the greater the product 𝐵 & 𝑓 is.

For such applications of GOES at frequencies higher than for their normal use, the core losses will be higher. Therefore, the core will be hotter, and the transformer design must be made so as to accept high core temperatures with coils at standard ones.

Moreover, it was proved in a recent experimental work [START_REF] Hamdinou | Potentiel des aciers électriques à grains orientés pour la réalisation de transformateurs compacts à forte densité de puissance[END_REF] that, with GOES wound cores operating in the kHz range, core losses are lower for rectangular voltages than for sine ones for the same peak flux density and frequency as presented in Fig. 1. This result proves that basic inverter commands producing rectangular voltages are better than complex Pulse Width Modulation (PWM) to eliminate several voltage harmonics. Figure 2 presents the chosen SST structure that connects a DC smart grid to the battery of an electric vehicle. The electrical energy comes from the DC smart grid during charging operations; the SST control system adapts the power flow to the battery characteristics. When economic conditions are favorable, the system can reverse the energy flow during short times to participate in grid stability. However, the charger global control system must guarantee a full charge at the programmed starting time of the vehicle. The two converters of the SST are active bridges; each bridge is made of two legs of two insulated gate bipolar transistors (IGBT). With a basic control, the voltages 𝑣 , and 𝑣 -are rectangular. For the angular frequency 𝜔, the output bridge command signals are delayed by 𝜑/𝜔 relative to the input bridge ones. Noting VA and VB the DC voltages at the input and output of the SST and for the transformer leakage inductance 𝜆 and neglected resistances, the power transferred between the SST input and output is given by (2) [START_REF] Roger | Solid-state transformers of smart high-power battery charger for electric vehicles[END_REF].

𝑃 = 𝑉 . 𝑉 / 𝜋𝜆𝜔 𝜑(𝜋 -|𝜑|) (2)
This relation shows that the power depends on the tuning variable 𝜑. The power flow goes toward the electric vehicle for 𝜑 > 0 and toward the smart grid for 𝜑 < 0. Relation (1) also shows that the product 𝜆𝜔 must remain low; the transformer leakage inductance must be as small as possible.

III. GOES WOUND CORE FOR MEDIUM FREQUENCY TRANSFORMERS

A. Testing bench and experimental results

A half-scale prototype was built to test the specific structure required to use a GOES wound core at a much higher temperature that of coils.

Figure 3 shows the GOES wound core alone whose total mass is 2.54 kg. The whole core combines two half-cores. To build such a half-core, a GOES long strip whose width corresponds to the core's (y-direction) is wound around a steel mandrel, whose shape defines the core windows. After the thermal annealing cycle at a temperature between 500°C and 600°C, the mandrel is removed, and the half core is cut into two pieces to make possible the winding assembly. After cutting the core, a machining operation is carried out for obtaining flat and parallel surfaces. The small defaults of these surfaces yield small residual airgaps, which must be as low as possible. An external fixing system must apply a force on the two parts of each half core.

GOES strips are made with a complex metallurgic process at high temperatures. The inorganic coating layer appears during this process providing natural insulation, which is a thin and hard layer (about 5 μm) that withstands temperatures up to 800°C. During the cooling phase, the higher expansion coefficient of the steel relative to that of the ceramics creates a tensile force inside the steel, which improves the GOES's magnetic characteristics. Consequently, the force between the upper and the lower parts of the half-core is an important parameter. For a too low force, residual microscopic airgaps create vibrations. Too large forces reduce the natural tensile force inside each strip, decreasing the magnetic performance. The metallic structure provides a separate fixing of the hot core and the cold coils creating a 5 mm thick space between the core and coil support. Two small centrifugal fans create a vertical air flux between the core and coil support. The thermal insulation between the core and the metallic fixing frame is made of mica spacers. Two pressure springs placed on the upper part of the threaded rods provide a controlled vertical force on the upper parts of the half-core and a degree of freedom for the core thermal expansion. The coil support is positioned on 4 rods screwed on the lower part of the fixing frame. The 4 rods of the upper part, with their small springs, provide constant pressure on the coil support despite the small vertical displacement of the metallic frame due to the core thermal expansion. There is no contact between the core and the coil support with this design. The prototype includes 6 thermocouples; 2 between the half cores in the central leg (Cc up -1 in Fig. 4, Cc down -2 in Fig. 4), two between the upper parts of the half-cores, and the mica spacers (H-core left -3 in Fig. 4, H-core right -4 in Fig. 4), 2 between the coil support and the wires (wire left -5 in Fig. 4, wire right -6 in Fig. 4). Two 6-turn coils are wound on the support; the primary is made with a Litz wire (20 enameled wires of diameter 0.6 mm), providing a negligible AC resistance and coil heating by the magnetizing current. The secondary coil is made with a standard wire because the current is negligible for voltage measurements. Figure 5 shows a synoptic diagram of the testing bench. The primary coil can be fed with the rectangular voltage produced by an inverter or by a quasi-sine one when a lowpass filter is used. The voltage waveform is measured at the secondary; the DC source-level adjusts the average flux density peak value. The inverter command tunes the frequency. The core losses are obtained from secondary voltage and the primary current by (3) considering an exact period T. The temporal signals are recorded with a numeric oscilloscope, and expression (3) is computed with Matlab. The flux density in the core is obtained by numeric integration of the secondary voltage. Figure 6 presents an example of a recorded curve for a rectangular voltage that corresponds to a peak flux density of 1 T. For this measurement point, the core losses are 320.5 W (126 W/kg) and the core central leg temperature is 260°C. The tests were aimed at proving the feasibility of the concept. Most of them were made at 4 kHz. Several investigations were made at lower frequencies. At 2 kHz, for instance, the acoustic level is very disruptive for the operator. Deeper investigations on the noise must be made in a further study for developing the new concept and for designing bigger medium-frequency transformers able to operate in a standard industrial context.

Figure 7 presents the variations in core losses with temperature for two similar wound cores made with 0.18-and 0.10-mm thick strips operating at 4 kHz and 1 T. This figure shows the importance of choosing a wound core made of GOES strips as thin as possible. This figure also shows a significant reduction in specific core losses with temperature, which presents another argument for the design proposed allowing a GOES wound core much hotter than coils. Table 1 presents the temperature measurements made on the prototype for two operating frequencies at the same flux density when the transformer is at a steady state (1.5 h after the inverter is switched on). These measurements show that the proposed structure made it possible a hot core transformer design with coils at standard temperatures far below the core ones.

Table 1. Temperature measurements at 𝐵 ! = 1T. The temperature of this table was measured for a prototype that operates at no-load; there are no winding losses The temperature increases are only caused by core losses. In an actual transformer operating at full load, Joule losses in conducting materials participate in the increase in winding temperatures. With a class C winding electrical insulation, able to operate up to 200°C, the Joule losses can be evacuated by a temperature elevation of about 100°C despite the presence of the hot core. This temperature difference is more or less the same as for standard designs with cold cores.

B. Interpretation of experimental results

Over 2 kHz, the skin effect in GOES strips must be considered. For a given average flux density in the GOES strip, the local distribution of flux density and eddy currents highly depends on the frequency and the material resistivity. As for all metals, the resistivity of GOES increases with temperature. A non-linear time-stepping finite element simulation was made to analyze these phenomena. The model is a small part of a strip inside the GOES wound core excited by a triangular flux that corresponds to rectangular voltages. The resistivity used by the model is given by ( 4) where 𝜌 20 is the GOES resistivity at 20°C (0.48 10 23 Ω. 𝑚); T, the core temperature and 𝛼 , the material temperature coefficient 22x10 -3 K -1 . The B(H) static curve is supposed to be unchanged.

ρ = ρ -4 [1 + α(T -20)] (4)
Figure 8 shows that for low temperatures the flux density at the strip center is much lower than at the borders. This effect is much lower at high temperatures. Figure 9 shows that the current density's absolute value is maximum at the strip borders and that it decreases strongly with temperature. Despite the higher resistivity at high temperature, the eddy current losses decrease strongly with temperature because the local eddy current losses 𝑝 78 (𝑥, 𝑡) = 𝜌 𝐽 -(𝑥, 𝑡) depends on the square of the current density. However, the eddy current losses are not alone. At 4 kHz, the experimental results of Fig. 7 show a significant decrease in global losses due to the decrease in eddy current losses despite the unknown part of the excess losses of the standard model [START_REF] Bertotti | General properties of power losses in soft ferromagnetic materials[END_REF].

IV. HIGH-CURRENT MEDIUM-FREQUENCY COIL DESIGN

Keeping coil AC resistances as low as possible is a major issue for designing high-efficiency compact mediumfrequency transformers. Litz wires are a complex and expensive component, he they are not compatible with a lowcost objective. The choice of coils made with copper or aluminum foils is for the proposes of a simple design. On the other hand, the foil coil structure has a high turn-to-turn capacitance, which provides an excellent voltage distribution between turns during the fast fronted voltages imposed by electronic converters. Coils can be made with a single foil, but the cross-section required may be large for high currents. In these conditions, a specific winding design is required to limit proximity effect.

A. Coils made with a single foil

Preliminary analytical, numerical, and experimental investigations on a reduced-size transformer are presented. Two 12-turn concentric coils are placed on a core. Each coil is made with one 0.7 mm thick copper foil. A thin polymer layer insulates the turns. The DC resistance is 1.2 mΩ for the external coil and 1.04 mΩ for the internal one because of the shorter average turn length. AC impedance measurements are made at room temperature, up to 20 kHz, with an Agilent 4294A impedance analyzer when the internal coil is shortcircuited. Figure 10 presents the experimental transformer made with standard technologies because, for in short-circuit tests the flux density is low, and the core remains cold. In this figure, a part of the pressboard carcass is removed to show the copper foils between the white insulating layers. Measurements are compared with analytical results and finite simulation ones. The analytical model of Dowell [START_REF] Dowell | Effects of eddy currents in transformer windings[END_REF], published in 2962, is compared to other ones in a recent study, which shows that it remains the most suitable for foil windings [START_REF] Gradinger | Enabling foil windings of medium-frequency transformers for high currents[END_REF].

