« Si j'ai appris une chose au cours de ma longue vie, c'est que toute notre science, confrontée à la réalité, apparaît primitive et enfantine -et pourtant, c'est ce que nous possédons de plus précieux. » « Ce qu'il y a de plus incompréhensible au monde, c'est que le monde est compréhensible. » Albert EINSTEIN

« Une théorie nouvelle ne triomphe jamais, ce sont ses adversaires qui finissent par mourir. » Max PLANCK kindness and for all the good times we have spent together during these three years. Moreover, even if I had not the chance to carry out an experiment with Mr. Hervé CLOEZ, I would like to thank him for the legacy he has left with the JOSEFA facility. I am deeply thankful to Mr. Benoit LACROIX for all the very good times we spent together during these three years.

In addition, I would like to express my gratitude to Mr. Jean François MAHIEU and Mr. Yannick MINEO for their king help in fixing informatics-related issues and to Mrs. Evelyne POITEVIN and Mrs. Stéphanie SANCHEZ for their profound kindness, availability and efficiency in any administrative process.

Working within the IRFM during these three years has been a real pleasure and I would therefore also like to thank those whose presences, discussions and sharing have been greatly appreciated: Mr. Quentin LE COZ, Mrs. Sylvie NICOLLET, Mr. Patrick PROCHET, Mr. Patrick DECOOL, Mr. Clément NGUYEN, Mr. Jérémy FAUCHEUX, Mr. Christophe ROUX, Mr. Jean-Louis MARECHAL, Mr. Patrick HERTOUT.

Finally, I am sincerely indebted to my friends and family who have provided me moral support during the whole PhD project. Especially to Mr. Aymeric FABIEN, Mr. Anthony KOENIG, Mr. Julien MOULLET, Mr. Maxime DUBOIS, Mrs. Margaux CHILETTI, Mrs. Florence CHILETTI and Mrs. Tina CHILETTI.

I conclude these acknowledgments by expressing my profound gratitude to my wife, Tina, for having supported me and accepted the several sacrifices associated with this project. For her help and patience in the everyday life and for her unwavering support since the beginning of this doctoral project.

Table of contents

Glossary :

𝑇 𝑐 : critical temperature (K)

I. Introduction

I.1 Superconductivity

Superconducting materials are known since more than a hundred years for their capacity to conduct current without resistance and to expulse magnetic field from their inner volume (Meissner effect). Nowadays, superconducting materials are used in several fields as medicine (ex: MRI, hadron therapy (see [START_REF] Quettier | Tests of a Prototype for Assessing the Field Homogeneity of the Iseult/Inumac 11.7 T Whole Body MRI Magnet[END_REF] and [START_REF] Felcini | Magnetic Design of a Superconducting Toroidal Gantry for Hadron Therapy[END_REF])), particle physics (ex: CERN, see [START_REF] Kate | The superconducting magnet system for the ATLAS detector at CERN[END_REF]), biology (ex: NMR), electronics (ex: Josephson junction), fusion (ex: superconducting tokamaks in operation) or even military application (ex: SMES for electromagnetic launcher, see [START_REF] Badel | SMES to supply an Electromagnetic Launcher[END_REF] and [START_REF] Sineglazov | Hybrid energy storage system design[END_REF]). In this thesis, we are focusing on low temperature superconductor as NbTi or Nb3Sn composite strands used for fusion magnets.

Discovered by H. Kammerlingh Onnes in 1911, superconducting state appears when specific materials are cooled down below a critical temperature named 𝑇 𝑐 (see Figure 1). The critical temperature of a material depends on its chemical composition and for most of them, named conventional superconductors, it is between 1 and 33 K. A cooling down with liquid helium is mandatory in order to achieve this range of temperature. Superconductivity is a quantum effect that is observable macroscopically. In "regular" superconductors, electrons are gathered in pairs (named Cooper pairs) and interact with the surrounding nucleus of atoms. Cooper pairs are moving like an electronic collective wave inside the material. That is why the material needs to be cooled down: the low temperature makes the nucleus of atoms more stable and slows down their vibration. It allows the wave of cooper pairs to propagate without collision and energy losses.

As we said, we need to reach a peculiar temperature in order to access the superconducting state of a material (no resistivity, no magnetic field inside the superconductor, see [START_REF] Tixador | Les supraconducteurs[END_REF]). Nevertheless, these two last assertions are not always true:

-If we force too much current density inside the superconducting material, it will go back to its normal state, this threshold in current density is defined by convention and named 𝐽 𝑐 .

-In case of type II superconductivity, which is characterizing most of superconductors used in applied superconductivity, the magnetic field is allowed to penetrate into the material. The material is in a mixed state presenting an array of cores in normal state where the magnetic field is present. Increasing the magnetic field, the array of cores densifies until 𝐵 𝑐 2 , where superconductivity disappears. 𝐵 𝑐 2 can be very high which makes these superconductors very attractive for applications.

All of these characteristics are described in the 3D curves in Figure 2. The critical surface of a material defines all the possible operating condition for the superconducting state to be present.

I.2 Superconductors (CICCs) and Fusion tokamaks

Approximatively at the same time of the discovery of superconductivity, A. Eddington explained that thermonuclear fusion is probably on going inside core of stars producing enough energy to sustain their gravitational pressure and shine across millions of kilometers. E. Rutherford demonstrates for the first time that fusion is accessible in laboratory by merging deuterium atoms. From this first attempt, the run to an experimental setup capable to achieve fusion at a larger scale (centimeter, meter) was started and in 1950, G.P. Thomson and M. Blackman patented a fusion reactor using a toroidal shape to confine the plasma. The first corresponding experimental setup was achieved in SSSR in 1969 and named tokamak for "toroidal chamber with magnetic system". Experimental fusion machines have taken several forms since their first realisation: toroidal, spherical, and even more complex (stellarator).

Several confinement systems are used to maintain the plasma such as magnetic confinement based on conventional (copper based) or superconducting (NbTi or Nb3Sn) magnet system. The confinement system using toroidal field coils (superconducting or not) is placed as shown in Figure 3.Initially, fusion reactors were promoted under merits of its low production of nuclear wastes and of the great availability of its fuel (deuterium and tritium from lithium). Superconducting magnets systems are absolutely essential for their low energy cost during operation in order to produce in very large volumes the large magnetic field able to confine the plasma.

The need for superconducting magnets in large fusion devices was already recognized from the beginning. In the middle of the 1970's, the initial major development programs were started based on the first definitions for the required magnet parameters. These programs followed two lines. Firstly was the development of conductor and magnet systems for small and medium size plasma devices such as T-7 and T-15 in the Soviet-Union [START_REF] Smirnov | Tokamak foundation in USSR/Russia1950-1990[END_REF], TRIAM in Japan [START_REF] Itoh | Construction of high magnetic field tokamak 'TRIAM-1' for nuclear fusion research[END_REF] and most prominently TORE SUPRA (now WEST) at CEA in France [START_REF] Aymar | Tore Supra. Basic design Tokamak system[END_REF]. Secondly, in the form of an international project, was initiated the development of a conductor and magnet arrangement with parameters pertinent to large fusion devices and their test in a dedicated special facility, the Large Coil Task (LTC) project (see [START_REF] Beard | The IEA Large Coil Task[END_REF]).

Following the successful execution of these projects, the design and construction of larger fusion devices with superconducting confinement magnets were started. Some of them are already in operation: the tokamaks EAST in China [START_REF] Wan | Progress of the EAST project in China[END_REF] and KSTAR in Korea [START_REF] Choi | The KSTAR tokamak[END_REF], the stellarator LHD in Japan [START_REF] Motojima | LHD Magnet System Design and Construction[END_REF], the SST1 tokamak in India [START_REF] Saxena Et S.-1 | Present status of the SST-1 project[END_REF] and the stellarator Wendelstein W7-X in Germany [START_REF] Erckmann | The W7-X project: scientific basis and technical realization[END_REF]. Finally, the international fusion community felt brave enough to start the development, in a worldwide effort, of ITER, a tokamaktype reactor [START_REF] Aymar | The ITER project: A physics and technology experiment[END_REF], [START_REF] Shimomura | Review of the ITER Project[END_REF] accompanied by a satellite tokamak JT-60SA, in Japan, with major contributions from Europe [START_REF] Ishida | Overview of the JT-60SA project[END_REF].

For the ITER coils, the requirements for high currents in the 70-80 kA range and for a very high voltages in operations, 10 to 20 kV to ground for the poloidal field and central solenoid systems inherent in the size of the magnetic systems, led to the selection of the cable-in-conduit-conductor (CICC, see [START_REF] Lue | Stability of cable-in-conduit superconductors[END_REF]) as the best choice for the conductors in the present state of the superconducting technology. Moreover, this type of conductor is well adapted to support fast heat deposition. The principle of CICC is not recent: M. Hoenig at MIT (USA) introduced it in 1975 (see [START_REF] Hoenig | Dense supercritical-helium cooled superconductors for large high field stabilized magnets[END_REF]) and all the superconducting tokamaks in operation except TORE SUPRA used this concept. The coil which pioneers this concept was the Westinghouse coil in the Large Task Coil described in [START_REF] Beard | The IEA Large Coil Task[END_REF], where Nb3Sn was the superconductor. The maximum performance of this magnet was unfortunately limited by some spreading out of a resistive phase in the magnet.

A modern CICC is basically made of several stages by cabling superconducting and copper strands and then by compacting the cable inside a conduit generally made of stainless steel. A CICC such as the one used for ITER is composed of several components, superconducting strands, copper strands, steel bandages (named wrapping), one or more helium channels and the steel encasing conduit, as shown in Figure 4 below. In a project like ITER the optimum composition of the conductor components is defined by the system design criteria.

The CICC was invented to benefit from the very high volumetric heat capacity of helium, about 500 times the volumetric heat capacity of metallic materials, limiting by the way the temperature excursions in the case of fast energy deposition. This occurs in tokamaks after a very fast decrease to zero of the plasma current when plasma disruption occurs. In this case a fast magnetic field variation, with a time constant of the order of 100 ms for example, affects the whole coil over lengths of several meters, creating losses in the superconducting strands. This is rather similar to the kind of event which can occur in high field test facilities, affecting the outer superconducting magnet of a hybrid magnet when the central copper magnet disrupts. The CICC offers an adequate solution to this problem by providing:

-A local helium reservoir -A very long wetted perimeter. The diameter of the ITER TF cable is 39.7 mm. It is made of 900 superconducting strands 0.82 mm in diameter with a void fraction in the cable of 30 %. This fine subdivision of the strands can be translated in a total of 2.3 meters of wetted perimeter in the cable section, facilitating a large heat transfer to the helium reservoir. -Small AC losses for the conductor by controlling its time constant through the contact resistance between strands (through cable void fraction/compaction).

The strands of CICC's are magnetically transposed thanks to the cabling. This transposition is not perfect but it ensures an optimal sharing of the current among the strands in inductive mode.

During operation, the helium mass flow circulating in the conductor limits the temperature increase due to the residual nuclear heating and to the AC losses generated by the varying magnetic fields during a plasma discharge. The central channel in some cables as shown in Figure 4 helps to keep the pressure drop at an acceptable level. The elementary brick of CICC for fusion superconducting magnets is the composite strand. "Composite" means the strand is a mixed assembly of copper and superconducting filament. Coupling losses are generated in the copper matrix during field and current variations, however twisting the filaments is effective to limit these coupling losses.

There are hundreds of strands in a CICC. The composites strands (cylindrical) are consisting of hundreds superconducting filaments (NbTi, Nb3Sn) twisted in a copper matrix with a twist pitch 𝑙 𝑝 0 (as shown in Figure 5). Filament diameter usually lies in the range of microns and they are usually located in what is known as the filamentary zone. Strands used in fusion machines are often composed of several layers, alternating resistive and filamentary zones. A resistive zone is mainly composed of copper and filamentary zone is mainly composed of twisted filaments of superconducting materials embedded in a normal matrix (see Figure 5). In this filamentary zone, we also find a resistive matrix usually made of copper filling every space between the filaments.

Architecture of the strand is different depending it is based on NbTi or Nb3Sn filaments. This is due to the fact that Nb3Sn is issued from a chemical reaction taking place in the strand once it is produced. Nb3Sn strands have to follow a peculiar thermal cycle to for the Nb3Sn phase to form in the strand. We can see the different architectures in Figure 7. The strand itself can generate losses even though it is in its superconducting state. Subject either to external magnetic field or to current variations, it will generate losses of two kinds: hysteresis losses and coupling losses. Hysteresis losses are due to the current flowing inside the superconducting filaments whereas coupling losses are due to current crossing normal zone to connect two superconducting filaments. These currents, crossing normal zone in a strand, as depicted in Figure 6, are due to current redistribution inside the strand. This current redistribution is due to the fact that, in order to shield its volume from the external magnetic field variation, the twist of filaments in the strand will generate current loops. All these current loops together, limit the external field penetration into the strand filamentary zone by creating a magnetic field inside the strand, which is opposed to the external one.

These composite strands response to an external magnetic field variation is described by the following differential equation (1): Extracted from [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]. 𝐵 𝑖 + 𝜏𝐵 ̇𝑖 = 𝐵 𝑎 [START_REF] Quettier | Tests of a Prototype for Assessing the Field Homogeneity of the Iseult/Inumac 11.7 T Whole Body MRI Magnet[END_REF] Where 𝐵 𝑎 is the external field applied on the superconducting strand (or cable) and 𝐵 𝑖 the internal field response to the applied one. 𝜏 is the time constant related to the current loop inside the strand.

This kind of currents generates, as described above, the so called coupling losses which are defined as:

𝑃 𝑐𝑜𝑢𝑝 𝑠𝑡𝑟𝑎𝑛𝑑 = 𝑛𝜏𝐵 ̇𝑖2 𝜇 0 (2)
where 𝑃 𝑐𝑜𝑢𝑝 𝑠𝑡𝑟𝑎𝑛𝑑 is the power per unit volume of strand with 𝑛 = 2 for a cylindrical composite. 𝑛 being a geometrical factor.

The relation between 𝐵 𝑖 and 𝐵 𝑎 can be illustrated for an applied sinusoidal field. If 𝐵 𝑎 = 𝐵 𝑚 sin(𝜔𝑡) + 𝐵 𝑜𝑓𝑓 , with 𝜔 = 2𝜋𝑓 the angular frequency, using the above first order differential equation (1), we obtain in complex notations:

𝐵 ̅ 𝑖 = 𝐵 𝑚 𝑒 𝑗𝜔𝑡 1 + 𝑗𝜔𝜏 . (3)
Then we can readily give the internal magnetic field amplitude |𝐵 ̅ 𝑖 | as:

|𝐵 ̅ 𝑖 | = 𝐵 𝑚 √1 + (𝜔𝜏) 2 . ( 4
)
The associated power density 𝑃 𝑐𝑜𝑢𝑝 𝑠𝑡𝑟𝑎𝑛𝑑 averaged over time (after a time long compared to 𝜏) will then be:

𝑃 𝑐𝑜𝑢𝑝 𝑠𝑡𝑟𝑎𝑛𝑑 = 𝐵 𝑚 2 2𝜇 0 𝑛𝜏𝜔 2 1 + (𝜔𝜏) 2 .
(

) 5
As it is widely used within the applied superconductivity community, we can also express the losses in terms of average losses per cycle 𝑄 per unit volume (of strand), this can be done very quickly multiplying 𝑃 by the period 𝑇 of the cycle. Using the expression of 𝑃 𝑐𝑜𝑢𝑝 𝑠𝑡𝑟𝑎𝑛𝑑 , we have:

𝑄 𝑐𝑜𝑢𝑝 𝑠𝑡𝑟𝑎𝑛𝑑 = 𝐵 𝑚 2 𝜇 0 𝜋𝑛𝜏𝜔 1 + (𝜔𝜏) 2 . ( 6
)
Of course, these considerations and formulae only concern the coupling losses generated by a strand subjected to a transverse field (as shown in Figure 6) and assume that the outer edge filaments are not saturated and that the composite is not carrying any transport current. In case of saturation, we would need to add the penetration losses, which are well described in [START_REF] Ciazynski | Distributions de courant et pertes à l'intérieur d'un composite multifilamentaire supraconducteur soumis à un champ magnétique variable[END_REF].

I.4 Considerations on losses in superconducting magnets for fusion

Presently, several experimental superconducting tokamaks are in operation. The largest superconducting tokamak ever built (JT-60SA) is in its final phase of assembly and commissioning and first operation is expected end in 2020. The first plasma discharges of the superconducting experimental reactor (ITER) are expected in 2025. Moreover, studies are devoted to the design of reactors producing electricity after ITER (DEMO, DTT). On the energetic plan, fusion is obviously very attractive. The energy consumption of, for example, 400 000 inhabitants is in the range of 500 MW of electric power. In order to produce this amount of energy we would only need to burn 175 kg of deuterium tritium in a fusion plant while we would have used either 1.35 megatons of coal, or 0.9 megaton of petrol, or 12.5 tons of uranium in a fission plant. With renewable energy, it corresponds to 35 𝑘𝑚 2 of solar panels or around 25 𝑘𝑚 2 of wind generator with a 20% load factor.

Using fusion plant instead of fission plant could offer nearly a carbon free electricity as fission but suppressing difficulties linked to fission as:

-Large available resources and suppression of the geopolitical tensions linked to fuel procurement. -Simplification of the problems associated with long lived waste.

-No uncontrolled chain reaction or runaway.

The demonstration of the feasibility of fusion was initiated in the eighties by the production of fusion power in JET and TFTR (see [START_REF] Gibson | The JET project[END_REF] and [START_REF] Meade | Results and plans for the Tokamak Fusion Test Reactor[END_REF]). The operation of existing superconducting tokamak (WEST, EAST, KSTAR, etc.) is preparing ITER even if it was not their initial goal. The main objective of the ITER project, which will deliver first plasmas in 2025, is to demonstrate at a representative scale the feasibility of energy production using a fusion reactor under the form of a tokamak. The objective is to produce thousands of plasma discharges (500 MW during 500 s).

Applied Superconductivity in MRI is mainly in DC field. During ITER scenarios, causing large field variations across the magnets, important losses are developed associated to temperature increase and reduction of temperature margins. This is a new very important challenge for applied superconductivity.

The ITER superconducting magnet system constitutes the largest ITER components, representing about 30% of the machine cost investment. This underlines the fact that designing the magnet system from specific design parameters in order to limit its AC losses remains a challenge. The objective is to keep the superconductors within their critical limits and to mitigate the power associated to the cryogenic system.