This well-known model is used to aid the design of foil coils. This model yields 𝑅 ." /𝑅 9" , the ratio between the AC and DC resistances of a 𝑛 5 -turn coil made with a conducting foil of thickness 𝑡 : . The skin depth 𝛿 in the conducting material depends on its conductivity 𝜎 , on the absolute permeability 𝜇 of the material (𝜇 = 𝜇 4 for Cu and Al) and the angular frequency 𝜔 (5). Dowell's formulae (6) introduce the variable 𝑋 = 𝑡 : 𝛿 ⁄ , providing a simpler equation.

𝛿 = V 2 𝜔𝜇𝜎 (5) 𝑅 ." 𝑅 9" = 𝑋 sinh(2𝑋) + sin(2𝑋) coxh(2𝑋) -cos(2𝑋) + 2𝑋 𝑛 5 --1 3 sinh(𝑋) + sin(𝑋) coxh(𝑋) -cos(𝑋) (6)
Figure 11 applies Dowell's model to the considered transformer made of two concentric foil coils. The red line in Fig. 13 is plotted using [START_REF] Olowu | Multiphysics and Multiobjective Design Optimization of High-Frequency Transformers for Solid-State Transformer Applications[END_REF]. The finite element simulation is made with the 2D model presented in Fig. 12, where air layers are introduced. The core permeability is supposed to be infinite, and the simulation is made with a dense mesh and a linear solver. Results are presented in Fig. 13 as measurements. This figure shows that Dowell's model is optimistic; it gives the lowest increase in AC resistance. Conversely, the 2D FEM model is very near the measurements. The FEM model was used with several horizontal and vertical distances, at 10 kHz, for the same coil sizes, varying the core geometry. For a vertical distance between the core and the coil ranging from 1.5 mm to 6.5mm, the AC resistance does not change (a decrease of 0.3%). Similar results are obtained when the change is applied to the air gap between coils. Conversely, increasing the horizontal distance between the core and the coil by 10 mm decreases the AC resistance by 30%; but the core window filling factor is lower.

Experimental results and simulations show that, at 4 kHz, the ratio 𝑅 ;8 𝑅 <8 ⁄ is about 8 for two concentric coils made with 0.7 mm thick foils. This high value is a strong limit for high-power applications. A possible solution is to split the two coils into several parts and interleave them. With such a solution, the magnetic fields at the foil borders (red points in Fig. 11) form several triangular curves of reduced magnitude rather than a single one, the proximity effect is lower. This interpretation can also be made by observing the lower 𝑛 5 term in the second line of Dowell's formulae ( 6) is squared; a reduction of this term reduces the Dowell's result despite the series connection or the elementary coils forming the primary and the secondary.

Lower cost aluminum foils have a higher resistivity, and the DC resistance is higher for the aluminum coils (𝑅 9" = 1.02 𝑚Ω for Al and 𝑅 9" = 0.66 𝑚Ω for Cu). The AC resistances must be compared with absolute values (Fig. 14). This figure shows that the higher resistivity of aluminum becomes an advantage over 2 kHz for this geometry. A thicker skin depth δ for aluminum foils than for copper ones explains this result.

B. Coils made of several thin foils connected in parallel

For high current applications (several hundred Amperes), a parallel connection of thin foils is necessary to provide a sufficient conductor cross-section with low AC resistances.

A hundred years ago, in 1914, Ludwig Roebel developed a design for low AC resistance copper cable, which takes his name. Its main idea was to split the current into several bars and ensure their transposition along the cable [START_REF] Roebel | Leiter fur Electrische Maschinen, Welcher aus Zwei Oder Mehr Gruppen von Teilleitern Besteht[END_REF]. These cables were used as stator bars in large turbogenerators [START_REF] Bennington | Transposition in turbogenerator coil sides short circuited at each end[END_REF]. Several decades later, papers were published describing methods of transpositions of cable strands for power electronic applications [START_REF] Wojda | Analytical winding size optimisation for different conductor shapes using Ampère's Law[END_REF][START_REF] Nigam | Multi-layer barrel-wound foil winding design[END_REF][START_REF] Pollock | The Design of Barrel-Wound Foil Windings with Multiple Layers Interchanged to Balance Layer Currents[END_REF].

Figure 15 presents the central fold of 5 conducting foils used to wind a 7-turn coil. The current in each foil was simulated with a 2D finite element model. Results show an imbalance between the currents in the 5 aluminum foils of the same length. This imbalance is due to the flux in the thickness of the insulating layer between the foils. To limit this drawback, anodized aluminum foils insulated on each side by a thin alumina layer (3 -6 μm) can be used [39]. The preliminary investigations on windings show that coils made with thin foils connected in parallel become a complex solution for high-current applications. However, these solutions cannot be considered for a low-cost application. Concentric wide foil coils with primary and secondary coils split into several parts, connected in series, and placed alternately in the global concentric winding is a better solution.

V. DETAILED STRUCTURE OF THE SMART CHARGER

The smart charger must be able to adapt its characteristics to a large panel of electric vehicles with various DC voltage requirements. The structure proposed in Fig. 16 has two output converters (bridges B and C). The 800 V DC smart grid imposes 𝑈 . at the bridge A input and the vehicle batteries 𝑈 / and 𝑈 " . The filtering capacitances 𝐶 . , 𝐶 / and 𝐶 " are connected very near the corresponding IGBT bridges to avoid over-voltages due to the fast fronted currents imposed by the IGBT fast switching.

The medium frequency transformer has two secondary windings connected to the AC sides of bridges B and C. For high voltage vehicles (500 -1000V), the outputs of bridges B and C are connected in series. For low voltage vehicles (250 -500 V), these outputs are connected in parallel to double the available current. The control system acts on the pulses widths and on the phase shifts 𝜑 / & 𝜑 " of the bridges B and C, relatively to the bridge A. As detailed in (2), these phase shifts define the transferred power. For a perfect transformer (equal leakage inductances), a single variable 𝜑 = 𝜑 / = 𝜑 " defines two equal powers. However, the control system can create a small difference between 𝜑 / and 𝜑 " to compensate the small difference in the transformer's leakage inductance.

The smart charger is designed for a rated power of 400 𝑘𝑊 (output mean currents < 𝑖 / > = < 𝑖 " > = 400 𝐴 for 𝑈 / = 𝑈 " = 500 𝑉 and < 𝑖 . > = 500 𝐴. Consequently, the medium frequency transformer must have a higher rated apparent power (~500 𝑘𝑉𝐴) because of the reactive power exchanged with the capacitances 𝐶 . , 𝐶 / and 𝐶 " due to the leakage inductances and the magnetizing one. The rated power is obtained when the positive and negative pulse widths are equal to half a period. Therefore 𝑉 ,#$% = 𝑈 . = 800𝑉 and 𝑉 -#$% = 𝑉 )#$% = 500𝑉 . The primary and secondary current are respectively 625 𝐴 #$% and 500 𝐴 #$% . The medium frequency transformer proposed structure is presented in Fig. 17. The 2 secondary coils are placed between the 3 coils connected in series forming the transformer primary. This winding structure is based on the results of the previous section; the interleave of primary and secondary coils reduces the proximity effect and allows the use of thicker folds. A large cross-section is obtained using wide foils (150 mm). A 2D finite element simulation computes the AC equivalent resistance of the part of the foil winding inside a core window. This equivalent resistance is seen from the primary side when the two secondary coils are connected in series and short-circuited. 8 solutions are considered (Cu and Al of thicknesses 1, 1.5, 2, and 3 mm). Results are presented in Fig. 18. This figure shows that, at 4 kHz, the best solutions, in decreasing order, are Cu -1.5 mm & Cu -2 mm. Two solutions are equivalent Cu -1 mm & Al -1.5 mm, with a resistance of 0.52 mΩ. Al -1 mm has a too high DC resistance, and for 3 mm folds, the proximity effect is too high. At 6 kHz, the best solutions are Cu -1.5 mm, Cu -1 mm, and Al -1.5 mm. The choice must be made comparing the costs, considering that, for the same thickness, an aluminum foil is approximately five times cheaper than the copper one. A low-cost solution can use aluminum foils of 1.5 mm thickness and 150 mm width. With this choice, the crosssection is 225 mm 2 and the current densities are 2.77 A/mm 2 for the primary and 2.22 A/mm 2 for the secondary. The foils are insulated by a class C (200°C) polymer layer a little wider than the foils and notched on each side to ensure good foil borders insulation. The mechanical structure is similar to the prototype one. The winding is fixed between the 8 pressure rods of the mechanical structure drawn in blue in Fig. 17. There is no contact between the hot core and the windings. Six small centrifugal fans create a vertical fresh air flow between the hot core and the coils. The GOES wound core is made of 0.1 mm thick GOES strips. The total section offered to the flux is 𝐴 " = 53 𝑐𝑚 -. At 1 T -4 kHz, the primary is made of three 4-turn coils connected in series to reach 800 V. Each secondary coil has 8 turns, and the no-load voltage is 533 V.

With this design, the mass of 5 concentric coils is estimated to 11 kg. The mass of the GOES wound core is 25 kg, hence the mass of the active parts is 36 kg. Adding 20% for the fixing system, the transformer mass is estimated to be 43 kg.

The core losses are estimated from measurements made at 4 and 6 kHz on the half-size prototype at the same flux density and frequency. The winding losses are estimated considering the equivalent resistances obtained by simulation (Fig. 18). The power factor is supposed to be 0.8. With these parameters, the performances of the transformer alone at 4 kHz are estimated in the first line corresponding line of table 2.