Since TORE SUPRA (CEA, Cadarache), now renamed WEST from its tungsten divertor upgrade, several other superconducting tokamaks are in operation. They are presented in Table 1 below: Starting from the first small superconducting magnet in 1962, unprecedented advances have been accomplished in applied superconductivity in particular in MRI. Regarding fusion, important challenges have been mastered such as the development of CICC to support very high currents (50-70 kA), very high voltage (10 kV), the description of AC losses generated by field variation and eliminated by the helium flowing inside the CICC. Connections of several superconducting coils have also been managed with the development of various connections techniques as the twin box developed at CEA Cadarache.

In a CICC, strands are cabled and twisted in stages, and stages are also twisted and cabled as bigger stages. Regarding AC losses, the crucial role of the twist and transposition pitches is well known but the general impact of the geometry and in particular of the contact resistances between the cable strands has still to be explored. The cost investment of these large machines is dependent for a non-negligible part on the conductor itself. Progresses are needed to optimize and to analytically predict the behaviour of the CICCs under field variations. Presently, the one time constant model is used in the fusion community to describe AC losses. Also numerical code (JackPot) or heuristic model (MPAS, for Multizone PArtial Shielding) exist but no analytical tools are capable of predicting the AC losses a CICC will generate with respect to its geometry, electrical contact distribution, and strand type. The objective of the thesis is to progress in this direction. Such a model presently exists at strand scale and it is named CLASS for Coupling Losses Algorithm for Superconducting Strand, developed by A. Louzguiti. This model (fully described in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]) will be shortly introduced in Section II.1.2 for further uses. Based on this first step, he developed a twostage cable model (see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]) where two cabling stages of a CICC (example: the two first ones of JT-60SA TF conductor, a triplet of triplet) magnetically and electrically interact giving the amount of coupling losses generated by the coupling of two cabling stages.

The different versions of the conductors used in ITER are tested in the European SULTAN test facility (CRPP, see [START_REF] Bruzzone | Upgrade of operating range for the SULTAN test facility[END_REF]). The test facility SULTAN inaugurated in 1992 in preparation of ITER has many objectives, among which:

-Critical properties characterisation of CICCs under representative magnetic field -Losses measurement in CICCs in harmonic field variations ranging from 0 to 10 Hz

The experimental coupling losses are fitted using the heuristic MPAS model [START_REF] Turck | A macroscopic model for coupling current losses in cables made of multistage of superconducting strands and its experimental validation[END_REF] (MPAS description will be done in Section II.1.1). The ITER scenarios are divided in small time steps into linear variations of magnetic fields, enabling to calculate the AC losses using the MPAS model and a model of hysteresis losses, then using a thermohydraulic code enables to check that the temperature margins are sufficient. This is the approach presently used in the ITER program to prepare the operation.

I.5 Thesis content and objectives

This thesis work aims at progressing in the analytical description of coupling losses under time varying field inside large superconducting cable in conduit conductors used for the magnet system of a fusion reactor. To illustrate this approach, the discussion about the presented models will be confronted to the modelling of cables of type similar to the TF conductor of JT-60SA.

The MPAS model (see [START_REF] Turck | A macroscopic model for coupling current losses in cables made of multistage of superconducting strands and its experimental validation[END_REF]), which relies essentially on the experimental data of coupling losses gathered by using test facility as SULTAN, is used for ITER to assess the magnetic parameters of the considered cable. This model is not able to predict AC losses but only to give a description on how there are distributed in the cable among each stage. However, the present thesis is in line with the work led in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF], where an analytical model capable to model any two-stage cable was developed. We name this model COLISEUM for COupling Losses analytIcal Staged cablEs Unified Model. This modelling of a two-stage cable is not sufficient to properly model or predict the coupling losses generated by a cable composed with five stages for example. Further development toward higher number of stages has therefore been carried out within the thesis work.

Both of the above models are based on analytical expression of coupling losses but COLISEUM is intrinsically predictive as needing cable-oriented inputs as cable geometry and conductances between the cable stages. Based on the geometrical description of the cable (strand radius, twist pitches, cabling stage radius) and on its electrical description, COLISEUM can compute and tentatively predict induced current and coupling losses of a two-stage CICC (see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]).

In its original state, COLISEUM could describe one-stage or two-stage cables but it has not been tested in a wide variety of parameters in order to check its consistency (check for divergence, limits of the model). We will start presenting briefly the CLASS model and the two versions of COLISEUM in their actual state (one-stage and two-stage description) for further use in the present work. MPAS will also be presented in two derived versions from its original formulation (see [START_REF] Turck | A macroscopic model for coupling current losses in cables made of multistage of superconducting strands and its experimental validation[END_REF]) named restricted and advanced. Of course, these versions of heuristic model (MPAS) are developed to improve the unique time constant model which has shown some weaknesses in coupling losses description of CICC especially in fast transient regimes (see section II.1.1.1).

Using both CLASS and COLISEUM, we will first demonstrate the capability of the one-stage COLISEUM to reproduce the magnetic behaviour (time constant and shielding coefficient) of a strand modelled by CLASS. By using COLISEUM to replicate the strand from CLASS, we will be able to integrate the strand in the modelling of a two-stage cable. This first step will allow us to show an original analytical model for a composite strand as if coupled in multiplets.

The consistency of the two-stage COLISEUM is also tested to consolidate the validity of its description. Range and value of coupling parameters (time constants and shielding coefficients) with respect to cabling parameters (twist pitch, conductances), absence of divergences in the coupling parameters, correspondence with the one-stage COLISEUM are points that will be checked in order to be perfectly confident in the COLISEUM for its iteration to a 𝑛-stage description.

COLISEUM will be presented and discussed in its initial formulation and then simplified t its only relevant driving variables, already at the two-stage level, making it accessible to a fully explicit analytical approach and, further, prepare its extension to the description of cables with more than two stages. The analytical extension of the COLISEUM will be managed and presented as well as some applicative cases to start exploring its predictive capability. Several demonstrations strengthen our modelling and are also discussed in this thesis work.

Also, various collaborations have been pursued during this thesis work. The first one was with INFLPR (RO) and I. Tiseanu on the study of tomographic images of CICC. This study led us to the analysis of the contact distribution of the CICC (JT-60SA TF type). We also started to develop an approach to reconstruct the contact network in the section of cable from tomographic images. Collaboration with the University of Twente has also been continued in the confrontation of both our model (COLISEUM and JackPot). This collaboration aims at confronting both models (JackPot and COLISEUM) in order for COLISEUM to strengthen the validity of its description.

The hypothesis in MPAS will also be discussed in order to provide this model sufficient degrees of freedom regarding the variety of cabling patterns existing among all CICCs.

During this thesis, several experimental campaigns have been led using CICC samples of the same type as the JT-60SA TF production cable. These measurements are led to benchmark our models and quantify the effect of the void fraction on coupling losses. The choice of JT-60SA TF is related to the fact that we have the samples ready to be used due to another previous experiment [START_REF] Zani | Progresses at CEA on EU demo reactor cryomagnetic system design activities and associated R&D[END_REF]. The outputs of the experimental campaigns we led during this thesis are twofold: on one side, we have explored and quantified the effect of various void fractions (different compaction rate) on the CICC sample AC performances. On the other side, we have started to constitute an experimental database in order to confront COLISEUM (once upgraded to an 𝑛-stage description) to the experimental data checking the validity of its predictions. A full description of the JOSEFA facility used to measure AC losses at CEA Cadarache will be given, with all the recent enhancements made on the facility to deliver a sinusoidal applied field. Several reminders will also be done regarding hysteresis and coupling losses modelling in order to describe the evaluation of AC losses from measured data.

The different measurements performed by using JOSEFA on the different versions of the JT-60SA TF conductor will allow us to confront the modelling given both by MPAS and an extended version of COLISEUM.

II. Models

Content: This part is dedicated in a first time to the brief presentation of the four existing models developed at CEA:

-MPAS for cable.

-CLASS for composite strands.

-One-stage model to simulate one isolated cable stage named one-stage COLISEUM.

-Two-stage model to simulate the coupling between two consecutive cabling stages, named two-stage COLISEUM.

In a second time, applications of models (COLISEUM) are shown using JT-60SA TF cable parameters. Preliminary comments are given on both MPAS model and especially on COLISEUM regarding the handling and the outcome of the COLISEUM model at different scales of a cable (strand, one-stage and two-stage). We will demonstrate that the strand model and the two versions of COLISEUM (one-stage and two-stage) can be used at the same scale (i.e. strand scale), to model either a strand using the strand model or the one-stage COLISEUM or to model a multiplet of strand using the two-stage COLISEUM.

Then, based on those results, the first phase of our work is presented with, as preliminary step, a reduction of the dimension of the system equation describing the two-stage COLISEUM from dimension four to dimension two.

Associated publication:

M. Chiletti, J.L.

II.1 Presentation of the models

II.1.1 The MPAS model

The MPAS model requires typically five couples of magnetic parameters to model the coupling experimental losses in a five-stage cable as a function of frequency: five time constants 𝜏 and five shielding coefficients 𝑛𝜅, one coupled (𝜏, 𝑛𝜅) per stage.

Using the initial description of the MPAS model in 2010 in [START_REF] Turck | A macroscopic model for coupling current losses in cables made of multistage of superconducting strands and its experimental validation[END_REF], an effort was produced to reduce for the MPAS users the number of free parameters of the model to ease and rationalize the modelling work. This was done introducing an iterative process between magnetic parameters. Two models are given below to illustrate this effort especially applied on JT-60SA TF type cable (which is particularly homogeneous in pattern): the restricted MPAS and the advanced MPAS.

II.1.1.1 Restricted MPAS

The MPAS model (in its original formulation) is presently spreading within the fusion community. This model, in tight relation with AC losses characterization on samples, offers a method with a good compromise between simplicity and reliability to characterize AC coupling losses in the operation of large superconducting tokamaks. It has been shown (see [START_REF] Turck | A macroscopic model for coupling current losses in cables made of multistage of superconducting strands and its experimental validation[END_REF]) that the classical model, with a unique time constant called "𝑛𝜏", is not adequate to calculate losses for fast magnetic variations such as disruption, Vertical Displacement Event or even the initial discharge of the central solenoid (CS).

MPAS was used for the first time for the estimation of the stability of JT-60SA TF conductor in case of a plasma disruption [START_REF] Turck | A macroscopic model for coupling current losses in cables made of multistage of superconducting strands and its experimental validation[END_REF], [START_REF] Duchateau | Stability of a cable in conduit conductor under fast magnetic field variations[END_REF]. MPAS has been applied for the estimation of the losses of the JT-60SA TF conductor [START_REF] Torre | Impact of a plasma disruption on the magnetic field variations and heat deposition in the TF conductor of JT-60SA[END_REF]. For ITER, MPAS has been used as soon as 2013 in particular for the evaluation of AC losses in the ITER CS, for the reference 15 MA scenario [START_REF] Bessette | Design of a Nb3Sn Cable-In-Conduit-Conductor to withstand the 60000 electromagnetic cycles of the ITER central solenoid[END_REF]. Another application was the estimation of the Minimum Quench energy (MQE) of CS and PF ITER conductors [START_REF] Bagni | Analysis of ITER NbTi and Nb3Sn CICCs experimental minimum quench energy with JackPot, MCM and THEA[END_REF]. It is now being used to estimate the losses of the CS Model coil under development in China within the CFETR reactor program [START_REF] Shi | Coupling loss characterisitics of Nb3Sn CIC conductor for CFETR CS model coil[END_REF].

The main advantage of the MPAS model lies in its simplicity, which makes its application straightforward, as soon as the AC coupling losses characterisation of the conductor is available out of a curve 𝑄(𝑓) obtained in experiments. The objective of this part of the thesis, in complement to the initial publication in Cryogenics [START_REF] Turck | A macroscopic model for coupling current losses in cables made of multistage of superconducting strands and its experimental validation[END_REF], is to recall the bases of the MPAS model, and to detail some practical rules for the derivation of the model. Since 2010, substantial experience has been accumulated for characterisation of JT-60SA and ITER conductors. The resolution of some difficulties associated to this work will be discussed and illustrated along this thesis.

For a multistage cable, its behaviour is represented in MPAS by a given number of domains (typically the number of stages + the basic strand). Each domain is dominated by one stage in the cabling system. It can be represented by only two parameters: a time constant τ and a shielding coefficient 𝑛𝜅. These domains behave as dipoles. That means that, when submitted to an applied sinusoidal field 𝐵 𝑎 , an effective internal field 𝐵 𝑖 is established, related to the external field by the linear differential equation [START_REF] Smirnov | Tokamak foundation in USSR/Russia1950-1990[END_REF]:

𝐵 𝑎 = 𝐵 𝑖 + 𝜏 𝐵 𝑖 ̇. (7)
That means also that, in each of these domains 𝑗, the coupling power loss per unit volume of cable can be calculated with the expression:

𝑃 𝑗 = 𝑛𝜅 𝑗 𝜏 𝑗 𝐵 𝑖 2 μ0 (8)
The total power dissipated in the cable is given by the sum of the contributions 𝑃 𝑗 over the number of domains 𝑗. The modelling is usually made with shielding coefficients 𝑛𝜅 𝑗 referring to the volume of superconducting strand rather than that of the whole cable. There is simply a factor 𝑟 = 𝑆 𝑐𝑎𝑏𝑙𝑒 𝑆 𝑐𝑜𝑚𝑝 between them, where 𝑆 𝑐𝑎𝑏𝑙𝑒 and 𝑆 𝑐𝑜𝑚𝑝 respectively stand for the circumscribed area of the cable and for the area of superconducting strand inside the cable. In the following, the parameters are therefore directly scaled to the composite volume, in practical uses (such as comparisons with experiments) the results are referred to unit of volume of superconducting strand.

An iterative process is presented in details here to determine the relation between the coefficients 𝜅 as fully constrained. Some enhancements of the model are described in the following thesis. It is implicitly considered that the cable is homogeneous, which means that the transverse resistivity is average and constant inside the cable. Also, a first step of simplification is brought about: we reduce the number of degrees of freedom of the model to the minimum, i.e. one shielding coefficient and one time constant. This model is called the restricted MPAS.

𝜅 𝑗 are the shielding coefficients in the volume of composites. Defining the last stage shielding coefficient 𝜅 5 , the other shielding coefficients are expressed by iteration as follows:

𝜅 4 = (1 - 𝜅 5 2𝑟 ) 𝜅 5 𝜅 3 = (1 - 𝜅 5 2𝑟 ) 𝜅 4 = (1 - 𝜅 5 2𝑟 ) 2 𝜅 5 𝜅 2 = (1 - 𝜅 5 2𝑟 ) 𝑘 3 = (1 - 𝜅 5 2𝑟 ) 3 𝜅 5 𝜅 1 = (1 - 𝜅 5 2𝑟 ) 𝑘 2 = (1 - 𝜅 5 2𝑟 ) 4 𝜅 5
The considered case is the one of a five-stage cable.

They can be iterated from the last stage shielding coefficient 𝜅 5 as the only free parameters (for shielding coefficients) using the following relation ( 9):

𝜅 𝑗-1 = (1 - 𝜅 5 2𝑟 ) 𝜅 𝑗 𝑜𝑟 𝑎𝑠 𝜅 𝑗 = 𝜅 5 (1 - 𝜅 5 2𝑟 ) 5-𝑗 (9)
We note that we can change the reference area of the shielding coefficients 𝜅 𝑗 from the area of superconducting material to the circumscribed area of the cable (𝑘 𝑗 ) expressed as follows:

𝜅 𝑗 = 𝑟 𝑘 𝑗 . ( 10
)
where 𝑟 = 𝑆 𝑐𝑎𝑏𝑙𝑒 𝑆 𝑐𝑜𝑚𝑝 .

Regarding time constants 𝜏 𝑗 , they are proportional to the square of its twist pitch length 𝑙 𝑝 𝑗 as already demonstrated for the case of the composite strand as in [START_REF] Wilson | Time-varying fields and A.C. losses[END_REF].

𝜏 𝑗 = 𝛼 𝑙 𝑝 𝑗 2 𝜌 𝑡 (11) 𝜏 𝑗-1 = 𝜏 𝑗 [ 𝑙 𝑝 𝑗-1 𝑙 𝑝 𝑗 ] 2 (12)
with 𝛼 a constant coefficient.

We stress that this version of MPAS called restricted MPAS, using equations ( 9) and [START_REF] Choi | The KSTAR tokamak[END_REF], is the version with the lowest number of degrees of freedom for this heuristic model, i.e. 𝜏 5 and 𝑛𝜅 5 .

For their characterization, all ITER conductors have been tested in SULTAN facility in the range 0-10 Hz [START_REF] Bruzzone | Upgrade of operating range for the SULTAN test facility[END_REF]. In the test facility, the conductor is submitted, in addition to the background magnetic field, to a sinusoidal perpendicular magnetic field excitation, such as: 𝐵 𝑎 ⃗⃗⃗⃗ (𝑡) = 𝐵 𝑜𝑓𝑓 𝑒 𝑥 ⃗⃗⃗⃗ + 𝐵 𝑚 𝑠𝑖𝑛(𝑡)𝑒 𝑦 ⃗⃗⃗⃗ . Both magnetic fields are perpendicular and in addition transverse to the conductor (see Figure 8).

𝐵 𝑜𝑓𝑓 , the background field created by the test facility, stays constant during the AC loss experiment and it can be equal to 0.

At a given frequency (or corresponding ), the total volumetric losses (per volume of composite and per cycle) 𝑄 𝑡𝑜𝑡 is measured. The experimental volumetric coupling losses 𝑄 𝑐𝑜𝑢𝑝 is obtained by subtracting the hysteresis losses 𝑄 ℎ𝑦𝑠𝑡 . 𝑄 ℎ𝑦𝑠𝑡 must be calculated from the effective filament diameter 𝑑 𝑒𝑓𝑓 and the critical properties of the current density ( 𝐽 𝑐 (𝐵 𝑖 , 𝑇) ) and not by extrapolation towards  = 0 of 𝑄 𝑡𝑜𝑡 (the total losses), which is very imprecise. This operation can be delicate in particular in case of partial magnetic field penetration.

𝑥 𝑦

𝑄 𝑐𝑜𝑢𝑝 = 𝑄 𝑡𝑜𝑡 -𝑄 ℎ𝑦𝑠𝑡 [START_REF] Motojima | LHD Magnet System Design and Construction[END_REF] In each cable domain 𝑗, the external applied field 𝐵 𝑎 and the internal field 𝐵 𝑖 associated with stage 𝑗: 𝐵 𝑖𝑗 , are linked by the following equation [START_REF] Saxena Et S.-1 | Present status of the SST-1 project[END_REF]. Note that the internal field 𝐵 𝑖𝑗 is different for each stage.

𝐵 𝑖𝑗 (𝑡) = 𝐵 𝑎 (𝑡) -𝜏 𝑗 𝐵 ̇𝑖𝑗 (14)
The coupling loss power per unit volume of composites for a given stage can be derived according to the following equation [START_REF] Erckmann | The W7-X project: scientific basis and technical realization[END_REF].

𝑃 𝑗 (𝑡) = 𝑛𝜅 𝑗 𝜏 𝑗 𝐵 ̇𝑖𝑗 2 𝜇 0 (15)
For a sinusoidal field excitation, solving [START_REF] Saxena Et S.-1 | Present status of the SST-1 project[END_REF]:

𝐵 𝑖𝑗 (𝑡) = 𝐵 𝑚 √ 2 𝜏 𝑗 2 +1 sin (𝜔𝑡 -δ) (16)
With:

tan (δ) = 𝜏 𝑗 (17)
Integrating 𝑃 𝑗 (𝑡) over a cycle for a given pulsation 𝜔 and a given stage 𝑗, gives the coupling energy per cycle and per unit volume of composite (from ( 15) and ( 16)):

𝑄 𝑐𝑜𝑢𝑝 𝑗 = 𝜅 𝑗 𝐵 𝑚 2  0 𝜏 𝑗 ( 2 𝜏 𝑗 2 + 1) . ( 18
)
Considering all five stages of the conductor the total coupling losses QMPAS per cycle and per unit volume of composite is:

𝑄 𝑀𝑃𝐴𝑆 (ω) = ∑ 𝑄 𝑐𝑜𝑢𝑝 𝑗 (ω) 5 j=1 . (19)
Similarly, for comparison with equation ( 19), the energy associated with 𝜏 𝑠𝑡 (𝜏 Single Time constant)

given by using the model with the single time constant can be calculated as in [START_REF] Wilson | Time-varying fields and A.C. losses[END_REF]:

𝑛𝜏 𝑠𝑡 = ∑ 𝜅 𝑗 𝜏 𝑗 5 𝑗=1 (20)
𝑸 𝒔𝒕 = 𝒏 𝑩 𝒎 𝟐  𝟎 𝝉 𝒔𝒕 ( 𝟐 𝝉 𝒔𝒕 𝟐 + 𝟏) . ( 21
)
The use of a multi-constant model is necessary as the single time constant model (Q_ST in Figure 9) is less efficient to model the experimental coupling losses generated in a CICC at high frequency as shown in Figure 9 below: Since its first development, MPAS has already evolved towards the above presented version (described with equations ( 9) and ( 12)) which is kind of a fully constrained MPAS.

During this thesis work, we will show that starting with a MPAS with the minimal number of free parameters and giving it more freedom by unrestricting one degree of freedom is a good option to fit coupling losses in a more general way than with the restricted MPAS.

II.1.1.2 Advanced MPAS

The development of the MPAS in the original paper (see [START_REF] Turck | A macroscopic model for coupling current losses in cables made of multistage of superconducting strands and its experimental validation[END_REF]), leads to quite general results. The only strong condition being that the basic time constants be well distant.

For the shielding coefficients, the cascade of shielding coefficients could be formulated with different basic shielding coefficient per stage based on recent results. As a matter of fact, it is evidenced in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] that the shielding coefficient should be larger for a sextuplet than for a quadruplet or a triplet, for instance.

Based on the above result, our intuition to provide MPAS more flexibility (with respect to its restricted version) was to unconstraint some shielding coefficients by using the hypothesis that each stage can be composed of a different number of elements (i.e. DEMO, JT-60SA) which affect their shielding coefficient differently. For instance, in the JT-60SA TF conductor all stages are triplets except the last stage which is a sextuplet. We thus modify the iteration rules for shielding coefficients based on the above assumptions that this rule cannot be the same when going from a triplet to a triplet or from a sextuplet to a triplet for example. As said earlier, that affects the modelling of the coupling losses generated in the cable. Thus the first enhancement is under the form of considering two intermediate variables called 𝜅 𝑎 and 𝜅 𝑏 instead of the only one in the restricted MPAS (i.e. 𝜅 5 ) . We recall that this index 5 refers to the fifth stage of the cable.

We have to note that 𝜅 𝑎 in this enhanced version of MPAS is similar to 𝜅 5 of the first version of MPAS, it represents the last stage shielding coefficient. 𝜅 𝑏 , as said earlier, represents the new degree of freedom of this advanced MPAS. For the computation of the shielding coefficient for the advanced MPAS, we have:

𝜅 5 = 𝜅 𝑎 𝜅 4 = 𝜅 𝑏 (1 - 𝜅 𝑎 2𝑟 ) 𝜅 3 = 𝜅 4 (1 - 𝜅 𝑏 2𝑟 ) = 𝜅 𝑏 (1 - 𝜅 𝑎 2𝑟 ) (1 - 𝜅 𝑏 2𝑟 ) 𝜅 2 = 𝜅 3 (1 - 𝜅 𝑏 2𝑟 ) = 𝜅 𝑏 (1 - 𝜅 𝑎 2𝑟 ) (1 - 𝜅 𝑏 2𝑟 ) 2 𝜅 1 = 𝜅 2 (1 - 𝜅 𝑏 2𝑟 ) = 𝜅 𝑏 (1 - 𝜅 𝑎 2𝑟 ) (1 - 𝜅 𝑏 2𝑟 ) 3 .
Redefining the iteration rules between shielding coefficients, the expression can be written as follows for a five-stage cable:

𝜅 𝑗 = 𝜅 𝑏 (1 - 𝜅 𝑎 2𝑟 ) (1 - 𝜅 𝑏 2𝑟 ) 4-𝑗 (22)
Following our above intuitions and hypotheses, we have to note that the expression [START_REF] Gibson | The JET project[END_REF] of the shielding coefficients depends on the cabling pattern of the considered cases, the above expression corresponding to the one of the JT-60SA TF cable where the difference is introduce between the fourth and fifth stages.

It is important to note that if we consider 𝜅 𝑎 = 𝜅 𝑏 = 𝜅 5 , we found that the above equation ( 22) is equal to equation ( 9) used for the restricted MPAS. Thus, the advanced MPAS contains the restricted MPAS in its modelling.

The iteration rule on the time constants is conserved with the ratio of consecutive twist pitches only. For the time constants, the main dimensional factor was given to the basic twist pitch lengths. In the development of section II.1.1, MPAS considers that the only varying parameters in between two-stage is the twist pitch and thus that it is this twist pitch which drives the relation between time constants as exposed in section II.1.1. The interstage conductances (or average resistivity as considered in MPAS) are considered constant in the whole cable.

We can already distinguish two approaches for MPAS:

-the initial one, called restricted MPAS, where time constants are iteratively derived through the ratio of twist pitches and shielding coefficients through equation ( 9) -the enhanced one, called advanced MPAS, where time constants are still derived through the ratio of twist pitches but shielding coefficient are now iteratively derived through equation [START_REF] Gibson | The JET project[END_REF].

In the restricted MPAS, the only free parameters are the last stage time constant 𝜏 5 and the last stage shielding coefficient 𝜅 5 (index 5 refers to the fact that we consider a five-stage cable).

In the advanced MPAS, we have one more free parameters. We can set the last stage time constant 𝜏 5 as in the restricted MPAS but shielding coefficients are set using 𝜅 𝑎 and 𝜅 𝑏 .

In order to be efficient in the probing of experimental data with the two versions of the MPAS model we deal with, we have developed a statistical approach (and the associated numerical tool) to study the parametric behaviour of both MPAS models with respect to the experimental data we measure during the thesis work (chapter IV and V).

Then, MPAS will be confronted to newly extended model as COLISEUM (see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]).

II.1.2 Coupling Losses Algorithm for Superconducting Strands (CLASS)

Starting from the basic element composing CICCsstrands -, we briefly present the Coupling Losses Algorithm for Superconducting Strand (CLASS), which is a fully analytical algorithm capable to predict the coupling losses in a composite strand (see Figure 10). The strand is designed as a layered composite alternating normal and superconducting zones. The analytical tools needed for the development of this model can be found in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] and in [START_REF] Louzguiti | Development of an Analytical-Oriented Extensive Model for AC Coupling Losses in Multilayer Superconducting Composites[END_REF]. Normal zones are mainly composed of copper and superconducting zones are composed of hundreds of superconducting filaments embedded in a matrix of normal metal (mainly copper) with a specific twist pitch. In some cases (as presented in Figure 10), a cupronickel barrier (CuNi) is implemented to limit coupling current between the resistive zones of the strands. This model can be used to simulate several types of layered composite strands as can be seen in Figure 11: This model predicts as many time constants as edges of filamentary zones. These edges are the surfaces where the coupling currents carried by superconducting filaments flow. Each surface current flows with its own time constant and shielding coefficient (𝜏, 𝑛𝜅).

The time constants are related to the current loops taking place inside the superconducting strand subjected to an external magnetic excitation. These currents loops inside the strand create an internal magnetic field 𝐵 𝑖 which shields the inner volume of the filamentary zone of the strand.

As shielding coefficient 𝑛𝜅 is related to the capability for the considered current loops (time constant 𝜏) to shield the inner volume. The parameters (𝜏, 𝑛𝜅) will be referred as magnetic parameters in the following.

Using this model several parameters are needed to solve the system of equations and evaluate these magnetic parameters, for example:

• 𝑅 𝑖 the radius of each zone starting from 𝑅 1 to 𝑅 𝑛 .

• Zone type: R for resistive and F for filamentary.

• Filament twist pitch length inside the strand named 𝑙 𝑝 0 .

• The expression of the applied magnetic field 𝐵 𝑎 (amplitude, frequency, phase).

• Transverse resistivity of each layer 𝜌 𝑡 𝑖 .

𝜌 𝑡 𝑖 is a homogenized variable and is used in the computation of the time constants 𝜏 and of the shielding coefficient of the strand 𝑛𝜅.

On top of that, several assumptions are used and recalled here:

• Invariant system along z-axis.

• Applied magnetic field 𝐵 𝑎 is transverse and spatially uniform within the composite.

• The composite does not carry any transport current 𝐼 𝑡 = 0.

• Superconducting filaments are not saturated 𝐸 𝑠 ⃗⃗⃗⃗⃗ = 0 ⃗ .

• Superconducting filaments are lightly twisted

( 2𝜋𝑅 𝑙 𝑝 ) 2 ≪ 1.
This approach is generalized and based on the previous analytical approach developed in [START_REF] Ciazynski | Distributions de courant et pertes à l'intérieur d'un composite multifilamentaire supraconducteur soumis à un champ magnétique variable[END_REF] and is in agreement with the state of the art concerning a simple strand model. In fact, with this approach we are able to see that the single time constant approach for a strand is not realistic in cases where strands show several edges of filamentary zones. They would have as many time constants as the number of edges of filamentary zones.

Based on the same principle of multiple superconducting layers shielding the interior volume of the strand, the one-stage model named one-stage COLISEUM is developed (see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]) using superconducting tubes twisted in tangency condition and interacting together to shield the cable stage.

CLASS cannot be coupled to another stage in order to form multiplets of composite. We will see in the following section II.3.1, using CLASS and the one-stage COLISEUM presented below, that this onestage model can be used to reproduce the strand behaviour and geometry. In the next sub-section, we will briefly present the one-stage COLISEUM in order to be able to use its equations and results in the following work.

II.1.3 One-stage COLISEUM

The one-stage COLISEUM is based on superconducting elements of radius 𝑅 𝑒𝑙𝑒𝑚 (red superconducting tubes + outer normal shells) assembled together in tangency conditions with a specific twist pitch (𝑙 𝑝 ) in order to form a cable stage. Superconducting elements (i.e. red superconducting tubes + outer normal shells) in this system can either represent a strand or a group of strands (as a petal (i.e. multiple strands twisted together in several cabling stages)) as seen in Figure 12. This model, developed in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] and in [START_REF] Louzguiti | Development of a New Generic Analytical Modeling of AC Coupling Losses in Cable-in-Conduit Conductors[END_REF], is capable to compute the induced currents 𝐼 𝑘 in these superconducting tubes taking into account their electric and magnetic interactions when subjected to a transverse sinusoidal magnetic field 𝐵 𝑎 . It can also predict the AC losses generated by such a system.

The current 𝐼 𝑘 = 𝐼 cos ( ) , which flows on the red superconducting shell as depicted in Figure 12, with amplitude 𝐼 which is a function of time only driven by the following equation ( 23):

𝐼 + 𝜏𝐼 ̇= 4𝜎𝑅 𝑐 sin ( 𝜋 𝑁 ) 2 ( 𝑙 𝑝 2𝜋 ) 2 𝐵 ̇𝑎 = 𝐼 𝑒𝑥𝑡 (23)
This model has been described in detail by A. Louzguiti in his thesis [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]. Based on the definition of the vector potential of each element, it can provide the magnetic parameters (time constant 𝜏, partial shielding coefficient 𝑛𝜅) of a one-stage system of twist pitch 𝑙 𝑝 , composed of 𝑁 superconducting elements of radius 𝑅 𝑒𝑙𝑒𝑚 , with a filamentary zone radius named 𝑅 𝑓 . All 𝑁 superconducting elements are twisted with the cabling radius 𝑅 𝑐 (blue stripped line). Thus, the circumscribed area of the system is 𝑅 𝑐𝑖𝑟𝑐 = 𝑅 𝑒𝑙𝑒𝑚 + 𝑅 𝑐 (black stripped line). Between each element, the current flows from one element to another with a conductance 𝜎 named interstage conductance of the considered stage generating the so called coupling losses.

Here also, the time constant 𝜏 is related to current loops taking place in the one-stage model. The current loops related to one-stage of a CICC (as described by the one-stage COLISEUM) is also characterized 𝐼 𝑘 𝑙 𝑝 by a shielding coefficient 𝑛𝜅 which can be seen as the capability for the considered cabling stage to shield its inner circumscribed volume from the applied field.

𝑅 𝑐𝑖𝑟𝑐 𝑙 𝑝 𝜎 𝑅 𝑒𝑙𝑒𝑚 𝑅 𝑐 𝑅 𝑓
A geometrical factor named 𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 is also used in the computation of these magnetic parameters in order to take full account of the peculiar twisted geometry of the system. Its analytical expression is not reported here because of its complexity and can be found in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF].

Thus we have:

𝜏 = { 𝜎𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 ( 𝑙 𝑝 2𝜋 ) 2 𝑓𝑜𝑟 𝑁 = 2 4𝜎𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 sin 2 ( 𝜋 𝑁 ) ( 𝑙 𝑝 2𝜋 ) 2 𝑓𝑜𝑟 𝑁 ≥ 3 (24)
and

𝑛𝜅 = { 𝜇 0 𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 𝜋 ( 𝑅 𝑐 𝑅 𝑐 + 𝑅 𝑒𝑙𝑒𝑚 ) 2 [1 -𝑠𝑖𝑛𝑐 ( 4𝜋𝑧 𝑙 𝑝 )] 𝑓𝑜𝑟 𝑁 = 2 𝜇 0 𝑁 2𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 𝜋 ( 𝑅 𝑐 𝑅 𝑐 + 𝑅 𝑒𝑙𝑒𝑚 ) 2 𝑓𝑜𝑟 𝑁 ≥ 3 (25)
Using cable parameters (𝜎, 𝑅 𝑐 , 𝑒𝑡𝑐) to determine these magnetic parameters, we are able to quantify the coupling losses per circumscribed cable area 𝑄 𝑣𝑜𝑙 generated in a single stage of CICC. This fully analytical model can be used either to predict coupling losses or to fit existing coupling losses data.

𝑃 𝑣𝑜𝑙 (𝑧) = 𝑛𝜅𝜏𝐵 ̇𝑖 2 𝜇 0 (26)
The formulation of the one-stage COLISEUM is based on a solid analytical basis and it appears that major variables such as the power per unit volume (equation 26) can be expressed under a form as the one of MPAS. This is a strong result of [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] that we will use in the following of this work to construct an analytical 𝑛-stages model close to the MPAS initial intuition.

The above one-stage model can be used to model any stage of a cable as a homogenous uncoupled stage where we only consider inter-elements coupling losses at the considered stage. It is the simplest analytical model to simulate inter-elements coupling losses in one cable stage. This one-stage model will also be used in the following to model a superconducting composite strand. Modelling a strand with this homogenized one-stage model will allow us to model analytically multiplets of composites with the two-stage COLISEUM.

It is important to note that modelling analytically multiplets of strands has remained so far due to the difficulty of handling the analytical description.

II.1.4 Two-stage COLISEUM

II.1.4.1 Presentation

The two-stage COLISEUM is a determining enhancement of the one-stage model. It describes a two-stage system using the same analytical methodology as used for the development of the one-stage COLISEUM (see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]). The analytical description and the calculations are quite hard to handle but the magnetic parameters (𝜏, 𝑛𝜅) of the two-stage system have been derived in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] and in [START_REF] Louzguiti | AC Coupling Losses in CICCs: Analytical Modeling at Different Stages[END_REF].

In Figure 13, index 1 refers to the substage parameters and index 2 refers to the super-stage parameters. 𝐼 1 and 𝐼 2 are the amplitudes of the currents 𝐼 𝑘 1 𝑘 2 circulating on the red superconducting shell of the superconducting element 𝑘 1 in the second stage 𝑘 2 in Figure 13 and used to compute the coupling currents inside the system, 𝑅 𝑒𝑙𝑒𝑚 and 𝑅 𝑐𝑖𝑟𝑐 stand respectively for the radius of a superconducting element of the first stage and for the radius of the circumscribed circle of the system. The superconducting elements in this model are the same as in the one-stage COLISEUM, they are named superconducting elements of radius 𝑅 𝑒𝑙𝑒𝑚 and composed with the red superconducting tubes plus an outer normal zone around.