At 6 kHz and with the same coils, the currents are unchanged, but the voltages increase with the same ratio as frequencies (6/4). The losses in the core and in the winding increase. However, the higher voltages with the same currents correspond to a higher power and a slightly better efficiency. Results are presented in the last line of table 2. Comparing to a standard dry transformer at 50 Hz, at 4 kHz, the specific power is 20 times higher.

Tables 3 gives the losses in bridges made of SKM 400 GM 17 E4 (1700V/400A) legs. The calculus is made using wellknown methos [START_REF] Seguier | Electronique de puissance -structures, commande, applications -dixième edition[END_REF]. Table 4 presents the global losses and the efficiencies of the whole smart charger for two frequencies.

VI. CONCLUSION

The paper presents theoretical and experimental investigations on a transformer made with low-cost GOES wound cores operating at medium frequencies (4 and 6 kHz). It proposes a practical implementation of the original concept of a hot core inside windings at standard temperatures. This concept uses the good performances of GOES at high temperatures. The mechanical structure includes an airflow in the small space between the core and windings. There is no direct contact between core and winding. This structure makes it possible a significant temperature difference between the core and the winding. The experimental part of the paper proves that, with the design proposed and airflows supplied by two small centrifugal fans, the core central leg can be at a temperature over 270°C inside a winding under 100°C when the transformer is at a steady state.

The paper also proposes a winding structure of concentric coils made with wide aluminum foils providing low AC resistance at 4 and 6 kHz. This low-cost design has a low environmental footprint; it offers a good solution for keeping low Joule losses, considering skin and proximity effects.

Comparing the new design proposed with the ones currently used ones for SST's transformers, the main disadvantage is the lower operating frequencies, which corresponds to slightly lower specific powers despite higher peak flux density capabilities. However, the new design has several advantages. It is based on low-cost and widely available components able to operate at high voltages and currents. It uses simple power electronic structures operating with a minimum number of switching per period and, consequently, reduced switching losses. The coil's structure provides an excellent voltage distribution allowing high voltage operations. With a good heat flow management and limited coil AC resistance, it is possible to obtain higher global efficiencies than conventional solutions operating at higher frequencies.

The paper presents a study that can be considered at the technology readiness level 3 (TRL3) that corresponds to a proof of concept. Tests have highlighted a technological barrier, related to acoustic noise, that must be overcome to reach the TRL4 level. Several research results are available for starting these further works. One of them deals with the influence of the GOES manufacturing process on transformer noise [START_REF] Lahn | Improved Transformer Noise Behavior by Optimized Laser Domain Refinement at ThyssenKrupp Electrical Steel[END_REF]. Another paper centered on noise measurement methods in the aircraft industry [START_REF] Kelley | Measurement of spacecraft power transformer acoustic noise[END_REF] provides an interesting approach. The paper [START_REF] Rossi | Vibration reduction of inductors under magnetostrictive and Maxwell forces excitation[END_REF] proposes a simple analytical model of the core vibration caused by the magnetic forces that can help the interpretation of further measurements. A more fundamental study. [START_REF] Shuai | Influence of Material Properties and Geometric Shape of Magnetic Cores on Acoustic Noise Emission of Medium-Frequency Transformers[END_REF] give more details on the core vibrations; it shows that the noise is dominant in directions perpendicular to the lamination layers of wound cores. This result will be helpful for defining the position of the acoustic insulation panels in the SST's electrical cabinet.

The following step is the construction of a full-scale lowcost battery smart charger placed in its industrial electrical cabinet. An experimental analysis of the thermal behavior will provide the rated power with a well-defined safety margin relative to the thermal temperature class of the winding electrical insulation system. Thermal performances and acoustic noise spectra will be the two major elements for choosing of the operating frequency.

Fig. 1 .

 1 Fig. 1. Specific losses in a wound core, made with 0.18 mm thick GOES sheets, for a rectangular and a sine voltage and the same peak flux density 1 T (experimental results) [17].

Fig. 2 .

 2 Fig. 2. Smart bidirectional charger structure based on an SST made of a medium frequency transformer placed between two IGBT bridges forming a Dual Active Bridge (DAB) structure.

Fig. 3 .

 3 Fig.3. GOES wound core before its installation in the transformer.

Figure 4

 4 Figure4presents the half-size prototype. The metallic structure provides a separate fixing of the hot core and the cold coils creating a 5 mm thick space between the core and coil support. Two small centrifugal fans create a vertical air flux between the core and coil support. The thermal insulation between the core and the metallic fixing frame is made of mica spacers. Two pressure springs placed on the upper part of the threaded rods provide a controlled vertical force on the upper parts of the half-core and a degree of freedom for the core thermal expansion. The coil support is positioned on 4 rods screwed on the lower part of the fixing frame. The 4 rods of the upper part, with their small springs, provide constant pressure on the coil support despite the small vertical displacement of the metallic frame due to the core thermal expansion. There is no contact between the core and the coil support with this design.

Fig. 4 .

 4 Fig. 4. Image of the half-size prototype aimed at measuring core losses at medium frequencies for several temperatures. The 6 thermocouples are inside the yellow circles.

Fig. 5 .

 5 Fig. 5. Synoptic diagram of the testing bench.

Fig. 6 .

 6 Fig. 6. Example of recorded voltage and current curves at 6 kHz for a wound core made with 0.10 mm thick GOES trips. The rectangular voltage imposed is 150 VPeak, corresponding to a peak average flux density of 1 T.

Fig. 7 .

 7 Fig. 7. Variations of specific core losses with temperature at 1 T -4 kHz for two GOES strip thicknesses.

1 .

 1 Cc up: upper part of the central core. 2. Cc down: lower part of the central core. 3. H-core left: upper part of the left half-core. 4. H-core right: upper part of the right half-core. 5. Wire left: upper part of the coi1 on the left. 6. Wire right: upper part of the coil on the right.

Figures 8 and 9

 9 Figures 8 and 9 give the local flux density B(x,t) and the current density J(x,t) along a line in the strip thickness. These curves are plotted after the transient state, at a time that corresponds to the peak average flux density. A simulation was made for rectangular voltages at 4 & 6 kHz and 20°C & 350°C.

Fig. 8 .

 8 Fig. 8. Distribution of the flux density inside the GOES strip thickness for Bm = 1T at 4 and 6 kHz for two temperatures.

Fig. 9 .

 9 Fig. 9. Distribution of the current density inside the GOES strip thickness for Bm = 1T at 4 and 6 kHz for two temperatures.

Fig. 10 .

 10 Fig. 10. Wound core transformer with 2 concentric 12-turn coils used for preliminary tests; The black line shows the separation between the primary and secondary coils.

Fig. 11 .

 11 Fig.11. 1D Dowell's model hypothesis applied to the experimental transformer with two 12-turn coils. The field is computed for a primary and secondary currents of 100A.

Fig. 12 .

 12 Fig.12. Finite element model of concentric coils made with copper foils.

Fig. 13 .

 13 Fig. 13. Resistance increase with frequency. Experimental, FEM and Dowell model for the experimental transformer geometry.

Fig. 14 .

 14 Fig. 14. Comparison of the Cu and Al AC resistances.

Fig 15 .

 15 Fig 15. Central fold used to build a 5-fold coil with an inversion the foils that provide the same length to foils when the coil is wound.

Fig. 16 .

 16 Fig. 16. Detailed power structure of the smart charger.

Fig. 17 .

 17 Fig. 17. Proposed structure for a low-cost medium frequency 500 kVA transformer based on a GOES wound core and aluminum foil coils.

Fig. 18 .

 18 Fig.[START_REF] Kauder | Performance Factor Comparison of Nanocrystalline, Amorphous, and Crystalline Soft Magnetic Materials for Medium-Frequency Applications[END_REF]. AC equivalent resistance of the part of the foil winding inside a core window for several foil materials and thicknesses.

Table 2 .

 2 Transformer main characteristics expected for 4 and 6 kHz.

	f	V1	I1	S	P	PC	PW	P/m	𝜂
	(kHz)	(V)	(A)	(kVA)	(kW)	(kW)	(kW)	(kVA/kg)	(%)
	4	800	625	500	400	3.85	1.35	11.6	98.7
	6	1200	625	750	600	4.3	1.59	17.4	99.0

Table 3

 3

			. Losses in the 3 converters.
	f	Bridge A	Bridges B and C	Total
	(kHz)	Pcond	Pcomm	Pcoond	Pcomm	(W)
		(W)	(W)	(W)	(W)
	4	1860	130	2800	132	4922
	6	1860	290	2800	200	5250
		Table 4. Global losses and efficiencies.
	f	Transformer	Converters	Total	Efficiency
	(kHz)	(kW)	(kW)	(kW)	(%)
		Core	Windings
	4	3.85	1.35	4.92	10.12	97.4
	6	4.3	1.6	5.25	11.15	98.1


Mauro Ettorre

David González Ovejero
email: david.gonzalez-ovejero@univ-rennes1.fr

High Speed and Resilient Wireless Interconnects in the Near Field

Keywords: non-diffractive waves, near-field communications, Bessel beams, millimeter-wave links

Bessel beams present the peculiar properties of being non-diffractive and self-healing within the near field of the radiating aperture. These properties are here exploited to establish robust near-field links with error-free transmission at W-band. These links are generated by using two non-diffractive launchers, consisting of spline-profile metallic horns. An appropriate design of the horn enables the generation of a Bessel beam within a distance, called non-diffractive range, which is typically in the near field of the launcher. This contribution experimentally demonstrates non-diffractive Bessel beams generated by a spline profile horn within the launcher's nondiffractive range. Besides, the self-healing feature has been tested by placing a metallic object in front of the transmitting horn. Finally, we have evaluated a near-field link by measuring its bit error rate (BER) with and without the metallic obstructions. An error-free communication has been obtained in both cases for a distance approximately twice the nondiffractive range of one horn and for a data stream of 1Gb/s using an on-off keying (OOK) modulation.