𝑅 𝑐𝑖𝑟𝑐 = 𝑅 𝑐 2 + 𝑅 𝑐 1 + 𝑅 𝑒𝑙𝑒𝑚 (27)
Initially, the derivation of the magnetic parameters of the two-stage COLISEUM presented in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] led to an infinite number of time constants and shielding coefficient. It was nonetheless shown in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] that it was possible to reduce them to only four without significant loss of information as the infinite matrix of time constants was strongly dominated by its diagonal and as only some spatial modes were initially excited (see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] for a complete explanation of the reduction process). A study in purely inductive regime presented in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] assessed the validity of this approach. The two-stage COLISEUM describes, with four time constants and four shielding coefficients, the induced currents and the coupling losses generated in such a system. As the elementary units of this two-stage model (the one-stage COLISEUM) are described with only one time constant and one shielding coefficient, we could have expected that this two-stage model could have been described with two time constants and two shielding coefficients, but unfortunately this is not the case. We mention this point in reason of the fact that in the already existing model MPAS [START_REF] Turck | A macroscopic model for coupling current losses in cables made of multistage of superconducting strands and its experimental validation[END_REF], the assertion is made that one time constant per stage of cable can describe the AC losses.

These four time constants are due to the fact that, as for the one-stage COLISEUM, for expressing the currents 𝐼 1 and 𝐼 2 circulating in the two-stage system we have to solve a differential equation through a diagonalization of a tridiagonal matrix of dimension four. The diagonalization is achieved by using numerical tools in order to derive the deposited power 𝑃 𝑣𝑜𝑙 and the magnetic parameters, 𝜏 and 𝑛𝜅 from the system equation. The fact that this model describes two stages with four time constants and that we need numerical tools to compute the magnetic parameters is inherent to its complex formulation. We will show in the following section that this complexity turns out to be apparent and can be overcome. This two-stage system can be represented by the following equation [START_REF] Chiletti | Analytical modelling of CICCs coupling losses: broad investigation of two-stage model[END_REF] as presented in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]: where the 𝜏 (𝛼 𝑘 )(𝛼 𝑘 ) are given by

[𝐼] + [𝜏][𝐼 ] = [𝑌]𝐵 ̇𝑎 (28)
𝜏 (𝛼 𝑘 )(𝛼 𝑘+𝑛 ) = 𝜎 1 𝜇 0 8𝜋 2 1 𝛼 𝑘 𝐸 𝑛 1 (𝛼 𝑘 )(𝛼 𝑘+𝑛 ) + 𝜎 2 𝜇 0 8𝜋 2 1 𝛼 𝑘 𝐸 𝑛 2 (𝛼 𝑘 )(𝛼 𝑘+𝑛 ) . ( 29
)
For the following, it is important to note that diagonal terms reflect the self-inductance whereas non diagonal terms reflect the mutual inductances between currents.

[𝐼] = [

𝐼 0 𝐼 1 𝐼 ̃2 𝐼 3 ]
[𝐼] are the time varying amplitudes of the induced currents for each spatial frequency. 𝐼 0 and 𝐼 3 are current amplitudes related to the development of the system equation in Fourier series in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]. After a reduction of the system dimension from infinity to four, we are left with four current amplitudes with 𝐼 0 and 𝐼 3 not directly induced by the external magnetic excitation as we can see in [𝑌] (equal to 0). 𝐼 1 and 𝐼 ̃2 are related to the currents induced at the first stage and second stage scales that are shielding the cable inner volume from the external magnetic excitation. Currents are computed by solving the above equation [START_REF] Chiletti | Analytical modelling of CICCs coupling losses: broad investigation of two-stage model[END_REF] and can be found in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]. It is important to note that the ̃ on 𝐼 2 is a change of notation regarding the initial development in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]. It reflects the fact that in the initial development found in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] and presented here above, the current amplitude 𝐼 ̃2 was rescaled to the substage scale to avoid having large differences between 𝐼 1 and 𝐼 2 (this explains the presence of the factor in equation ( 28)). We will show in the following subsection II.2.2.4 that using a change of variable (that simply consists in multiplying 𝐼 ̃2 by 𝑁 1 ), we are able to rescale the current amplitude 𝐼 2 to the superstage scale; this is an important step before the iterative process used to model a cable with 𝑛-stages.

In order to lighten the future expressions we will use the following notation for any 𝑘 ∈ ℤ

𝛼 𝑘 = 2𝜋 𝑙 𝑝 1 + 2𝜋(𝑘 -1) ( 1 𝑙 𝑝 2 - 1 𝑙 𝑝 1
).

𝐸 𝑛 𝑖 factors in the expression of time constants and all intermediate expressions are gathered in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF].

Assuming that [𝜏] is a diagonalizable matrix, we can express it as:

[𝜏] = [𝑉][𝜏] 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 [𝑉] -1 (30)
Where [𝜏] 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 is the diagonal matrix containing the eigenvalues of [𝜏] (time constants of the considered system) and [𝑉] is the matrix containing the ordered eigenvectors of [𝜏] corresponding to the eigenvalues [𝜏] 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 .

From now on 𝜏 (𝛼 𝑖 )(𝛼 𝑗 ) notation from equation (28) will be simplified by using 𝜏 𝑖𝑗 . This double index notation refers to elements in the time constant matrix [𝜏] before diagonalization and using 𝜏 𝑖 with a single index will refer to coupled time constant in the diagonal basis (diagonal of [𝜏] 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 ).

By analogy with the calculation presented above in the composite strand section, it is possible to express the coupling losses 𝑄 𝑐𝑜𝑢𝑝 per cycle of magnetic excitation 𝐵 𝑚 sin(𝜔𝑡) per unit volume of conductor envelope . It has been shown in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] that the losses can be written in a simple form by a sum of four terms, each term with a coupled time constant 𝜏 𝑗 and a shielding coefficient 𝑛𝜅 𝑗 :

𝑄 𝑐𝑜𝑢𝑝 (𝜔) = ∑ 𝑛𝜅 𝑗 𝐵 𝑚 2 𝜇 0 𝜋𝜔𝜏 𝑗 1 + (𝜔𝜏 𝑗 ) 2 3 𝑗=0 (31)
With

𝑛𝜅 𝑗 = 2𝜇 0 ∑ ∑ 𝛿 𝑘 [𝑉] 𝑘𝑗 [𝑉] 𝑘𝑙 [𝑌 𝑏 ] 𝑙 [𝑌 𝑏 ] 𝑗 𝜏 𝑗 + 𝜏 𝑙 3 𝑙=0 3 𝑘=0 (32)
where [𝑉] and [𝑌 𝑏 ] are obtained after diagonalizing [𝜏], and [𝑌 𝑏 ] is defined as

[𝑌 𝑏 ] = [𝑉] -1 [𝑌] .
Each terms of the sum of equation ( 31) only depends on a couple of magnetic parameters (𝜏, 𝑛𝜅) which can be viewed as a contribution to the coupling losses. Here for this two-stage model, we have four contributions to the coupling losses.

The (𝛿 𝑗 ) 0≤𝑗≤3 are defined in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] and given below:

{ 𝛿 0 = 𝑁 2 𝑁 1 𝛼 0 2 8𝜎
We also recall that for this two-stage model, we have the deposited power per unit volume:

𝑃 𝑣𝑜𝑙 = 𝛿 0 𝐼 0 2 + 𝛿 1 𝐼 1 2 + 𝛿 2 𝐼 ̃2 2 + 𝛿 3 𝐼 3 2 (34)
This two-stage system is a step toward the modelling of complete CICC (multistage cable). It simulates a multiplet of multiplet that we call here first stage and second stage.

Using this two-stage model, we can quantify the effect of coupling between two neighbouring stages. Iteration of the method presented in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] to model 𝑛-stages cable would be analytically too complex to handle. We will present in section III, another way to iterate the modelling to describe 𝑛-stage cables.

The coupling between two-stage is hidden in the time constants and shielding coefficients of the system. The second stage will shield a part of the inner volume of the cable, thus the substage will not see the same external excitation as the super-stage does because a part of it is already shielded.

At the end in this two-stage model, the coupling losses can be written under a form similar to the one of MPAS, for the formulation of AC losses (see equation [START_REF] Torre | Impact of a plasma disruption on the magnetic field variations and heat deposition in the TF conductor of JT-60SA[END_REF] and equation ( 18)).

II.2 Applications of COLISEUM II.2.1 Coupling between two stages: application of two-stage COLISEUM

In order to better understand how the coupling between two stages works, we will study the two-stage COLISEUM. For a given geometry, current loops inside a cable composed of several stages of twisted strands are driven by the conductance/resistance at each contact point between two strands. We will try to understand the behaviour of this system while uncoupling it electrically in order to check its consistency with respect to its uncoupled version (one-stage COLISEUM). We will show how this twostage system behaves while uncoupled. To illustrate, we consider that a stage is electrically uncoupled when its associated transverse conductivity is set to zero.

From now on, we choose to use the cabling parameters of JT-60SA TF conductor [START_REF] Zani | Starting EU production of strand and conductor for JT-60SA TF coils[END_REF] unless otherwise indicated. The choice to model JT-60SA TF cable is related to further experimental work (section IV). We start by studying the effect of coupling of two stages by running a two-stage COLISEUM on the system depicted in Figure 13 with the following parameters (see Table 2) representing the two first stages of the JT-60SA TF cable, a triplet of triplet (3x3): 13. The range value chosen for 𝜎 ′ 𝑠 is realistic regarding the work done by the University of Twente on bundle resistance measurement (see in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] and [START_REF] Nijhuis | Change of interstrand contact resistance and coupling losses in various prototype ITER NbTi conductors with transverse loading in the Twente Cryogenic Cable Press up to 40000 cycles[END_REF]). The magnetic parameters (equation ( 29) and [START_REF] Bagni | Analysis of ITER NbTi and Nb3Sn CICCs experimental minimum quench energy with JackPot, MCM and THEA[END_REF]) given by the two-stage model are given in Table 3.

Table 3: Magnetic parameters of the two first stages of the JT-60SA TF

These four couples of magnetic parameters are related to the two-stage system in Figure 13. They take into account the coupling effect between stage 1 (sub-stage) and stage 2 (super-stage). We notice that two time constants are non-contributing because of their very low shielding coefficient.

In order to segregate the proper effect of each stage alone (first stage or second stage), we uncouple the systems by taking in the program the numerical limit close to zero alternatively for 𝜎 1 then for 𝜎 2 as done below in The analytical description of the two-stage model is based on the one-stage COLISEUM where two independent stages are coupled through coupling current and specific rules of currents redistribution which can be found in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]. Thus, we will demonstrate in the next section that taking the limit on both side (𝜎 1 → 0 and 𝜎 2 → 0), we can find the magnetic parameters (𝜏, 𝑛𝜅) of each independent stage (substage and super-stage as depicted in Figure 14 and Figure 15) as if they were simulated with the onestage COLISEUM alone.

Taking a limit and not the absolute value of zero is related to the fact that when 𝜎 = 0, the considered stage (first stage or second stage) is cancelled because its differential equation is reduced to 𝐼 = 0.

Taking the limit with numerical tools is related to the fact that analytical diagonalization of the [𝜏] matrix is not possible and we have to perform the diagonalization with each tested 𝜎. A part of the following work will be to study the weight of these two supplementary magnetic parameters given by the twostage COLISEUM in the description of coupling losses. We can note from Figure 14 that the sum of the product of uncoupled magnetic parameters is equal to the one of coupled magnetic parameters up to the numerical precision:

∑ 𝑛𝜅 𝑖 𝜏 𝑖 = 𝑏 𝑖=𝑎 ∑ 𝑛𝜅 𝑗 𝜏 𝑗

Coupled Approach

Uncoupled Approach

𝑙 𝑝 2 𝑙 𝑝 1 𝜎 1 𝜎 2 𝑥 𝑦 𝑧
The quality of the agreement is very good. It is important to note that looking at the ∑ 𝑛𝜅𝜏 gives us indications on the behaviour of coupling losses in the low frequency regime. We will demonstrate this relation in the next section. We precise that the values of magnetic parameters exposed in Figure 14 are given with limited precision, i.e. a check with those values would artificially show a deviation from the strict equality. It is stressed that this was numerically verified and will be demonstrated in section II.3.3.

The above equality confirms the trend seen in Figure 15 below, as in its uncoupled description, only the conductances are set to zero, thus the system become equivalent to an uncoupled system with no mutual inductances.

We can see how coupling two sub-systems (first stage and second stage) can modify the respective uncoupled time constants and shielding coefficients as shown in Figure 14 and Figure 15. In fact, coupling spreads the uncoupled time constants from each other by increasing the difference between the two time constants. Coupling also increases the second stage shielding coefficient and decreases the first stage one. It appears, among all simulations we ran, that this tendency of modification of the magnetic parameters while coupling two sub-systems is a strong trend characteristic; in the next section we will demonstrate that the spreading of the time constant under coupling is predictable for two-stage system.

We can explain this modification of the magnetic parameters with the supplementary mutual inductance due to the geometrical proximity of several cabling stages. This supplementary inductance modifies the time constants of both uncoupled subsystems to give the two coupled time constants depicted in blue in Figure 15. The shielding coefficients are also modified because the time constants and the matrix of eigenvector used for their computation are modified due to the effect of coupling.

First, among all simulations run in the coupled case, we will show in section II.2.2 that among the four couples time constants shielding coefficients considered initially by the two-stage COLISEUM, two of them are almost always negligible. Also in both uncoupled cases (𝜎 1 → 0 and 𝜎 2 → 0 ) we are left with only one time constant and one shielding coefficient. We will show in the next section, that the two extra time constants (𝜏 0 and 𝜏 4 ) are non-contributing to the coupling losses of this two-stage model. That is why we do not report the four time constants for each simulation in Figure 14 but only the contributing ones. The following chapter will explore the parametric domains in which this reduction occurs.

II.2.2 Crosscheck of the one-stage COLISEUM and JackPot

In this section, we present the continuation of the collaboration with Twente University on the crosscheck of JackPot (see [START_REF] Van Lanen | JackPot: a novel model to study the influence of current nonuniformity and cabling patterns in cable-in-conduit conductors[END_REF]) and COLISEUM. We recall that in the previous thesis work (see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]), some comparison have been already made between the two-stage model COLISEUM and a two-stage version of JackPot. The considered case was the one of the two last stages of JT-60SA TF conductor, and it showed a 40% discrepancy regarding the result. At the time however, only one point 𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 (𝑓) was considered for comparison at a peculiar frequency. We decided to start over the comparison work by comparing the result on a one-stage version of COLISEUM and a one-stage version of JackPot. The considered case is the one of the fourth stage of JT-60SA TF conductor. We chose to model the coupling losses from 0 to 50 Hz with logarithmical increase in order to get a global view of the agreement between both models at different frequencies.

In order to get a solid basis, we started to simulate a triplet of elements. Once the geometrical and electrical parameter have been chosen we both can predict the coupling losses inside a CICC. We explored the one-stage model developed in CEA trying to simulate the fourth stage of JT-60SA TF cable. In order to stay closer to the real geometry of this fourth stage, we used values of cabling radius and twist pitch length issued from the analysis of the tomography images (see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]). The configuration was the following:

-Number of elements 𝑁 = 3 -Cabling radii 𝑅𝑐 = 2.31 𝑚𝑚 -Element radii 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 = 𝑅 𝑠𝑡𝑟 = 2.05 𝑚𝑚 -Filamentary zone radii 𝑅 𝑓 = 𝛼𝑅 1 𝑤𝑖𝑡ℎ 𝛼  [0,1] -Twist pitch 𝑙 𝑝 = 185.2 𝑚𝑚 -Conductance 𝜎 𝑙 = 2.76. 10 7 𝑆. 𝑚 -1
We chose = for the first test but this value can be modified easily. In the one-stage COLISEUM, superconducting elements are placed using tangency condition between each other (see Figure 12), we can see that contacts between neighbouring elements are continuous along the 𝑍-axis and that 𝜎 is the same between each neighbouring superconducting elements. In JackPot, interpenetration of the superconducting elements is mandatory to define the conductance between each other so they aren't tangent anymore.

Regarding Boundary conditions:

• Conductor is subjected to sinusoidal transverse magnetic field: 𝐵 𝑎 = 𝐵 𝑚 sin(𝜔𝑡) + 𝐵 𝑜𝑓𝑓 with 𝐵 𝑚 = 1 𝑇, 𝐵 𝑜𝑓𝑓 = 0 𝑇 and 𝑓 logarithmically increasing from 0.01 Hz to 50.0 Hz. We choose to start with a null background field 𝐵 𝑜𝑓𝑓 = 0 𝑇. • In the same way, temperature isn't a parameter in our model so JackPot can use the value of 4.5 K. • Transport current is considered as zero, 𝐼 𝑡 = 0 𝑘𝐴. Jackpot:

• Any scaling law parametrization can be chosen because it does not act in COLISEUM model.

• For strand radius, it is proposed that JackPot takes a value extracted from the set of cabling radii provided by our analysis of tomographic images, this gives: 𝑅 𝑠𝑡𝑟 = 2.05 𝑚𝑚. • In order to have the maximum of the 𝑄(𝑓) curve closer to 50 Hz (much above if not) we choose to put 𝜎 𝑙 = 2.76. 10 7 𝑆. 𝑚 -1 between neighbouring elements. It is not a problem for Twente University from 𝜎 𝑙 to recover the inter-strand contact resistivity they need to run JackPot.

In Figure 16, we can found the comparison of the coupling losses prediction as given by JackPot in black plain line and given by the one-stage COLISEUM in blue and red plain line. Blue line is the full curve given by the one-stage COLISEUM whereas the red line is the low frequency approximation of the one-stage COLISEUM (low frequency approximation can be found in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]).

We can see that we are in very good agreement on the whole frequency domain until 50 Hz.

It is interesting to note that in order for COLISEUM to be in agreement with the AC losses given by Twente University, we started with The effect is non-negligible as we can see above. The reason of this peculiar value to be in agreement with JackPot is not so clear at this time but it is shown here that in JackPot, the current seems to flow on a tube of diameter close to the one found with COLISEUM to fit (i.e. 𝑅 𝑓 𝐽𝑎𝑐𝑘𝑝𝑜𝑡 = 0.29 𝑅 𝑠𝑡𝑟 ).

We can also take a look at the relative error between the coupling losses given by the one-stage COLISEUM and the one-stage JackPot in Figure 18.

Regarding the first try in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF], where the predicted coupling losses of the two-stage COLISEUM had been compared to the one of the two-stage JackPot, we are here in very good agreement. Here we scan from 0 to 50 Hz. Also, the discrepancy found in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] could be explained due to the use of a wrong 𝑅 𝑓 𝑅 ratio but has not been verified properly. This work should be continued to demonstrate the compatibility of two-stage COLISEUM and JackPot. Moreover, COLISEUM will be extended in the future to an 𝑛-stage modelling as JackPot does, allowing more comparison on full cable geometry. )

II.2.3 Preliminary comments on the models

Regarding the advanced MPAS, and considering the shielding coefficient side of the problem, it can be demonstrated that by using the simple following ratio of equation ( 25) to compute the shielding coefficient of a stage as done with the one-stage COLISEUM (uncoupled COLISEUM, see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]), we can reach a similar iterative rule for shielding coefficients:

𝜅 𝑖 = 𝛾 𝑖 𝑁 𝑖 𝑅 𝑐 𝑖 2 (𝑅 𝑐 𝑖+1 + 𝑅 𝑒𝑙𝑒𝑚 𝑖+1 ) 2 𝛾 𝑖+1 𝑁 𝑖+1 𝑅 𝑐 𝑖+1 2 (𝑅 𝑐 𝑖 + 𝑅 𝑒𝑙𝑒𝑚 𝑖 ) 2 𝜅 𝑖+1
It is very interesting to note that uncoupled shielding coefficients are related through the ratio of consecutive number of elements constituting the stages 𝑁 but also through the ratio of consecutive cabling radii 𝑅 𝑐 𝑖 2 , or also with the ratio of 𝛾 𝑖 the geometrical factor of the model (see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]).

As we have seen with its recent enhancement presented above, advanced MPAS tends toward the same kind of description than COLISEUM, by using the results from the one-stage COLISEUM to confidently unconstraint one of its shielding coefficients. We can also note that the interstage conductances for the one-stage COLISEUM or MPAS do not play any role in the computation of the shielding coefficient.

We stress out that MPAS aims to model a cable with coupled stages. Here above, we have briefly compared it to the one-stage COLISEUM in order to illustrate what is taken into account in both models. It is obvious that expressing an analytical ratio of coupled shielding coefficient from two-stage COLISEUM is out of reach by using explicit analytical approach due to the complexity of the computation as we would have to handle long expressions like equation [START_REF] Duchateau | Stability of a cable in conduit conductor under fast magnetic field variations[END_REF]. Contrariwise, by using the coupled COLISEUM, we can numerically compute the ratio of coupled time constants which will provide a comparison with the time constant ratio used in MPAS (twist pitch ratio). The ratio of coupled time constants given by COLISEUM could be used to guide MPAS through a new set of iterative rules for the determination of time constants.

A noticeable point concerning the two-stage COLISEUM is that we have observed that:

∑𝑛𝜅 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝜏 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 = ∑𝑛𝜅 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝜏 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑
This equality is not analytically demonstrated yet but is true for every simulation we have ran during this thesis work. We will demonstrate this equality at the two-stage level in the section II.3.3. Indeed, even knowing the five uncoupled time constants and shielding coefficients of a cable, we would not be able to deduce the five coupled time constants and five shielding coefficients with the equation above (only one equation of ten unknowns) (five coupled time constants and five shielding coefficients) that cannot be solved.

At this point, it can be simply remarked that in the low frequency regime, firstly, all 𝐵 ̇𝑖 are equal to 𝐵 ̇𝑎, and secondly all terms in currents involving inductances are negligible. As a result, the loss power is driven only by the mesh of resistances. On a more intuitive ground, we can say that the coupled cable, in the low frequency regime has a resistance mesh equivalent to the superposition of the uncoupled cables.

At this stage, we can also make some preliminary comments on COLISEUM we will use in the following to enlighten the future analytical development:

-One-stage COLISEUM can be used to determine the time constants and shielding coefficients of each consecutive stage of a cable. This approach is similar to the one used by MPAS and will help us to understand how to iterate the two-stage model presented below.

-We can note that although the system describing the two-stage COLISEUM has already been reduced to four dimensions in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] it appears that the current amplitudes 𝐼 0 and 𝐼 3 are not directly excited by the external magnetic field as we can see in equation ( 28) above (as [𝑌] 0 = [𝑌] 3 = 0). The system equation ( 28) is therefore thought to be possibly reducible to dimension two without major loss of information on the resulting coupling losses or on the magnetic parameters themselves.

-Looking at the component of each eigenvector, we can have an insight on the ponderation in the new basis after the diagonalization of the system (coupling of the two subsystems). We can distinguish the contribution of each dimension from the initial basis, to the new one. In this way we can determine either if a time constant is related to the current loops of the first stage or of the second stage.

-In the time matrix [𝜏], diagonal terms 𝜏 𝑖𝑖 are related to self-inductance (action of one-stage to itself) and other terms 𝜏 𝑖𝑗 (with 𝑖 ≠ 𝑗) are related to the mutual inductance between first stage and second stage (action of one stage to its neighbouring stage). Thus, we will see that using these mutual inductance terms we will be able to analytically uncouple the two-stage system more efficiently than going for the numerical limit of the system (i.e. alternatively both 𝜎 → 0).

-COLISEUM development is based on the idea of making a solid analytical basis for the prediction of coupling losses staying close to the MPAS intuition. As we can see above in the model presentation, in order for MPAS and COLISEUM to be in agreement, we would have to reduce the number of magnetic parameters describing the two-stage COLISEUM to only two couples of time constants/shielding coefficients.

All the previously quoted equations are very important to well define the physical systems we are going to manipulate all along this thesis work and also as a toolbox for the analytical calculation developed below.

II.3 COLISEUM: Models overlap and cross-checks

Using the three models previously presented (CLASS for strands, one-stage COLISEUM for N-uplet and two-stage COLISEUM for 𝑁 2 of 𝑁 1 -uplets) we will examine the global consistency between each part of the model, regarding the simulation of a same system, using two different approaches.

Describing several cabling stage has to deal with scale change as the first stage of a cable (multiplet of composites) is in the range of the millimeter whereas the last stage of the cable (which is also a multiplet of petals) lies in the range of several centimeters.

Our goal here, in a first step, is to model fully analytical multiplets of strands. In order to do so, we would like to model a two-stage cable where the first stage is the strand and the second stage is a multiplet of strands. The strand is initially modelled by CLASS which is not designed to be embedded in COLISEUM as such, thus we decided to model an equivalent strand using the one-stage COLISEUM and CLASS. We will show that our choice is driven by the fact that the one-stage and the two-stage COLISEUM are intrinsically compatible. Once our strand will be modelled with the one-stage COLISEUM, we will embed it in the two stage COLISEUM as the first stage of the system.

The global consistency between the description given by the one-stage and the two-stage COLISEUM will also be checked in order to be sure that the integration of the strand, modelled using a one-stage COLISEUM, as the substage of a two-stage system is reliable and coherent.

In order to achieve this multiplet of strands modelling, we start by reproducing the magnetic behaviour of the strand given by CLASS using the one-stage COLISEUM. Second, we check the consistency between the one-stage and the two-stage COLISEUM in order for us to be sure that we can embedded the one-stage modelling of the strand into the two-stage model.

II.3.1 CLASS vs one-stage COLISEUM

We here check to which extent a composite strand can be simulated with a one-stage COLISEUM and be consistent with the description of the strand given by CLASS. The ultimate goal of this exercise is to embed the strand modelling in COLISEUM as CLASS and COLISEUM are not designed to be directly coupled. This is the first step for the implementation of the modelling of a multiplet of composites. As seen in the CLASS model, superconducting strands are described using as many time constant as the number of edges of filamentary zones, i.e. two in the case of JT-60SA TF strand as shown in Figure 19. However, for the specific case of JT-60SA TF strand, only one time constant is dominant as we can see in the following example. This case is representative of most strands under use, with copper matrix.

JT-60SA TF strand diameter is 0.81 𝑚𝑚 with the twist pitch of its filament equal to 𝑙 𝑝 = 15 𝑚𝑚. External radius of the filamentary zone is taken equal to 0.327 𝑚𝑚 whereas its internal radius is equal to 0.2 𝑚𝑚. Starting from this simulation, we can see that on account of their shielding coefficient (𝑛𝜅) only one time constant contributes to AC losses (as already shown in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]). We will take 𝜏 𝐶𝐿𝐴𝑆𝑆 = 18.2 𝑚𝑠 and its shielding coefficient (𝑛𝜅 𝐶𝐿𝐴𝑆𝑆 = 1.24) as the only one which matters to describe losses (see equation ( 8)). In the following, the inner NZ (𝑁𝑍 1 ) is replaced with a filamentary zone of the same type as FZ. This replacement eliminates the time constant of 38.9 ms which is reasonable with respect to its very low shielding coefficient (i.e. 0.06). The error made on the coupling losses prediction is thus negligible (below 1%). We remind that here 𝑛𝜅 is referred to the total strand area as usually done in the fusion community.

Hence, due to the fact that we only consider the relevant time constant and shielding coefficient given by CLASS (see Table 5), the simplified composite simulated with CLASS is depicted in Figure 20 below. We can also note that the real strand geometry simulated with CLASS takes into account the CuNi barrier present in the JT60SA TF strand through the use of a specific transverse resistivity for filaments coupled through the outer shell.

𝝉 (𝒎𝒔)

18.2 38.9 𝒏𝜿 1.24 6.00.10 -2 Using the fact that in a composite strand, currents shield the inner volume of the filamentary zone by flowing in superconducting filaments located on the edge of the filamentary zone, thus we chose to represent the strand using the one-stage COLISEUM where currents will flow on average on the circle of radius 𝑅 𝑐 as shown in Figure 21.

For the one-stage COLISEUM representation of the strand, we use 𝑁 superconducting tubes assembled as a 𝑁-uplet as depicted in Figure 21. The cabling radius of these superconducting elements is taken equal to the external filamentary zone radius of the real strand (𝑅 𝑐 = 𝑅 𝑓 𝑠𝑡𝑟𝑎𝑛𝑑 ). In this way, in both models, on average current flows on a circle with equal dimension.

In doing so, the total induced current flows, on average on a circle of same radius as the original filamentary zone. Depending on the number of elements 𝑁 to consider, we can compute their radius 𝑅 𝑒𝑙𝑒𝑚 = 𝑅 𝑐 sin ( 𝜋 𝑁 ) = 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 sin ( 𝜋 𝑁 ) . All elements are tangent to each other, as can be seen in Figure 21.

The twist pitch is taken equal to the one of the JT-60SA TF strand, so 𝑙 𝑝 = 15 𝑚𝑚.

In order to simulate the magnetic behaviour of the strand we have to fit its two magnetic parameters (𝜏, 𝑛𝜅) given by CLASS with the one-stage COLISEUM. The cabling radius 𝑅 𝑐 is constrained as we force it to be equal to 𝑅 𝑓 𝑠𝑡𝑟𝑎𝑛𝑑 . Our only free parameter is 𝑁 , which constrains the 𝑅 𝑒𝑙𝑒𝑚 we have to They can be determined by using the expressions of shielding coefficient 𝑛𝜅 and the time constant 𝜏 of the one-stage COLISEUM. And making them coincide with the values given by CLASS in Table 5.

Using equation ( 25):

𝑛𝜅 = 𝜇 0 𝑁 2𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 (𝑅 𝑐 , 𝑅 𝑓 , 𝑙 𝑝 , 𝑁)𝜋 ( 𝑅 𝑐 𝑅 𝑐 + 𝑅 𝑒𝑙𝑒𝑚 ) 2 = 1.24
We can find (using dichotomy) the needed 𝑅 𝑓 to obey the following equation:

𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 (𝑅 𝑐 , 𝑅 𝑓 , 𝑙 𝑝 , 𝑁) = 𝜇 0 𝑁 2𝑛𝜅𝜋 ( 1 1 + sin ( 𝜋 𝑁 ) ) 2
Once 𝑅 𝑓 is derived, we can find 𝜎 by using the expression of the time constant (equation ( 24)) in the one-stage COLISEUM and make it coincide with the one of the real strand:

𝜏 = 18.25 𝑚𝑠 = 4𝜎𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 sin 2 ( 𝜋 𝑁 ) ( 𝑙 𝑝 2𝜋 ) 2
Which gives:

𝜎 = 𝜏 4𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 (𝑅 𝑐 , 𝑅 𝑓 , 𝑙 𝑝 , 𝑁) sin 2 ( 𝜋 𝑁 ) ( 𝑙 𝑝 2𝜋 )
The number of elements used to reproduce the strand magnetic behaviour is variable and we have tested several cases depicted in the Table 6 below. The greater 𝑁 is, the more time consuming the computation is. The time consumption of the above process increases linearly with 𝑁 and is described below in Figure 22.

We choose to maintain the cabling radius equal to the dimension of the filamentary zone radius of the real strand, which gives different radii for the circumscribed circle, depending on the chosen number of elements 𝑁. Each case (different 𝑁) is reported in Table 6 to show the consistency of the modelling.

Considering geometrical parameters fixed (𝑅 𝑐 , 𝑙 𝑝 ) and the constraint rules applied on both 𝑛𝜅 and 𝜏, the only free parameter at the end is the number 𝑁 of elements to use to adjust the circumscribed area given by the one-stage system to the one of the strand. The choice of 𝑁 = 13 gives the best agreement to find 𝑅 𝑐 + 𝑅 𝑒𝑙𝑒𝑚 close to 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 (by coincidence strictly equal for this case). In this way, the strand simulated with the one-stage COLISEUM is very close to the real strand in terms of geometry. That is why we choose to stay with 𝑁 = 13 for the rest of this manuscript when we will simulate the JT-60SA TF composite strand with a one-stage or a twostage model. We note that 𝑁 depends on the geometry of the considered strand.

By using 𝑁 = 13, we have constructed a system of external dimension strictly equal to the one of the strand, and thus we have the same reference area for our simulations.

As time constants are related to current loop characteristics (geometry, conductances), we have to reproduce these characteristics to model the strand using the one-stage COLISEUM.

We can show analytically that we cannot achieve both 𝑅 𝑐 = 𝑅 𝑓 𝑠𝑡𝑟𝑎𝑛𝑑 and 𝑅 𝑐 + 𝑅 𝑒𝑙𝑒𝑚 = 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 with 𝑁 integer for any 𝑅 𝑓 𝑠𝑡𝑟𝑎𝑛𝑑 . Starting with the strongest condition:

𝑅 𝑐 = 𝑅 𝑒𝑙𝑒𝑚 sin ( 𝜋 𝑁 ) = 𝑅 𝑓 𝑠𝑡𝑟𝑎𝑛𝑑 = 𝑟 𝑓 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 ⇔ 𝑅 𝑓 𝑠𝑡𝑟𝑎𝑛𝑑 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 = 𝑟 𝑓 (35)
With 𝑟 𝑓 a coefficient within ]0,1] representing the proportion of the filamentary zone radius with respect to the strand radius in the original strand, and:

𝑅 𝑐 + 𝑅 𝑒𝑙𝑒𝑚 = 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 (36)
We get:

{ 𝑅 𝑒𝑙𝑒𝑚 + 𝑟 𝑓 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 = 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 𝑅 𝑐 = 𝑟 𝑓 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 (37)
{ 𝑅 𝑒𝑙𝑒𝑚 = (1 -𝑟 𝑓 )𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 𝑅 𝑒𝑙𝑒𝑚 𝑟 𝑓 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 = sin ( 𝜋 𝑁 ) (38)
{ 𝑅 𝑒𝑙𝑒𝑚 + 𝑟 𝑓 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 = 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 𝑁 = 𝜋 sin -1 ( 1 -𝑟 𝑓 𝑟 𝑓 ) = 12.43 𝑤𝑖𝑡ℎ 𝑟 𝑓 = 0.8 (39)
The coefficient used for the simulated strand above is 𝑟 𝑓 =0.807 and is extracted from image analysis of Figure 19. We recall that using 𝑅 𝑐 = 𝑅 𝑓 𝑠𝑡𝑟𝑎𝑛𝑑 with this 𝑟 𝑓 we strictly obtain:

𝑅 𝑐 + 𝑅 𝑒𝑙𝑒𝑚 = 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 𝑤𝑖𝑡ℎ 𝑁 = 13 (40)
It is important to note that the above result that best fits a strand depends on the radius of the initial tube taken to simulate the considered strand (here

𝑅 𝑓 𝑠𝑡𝑟𝑎𝑛𝑑 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 = 0.807).
The result we have found in Table 6 as the solution to reproduce the strand magnetic behaviour is unique as we have constrained the parameters 𝑅 𝑐 , 𝜏 and 𝑛𝜅.

This advantage of the modelling of the strand with a one-stage COLISEUM is twofold. On one hand, it allows to check the consistency of both approaches (strand model vs one-stage model) when used on the same physical system, i.e. the strand. COLISEUM can handle different scales in its modelling, i.e. the one-stage COLISEUM, initially developed to describe multiplets of superconducting tubes can be used at the strand scale, where the basic elements can be seen as group of filaments or at the last stage scale (petals). On the other hand, it is the first step of the modelling of the multiplets of composites. Now we will be able to use this strand (modelled with the one-stage COLISEUM) as the first stage of a two-stage system to model analytical multiplets of composite strands.

This modelling of the strand using the one-stage COLISEUM and later the two stage COLISEUM is a way to model coupling losses inside a composite strand and between several composite strands (intra and inter composite losses) (see section II.3.4).

The analytical modelling of multiplets of composites inside a self-consistent approach remained out of common use for cable simulation. In the following, it is shown that now it becomes possible to compute analytically the magnetic parameters of multiplets of composites by using the above modelling of a strand and the two-stage COLISEUM (see section II.2.4).

Here above, we have presented an efficient method to model a composite strand with the one-stage COLISEUM approach, this method can be seen as added to our toolbox to embed composite strand in the modelling of coupling losses in large cable in conduit conductor (CICC).

II.3.2 Reduction to a two time constant system for the two-stage COLISEUM

II.3.2.1 Parametric study

The two-stage COLISEUM described a two-stage system with four time constants and four shielding coefficients. First thing, we have seen when applying the two-stage model that in the chosen case at least two shielding coefficients among the four are always nearly equal to zero, indicating that the model might be reducible to a two time constants model. In this way, we will have one couple (𝜏, 𝑛𝜅) per stage as in the MPAS description. Among all simulations we have run during this thesis work, the fact that two shielding coefficients are nearly equal to zero appears to be a strong trend characteristic.

The coupling losses for such a system when subject to a sinusoidal field variation 𝐵 𝑎 writes:

𝑄(𝜔) = ∑ 𝑛𝜅 𝑗 𝐵 𝑚 2 𝜇 0 𝜋𝜔𝜏 𝑗 1 + (𝜔𝜏 𝑗 ) 2 3 𝑗=0 (41)
with 𝐵 𝑎 = 𝐵 𝑚 sin(𝜔𝑡) + 𝐵 𝑜𝑓𝑓 and 𝜔 = 2𝜋𝑓 the pulsation. We note that 𝐵 𝑚 and 𝐵 𝑜𝑓𝑓 are parallel in this case as opposed to the two fields presented in SULTAN in Figure 8.