I. INTRODUCTION

Bessel-beams are exact solutions of the Helmholtz equation with a transverse field profile taking on a Bessel function [1]. In the ideal case, the transverse profile of the field does not change with the propagation distance, which provides the non-diffractive feature. Such an ideal solution requires an infinite amount of energy and an unbounded radiating aperture. However, devices with a finite radiating aperture can be conceived to generate a non-diffractive Bessel beam in a limited region of space that is called non-diffractive range and typically lies in their near field of the launcher. In addition, Bessel beams present the peculiar property of selfhealing, which means that they can reconstruct themselves after an obstruction while propagating [START_REF] Bouchal | Self-reconstruction of a distorted non-diffracting beam[END_REF]. Such a feature will be here exploited to establish a robust near-field link resilient to metallic objects obstructing the line of sight (LOS) between the transmitter and the receiver.

During recent years, the generation of Bessel beams at millimeter waves has attracted a lot of attention and one may find different types of launchers in the open literature. In particular, it is worth mentioning radial waveguides loaded by metasurfaces [START_REF] Ettorre | Generation of propagating Bessel beams using leaky-wave modes[END_REF]- [START_REF] Fuscaldo | Higher-order leaky-mode Bessel-beam launcher[END_REF], radial line slot arrays (RLSA) [START_REF] Mazzinghi | Large depth of field pseudo-Bessel beam generation with a RLSA antenna[END_REF], [START_REF] Pavone | Analysis and design of Bessel beam launchers: Transverse polarization[END_REF], arrays of patches [START_REF] Lemaitre-Auger | Generation of Bessel beams by two-dimensional antenna arrays using sub-sampled distributions[END_REF], lenses [START_REF] Chiotellis | Experimental demonstration of highly localized pulses (X waves) at microwave frequencies[END_REF], [START_REF] Chiotellis | X wave radiator implemented with 3-D printed metamaterials[END_REF], and spline-profile horns [START_REF] Pakovic | A fast and accurate method of synthesizing Xwave launchers by metallic horns[END_REF]. The non-diffractive behavior of Bessel beams and their capability to provide a robust link may be used in various applications such as wireless personal area networks (WPAN), kiosk downloading, future data centers, wireless power transfer and near-field chip-to-chip communications [START_REF] Alsolamy | Cylindrical vector beams for wireless power transfer[END_REF]- [START_REF] Pakovic | Bessel-Gauss beam launchers for wireless power transfer[END_REF]. This work explores the possibility of establishing nearfield links for robust communications resilient to metallic obstructions at W-band (75-110 GHz) by exploiting the peculiar non-diffractive and self-healing properties of Bessel beams. Metallic spline-profile horns, as those described in [START_REF] Pakovic | A fast and accurate method of synthesizing Xwave launchers by metallic horns[END_REF], will be used as transmitting and receiving devices in the link to avoid dielectric losses at such high frequencies. The adopted launchers present a radiating aperture with a diameter of 30 mm (9 0, with 0 being the free space wavelength at 90 GHz). The horns synthetize the transverse electric TE02 mode of a circular waveguide over the radiating aperture to generate a non-diffractive Bessel beam. Such a Bessel beam is characterized by an axicon angle of 15° and a non-diffractive range of about 60 mm (18 0) [1], [START_REF] Ettorre | Generation of propagating Bessel beams using leaky-wave modes[END_REF].

The transmitting and receiving horns have been fabricated by additive manufacturing using selective laser melting (SLM) and tested at IETR in the 75-105 GHz band. The selfhealing feature of the beams has been also validated by considering a metallic obstacle with a diameter a0 =7 mm at a distance d =9 mm in front of the transmitting aperture. Finally, a near-field link has been tested with and without the metallic obstruction over a range of distance from 15 mm to 110 mm. We have adopted an 80 GHz carrier and an on-off keying (OOK) modulation to transmit a 1 GB/s signal. The bit error rate (BER) of the link has been measured for the parameters above, showing that an error-free communication may be established both with and without an obstacle within the 30-84 mm range. This is the first reported example of the capability of Bessel beams to establish a resilient and robust link for near-field communications in the millimeter wave (mm-wave) range.

The paper is organized as follows. Section II illustrates the design of the spline-profile horns, and presents full-wave simulations and measurements that validate the generation of non-diffractive Bessel beams and their self-healing property in the band of interest. Section III introduces the nondiffractive near-field link and reports the measured BER for a link with and without a metallic obstruction within the LOS. Finally, Section IV presents our concluding remarks.

II. BESSEL BEAM LAUNCHER

1 shows the metallic launcher used to generate Bessel beams at W-band. It consists of two parts: a Marié transducer and a spline profile horn. The former presents a standard WR-10 rectangular waveguide input and transforms the TE10 mode into the output TE01 circular waveguide mode, which acts as input for the spline horn. The profile of the horn is designed as described in [START_REF] Pakovic | A fast and accurate method of synthesizing Xwave launchers by metallic horns[END_REF] to synthetize the TE02 circular waveguide mode over the radiating aperture with a 30 mm diameter. The imposed aperture field distribution provides a Bessel beam with an axicon angle of 15° and a non-diffractive range of about 60 mm (2 times the launcher's diameter). The total length of the spline profile of the horn is about 48 mm, while the total length of the launcher is 104 mm. The metal-only Bessel beam launcher has been manufactured at IETR by SLM using an aluminum alloy (AlSi10Mg). The non-diffractive and self-healing capabilities of the Bessel beam launcher has been validated by measuring the transverse near-field profile with and without obstacle in the 75-110 GHz band. Fig. 2 shows the experimental set-up used to measure the Ey component of the electric field in the xoz plane. The transmitted mm-wave signal is photonically generated. The photonic generation module in Fig. 2 provides two amplitude modulated optical tones around 1550 nm, whose separation corresponds to the desired mm-wave frequency. The two optical wavelengths are modulated by a 100 kHz sinusoidal waveform from a lock-in amplifier. In turn, the optical fiber output of the photonic generation module is used to illuminate a commercial photo-mixing device, which provides an amplitude modulated beat-note. Then, the WR-10 output of the photo-mixing block excites the launcher described above. The fields are sampled by a nearfield probe connected to a zero bias Schottky barrier diode detector (ZSBD) and placed in a 3-axis motorized. The RF output of the ZSBD is extracted with a bias-tee connected to the lock-in amplifier input, which measures the detected relative amplitude. For the sake of brevity, the following figures only report the simulated and measured E-field profiles at 90 GHz (the central frequency). However, it is important to note that a similar agreement has been obtained from 75 to 105 GHz. First, Fig. 3 presents the normalized magnitude of the electric field Ey, obtained for the LOS case, i.e., when the obstacle in Fig. 2 is absent. Fig. 3(a) shows full-wave simulation data, whereas Fig. 3(b) showcases the experimental results. There is a very good agreement between simulations and measurements. Since the transverse profile of the radiated beam is preserved in the near field of the launcher, its nondiffractive behavior up to z = 60 mm is also confirmed. The reminder of this section deals with the experimental demonstration of the self-healing feature of Bessel beams. To illustrate the non-LOS (NLOS) case, we place a circular metallic obstacle with diameter a0 = 7 mm and thickness t0 = 0.4 mm at a distance d = 9 mm from the launcher's aperture and along the horn's axis. Fig. 4 presents the normalized magnitude of the E-field in presence of the obstacle, which position is accurately controlled using a 4-axis micropositioner (as sketched in Fig. 2). In this case, each sub-figure is normalized with respect to the maximum value along the dashed white line. Such line represents the shadow depth boundary and provides the distance at which the beam heals. Fig. 4(a) and Fig. 4(b) show the results obtained from fullwave simulations and the experimental ones, respectively. The agreement is also very good in this NLOS case and a clear non-diffractive profile can be recognized both in Fig. 4(a) and Fig. 4(b) after the shadow depth. Indeed, the fields measured beyond the shadow depth are similar to those of the nonobstructed launcher reported in Fig. 3.

III. NON-DIFFRACTIVE LINK

The Bessel beam launcher described and characterized in Section II is used in this section as transmitting and receiving device in a near-field link. In other words, the near-field probe in Fig. 2 is replaced with a second launcher. The two launchers were axially aligned and placed at a relative distance ranging s from 15 mm to110 mm. Moreover, in order to measure the BER, a pulse pattern generator provides the input for the photonic generation module in Fig. 2 and the output of the ZSBD's bias tee is connected to an error detector.

The pulse pattern generator is used to generate a 1 Gb/s non return to zero (NRZ) OOK signal, which modulates a 80 GHz carrier. The BER has been then measured at the receiving side with and without the obstacle obtruding the link. Fig. 5(a) shows a picture of the measurement set-up and Fig. 5(a) presents the obtained BER versus the distance s between launchers. An error-free transmission is achieved in the range 30-84 mm in both cases. Such results validate, for the first time, the capability of non-diffractive beams to provide a robust and resilient link for near-field communications. It is worth mentioning that higher data rates can be achieved by using a photo-mixing device with a more stable response in the band of interest and an optimized receiver.

IV. CONCLUSION

This contribution reports, for the first time, the use of Bessel beams to establish a robust near-field link with errorfree high data rate transmission at W-band. Such link is also resilient to obstructions in the LOS and can be readily applied to near-field communications. The transmitter and the receiver of the reported link are based on metallic spline-profile horns fabricated by additive manufacturing. The operation of the horn as non-diffractive launchers and the self-healing feature of the generated beam have been experimentally validated in the band 75-105 GHz. In addition, a near-field link has been characterized using the horns; we have shown the capability to transmit a 1 Gb/s data stream based on an OOK modulation. In a range of 30 to 84 mm a free error communication has been achieved with and without an obstacle. The distance range is limited by the axicon angle chosen for the Bessel beam. However, smaller axicon angles can be synthesized paving the way to a larger operating distances. In addition, one can use more sophisticated photo-mixing hardware and/or adopt higher order modulations to better exploit the Bessel beam launchers' large bandwidth and increase the data rate.