At low frequency, the above expression writes:

𝑄(𝜔) = 𝐵 𝑚 2 𝜋𝜔 𝜇 0 [𝒏𝜿 𝟎 𝝉 𝟎 + 𝒏𝜿 𝟏 𝝉 𝟏 + 𝒏𝜿 𝟐 𝝉 𝟐 + 𝒏𝜿 𝟑 𝝉 𝟑 ] (42)
This expression represents the low frequency losses for a two-stage system described by COLISEUM. We can see that these losses are driven by four products 𝑛𝜅𝜏 (time constants and their related shielding coefficients). Defining 𝑎 𝑗 = 𝑛𝜅 𝑗 𝜏 𝑗 ∑ 𝑛𝜅 𝑖 𝜏 𝑖 𝑖 × 100 which can be seen as weights in % representing the 𝑛𝜅𝜏 contribution to the losses/power, we are able to distinguish the number of relevant contributions to the coupling losses regarding a certain threshold. The point here is to quantify the number of relevant 𝑛𝜅𝜏 among the four that contribute to coupling losses in order to perform a clean reduction of the system equation without information losses. Taking an example of simulation with the values from Table 2 (triplet of triplet, two first stage of JT-60SA TF). We can see in Table 7 the four initial contributions to coupling losses given by the two-stage COLISEUM for the two first stage of JT-60SA TF conductor. Taking only the two last lines will give us a result with 99.96% accuracy regarding the full approach. In order to have a clear eye on this possible reduction to two time constants, we choose to probe among the four given by COLISEUM, the number of products 𝑛𝜅𝜏 contributing to the coupling losses of this two-stage system in a large domain of parameters.

In order to do so, we explore the space

( 𝜎 2 𝜎 1 , 𝑙 𝑝 2 𝑙 𝑝 1
) which is representative of various cable patterns: short vs long twist pitch, low vs high void fraction or strand type (Chrome coated, CuNi barrier) (see [START_REF] Nijhuis | Optimization of ITER Nb3Sn CICCs for coupling loss, transverse electromagnetic load and axial thermal contraction[END_REF]). We gather results under the form of cartographies (see Figure 23) where colours indicate the number of contributions above a defined threshold "𝑠", among the four initial ones. If we choose 𝑠 = 1 %, in Table 7, two among the four contributions are not taken into account in the coupling losses computation, we are left with only two contributions (𝑎 𝑗 < 1 %). If we chose 𝑠 = 3 %, we are left with one shielding coefficient and time constant to describe the system (𝑎 𝑗 < 3 %). In both cases, we respectively make an error about 3.8.10 -3 % and 2.57 % regarding the four initial contributions. ratio are chosen in a realistic range regarding the design of the cable in conduit conductor (CICC) nowadays (see [START_REF] Nijhuis | Optimization of ITER Nb3Sn CICCs for coupling loss, transverse electromagnetic load and axial thermal contraction[END_REF], [START_REF] Bruzonne | 30 years of conductors for fusion: a summary and perspectives[END_REF]).

𝑙 𝑝 2 𝑙 𝑝 1
ratio is taken greater than one, as second stage twist pitch length is conventionally always greater than first stage twist pitch length, and up to four to represent both short twist pitch cables and long twist pitch cables. ratio is taken from 0.5 to 6 as we do not know for sure if successive interstage conductances are increasing or decreasing.

As we can see in the above Figure 23, taking the threshold to 10 % show us that only 2 time constants (or 1 in a great part of the domain) are needed to get at maximum a 20 % (or 30 % if we are left with one contribution) accurate coupling losses prediction. We decrease the threshold to probe the value where a third contribution will rise.

Decreasing the threshold from 10 % to 4 %, and then to 1 %, shows us that the coupling losses are mainly driven by two time constants (or 3 in a very small domain close to

𝑙 𝑝 2 𝑙 𝑝 1
= 1 which is quite unrealistic). Decreasing the threshold lets appear that the third contribution zone is small compared to the two already existing contributions. The error made on coupling losses with the 1% threshold by taking two contributions instead of the four initials is at maximum 2 %. Actually in Figure 23, we do not depict for each point of the simulation the value of the magnetic parameters thus we can simply make comments on the maximum of error made on the computation of coupling losses. As depicted in Table 7 and as said earlier, taking a threshold 𝑠 = 1 % takes out two contributions and intrinsically brings a maximum of 2 % error on the predicted coupling losses. But, the contributions we take out of the prediction are much smaller than the maximum of 2 % when we take a closer look to the magnetic parameters we take out as in Table 7. We will show in the following subsections that this trend is confirmed on the whole surface of Figure 23 and strengthens our intuition that the two-stage COLISEUM can be reduced to a system of dimension two.

The two time constants that are non-contributing (Table 7 and Figure 23) to the coupling losses are not null but they do not contribute due to their very low associated shielding coefficient. Using these insights on the two-stage model we clearly see a first step toward the reduction of the model to a two dimensions system without much loss of information. In the following subsections, we will establish that it is possible to reduce the dimension of the system by showing that the magnetic parameters we obtain after reduction are still very close to the contributing ones obtained with the initial four dimension modelling.

II.3.2.2 A simple two-stage system: Analytic expressions

In order to illustrate and consolidate the reduction, we investigate an analytical two-stage model. Following an idea and calculations of B. Turck [START_REF] Turck | Private communication[END_REF] we consider a doublet of doublets (𝑁 1 = 𝑁 2 = 2) with 𝑙 𝑝 2 = 2𝑙 𝑝 1 in order to carry out an easy and instructive analytical calculation. We consider only a magnetic interaction between current loops but no mixing of currents. We only consider half a pitch of the second stage in Figure 24. In COLISEUM, the calculus is managed in the cross section of the strand, here the calculus is from a top view, as for slab, but is correct to assess the range and behaviour of the time constants. This analytical system is driven by the two following equations [START_REF] Nijhuis | Optimization of ITER Nb3Sn CICCs for coupling loss, transverse electromagnetic load and axial thermal contraction[END_REF]:

𝑑𝜙 1 𝑒𝑥𝑡 𝑑𝑡 = (𝐿 1 𝑑𝐼 1 𝑑𝑡 + 𝑅 Ω 1 𝐼 1 + 𝑀 21 𝑑𝐼 2 𝑑𝑡 ) & 𝑑𝜙 2 𝑒𝑥𝑡 𝑑𝑡 = (𝐿 2 𝑑𝐼 2 𝑑𝑡 + 𝑅 Ω 2 𝐼 2 + 4𝑀 12 𝑑𝐼 1 𝑑𝑡 ) ( 43
)
1 and 2 are the index referring respectively to first-stage elements and to second-stage elements. 𝐿 𝑖 is the inductance of circuit 𝑖, 𝑀 𝑖𝑗 is the mutual inductance of circuit 𝑖 over 𝑗 and 𝑅 Ω 𝑖 is the resistance of the current loop in circuit 𝑖. 𝜙 𝑖 is the embraced flux for one loop of current in the circuit 𝑖 (hatched area in Figure 24).

We can assume for the following part that 𝑀 = 𝑀 21 = 𝑀 12 .

Using Laplace transform for a step function 𝜙 1 (𝑝) = 𝜙 1 𝑝 gives us:

{ 𝜙 1 (𝑝) = (𝐿 1 𝑝 + 𝑅 Ω 1 )𝐼 1 + 𝑀𝑝𝐼 2 𝜙 2 (𝑝) = (𝐿 2 𝑝 + 𝑅 Ω 2 )𝐼 2 + 4𝑀𝑝𝐼 1 (44)
We can also consider that 𝐿 2 = 4𝐿 1 , so we have

{ 𝜙 1 (𝑝) = (𝐿 1 𝑝 + 𝑅 Ω 1 )𝐼 1 + 𝑀𝑝𝐼 2 4𝜙 1 (𝑝) = (𝐿 2 𝑝 + 𝑅 Ω 2 )𝐼 2 + 4𝑀𝑝𝐼 1 (45)
Which gives:

{ 𝐼 2 = 𝜙 1 -(𝐿 1 𝑝 + 𝑅 Ω 1 )𝐼 1 𝑀𝑝 𝑓𝑟𝑜𝑚 𝑓𝑖𝑟𝑠𝑡 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝐼 2 = 4𝜙 1 -4𝑀𝑝𝐼 1 (𝐿 2 𝑝 + 𝑅 Ω 2 ) 𝑓𝑟𝑜𝑚 𝑠𝑒𝑐𝑜𝑛𝑑 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (46)
The two currents 𝐼 1 and 𝐼 2 are:

{ 𝐼 1 = 𝜙 1 ((4𝑀 -𝐿 2 )𝑝 -𝑅 Ω 2 ) 𝑝 2 (4𝑀 2 -𝐿 1 𝐿 2 ) -𝑝(𝐿 1 𝑅 Ω 2 + 𝐿 2 𝑅 Ω 2 ) -𝑅 Ω 1 𝑅 Ω 2 𝐼 2 = 4𝜙 1 ((𝑀 -𝐿 1 )𝑝 -𝑅 Ω 1 ) 𝑝 2 (4𝑀 2 -𝐿 1 𝐿 2 ) -𝑝(𝐿 1 𝑅 Ω 2 + 𝐿 2 𝑅 Ω 1 ) -𝑅 Ω 1 𝑅 Ω 2 (47)
The time constants related to this analytical system are given by the roots of the denominator, using 𝑀 = 𝐿 1 2 :

𝜏 ± = 6𝐿 1 (4𝑅 Ω 1 + 𝑅 Ω 2 ) ∓ √16𝑅 Ω 1 2 -4𝑅 Ω 1 𝑅 Ω 2 + 𝑅 Ω 2 2 (48)
It is seen that the system which consists of two type of loops (1 and 2), with respective basic "uncoupled" time constants for the first stage

𝜏 1 = 𝐿 1 𝑅 Ω 1
and for the second stage

𝜏 2 = 𝐿 2 𝑅 Ω 2
, is ruled by a set of two new time constants 𝜏 + and 𝜏 -, after coupling. In fact the above expressions of 𝜏 ± can be written in form only function of the ratio

𝜏 2 𝜏 1
as follow:

𝜏 ± 𝜏 1 = 3 2 (1 + 𝜏 2 𝜏 1 ∓ (1 - 𝜏 2 𝜏 1 -( 𝜏 2 𝜏 1 ) 2 )) (49)
This system of two-stage is driven by only two time constants, as in MPAS. The variations of

𝜏 ± 𝜏 1 against conductance ratios ( 𝜎 2 𝜎 1 = 𝑅 Ω 1 𝑅 Ω 2
) are shown in Figure 25.

We have to note that the asymptote in blue is the curve

𝜏(𝜎 1 →0) 𝜏(𝜎 2 →0) 𝑣𝑠 𝜎 2 𝜎 1
.

It appears that for the two above coupled time constants 𝜏 ± , while taking the limit of 𝜎 2 either to 0 or to ∞, one time constant

(𝜏 + ) is alternatively equal to 𝜏 1 ( if 𝜎 2 = 0 ) or to 𝜏 2 ( if 𝜎 2 → ∞ ).
The other time constant is null when 𝜎 2 → 0 or goes to a finite value with a null shielding coefficient when 𝜎 2 → ∞ as shown in Table 4.

As 𝜏 ± only depends on conductances (resistances), we will explore in the next subsection the variation of the four time constants given by the two-stage COLISEUM and compare them to the variation of the two time constants of the above model. 1

II.3.2.3 Outcomes from COLISEUM

We choose to use parameters of Table 2 to simulate the two first stages of JT-60SA TF conductor as done above in section II.1.3. But this time, we will check the variation of the four coupled time constants given by the two-stage COLISEUM regarding the conductance ratio as done for the doublet of doublet in Figure 25 in order to check for similarities with the previously presented model of section II.3.2.2.

In COLISEUM, for the case of the two first stages of JT-60SA TF, it is evidenced that two time constants among the four (yellow and blue in Figure 26) behave like the two obtained from the analytical model (black in Figure 25). In fact, two time constants are varying with the conductance ratio in the same way the time constants do in Figure 26.

The two time constants 𝜏 0 and 𝜏 3 are related to current flowing inside the cable and we show that they do not vary with the conductances ratio which is quite unintuitive. It is legitimate to think that time constants reflecting the electromagnetic behaviour of the system should depend on the conductance ratio (electric parameter). On the other side, the two time constants that do not vary are also the two with very small shielding coefficient seen in Figure 14. Using this, we can argue that regarding their shielding coefficient they are not contributing or seen as higher order correction to the total losses. A similar argument is already used in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] to reduce the dimension of the system from infinity to four. As the initial calculation for current is a Fourier development, we obtain an infinite number of contributions that are chosen to be ignored due to their very low contribution to the computation of the magnetic parameters of the two-stage system. 2.

1 1 𝜏 0 /𝜏 1 𝜎 2 =0 𝜏 1 /𝜏 1 𝜎 2 =0 𝜏 2 /𝜏 1 𝜎 2 =0 𝜏 3 /𝜏 1 𝜎 2 =0 𝜏 2 𝜎 1 =0 𝜏 1 𝜎 2 =0
We have already mentioned the fact that among the four shielding coefficients we have in the two-stage COLISEUM two of them are null. We now look at the variation of the value of the four shielding coefficients related to each time constant in Figure 27 below. It is also another mean to justify the reduction of the dimension of the system as if two shielding coefficients among the four are always null, we will be confident in the fact that these two couples of magnetic parameters we remove do not contribute to the losses as shown in equation [START_REF] Bruzonne | 30 years of conductors for fusion: a summary and perspectives[END_REF] and in Table 7.

The two unvarying time constants of Figure 26 are the two ones related to null shielding coefficient (𝑛𝜅 0 and 𝑛𝜅 3 ). It is also very interesting to note that the shielding coefficients variation go through non negligible variations around the value log ( We can also note an interesting fact, in Figure 27, the sum of 𝑛𝜅 1 and 𝑛𝜅 2 is constant with respect to the variation of the logarithmic ratio of conductances. We have

𝑛𝜅 1 + 𝑛𝜅 2 = 1.013
as we can already see in the coupled case of Figure 14.

Using the two previous arguments (statistical with Figure 23 and analytical with Figure 23), it is trustworthy to try to reduce our system description from the initial four-time constants description to two time constants without losing too much information on the system.

In the next sub-section, we analytically reduce the number of dimensions of the two-stage COLISEUM from four to two and show that this reduction is safe as we will demonstrate that the two removed dimension are small corrections regarding the two remaining contributions.

II.3.2.4 Dimensions reduction approach for the two-stage COLISEUM.

In order to consolidate this reduction of the number of time constants, we will look with the equations of the system. The two-stage model is governed by equation [START_REF] Breschi | Analysis of AC losses in a CS conductor sample for ITER project[END_REF] written below in dimension four where all terms are already defined in section II.1.4:

[ 𝐼 0 𝐼 1 𝐼 ̃2 𝐼 3 ] + ( 𝜏 (𝛼 0 )(𝛼 0 ) 𝜏 (𝛼 0 )(𝛼 1 ) 0 0 𝜏 (𝛼 1 )(𝛼 0 ) 𝜏 (𝛼 1 )(𝛼 1 ) 𝜏 (𝛼 1 )(𝛼 2 ) 0 0 𝜏 (𝛼 2 )(𝛼 1 ) 𝜏 (𝛼 2 )(𝛼 2 ) 𝜏 (𝛼 2 )(𝛼 3 ) 0 0 𝜏 (𝛼 3 )(𝛼 2 ) 𝜏 (𝛼 3 )(𝛼 3 ) ) [ 𝐼 ̇0 𝐼 ̇1 𝐼 ̃2 𝐼 ̇3] = [ 0 4𝜎 1 𝑅 𝑐 1 sin 2 ( 𝜋 𝑁 1 ) /𝛼 1 2 4 𝜎 2 𝑁 1 𝑅 𝑐 2 sin 2 ( 𝜋 𝑁 2 ) /𝛼 2 2 0 ] 𝐵 ̇𝑎 (50)
In order to get rid of the contribution from 𝐼 0 and 𝐼 3 , which we already have shown that they do not contribute to the modelling of coupling losses in a large domain, the above system (50) can be reduced to a two-dimensions system (red brackets), as follows:

[ 𝐼 1 𝐼 ̃2] + ( 𝜏 (𝛼 1 )(𝛼 1 ) 𝜏 (𝛼 1 )(𝛼 2 ) 𝜏 (𝛼 2 )(𝛼 1 ) 𝜏 (𝛼 2 )(𝛼 2 ) ) [ 𝐼 ̇1 𝐼 ̃2 ] = [ 4𝜎 1 𝑅 𝑐 1 sin 2 ( 𝜋 𝑁 1 ) /𝛼 1 2 4 𝜎 2 𝑁 1 𝑅 𝑐 2 sin 2 ( 𝜋 𝑁 2 ) /𝛼 2 2 ] 𝐵 ̇𝑎 (51)
We will show in the next sub-section II.3.2.5 that after diagonalization of equation ( 51), we get two time constants and two shielding coefficients without losing too much information on the extracted magnetic parameters (𝜏, 𝑛𝜅). These two magnetic parameters are those that show an expected behaviour as for the time constants in Figure 25, i.e., they vary with the conductances ratio. We are confident that the two other time constants that are excluded in this process are the two that do not vary with the conductance ratio as shown in Figure 23. This can be expected as in the second member of equation [START_REF] Breschi | Analysis of AC losses in a CS conductor sample for ITER project[END_REF], the vector contains 0 on the first and the last line, which are related to the unvarying time constant of Figure 26. This point will be further verified after the analytical development of the system in the next section II.3.2.5. The two time constants we are left with would be the ones related to the coupled two-stage system.

Another important point is that with four dimensions the system could not be expressed analytically. As a matter of fact, matrices with dimension above three are not analytically diagonalizable and we have to use a numerical tool in order to implement the diagonalization of the tridiagonal fourdimension matrix. Here with our reduction to a two dimensions system, we can analytically express the two time constants of this two-stage system.

We can also point out that with this reduction of the two-stage model, we are now in line with the MPAS description attributing one couple (time constant, shielding coefficient) to each stage of a cable. We will show in the next sub-section II.3.2.5 how to deal with the mathematical reduction of the system and with the computation of the magnetic parameters in dimension two.

II.3.2.5 Analytical development and determination of τ values and nk values for the two-stage COLISEUM

The second equation of system (51) will be redefined which is not unavoidable for this analytical development but will be of great interest for the generalization of the model in the next section III. We multiply both sides by 𝑁 1 as follows and as already mentioned in section II.1.4:

𝑁 1 𝐼 ̃2 +
Where we explicitly mentioned the conductance dependence of each element of the matrix to notice that the first line only depends on first stage conductance (𝜎 1 ) and the second line of the matrix only depends on the second stage conductance (𝜎 2 ).