Fig. 1 .

 1 Fig. 1. 3D-view of the interior of the Bessel beam launcher and its modal operation. The launcher is fed by a standard WR-10 rectangular waveguide. A Marié transducer converts the guided mode from the input rectangular waveguide to the circular one used for the horn. All the reported dimensions are in millimeters.

Fig. 2 .

 2 Fig. 2. Schematic representation of the experimental set-up used to sample the near field in the non-diffractive range.

Fig. 3 .

 3 Fig. 3. Normalized electric field in the xoz plane (see Fig. 2) at 90 GHz for the LOS configuration: (a) Simulation. (b) Measurement.

Fig. 4 .

 4 Fig. 4. Normalized electric field in the xoz plane (see Fig. 2) at 90 GHz for the NLOS configuration: (a) Simulation. (b) Measurement.

Fig. 5 .

 5 Fig. 5. (a) Picture of the non-diffractive near field link . (b) Measured BER of the non-diffractive link with (dashed green line) and without (solid blue line) a metallic obstacle within the operating range.

ACKNOWLEDGMENT

This work has received support from the French National Research Agency (ANR) through the project LinkAll with reference ANR-21-CE25-0014, and from the French Government through the Labex CominLabs Excellence Laboratory, managed by ANR in the Investing for the Future Program under Reference ANR-10-LABX-07-01.


D Gonzalez-Ovejero
email: david.gonzalez-ovejero@univ-rennes1.fr

O De Sagazan

X Morvan

C Bilitos

J Ruiz-Garcia

A Mahmoud

M Ettorre

D González-Ovejero

J Ruiz-García

Modulated metasurface antennas and beam

HAL is

I. INTRODUCTION

Operators are currently deploying millimeter-wave 5G cellular networks for reliable and low-latency communications, enhanced mobile broadband and massive machine-type communications [START_REF] Rappaport | Millimeter wave mobile communications for 5G cellular: It will work![END_REF]. However, ways forward at sub-THz frequencies (above 100 GHz) are already under study to cope with the race to data rate [START_REF]Wireless communications and applications above 100 GHz: Opportunities and challenges for 6G and beyond[END_REF] and some standards have been already proposed. One of the key challenges will be the deployment of a large number of small cells and the use of millimeter-waves and sub-THz frequencies to enable ultra-large bandwidth wireless, with seamless connectivity to the core network and to the cloud. Directive antennas with multi-beam or beam steering capabilities will be needed to compensate path-loss at these high frequencies and to increase the energy efficiency, by minimizing the amount of power radiated into directions in which it will not be received. Besides satisfying the link budget and providing broad bandwidths, antennas must be also amenable for integration on the chassis of vehicles, in smart urban furniture, and on buildings and rooms' walls. On account of their low-profile, gradient or modulated metasurfaces (MTSs) are excellent candidates to address all these challenges.

II. MODULATED METASURFACE ANTENNAS

Aperture antennas based on modulated metasurfaces gradually radiate the power carried by a surface wave (SW), owing to the SW interaction with a periodically modulated impedance boundary condition (IBC) [START_REF] Faenzi | Metasurface antennas: New models, applications and realizations[END_REF]. This mechanism enables an accurate control of the aperture fields and allows one to generate directive beams by means of low-profile and lightweight structures. Recent realizations of modulated MTS antennas have proven that they can meet challenging specifications such as high-gains, dual-polarization [START_REF] Tellechea | Dual circularly polarized broadside beam metasurface antenna[END_REF], multiple beams [START_REF] González-Ovejero | Multibeam by metasurface antennas[END_REF], and dual-band [START_REF] Faenzi | Overlapped and sequential metasurface modulations for bi-chromatic beams generation[END_REF][START_REF] Li | Multiwavelength multiplexing hologram designed using impedance metasurfaces[END_REF] or broadband operation [START_REF] Faenzi | Wideband active region metasurface antennas[END_REF].

To date, modulated metasurface antennas have been mostly fabricated and tested in the microwave range. Such realizations consist of sub-wavelength patches printed on dielectric substrates and fabricated by standard PCB technology [START_REF] Faenzi | Metasurface antennas: New models, applications and realizations[END_REF]. In this invited presentation, we will discuss some specific challenges of mm-wave and sub-THz modulated metasurface antennas. Special attention will be paid to material losses, and to the high accuracy and strict tolerances required in the sub-THz range. In order to avoid dielectric losses at such high frequencies, we have recently explored the use of metal-only metasurfaces [START_REF] González-Ovejero | Additive manufactured metalonly modulated metasurface antennas[END_REF][START_REF] González-Ovejero | Metasurface antennas: Flat antennas for small satellites[END_REF]. These structures can be fabricated by metal additive manufacturing [START_REF] González-Ovejero | Additive manufactured metalonly modulated metasurface antennas[END_REF] in the millimeter-wave range or by Deep Reactive Ion Etching (DRIE) above 100 GHz [START_REF] González-Ovejero | Metasurface antennas: Flat antennas for small satellites[END_REF]. Fig. 1 shows two recent realizations at IETR. Besides benefiting from reduced losses, metal-only structures can withstand harsh environments with large thermal ranges and/or high radiation levels.

III. BEAMFORMING BY MODULATED METASURFACES

Gradient metasurfaces can be also applied to design planar devices for guided SWs. By modulating the equivalent refractive index, one can address the SW rays along curvilinear paths in order to split an impinging SW beam [START_REF] Martini | Flat optics for surface waves[END_REF] or to obtain planar wave-fronts. Among the latter structures, one finds Luneburg lenses (LL) [START_REF] Bosiljevac | Non-uniform metasurface luneburg lens antenna design[END_REF]. LL enable the generation of multiple fan beams: one per each source in the focal circumference. Nevertheless, to provide pencil beams, an antenna should be placed after the lens and on the same plane. Therefore, the total footprint [START_REF] Li | Dual-physics manipulation of electromagnetic waves by system-level design of metasurfaces to reach extreme control of radiation beams[END_REF] will be much larger than in two-layered pillbox systems [START_REF] Ettorre | Multi-beam multi-layer leaky-wave SIW pillbox antenna for millimeter-wave applications[END_REF], where the beam-former is placed in te bottom layer and the radiating aperture in the top one.

To overcome this limitation, one can use reflecting Luneburg lenses (RLLs), which benefit from the compactness of double-layered pillbox beam-formers and the full azimuthal scanning of Luneburg lenses [START_REF] Ruiz-García | Reflecting luneburg lenses[END_REF]. RLLs consist of two vertically stacked parallel plate waveguides (PPWs) of circular shape, of which the bottom one is filled with a graded index (GRIN) medium with azimuthal symmetry. The rays launched by a source in the bottom PPW follow curvilinear paths such that they emerge collimated in the top PPW after encountering a reflecting boundary. Fig. 2 shows the GRIN profile required to obtain this behavior, along with the aforementioned rays trajectories (white lines in the bottom layer, black lines in the top one). Owing to the lens' symmetry, one can generate plane waves with arbitrary directions by simply changing the azimuthal position of the source in the bottom layer. Moreover, multiple beams can be achieved by accommodating several sources in the focal circumference or the inner focal region where the rays concentrate in the bottom PPW.

Fig. 1 :

 1 Fig. 1: (a) Metal-only modulated metasurface antenna at Ka-band fabricated by metal additive manufacturing. (b) Modulated metasurface antenna at 240 GHz fabricated by DRIE and metallized sputtering Au.

Fig. 2 :

 2 Fig. 2: (a) Ray trajectories in a RLL and color map with the refractive index profile of the GRIN medium. (b) Realization by CNC milling of a RLL where the GRIN medium is implemented by a bed-of-nails.

th International Congress on Artificial Materials for Novel Wave Phenomena -Metamaterials 2022 Siena, Italy, Sep. 12 th -17 th 2022

ACKNOWLEDGEMENT This work has been supported by the European Union through the European Regional Development Fund (ERDF), the French Region of Brittany, Ministry of Higher Education and Research, Rennes Métropole and Conseil Départemental 35, through the CPER Project STIC & Ondes. D. González-Ovejero's work has received a French government support granted to the Labex CominLabs excellence laboratory and managed by the National Research Agency in the "investing for the future" program under reference ANR-10-LABX-07-01.

IV. CONCLUSION

In this paper, we will review the application of modulated metasurfaces to the design of aperture antennas and beam-formers. During the presentation, we will mainly focus on high-gain and multi-beam capabilities. Then, we will present the generation of multiple pencil beams by combining planar beam-forming lenses with modulated metasurface apertures. Such structures can provide several pencil beams with high gains and, hence, enable scanning by beam switching. Finally, fabrication and testing aspects will be presented with emphasis on structures operating above 100 GHz.


Xinghan Liu

Emiliano Lorini

Antonino Rotolo

Giovanni Sartor

Modelling and Explaining Legal Case-Based Reasoners Through Classifiers

Keywords: Case-based reasoning, Modal Logic for classifiers, Explainable AI

 has developed the factor-based models of precedent into a theory of precedential constraint. In this paper we combine binary-input classifier logic (BCL) to classifiers and their explanations given by Liu & Lorini [13,14] with Horty's account of factor-based CBR, since both a classifier and CBR map sets of features to decisions or classifications. We reformulate case bases in the language of BCL, and give several representation results. Furthermore, we show how notions of CBR can be analyzed by notions of classifier explanation.

Introduction

This paper brings together two lines of research: factor-based models of case-based reasoning (CBR) and the logical specification of classifiers.