The dissipated power per unit volume is as follow, reduced from equation [START_REF] Wilson | Time-varying fields and A.C. losses[END_REF]:

𝑃 𝑣𝑜𝑙 = 𝑁 2 𝑁 1 𝛼 1 2 8𝜎 1 sin 2 ( 𝜋 𝑁 1 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2 𝐼 1 2 + 𝑁 2 𝛼 2 2 8𝜎 2 sin 2 ( 𝜋 𝑁 2 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2 𝐼 2 2 = 𝛿 1 𝐼 1 2 + 𝛿 2 𝑁 1 2 𝐼 2 2
Here, it is important to note that this change of variable only affects the description of the second stage and doesn't affect the first stage description. As already said earlier, the current 𝐼 ̃2 used in the initial description of the two-stage system does reflect the physical current amplitude up to a factor 𝑁 1 .

As we can see in the expression of 𝑃 𝑣𝑜𝑙 the contribution of the current amplitude 𝐼 1 is taken 𝑁 2 𝑁 1 times into account and the contribution of the current amplitude 𝐼 2 is taken 𝑁 2 times into account which is inline with the number of elements present in each stage of the two-stage cable.

The above time constant matrix allows us to derive analytically the two coupled time constants of the system by diagonalization as opposed to the previous version in dimension four, which could not be analytically solved.

The two eigenvalues of the matrix [𝜏] can be computed as:

𝜏 1 = 𝑇𝑟([𝜏]) -√(𝑇𝑟[𝜏]) 2 -4 det[𝜏] 2 𝑎𝑛𝑑 𝜏 2 = 𝑇𝑟([𝜏]) + √(𝑇𝑟[𝜏]) 2 -4 det[𝜏] 2
Where

We consider 𝜏 22 > 𝜏 11 which is a reasonable assumption as 𝜏 22 is the time constant of the uncoupled second stage and 𝜏 11 the time constant of the uncoupled first stage. Also assuming that all components of the [𝜏] matrix are positive numbers, we can easily demonstrate that the coupled time constant will always be segregated as depicted in Figure 15.Thus we always have:

𝝉 𝟐 > 𝝉 𝟐𝟐 𝑎𝑛𝑑 𝝉 𝟏 < 𝝉 𝟏𝟏 (55)
Which demonstrate that the behaviour of coupled time constants with respect to their uncoupled value are always as depicted in Figure 15.

In order to compute the value of the shielding coefficients, we have to carry out the calculation of the eigenvectors related to each time constant. These eigenvectors are paired with eigenvalues (the coupled time constants) such as:

[𝜏] ( 𝑥 2 𝑦 2 ) = 𝜏 2 ( 𝑥 2 𝑦 2 ) and [𝜏] ( 𝑥 1 𝑦 1 ) = 𝜏 1 ( 𝑥 1 𝑦 1 ) (56)
Where 𝑥 and 𝑦 with index 1 and 2 are the component of the eigenvector respectively related to 𝜏 1 and 𝜏 2 . Once ( 56) is developed, we get:

{ (𝜏
Adding to this a norm condition on both eigenvectors, we obtain two new systems of equations:

{ 𝑥 2 2 + 𝑦 2 2 = 1 𝑥 2 = 𝐶 2 𝑦 2 and { 𝑥 1 2 + 𝑦 1 2 = 1 𝑥 1 = 𝐶 1 𝑦 1 (59)
The resolution of the above systems [START_REF] Zani | Completion of TF strand production and progress of TF conductor manufacture for JT-60SA project[END_REF] gives:

( 𝑥 2 𝑦 2 ) = ( 𝐶 2 √1+𝐶 2 2 1 √1+𝐶 2 2
)

and ( 𝑥 1 𝑦 1 ) = ( 𝐶 1 √1+𝐶 1 2 1 √1+𝐶 1 2 ) (60)
Hence, we can write:

[𝜏] = [𝑉] ( 𝜏 1 0 0 𝜏 2 ) [𝑉] -1 (61)
Where [𝑉] is the eigenvector passage matrix defined by concatenation of the two eigenvector as:

[𝑉] = ( 𝐶 1 √1 + 𝐶 1 2 1 √1 + 𝐶 1 2 𝐶 2 √1 + 𝐶 2 2 1 √1 + 𝐶 2 2 ) (62)
Is the matrix composed of the concatenation of both eigenvector of equation (62). We can also defined 𝑌 as:

𝑌 = ( 4𝜎 1 𝑅 𝑐 1 sin 2 ( 𝜋 𝑁 1 ) 1 𝛼 1 2 4𝜎 2 𝑅 𝑐 2 sin 2 ( 𝜋 𝑁 2 ) 1 𝛼 2 2
) and

𝑌 𝑏 = [𝑉] -1 𝑌. ( 63
)
Here it is important to note that each time constant is related to one column of the matrix 𝑉 through the relation eigenvalue-eigenvector. As the eigenvector matrix described the transformation from the initial basis to the diagonal one, we can tell for each eigenvalue, regarding the two components of each eigenvector, if a coupled time constant is related to only the first stage or only to the second stage or to a linear combination of both uncoupled stages.

As done in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] and presented earlier in this sub-section, the power losses per unit of volume of cable when reduced to a two-dimension system writes:

𝑃 𝑣𝑜𝑙 = 𝛼 1 2 𝑁 1 𝑁 2 8𝜎 1 sin 2 ( 𝜋 𝑁 1 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2 𝐼 1 2 + 𝛼 2 2 𝑁 2 8𝜎 2 sin 2 ( 𝜋 𝑁 2 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2 𝐼 2 2 = 𝜆 1 𝐼 1 2 + 𝜆 2 𝐼 2 2 (64)
Where we defined 𝜆 1 = 𝛿 1 and 𝜆 2 = 𝛿 2 /𝑁 1 2 . We can note that in the following, 𝛿 symbols from the notation found in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] will be intentionally replaced as they are redefined here above using the 𝜆 ′ 𝑠.

Adapting the calculation of the 𝑛𝜅 performed in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] we can derive the expressions of the two related shielding coefficients. For this analytic-oriented version of the two-stage COLISEUM, we get:

𝒏𝜿 𝟏 = 𝟐𝝁 𝟎 ∑ ∑
Now that we have given the analytical expression of the magnetic parameters of this rationalized version of the two-stage COLISEUM we can assess the analytical limit of the system when one of the two 𝜎 is put to zero. These values are the proper magnetic parameters of the second stage. We know that 𝜏 2 and 𝑛𝜅 2 are directly coming from the uncoupled second stage because of its associated eigenvector:

( 𝑥 2 𝑦 2 ) = ( 0 1
).

Using the same methodology, in the case 𝜎 2 = 0:

System equations ( 52

implying that 𝑉 = ( 1 0 𝐶 2 √1+𝐶 2 2 1 √1+𝐶 2 2 ) -𝑌 𝑏 = ( 0 4𝜎 1 𝑅 𝑐 1 sin 2 ( 𝜋 𝑁 1 ) /𝛼 1 2)
Hence,

{ 𝜏 2 = 𝜏 11 𝜏 1 = 0 and { 𝑛𝜅 2 = 2𝜇 0 𝑁 1 𝑁 2 𝑅 𝑐 1 2 𝜎 1 sin 2 ( 𝜋 𝑁 1 ) 𝛼 1 2 𝜏 11 𝜋𝑅 𝑐𝑖𝑟𝑐 2 𝑛𝜅 1 = 0 (68)
These values are the proper magnetic parameters of the first stage for the same reason as above.

We can see in Figure 28 below that the two time constants behave as the two exposed in Figure 26.

We clearly see that the two above time constants behave as presented in Figure 26 in dimension four and we will show in the following that their value is not so much affected by the suppression of the two extra dimensions of the initial system (𝐼 0 and 𝐼 3 ). In Figure 29, we look at the two shielding coefficients we are left with after this reduction, i.e; 𝑛𝜅 1 and 𝑛𝜅 2 : The two above figures showed that reducing the dimension of our system by cancelling 𝐼 0 and 𝐼 3 supresses the two time constants with unexpected behaviour from Figure 26 (𝜏 0 and 𝜏 3 ) and their related null shielding coefficient from Figure 27 (𝑛𝜅 0 and 𝑛𝜅 3 ). The two couples of magnetic parameters we are left with in Figure 28 and Figure 29 are in correspondence (they overlap) with the ones shown in Figure 26 and Figure 27.

The reduction of the two-stage COLISEUM to a two dimensions system, allowing fully analytical calculation of the magnetic parameters (𝜏, 𝑛𝜅) of the two-stage model, is established. We have to push the verification of the coherence of the reduction by checking if the removed dimensions, do not play any role in the computation of time constants and shielding coefficients separately as shown above (Figure 28 and Figure 29) but in a broader domain of variation as done in Figure 23.

It is also interesting to notice that going from the coupled to the uncoupled case, the coupled system of equation become two independent equation as described below:

Coupled: Uncoupled: We know that these uncoupled equations (right) are the one given in the formulation of the one-stage COLISEUM. By intuition, we see that in order to switch from the coupled to the uncoupled case we have to set to zero the terms of mutual inductance (𝜏 𝑖𝑗 with 𝑖 ≠ 𝑗) in both equations of the coupled case. The coupling between first and second stage in a two-stage system is represented with the non-diagonal 𝜏 𝑖𝑗 . We can demonstrate that the intuition to set both mutual inductance terms to 0 in the initial coupled system will lead to uncoupled systems described simultaneously instead of two simulations with either 𝜎 1 = 0 or 𝜎 2 = 0.

{
If 𝜏 12 = 0 and 𝜏 21 = 0 in equation ( 52), we have:

𝜏 1 =
Using this method in order to uncouple two coupled systems is strictly equivalent to taking the limit of both 𝜎 to zero but more efficient as we obtain both limits in one computation. Also the fact that both stages are "alive" and uncoupled whereas using the method of sigma, in both cases (𝜎 1 = 0 or 𝜎 2 = 0 ) one stage systematically disappears. This way to uncouple systems will be useful once the model will be generalized to a greater number of stages.

The advantage of this "mutual inductance to zero" method is that the two stages are described from the coupled to the uncoupled cases at once (see Figure 30 and Figure 31) whereas using the limit of 𝜎 to zero, we artificially cancel one of the two uncoupled stages.

A coefficient 𝜉 is used numerically to put the mutual inductance terms 𝜏 𝑖𝑗 (𝑖 ≠ 𝑗) to zero by multiplication in the time matrix of equation ( 52) as follows:

[𝜏] = ( 𝜏 11 (𝜎 1 ) 𝜉 𝜏 12 (𝜎 1 )/𝑁 1 𝜉 𝑁 1 𝜏 21 (𝜎 2 ) 𝜏 22 (𝜎 2 ) )

If 𝜉 = 1 , the two-stage are fully coupled and if 𝜉 = 0, they are fully uncoupled. In between these two values, the two sub systems are partially coupled. The two curves of Figure 30 and Figure 31 illustrate the variation of the magnetic parameters with respect to the mutual inductance between the two considered stages. It is interesting to note that, in terms of magnetic parameters variations, they are both monotonic. This modelling of the coupling-uncoupling phenomenon allows to check that for this case of the two first stage of JT-60SA TF, the magnetic parameters do not pass by an extremum value while uncoupling subsystems. Time constants and shielding coefficients are enclosed between their coupled and uncoupled value; shown in Figure 15. This information on the magnetic parameters variation could be used to enhance the MPAS modelling of coupling losses (e.g. setting range for fitting parameters).

Reducing the initial four dimensions system to a two dimensions one allows us to perform the analytical calculation of the magnetic parameters of the two-stage COLISEUM. ) contribute in the calculation of the relevant magnetic parameters (the ones which do not have a null shielding coefficient, see equation [START_REF] Bagni | Analysis of ITER NbTi and Nb3Sn CICCs experimental minimum quench energy with JackPot, MCM and THEA[END_REF] and Table 3).

In order to be sure that the reduction is reliable, we have to check if the computation of the magnetic parameters (𝜏, 𝑛𝜅) in dimension two gives for the two conserved dimensions, the same results as if it were simulated with the initial four dimensions system. Using this fact, we can be sure that the reduction of the system dimension is reliable because the two removed dimensions do not contribute to the computation of the magnetic parameters we are left with (see Figure 33 and Figure 32). The first level of verification was to check the number of contributions (product 𝑛𝜅𝜏) to the coupling losses among the four initial dimensions in two stage COLISEUM to legitimate our reduction. Now we compare the value of magnetic parameters, obtained using on one side the reduced two dimension COLISEUM, to check if the removed dimensions do not affect the computation of their values initially computed with the initial four dimension COLISEUM. We can clearly see that removing the two extra dimensions (𝐼 0 𝑎𝑛𝑑 𝐼 3 ) does not affect too much the value of the two computed magnetic parameters. Some cases (points) show a greater error but also stay in the same order of magnitude as shown for the whole surface of Figure 32 and Figure 33. The relative error made on the computation of the time constants of the system is below 5.10 -4 % for the explored range of parameters. Same observations hold for the shielding coefficients, with a relative error below 2.10 -3 %. The explored range of parameters is assumed to be large enough to be representative of various cable configurations as presented in the parametric study of section II.3.2.1. This last point confirms that the reduction of the system dimension is safe and robust and that the two-stage COLISEUM updated approach (two dimensions) is not degrading the model reliability in its reduced version.

Thus, at this stage we can assess that we have strengthened the existing two-stage COLISEUM by allowing it to be represented under explicit analytical expressions of its magnetic parameters (formerly performed using numerical tools) and also bringing it closer to the MPAS description of AC losses (one set of magnetic parameters per cable stage). 2.

II.3.3 Two-stage COLISEUM vs one-stage COLISEUM

In this section we will check the consistency between the description given by the uncoupled two-stage COLISEUM and by the one-stage COLISEUM. Both uncoupled sub-systems of the two-stage COLISEUM (first and second stage) will be simulated by a one-stage COLISEUM and compare (in terms of magnetic parameters) to the uncoupled description given by the two-stage COLISEUM. In this way, we will be sure that a system modelled with the one stage COLISEUM (i.e. the strand from section II.3.1) can be assembled in multiplet using the two-stage COLISEUM.

Starting from the two-stage modelling of Figure 14, we use the one-stage COLISEUM to check if both uncoupled sub-systems of the two-stage system can be modelled using the one-stage approach. As said earlier, this crosscheck between one-stage and two-stage COLISEUM is a way to be sure that we can include the strand (as a one-stage system) in the two-stage COLISEUM and that the modelling will be correctly defined once coupled in multiplet.

The radius 𝑅 𝑒𝑙𝑒𝑚 of 0.405 mm is the radius of the JT-60SA TF composite. 𝑅 𝑓 , the radius of the basic tube is taken equal to the radius of the filamentary zone of JT-60SA TF strand. The ratio The two-stage system can be seen as composed of two one-stage systems. Parameters and dimensions for the two systems (bottom part of Figure 34 We find that the time constant is strictly equal in both cases to the value of 1.68 𝑚𝑠 as seen in the bottom part of Figure 34. The shielding coefficients are also in good agreement when referred to the same reference area (circumscribed area of the first stage). In the case where the first stage is described by the uncoupled two-stage COLISEUM, its shielding coefficient is referred to the circumscribed area of the two-stage system. Once referred to the circumscribed area of the first stage only, we clearly see the agreement:

𝑛𝜅 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑓𝑖𝑟𝑠𝑡 𝑠𝑡𝑎𝑔𝑒 = 0.611 × 1 3 ( 𝑅 𝑐𝑖𝑟𝑐 𝑠𝑢𝑝𝑒𝑟𝑠𝑡𝑎𝑔𝑒 𝑅 𝑐𝑖𝑟𝑐 𝑠𝑢𝑏𝑠𝑡𝑎𝑔𝑒 ) 2 = 0.949
As seen in Figure 34, using these parameters with the one-stage COLISEUM evidences that the uncoupled substage in the two-stage COLISEUM is consistent with the one-stage COLISEUM used to model the substage alone.

The slight difference between the two values of 𝑛𝜅 comes from the magnetic interaction between the 3 substages that exists in the uncoupled two-stage model but not in the one-stage model. Indeed, even though the second and first stages are uncoupled, the three first stages still feel one another magnetically speaking (i.e. through the magnetic field variations they produce when shielding their volume from the external field variation 𝐵 ̇𝑎). This effect can be seen as a collective demagnetization effect which thus tends to increase the corresponding 𝑛𝜅 value without modifying the time constant.

System 2:

In order to compute the magnetic parameters of the second stage alone, as simulated with the one-stage COLISEUM, we need to select a value for the superconducting tube radius 𝑅 𝑓 2 to be used in the onestage model. In order to determine 𝑅 𝑓 2 we use the value of the shielding coefficient found in the case of the uncoupled super-stage simulated with the two-stage COLISEUM: We can determine numerically, using dichotomy, the value of 𝑅 𝑓 2 in order, for the one-stage COLISEUM, to have the same shielding coefficient and time constant of the uncoupled super-stage.

𝑛𝜅 𝑑𝑒𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑠𝑢𝑝𝑒𝑟𝑠𝑡𝑎𝑔𝑒 = 0.
Using the geometrical parameters of system 2 gathered below, we determine ( 𝑅 𝑓 𝑅 𝑒𝑙𝑒𝑚 ) 2 as described above and found 0.68.

We also show that this value can also be found using the magnetic parameters of the above uncoupled system 1 (simulated using the one-stage COLISEUM) as follows:

( 𝑅 𝑓 𝑅 𝑒𝑙𝑒𝑚 ) 2 = √ 𝑛𝜅 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑓𝑖𝑟𝑠𝑡 𝑠𝑡𝑎𝑔𝑒 2 = √ 0.949 2 = 0.68
This relation is due to the assumption that the system 1 modelled with the one-stage COLISEUM can be homogenized in a round composite with a filamentary zone radius equal to the above 0.68 value. This result was not evident and comforted us in the global consistency of our different models. Once again, we can see that the uncoupled description of the super-stage is accurate in the two-stage COLISEUM as it gives a result in agreement with the homogenized model. Here, the time constant of the second stage isolated (simulated with the one-stage COLISEUM) is strictly equal to the time constant of the uncoupled second stage (simulated with the two-stage COLISEUM) as seen in Figure 34. It shows the explicit correspondence between the diagonal terms of the [𝜏] matrix and the expression of time constants given by the one-stage COLISEUM for each stage of the cable:

𝜏 𝑖𝑖 = 𝜏 𝑖 𝑓𝑟𝑜𝑚 𝑜𝑛𝑒-𝑠𝑡𝑎𝑔𝑒 𝐶𝑂𝐿𝐼𝑆𝐸𝑈𝑀
This is evident as we have used the shielding coefficient of the uncoupled second stage to set the equivalent filamentary zone radius of the system 2. We have also shown that it is equivalent to use the shielding coefficient of the uncoupled second stage given by the two-stage COLISEUM or the shielding coefficient of the isolated first stage given by the one-stage COLISEUM. Shielding coefficients of uncoupled second stages are equal as already referred to the same circumscribed area, the one of the second stage.