Logical approaches to classifiers capture the connection between features and outcomes in classifier systems. They are well-suited for modeling and computing a large variety of explanations of a classifier's decisions [START_REF] Shih | Formal verification of bayesian network classifiers[END_REF][START_REF] Darwiche | On the reasons behind decisions[END_REF][START_REF] Ignatiev | Abduction-based explanations for machine learning models[END_REF][START_REF] Ignatiev | From contrastive to abductive explanations and back again[END_REF][START_REF] Audemard | On the computational intelligibility of boolean classifiers[END_REF][START_REF] Liu | A logic for binary classifiers and their explanation[END_REF], e.g., prime implicants, abductive, contrastive and counterfactual explanations. Consequently, they enable detecting biases and discrimination in the classification process. They can thus contribute to provide controllability and explainability over automated decision-making (as required, e.g., by Art. 22 GDPR and by Art. 6 ECHR relative to judicial decisions).

Factor-based reasoning [START_REF] Ashley | Modeling Legal Argument: Reasoning with Cases and Hypotheticals[END_REF][START_REF] Vincent Aleven | Using background knowledge in case-based legal reasoning: a computational model and an intelligent learning environment[END_REF] is a popular approach to precedential reasoning in AI&law research. The key idea is that a case can be represented as a set of factors, where a factor is a legally relevant aspect. Factors are assumed to have a direction, i.e., to favor certain outcomes. Usually both factors and outcomes are assumed to be binary, so that each factor can be labelled with the outcome it favors (usually denoted as π, the outcome requested by the plaintiff, and δ , the outcome requested by the defendant). The party which is interested in a certain outcome in a new case can support her request by citing a past case that has the same outcome, and shares with the new case some factors supporting that outcome. The party that is interested in countering that outcome can respond with a distinction, i.e., can argue that some factors which supported that outcome in the precedent are missing in the new case or that some additional factors against that outcome are present in the new case. Horty [START_REF] Horty | The result model of precedent[END_REF][START_REF] Horty | A factor-based definition of precedential constraint[END_REF] has developed the factor-based models of precedent into a theory of precedential constraints, i.e., of how a new case must be decided, in order to preserve consistency in the case law. In [START_REF] Horty | Rules and reasons in the theory of precedent[END_REF][START_REF] Horty | Reasoning with dimensions and magnitudes[END_REF], he takes into account the fact that judges may also provide explicit reasons for their choice of a certain outcome. This leads to the distinction between the result and the reason model of precedents. In the first model, the message conveyed by the case is only that all factors supporting the case-outcome (pro-factors) outweigh all factors against that outcome (con-factors). In the second, the message is that the factors for the case outcome indicated by the judge outweigh all factors against that outcome.

In this paper we shall combine Liu & Lorini's modal logic approach to classifiers and their explanations [START_REF] Liu | A logic for binary classifiers and their explanation[END_REF][START_REF] Liu | A unified logical framework for explanations in classifier systems[END_REF] with Horty's account of factor-based CBR. The combination is based on the fact that both a classifier and CBR map sets of features to decisions or classifications. In this way, our contribution is at least twofold.

First, we explore the relation between two apparently unrelated reasoning systems. While the connection between CBR and reasoning about classifier systems is of interest in itself, we believe that, through this relation, new research perspectives can be offered, since we could in the future investigate CBR by exploiting several techniques and results from modal logic. We will see that the challenge of this paper is to adapt the formal representation of a classifier to the bidirectionality of factors in the HYPO model. Once this is solved, we can provide a logical model and a semantics for factor-based CBR.

Second, we investigate the idea of normative explanation: While the literature on the concept of explanation is immense, the AI community is now paying attention to it due to the development of explainable AI (XAI) [15,[START_REF] Atkinson | Explanation in ai and law: Past, present and future[END_REF]. Our paper, by connecting CBR and reasoning about classifier systems, explores different notions of explanation in law, such as abductive and contrastive explanations for the outcome suggested by the case-based reasoner. Our model allows for building explainable case-based reasoners, which could also be deployed to reproduce and analyze the functioning of opaque predictors of the outcome of cases. We import notions such as prime implicant and contrastive explanation in the domain of XAI for classifiers to showcase how to analyze CBR in the field of XAI.

The the paper is organised as follows. Section 2 presents Horty's models of CBR. Section 3 introduces the notion of classifier model (CM) for the binary-input classifier logic BCL. Section 4 studies the connection between CBR and classifier models.Section 5 shows that notions for classifier explanation in XAI help study case base. Finally, Section 6 discusses related work and concludes. Proofs and the axiomatics are in the appendix. 1

Horty's Two Models of Case-Based Reasoning

In this section we account for the two models of case-based reasoning / precedential constraint proposed by Horty. We simply say result model for "the factor-based result model of precedential constraint" and reason model for "the factor-based reason model of precedential constraint".

Let Atm 0 = Plt ∪ Dfd, where Plt and Dfd are disjoint sets of factors favoring the plaintiff and defendant respectively. In addition, let Val = {1, 0, ?} where elements stand for plaintiff wins, defendant wins and indeterminacy respectively. Let Dec = {t(x) : x ∈ Val} and read t(x) as "the actual decision/outcome (of the judge/classifier) takes value x". An outcome t(1) or t(0) means that, the judge is predicted to decide for the plaintiff or for the defendant (the classifies "forces" one of the two outcomes). The outcome t(?) means either outcome would be consistent: the judge may develop the law in one direction or the other. This reflects the incompleteness nature of CBR. We use Atm to denote Atm 0 ∪ Dec.

We call s ⊆ Atm 0 a fact situation. A set of atoms X is called a reason for an outcome (decision) x if it a set of factors all favoring the same outcome: X ⊆ Plt is a reason for 1 and X ⊆ Dfd is a reason for 0. A (defeasible) rule consist of a reason and the corresponding outcome: X → x is rule, if X ⊆ Plt and x = 1, or X ⊆ Dfd and x = 0. For readability, we make a convention that, for x ∈ {0, 1}, let x = 1x and x = x. Moreover, let Atm x 0 = Plt if x = 1, and Atm x 0 = Dfd if x = 0. In the reason model, a precedent case (precedent) is a triple c = (s, X, x), where s ⊆ Atm 0 , X ⊆ Atm x 0 , x ∈ {0, 1}. In plain words, s ∩ Atm x 0 contains all pro-factors in s for x, while s ∩ Atm x 0 all con-factors in s for x. X is the reason of the case, namely a subset of the pro-factors which the judge considers sufficient to support that outcome, relative to all con-factors in the case.

A case base CB (for reason model) is a set of precedential cases. When the reason contains all pro-factors within the situation (i.e., when c = (s, s∩Atm x 0 , x)) all such factors are considered equally decisive. If a case base only contains cases of this type, we obtain what Horty calls "the result model", and note such a case base CB res . 2 The class of all CBs and CB res s are noted CB and CB res respectively. Example 1 (Running example). In the paper we refer to the following running example taken from [START_REF] Prakken | A formal analysis of some factor and precedentbased accounts of precedential constraint[END_REF]. Let us assume the following six factors, each of which either favors the outcome 'misuse of trade secrets' ('the plaintiff wins') or rather favors the outcome no misuse of trade secrets ('the defendant wins'): the defendant had obtained the secret by deceiving the plaintiff (π 1 ) or by bribing an employee of the plaintiff (π 2 ), the plaintiff had taken security measures to keep the secret (π 3 ) , the information is obtainable elsewhere (δ 1 ), the product is reverse-engineerable (δ 2 ) and the plaintiff had voluntarily disclosed the secret to outsiders (δ 3 ). Hence in our running example

Atm = {π 1 , π 2 , π 3 , δ 1 , δ 2 , δ 3 , t(0), t(1), t(?)} Let us consider a case base CB ex = {c 1 , c 2 ) where c 1 = ({π 1 , π 3 , δ 1 , δ 3 }, {π 1 }, 1); c 2 = ({π 2 , δ 1 , δ 3 }, {δ 3 }, 0), which means: • c 1 has factors (fact situation) s 1 = {π 1 , π 3 , δ 1 , δ 3 }, reason {π 1 } and outcome 1; • c 2 has outcome δ , factors s 2 = {π 2 , δ 1 , δ 3 }, reason {δ 3 } and outcome 0
A case base can be inconsistent when two precedents map the same fact situation to different outcomes. Another scenario is that a consistent case base becomes inconsistent after update, namely after expanding it with some new case. Hence maintaining consistency is the crucial concern of case-based reasoning. But first of all, one need to define these notions. The following definitions, except symbolic difference, are based on [START_REF] Horty | Rules and reasons in the theory of precedent[END_REF][START_REF] Prakken | A formal analysis of some factor and precedentbased accounts of precedential constraint[END_REF].

Definition 1 (Preference relation derived from a case). Let c = (s, X, x) be a case. There is more than one way to satisfy the precedential constraint, depending on how the precedents in CB interacts with the new case. The requirement of consistency dictates the outcome when the "a fortiori" constraint applies: if reason X for x outweighs (i.e., is stronger that) reason s ∩ Atm x 0 , a fortiori any superset of X outweighs any subset of s ∩ Atm x 0 , so that only by deciding for x rather than for x consistency is maintained. 3Example 2 (Running example). Let us consider two fact situations according to case base CB ex running example.

• In s 3 = {π 1 , π 3 , δ 1 }, only a decision for 1 in s 3 is consistent with CB ex , since a decision for 0 would entail that {δ 1 } > CB ex {π 1 }, contrary to the preference {π 1 } > CB ex {δ 1 }, which is derivable from c 1 . • In s 4 = {π 2 , δ 2 } both (s 4 , {π 2 }, 1) and (s 4 , {δ 2 }, 0) are consistent with CB ex , since neither {π 2 } > CB ex {δ 2 } nor {δ 2 } > CB ex {π 2 }.

Classifier Model of Binary-input Classifier Logic

In this section we introduce the language and semantics of binary-input classifier logic BCL first appeared in [START_REF] Liu | A logic for binary classifiers and their explanation[END_REF]. Recall that Atm = Atm 0 ∪ Dec, where Atm 0 = Dfd ∪ Plt, and Dec = {t(x) : x ∈ Val = {0, 1, ?}}. The modal language L (Atm) of BCL is defined as:

ϕ ::= p | t(x) | ¬ϕ | ϕ ∧ ϕ | [X]ϕ,
where p ranges over Atm 0 , t(x) ranges over Dec, and X is a finite subset of Atm 0 . 4Operator X is the dual of [X] and is defined as usual: X ϕ = def ¬[X]¬ϕ. Finally, for any X ⊆ Y ⊆ Atm 0 , the following definition syntactically expresses a valuation on Y s.t. all variables in X are assigned as true, while all the rest in Y are false.

cn X,Y = def p∈X p ∧ p∈Y \X ¬p.
The language L (Atm) is interpreted relative to classifier models defined as follows.

Definition 5 (Classifier model). A classifier model (CM) is a pair C = (S, f ) where:

• S ⊆ 2 Atm 0 is
(C, s) |= p ⇐⇒ p ∈ s, (C, s) |= t(x) ⇐⇒ f (s) = x, (C, s) |= ¬ϕ ⇐⇒ (C, s) |= ϕ, (C, s) |= ϕ ∧ ψ ⇐⇒ (C, s) |= ϕ and (C, s) |= ψ, (C, s) |= [X]ϕ ⇐⇒ ∀s ∈ S : if (s ∩ X) = (s ∩ X) then (C, s ) |= ϕ.
A formula ϕ of L (Atm) is said to be satisfiable relative to the class CM if there exists a pointed classifier model (C, s) with C ∈ CM such that (C, s) |= ϕ. It is said to be valid if ¬ϕ is not satisfiable relative to CM and noted as |= CM ϕ.

We can think of a pointed model (C, s) as a pair (s, x) in f with f (s) = x. The formula [X]ϕ is true at a state s if ϕ is true at all states that are modulo-X equivalent to state s. It has the selectis paribus (SP) (selected things being equal) interpretation "features in X being equal, necessarily ϕ holds (under possible perturbation on the other features)".

[Atm 0 \ X]ϕ has the standard ceteris paribus (CP) interpretation "features other than X being equal, necessarily ϕ holds (under possible perturbation of the features in X)". Notice when X = / 0, [ / 0] is the S5 universal modality since every state is modulo-/ 0 equivalent to all states, viz. (C, s) |= [/ 0]ϕ ⇐⇒ ∀s ∈ S, (C, s ) |= ϕ.

Representation between Consistent Case Base and CM

In this section we shall show that the language of case bases can be translated into the language L (Atm); hence case bases can be studied by classifier models. More precisely, a case base is consistent iff its translation, together with the following two formulas that we abbreviate as Compl and 2Mon, is satisfiable in the class CM:

Compl = def X⊆Atm 0 / 0 cn X,Atm 0 2Mon = def x∈{0,1},X⊆Atm x 0 ,Y ⊆Atm x 0 / 0 (cn X∪Y ,Atm 0 ∧ t(x)) → Atm x 0 ⊇X ⊇X,Y ⊆Y [ / 0](cn X ∪Y ,Atm 0 → t(x))
According to Compl, every possible situation description must be satisfied by the classifier, where a situation description is a conjunction of factors (those being present X) and negations of factors (those being absent, Atm 0 \ X).

2Mon introduces a two-way monotonicity, which is meant to implement the a fortiori constraint: if the classifier associates a situation s to an outcome x, then it must assign the same outcome to every situation s such that both (a) s includes all factors for x that are in s and (b) s does not include factors for x that are outside of s. This formula is meant to maintain consistency with respect to the preference relation, as Definition 1 indicates: if a case including reason X for x and factors Y for x, has outcome x, it means that X > Y . Thus it cannot be that outcome x is assigned to a situation s including both a superset X ⊇ X of factors for x and only a subset Y ⊆ Y of factors for x. In fact, if X > Y , then is must be the case that also X > Y , while a decision for x would entail that X < Y .

Let

CM prec = {C = (S, f ) ∈ CM : ∀s ∈ S, (C, s) |= Compl ∧ 2Mon}
, where CM prec means the class of CMs for precedent theory. Satisfiability and validity relative to CM prec are defined in an analogous way as CM.

To translate a result-model case-base CB res into a classifier model (C, f ), we need to ensure that all precedents in the case-base are satisfied by the classifier, with regard to both their factors and their outcome.

Definition 7 (Translation of case base for result model). The translation function tr 1 maps each case from a case base CB res to a corresponding formula in the language L (Atm). It is defined as follows:

tr 1 (s, s ∩ Atm x 0 , x) = def / 0 (cn s,Atm 0 ∧ t(x)).
We generalize it to the entire case base CB res as follows:

tr 1 (CB res ) = def (s,s∩Atm x 0 ,x)∈CB tr(s, s ∩ Atm x 0 , x).
Therefore, in the result model a precedent (s, s ∩ Atm x 0 , x) is viewed as a situation s being classified by f as x.

Example 3 (Running example). The case

({π 1 , π 2 , δ 1 }, {π 1 , π 2 }, 1}) is translated as / 0 (π 1 ∧ π 2 ∧ δ 1 ∧ ¬π 3 ∧ ¬δ 2 ∧ ¬δ 3 ∧ t(1)), which means that f (π 1 , π 2 , δ 1 ) = 1
In translations for the reason model we need to capture the role of reasons. This is obtained by ensuring that for every case (s, X, x), not the fact situation s directly, but the one consisting only of reason X and all x-factors in s (i.e. s ∩ Atm x 0 ) is classified as x. It reflects that the precedent finds x-factors in s outside of X dispensable for the outcome.

Definition 8 (Translation of case base for reason model).

The translation function tr 2 maps each case from a case base CB to a corresponding formula in the language L (Atm). It is defined as follows:

tr 2 (s, X, x) = def / 0 (cn X∪(s∩Atm x 0 ),Atm 0 ∧ t(x)).
We generalize it to the entire case base CB as follows:

tr 2 (CB) = def (s,s∩Atm x 0 ,x)∈CB tr 2 (s, s ∩ Atm x 0 , x).
Notice that the function tr 1 for the result model is a special case of the function tr 2 for the reason model, since ((s ∩ Atm x 0 ) ∪ (s ∩ Atm x 0 ) = s. Fact 1. tr 1 (s, s ∩ Atm x 0 , x) = tr 2 (s, s ∩ Atm x 0 , x). The formulas 2Mon and Compl require that the outcome x supported by reason X in a precedent is assigned to all possible cases including X that do not contain additional factors against x. If both formulas are satisfiable then the case base is consistent, as stated by the following theorem.

Theorem 1. Let CB ∈ CB be a case base. Then, CB is consistent iff tr 2 (CB) is satisfiable in CM prec .
In light of the theorem and the fact above, the representation of a case base for result model turns to be a corollary.

Corollary 1. Let CB res ∈ CB res be a case base for the result model. Then, CB res is consistent iff tr 1 (CB res ) is satisfiable in CM prec .

Similarly, the precedential constraint can also be represented as a corollary.

Corollary 2. Let CB ∈ CB be a consistent case base and (s , X, x) a case. Updating CB with (s , X, x) meets the precedential constraint, iff tr 2 (CB) ∧ tr 2 (s , X, x) is satisfiable in CM prec . Example 4 (Running example). Case c 3 = ({π 1 , π 2 , δ 2 }, {δ 2 }, 0) is incompatible with the CB ex . According to tr 2 (CB ex ∪ {c 3 }), 2Mon and Compl, the fact situation {π 1 , π 2 , δ 1 } should be classified both as 1, based on CB ex , and 0, based on c 3 .

Explanations

The representation results above pave the way to providing explanations for the outcomes of cases. For this purpose it is necessary to introduce the following notations. Let λ denote a conjunction of finitely many literals, where a literal is an atom p (positive literal) or its negation ¬p (negative literal). We write λ ⊆ λ , call λ a part (subset) of λ , if all literals in λ also occur in λ ; and λ ⊂ λ if λ ⊆ λ but not λ ⊆ λ . We write Lit(λ ), Lit + (λ ), Lit -(λ ) to mean all literals, all positive literals and all negative literals in λ respectively. By convention is a term of zero conjuncts. In the glossary of Boolean classifier (function), λ is called a term or property (of the instance s). The set of terms is noted Term. A key role in our analysis is played by the notion of a (prime) implicant, i.e., a (subset-minimal) term which makes a classification necessarily true. Definition 9 (Implicant (Imp) and prime implicant (PImp)). We write Imp(λ , x) to mean that λ is an implicant for x and define it as Imp(λ , x) = def [ / 0](λ → t(x)). We write PImp(λ , x) to mean that λ is a prime implicant for x and define it as

PImp(λ , x) = def [ / 0] λ → t(x) ∧ p∈Atm(λ ) Atm(λ ) \ {p} ¬t(x) .
According to the definition, λ being an implicant for x means that any state s verifying λ is necessarily classified as x (necessity); and λ being a prime implicant for x means that any proper subset of λ is not an implicant for x (minimality). 5 Implicants explain the classifier in the sense that to know an implicant satisfied at a state is to know the classification of the state.

Intuitively, for a case base containing precedent (s, X, x) to be consistent, s must be incompatible with every prime implicant λ for x. To guarantee that, either λ must have some literal ¬p, where p is in X and hence is true at s; or λ must have some literal p, where p / ∈ s ∩ Atm x 0 and hence is false at s.

δ 3 }) = 1, contrary to c 2 .
In XAI, people [START_REF] Shih | Formal verification of bayesian network classifiers[END_REF][START_REF] Darwiche | On the reasons behind decisions[END_REF][START_REF] Ignatiev | Abduction-based explanations for machine learning models[END_REF]] also focus on "local" (prime) implicants, namely (prime) implicants true at a given state. We adopt the definitions of abductive explanations in [START_REF] Ignatiev | Abduction-based explanations for machine learning models[END_REF][START_REF] Huang | Tractable explanations for d-dnnf classifiers[END_REF], and express these notions in L (Atm) as follows:

Definition 10 (Abductive explanation (AXp) and weak abductive explanation (wAXp) ). We write AXp(λ , x) to mean that λ abductively explains the decision x and define it as AXp(λ , x) = def λ ∧ PImp(λ , x). We write wAXp (λ , x) to mean that λ weak-abductively explains the decision x and define it as wAXp (λ , x) = def λ ∧ Imp(λ , x).

The proposition below states that to be the reason (of a fact situation) is to be the positive part of some weak AXp of that situation. Notice a reason is not always the positive part of some AXp, since reasons in precedent do not in general respect minimality. In fact, we always know one weak AXp for a precedent (s, X, x) in a consistent case base, i.e., the conjunction of all factors in X and negations of all x-factors that are in s. The idea of contrastive explanation is dual with abductive explanation, since it points to a minimal part of a situation whose change would falsify the current decision, and the duality between their weak versions is similar [START_REF] Huang | Tractable explanations for d-dnnf classifiers[END_REF]. A conjunction of literals λ is a contrastive explanation for outcome x in situation s, if the following conditions are satisfied: (a) λ is true at s, and s has outcome x, (b) if all literals in λ were false then the outcome would be different, (c) λ is the subset-minimal literals satisfiying (a) and (b). A weak contrastive explanation is only based on conditions (a) and (b).

Definition 11 (Contrastive explanation (CXp) and weak contrastive explanation (wCXp)). We write CXp(λ , x) to mean that λ constrastively explains the decision x and define it as

CXp(λ , x) = def λ ∧ Atm 0 \ Atm(λ ) ¬t(x) ∧ p∈Atm(λ ) [(Atm 0 \ Atm(λ )) ∪ {p}]t(x).
We write wCXp (λ , x) to mean that λ weak-contrastively explains the decision x and define it as wCXp (λ , x)

= def λ ∧ t(x) ∧ Atm 0 \ Atm(λ ) ¬t(x).
Intuitively speaking, we can test whether λ is a wCXp of situation s having outcome x by "flipping" its positive literals to negative, and negative to positive, and observe if the resulting state is classified differently from x. CXp is the subset-minimal wCXp.

Weak CXps can be used to study the preferences between reasons in a case base. The next proposition indicates that given a precedent (s, X, x), if the absence of Y at s, by itself alone can weakly contrastively explain x, then Y is "no weaker than" X in CB. Example 7. Let C ∈ CM prec be a model of tr 2 (CB ex ). Since {δ 3 } < CB ex {π 1 }, we have (C, s 2 ) |= wCXp (cn / 0,{π 1 } , 0). Indeed f (π 1 , δ 1 , δ 3 ) = 0 by 2Mon according to s 1 .

Related Work and Conclusion

In this paper, we have shown that through the concept of classifier a novel logical model of factor-based case-based reasoning can be provided, which allows for a rigorous analysis of case bases and of the inferences they support.

As noted in the introduction, our work is based upon the case-based reasoning models of HYPO and CATO [START_REF] Ashley | Modeling Legal Argument: Reasoning with Cases and Hypotheticals[END_REF][START_REF] Vincent Aleven | Using background knowledge in case-based legal reasoning: a computational model and an intelligent learning environment[END_REF] and upon the analysis of precedential constraint by Jeff Horty [START_REF] Horty | Rules and reasons in the theory of precedent[END_REF][START_REF] Horty | A factor-based definition of precedential constraint[END_REF]. Further approaches exist that make use of logic in reasoning with cases. For instance, [START_REF] Prakken | Modelling reasoning with precedents in a formal dialogue game[END_REF] provided a factor-based model based on formal defeasible argumentation. More recently [START_REF] Zheng | Case-based reasoning with precedent models: Preliminary report[END_REF][START_REF] Zheng | Precedent comparison in the precedent model formalism: theory and application to legal cases[END_REF] represent precedents as propositional formulas and compare precedents by (propositional) logical entailment.

However, this propositional representation does not fully use the power of logic, in the sense that it does not provide a proof theory (axiomatics) for reasoning with precedents. By contrast, besides the semantic framework presented here, we can make syntactic derivations of properties of CBR using the axiomatics of BCL (see in Appendix).

Moreover, our representation results allow for exploring different notions of explanation, such as abductive and contrastive explanations. We can accordingly explain why a case-based reasoning suggests a particular outcome (rather then a different one) in a new case. Thus, out model could be used to build explainable case-based reasoners,

  a set of states (or fact situations), and • f : S -→ Val is a decision (or classification) function. The class of classifier models is noted CM. A pointed classifier model is a pair (C, s) with C = (S, f ) a classifier model and s ∈ S. Formulas in L (Atm) are interpreted relative to a pointed classifier model, as follows. Definition 6 (Satisfaction relation). Let (C, s) be a pointed classifier model with C = (S, f ) and s ∈ S. Then:

Proposition 2 .

 2 Let CB be a consistent case base, (s, X, x) ∈ CB, and C ∈ CM prec s.t. (C, s) |= tr 2 (CB). Then ∃λ ∈ Term s.t. Atm(Lit + (λ )) = X and (C, s) |= wAXp (λ , x).

Proposition 4 .

 4 Let CB be a consistent case base and (s, X, x) ∈ CB, andC ∈ CM prec s.t. (C, s) |= tr 2 (CB). If (C, s) |= wCXp (cn / 0,Y , x), then it is not the case that Y < CB X.

  Definition 2 (Preference relation derived from a case base). Let CB be a case base. Then the preference relation < CB derived from CB is s.t. for any two reasons Y,Y favoring x and x respectively, Y < CB Y iff ∃c ∈ CB s.t. Y < c Y . Definition 3 ((In)consistency). A case base CB is inconsistent, if there are two reasons Y,Y s.t. Y < CB Y and Y < CB Y . CB is consistent if it is not inconsistent.

	Definition 4 (Precedential constraint). Let CB be a consistent case base, X is a reason
	for x in CB and applicable in a new fact situation s , i.e. X ⊆ s . Updating CB with the new
	case (s , X, x) meets the precedential constraint, iff CB ∪ {(s , X, x)} is still consistent.

Then the preference relation < c derived from c is s.t. for any two reasons Y , Y favoring x and x respectively, Y <

c Y iff Y ⊆ s ∩ Atm x

0 and X ⊆ Y .

  Proposition 1. Let CB be a consistent case base and (s, X, x) ∈ CB, and C ∈ CM prec s.t. (C, s) |= tr 2 (CB). Let C = (S, f ) ∈ CM prec and tr 2 (CB ex ) is satisfiable in C. Obviously π 1 cannot be PImp for 0, otherwise f (s 1 ) = 0, contrary to c 1 . Also ¬δ 2 ∧ π 2 cannot be PImp for 1, otherwise f ({π 2 , δ 1 ,

	Then, ∀λ ∈ Term,, if (C, s) |= PImp(λ , x), then either
	X ∩ Atm(Lit -(λ )) = / 0 or s ∩ Atm x 0	Atm(Lit + (λ )).
	Example 5.

  Proposition 3. Let CB be a consistent case base, (s, X, x) ∈ CB, and C ∈ CM prec s.t. (C, s) |= tr 2 (CB). Then we have (C, s) |= wAXp (cn X,X∪(s∩Atm x 0 ) , x). Let C ∈ CM prec be a model of tr 2 (CB ex ). Then we have (C, s 1 ) |= wAXp (π 1 ∧ ¬δ 2 , 1) and (C, s 2 ) |= wAXp (δ 2 ∧ ¬π 1 ∧ ¬π 2 , 0). Notice that (C, s 2 ) |= ¬wAXp (δ 2 , 0), because e.g. (C, s 1 ) |= δ 2 ∧ ¬t(0).

	Example 6.

The paper with appendix is available here: https://arxiv.org/abs/2210.11217.X. Liu et al. / Modelling and Explaining Legal Case-Based Reasoners Through Classifiers

So we view result model as a special kind of reason model, as[8, p. 25] also mentioned.X. Liu et al. / Modelling and Explaining Legal Case-Based Reasoners Through Classifiers

We generalize a fortiori constraint from only acting on result models in[START_REF] Horty | Rules and reasons in the theory of precedent[END_REF] to also on reason models in the same manner as viewing a result models as a special reason model, whose reason contains all pro-factors.

Atm is finite since the factors in case-based reasoning are supposed to be finite. Notice p ranging over Dfd ∪Plt, i.e. p can be some δ or some π. X can denote a reason (an exclusive set of plaintiff/defendant factors), or any subset of Atm 0 , which is clear from the context. Last but not least, p and t(x) have different statuses regarding negation: ¬p means that the input variable p takes value 0, but ¬t(x) merely means the output does not take value x: we do not know which value it takes, since the output is trinary.

Notice that we have not fully used the expressive power of [X]ϕ and X ϕ until now for minimality. The intuitive meaning of Atm(λ ) \ {p} ¬t(x) in the formula is that even if we just perturb one variable p in λ from its actual value, the classification will possibly no longer be x.

which could also be deployed to reproduce and analyze the functioning of opaque predictors of the outcome of cases. Thus, by bringing CBR into the broader context of classifier systems, we connect three lines of research: legal case-based reasoning, AI&Law approaches on to explanation [3], techniques and results developed in the context of XAI.

In future work we will examine more deeply the relation between classifiers, explanations, and reasoning with legal precedents. Interesting developments pertain to addressing analogical reasoning beyond the a fortiori constraint considered here and to deploying ideas of explanation to extract knowledge out of cases (e.g., to determine the direction of factors and the way in which they interact).