According to Figure 34 (and to our several tests on different type of geometries) we see that the correspondence between the one-stage and uncoupled two-stage models is well established and the consistency between them is clearly exposed. We are confident that we will succeed to model analytical multiplet of strand where the strand will be modelled as the first stage of the two-stage COLISEUM.

Simulating the same system (first stage or second stage) by using the uncoupled two-stage COLISEUM or two separate one-stage COLISEUM is almost equivalent. We obtain very close magnetic parameters in the two approaches:

𝑵

One-stage COLISEUM ≈ Uncoupled two-stage COLISEUM

As the two-stage COLISEUM has been developed following the same methodology as for the one-stage COLISEUM, the precedent observation seems logical but has to be verified.

Coupled two-stage COLISEUM ≈ Coupling two one-stage COLISEUM.

Sum of 𝒏𝜿𝝉 :

The demonstration of the conservation of the sum of the product of magnetic parameters for the uncoupled and coupled case is given here to show the reliability of our model reduction.

Knowing that the uncoupled two-stage COLISEUM is equivalent to the one-stage COLISEUM, we can use the analytical expression of the one-stage model to say that the sum of the product of its magnetic parameter is equal to:

∑ 𝑛𝜅𝜏 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 = 𝜇 0 𝑁 2 (2𝜎 1 𝑁 1 𝑅 𝑐 1 2 sin 2 ( 𝜋 𝑁 1 ) ( 𝑙 𝑝 1 2𝜋 ) 2 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2 + 𝜇 0 (2𝜎 2 𝑁 2 𝑅 𝑐 2 2 sin 2 ( 𝜋 𝑁 2 ) ( 𝑙 𝑝 2 2𝜋 ) 2 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2
The multiplicative terms 𝑁 2 for the first term of the above expression is because in a two-stage system we have 𝑁 2 times the first stage in the contribution to coupling losses.

Looking at equation [START_REF] Itoh | Construction of high magnetic field tokamak 'TRIAM-1' for nuclear fusion research[END_REF], and in the low frequency regime, the equality between sum of 𝑛𝜅𝜏 products looks like an power conservation rule as the dissipated power in an uncoupled cable where each stage is isolated from the surrounding others is equal to the dissipated power in the same cable where each stage is coupled to its neighbouring stages. The same amount of power is redistributed inside the cable regardless if it is coupled or uncoupled. As said earlier, on a more intuitive ground, we can say that the coupled cable, in the low frequency regime has a resistance mesh equivalent to the one of an uncoupled cable.

In the case of the two coupled stages and in the low frequency regime, we can neglect the magnetic coupling between currents induced in all elements. This implies that 𝐼 ̇ in the differential equation of the system (equation ( 52)) can be neglected so we have:

𝐼 1 = 4𝜎 1 𝑅 𝑐 1 𝐵 ̇𝑎 sin 2 ( 𝜋 𝑁 1 ) ( 𝑙 𝑝 1 2𝜋 ) 2 𝑎𝑛𝑑 𝐼 2 = 4𝜎 2 𝑅 𝑐 2 𝐵 ̇𝑎 sin 2 ( 𝜋 𝑁 2 ) ( 𝑙 𝑝 2 2𝜋 ) 2
Thus, we have the instant coupling power per unit volume of cable using the limit of equation ( 64):

𝑃 𝑣𝑜𝑙 (𝑧) = [ 𝑁 2 (2𝜎 1 𝑁 1 𝑅 𝑐 1 2 sin 2 ( 𝜋 𝑁 1 ) ( 𝑙 𝑝 1 2𝜋 ) 2 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2 + (2𝜎 2 𝑁 2 𝑅 𝑐 2 2 sin 2 ( 𝜋 𝑁 2 ) ( 𝑙 𝑝 2 2𝜋 ) 2 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2 ] 𝐵 ̇𝑎 2
Where the expression between brackets is clearly identified as a sum of product of magnetic parameters. We recall that this expression is derived from the analytical expression of the coupling power in the case of low frequency regime found in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]. This demonstration ensures that the result we found numerically at first, is easily evidenced analytically already at the two stage level. The reduction of the system dimension allows performing the demonstration of this conservation rule giving a strong analytical foundation to the two-stage COLISEUM.

II.3.4 Global overlap: Application to a multiplets of Composite strands

Now we have checked that the strand, firstly modelled with CLASS, can be simulated by using the onestage COLISEUM and that the one-stage and the uncoupled two-stage COLISEUM are in agreement. We will try to apply these two points to model a multiplet of composites.

The following modelling of strand multiplets is twofold:

-Check if the consistency of the models is global, from strand to two-stage model, i.e. modelling the strand behaviour with the two-stage system where the strand is the first stage of the system. This first stage is modelled using the methodology presented above in section II.2.1 to reproduce the strand magnetic parameters with the one-stage COLISEUM. It is the elementary building block of our simulation. -Exhibit the analytical computation of multi-scale coupled multiplets of strands which is expected to be an original approach in the fusion community (the analytical formulation).

This system represents a triplet of strands (see Figure 35), where strands are modelled with thirteen tubes sub-elements. This sub-system has been tested and constructed (see section II.3.1) in order to have the same geometric and magnetic parameters than the strand presented in Table 6.

𝑵

$ $ $

We use the parameters of Table 6 to constitute our first stage as given in Table 10, we can note that the first stage is composed of 13 elements and its circumscribed area is equal to the strand one. We can also notice that the cabling radius of the first stage is equal to the filamentary zone radius of the real strand as shown in section II.3.1, thus both the real strand and the strand constructed with the one-stage COLISEUM have the same reference area and magnetic behaviour.

𝜎 1 is not free and determined in section II.3.1 to replicate the strand behaviour while 𝜎 2 is free but taken in a realistic range for cable stages. The magnetic parameters for this two-stage system are gathered in Table 11 below: Each couple of magnetic parameters (𝜏, 𝑛𝜅) can be discriminated to properly see to which stage it is related to as shown in section II.3.2.5.

Here we must recall that all 𝑛𝜅 values refer to the area inside the circle circumscribing the second stage.

We can also note that we have:

∑𝑛𝜅 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝜏 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 = ∑𝑛𝜅 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝜏 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 = 16.260 (… 𝑢𝑝 𝑡𝑜 17 𝑑𝑖𝑔𝑖𝑡𝑠) 𝑚𝑠
Now, to check that point, in the uncoupled approach, the shielding coefficient of the first stage is 0.806 refered to the circumscribed area of the system in Figure 35. We precise that the values of magnetic parameters exposed in Table 11 are given with limited precision, i.e; a check with those values would artificially show a deviation from the strict equality. It is stressed that this was numerically verified and demonstrated in section II.3.3.

In order to scale to the same reference area as for the 𝑛𝜅 presented in Table 5 we have to rescale this 0.806 to the reference area of the strand:

𝑛𝜅 𝑠𝑢𝑏𝑠𝑡𝑎𝑔𝑒 = 0.806 × (𝑅 𝑒𝑙𝑒𝑚 + 𝑅 𝑐 1 + 𝑅 𝑐 2 ) 2 3(𝑅 𝑒𝑙𝑒𝑚 + 𝑅 𝑐 1 ) 2 = 0.806 × 1.548 = 1.24
which is consistent with the shielding coefficient of the strand modelled by CLASS (see subsection II.3.1). Uncoupling this multiplet of composites, we recover the individual strand behaviour at the uncoupled first stage level (same time constant 𝜏 and shielding coefficient 𝑛𝜅). This is a second verification that comforts us in the modelling of multiplet of composites with as a two-stage system.

The time constant corresponding to the second stage, i.e. the stage of multiplets of composite, is now expressed in both uncoupled and coupled situations.

We can see in Figure 36 and Figure 37 below, the change in the magnetic parameters distribution between two multiplets of composites where we have increased arbitrarily the second stage conductivity by a factor 10. This little exercise can be viewed as a slight compaction of the triplet, changing only the interstage conductivity of the multiplet and not the conductivity inside the strand, which seems quite a reasonable assumption. Multiplying the super-stage conductance by a factor 10, we obtain the following magnetic parameters of Table 12:

𝒏𝜿

Table 12: Magnetic parameters of the triplet of strands from Table 8 modelled with the two stage COLISEUM using 𝜎 2 = 6.10 8 𝑆. 𝑚 -1 .

The table of results can also be represented under the form of bar graph (see Figure 34). 11.

𝒏𝜿 𝝉(𝒎𝒔) 𝒏𝜿 𝝉(𝒎𝒔)

Figure 37: Time constants dynamic for the previously presented multiplet of composite with 𝜎 2 = 6.10 8 𝑆. 𝑚 -1 . Value of the times constants and shielding coefficient are given in the above Table 12.

Which also verify:

∑𝑛𝜅 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝜏 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 = ∑𝑛𝜅 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝜏 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 = 30.

(… 𝑢𝑝 𝑡𝑜 17 𝑑𝑖𝑔𝑖𝑡𝑠) 𝑚𝑠

It is now possible with this two-stage system to model analytically and to study the effect of coupling on the magnetic parameters of a multiplet of strands. We precise that the values of magnetic parameters exposed in Table 12 are given with limited precision, i.e. a check with those values would artificially show a deviation from the strict equality. It is stressed that this was numerically verified and demonstrated in section II.3.3.

It is also interesting to note that the triplet of composites (second stage in the above system of Figure 35) can be modelled as isolated by using the one-stage COLISEUM. Of course, as described above in section II.3.3, the shielding coefficient found in the uncoupled case at the second stage level depends on an effective filamentary zone radius to feed our one-stage COLISEUM (𝑅 𝑓 ). We can determine numerically, using dichotomy, the value of 𝑅 𝑓 in order, for the one-stage COLISEUM, to mimic the shielding coefficient and time constant of the uncoupled super-stage. Using the geometrical parameters of the super stage gathered below in Table 13, we find

𝑛𝜅
𝑅 𝑓 𝑅 𝑒𝑙𝑒𝑚 = 0.8 .
We stress that once again, this value is also in agreement with the value of filamentary zone given by the homogenized first stage simulated with the one stage COLISEUM. The shielding coefficient of the first stage given by the one stage COLISEUM being 1.24 we found that:

√ 𝑛𝜅 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑓𝑖𝑟𝑠𝑡 𝑠𝑡𝑎𝑔𝑒 2 = √ 1.244 2 = 0.79 ≈ 0.8
Table 13: geometric parameters of the triplet of strands modelled with the one-stage COLISEUM.

We found for this system, using the one-stage model, 𝜏 = 1.69 𝑚𝑠 and 𝑛𝜅 = 0.930, almost the same magnetic parameters found in the uncoupled two-stage approach for the second stage. We have shown that a composite strand can be modelled with the strand model (CLASS) but also the one-stage COLISEUM and then, it can be embedded in the two-stage COLISEUM once the one-stage equivalence of the strand is established in section II.3.1.

The Figure 38 below shows the geometrical correspondence between the multiplet of composites modelled with the two-stage COLISEUM (strand + first stage of the cable) and the multiplets of composites with the one-stage COLISEUM (only the first stage is modelled).

The two columns below summarize our above work. We have seen that strands can be modelled using a strand model (CLASS), a one-stage COLISEUM or even as the first stage of a two-stage COLISEUM.

As said earlier, this two-stage model allows us to handle scale changes as we can simulate the two first stages of a cable or also its two last stages that are very different in terms of geometry.

We have also demonstrated that uncoupling the two-stage COLISEUM is in agreement with the description given by the one-stage COLISEUM. In the above sections, MPAS, CLASS and COLISEUM have first been presented in their actual state with the last enhancement (advanced MPAS, reduction of the dimension of COLISEUM, first multiplet of strand modelling). We also have presented how to handle each of these models in order to assess the range of coupling losses in a cable (for MPAS) and in two-stage cable (for COLISEUM). We check with success that the three models are well defined and consistent with each other. The uncoupled approach (𝜎 𝑖 → 0 or 𝜏 𝑖𝑗 = 0 with 𝑗 ≠ 𝑖) in the two-stage COLISEUM is perfectly consistent with the one-stage approach as shown in the last section. Finally, we have demonstrated that a composite strand can be modelled using the one-stage COLISEUM and then be embedded in the two-stage COLISEUM allowing us to model multiplet of composite strands. We can now think to better understand how multiplet of composites behave and how the magnetic parameters (𝜏, 𝑛𝜅) of the strand are modified through the coupling of several strands. This global consistency in the description of stages using different models for different scales shows the robustness in the elaboration of the two-stage COLISEUM.

CLASS

As depicted in Figure 14 and Figure 15, coupling two uncoupled systems modifies magnetic parameters of the system. A kind of a dynamics can be seen above all the tests we have done and is demonstrated for time constants in section II.3.2.5. At first, we have two separate uncoupled system (described either with the one-stage COLISEUM or the uncoupled two-stage COLISEUM) that we couple through the two-stage COLISEUM. The resulting time constants (the coupled ones) are separated (increasing Δ𝜏 as demonstrated in section II.3.2.5) and shielding coefficients are redistributed over the two-coupled stages.

III. Extension of the model

In this section, we will demonstrate how we iterate the two-stage COLISEUM to simulate a full cable geometry (all cabling stages). The iterative method have been chosen among others as the direct analytical extension of the two-stage modelling by taking into account all interaction between the different cabling stages seemed out of reach analytically speaking.

III.1 𝑛-stages COLISEUM iteration

III.1.1 First step

For clarity of the text and equations, 𝐼 𝑖 is referring to the current amplitude of the considered stage 𝑖. The first step of COLISEUM extension is already described in section II.3.2 with the reduction of the number of time constants of the system equation. The matrix equation below (equation ( 74)) represents the two-stage COLISEUM after its analytical reduction where we dropped the notation 𝜏 𝑖𝑗 (𝜎) for 𝜏 𝑖𝑗 .

[

We recall that this model can provide the amplitude of the induced currents inside a two-stage cable, and the total deposited power per unit volume in the system. After diagonalization of the above equation (74), it remains two time constants and two shielding coefficients describing these coupling losses (see section II.3.2.5). We have shown that these two magnetic parameters have the expected behaviour seen in Figure 25 and Figure 26 (variation with conductances ratio). The two other time constants that are omitted in this process are proven to be higher order correction of the coupling losses (as seen in Figure 23 and Figure 32) and do not vary with the conductances ratio. The two remaining time constants we are left with describe the coupling between the two considered stages (if both 𝜎 ≠ 0) as seen in Figure 15.

We consider two two-stage systems:

-System A, composed of a triplet of triplets of composite strands, i.e. stage 1 and stage 2.

-System B, composed of a triplet of triplets, where the basic element is a homogenized second stage, i.e. stage 2 and stage 3.

See Figure 39 and Table 18 for the respective geometrical parameters of systems A and B.

We consider the three first stages of a cable with two independent modelling using the two-stage COLISEUM. These separated modellings only consider the coupling between stages 1 and 2 on one side and the coupling between stages 2 and 3 on the other side. We would like to find the best way to combine the two systems in order for the second stage to interact with both first stage and third stage at the same time (i.e. through the same differential equation).

We use this separate approach to show how to couple both systems (A and B) to represent a larger system which will take into account the first neighbour coupling of different stages through only one calculation.

At this step, the only thing we can do to model a cable with three consecutive stages is two independent calculations using the two-stage COLISEUM (stages 1 and 2 on one hand, stages 2 and 3 on another hand), thus giving a total of four time constants and four shielding coefficients for these three consecutive stages. This is not consistent with the MPAS philosophy we tend to approach that consider one couple of magnetic parameter per stage for the coupling losses description. It is also deviating from the description given by the reduced two-stage COLISEUM giving also one couple of magnetic parameters per stage of cable.

To manage this out, we show how to handle the explicit coupling of three consecutive stages using the two-stage COLISEUM as building block of the iteration.

We can note that the second stage of the system A is also the first stage in the system B (by construction) as depicted in the following Figure 39.

System A is governed by equation ( 75): Subscripts on the time constant matrices are here to refer to the considered system (see Figure 39).

[
It is important to note that system B filamentary zone radius named 𝑅 𝑓 𝐵 is defined using the uncoupled shielding coefficient of the second stage of system A going through the process described in section II.3.3. This uncoupled shielding coefficient of the second stage of system A allows us to define an equivalent filamentary zone radius for the first stage of system B, in order for both sub-systems to be in agreement with the uncoupled modelling. This methodology ensures that both uncoupled second stage of system A and uncoupled first stage of system B will be describing the same system, i.e. the second stage of the considered three-stage cable.

First of all, as the second stage of system A is described by the same parameters as the first stage of the system B, we thus have 𝜏 22 𝐴 = 𝜏 22 𝐵 by definition and construction. In order to have this strict equality the filamentary zone radius in system B has to be determined using the method presented in section II.3.3 (using the uncoupled shielding coefficient of the second stage of system A).

Once 𝑅 𝑓 𝐵 determined, we have by construction 𝜏 22 𝐴 = 𝜏 22 𝐵 and if we take a closer look at the equations describing this stage, i.e. the second stage of the considered three-stage cable, in the two different systems (equation ( 75) and ( 76)), we have:

𝑁 1 𝜏 21 𝐼 ̇1 + 𝐼 2 + 𝜏 22 𝐼 ̇2 = 4𝜎 2 𝑅 𝑐 2 sin 2 ( 𝜋 𝑁 2 ) 𝐵 ̇𝑎/𝛼 2 2 (77)
for the super-stage of the system A.

And:

𝐼 2 + 𝜏 22 𝐼 ̇2 + 𝜏 23 𝑁 2 𝐼 ̇3 = 4𝜎 2 𝑅 𝑐 2 sin 2 ( 𝜋 𝑁 2 ) 𝐵 ̇𝑎/𝛼 2 2 (78)
for the substage of the system B, where we dropped both A and B index for 𝜏 22 as they are equal.

Equation (77) describes the coupling between the first and second stages of a cable whereas equation (78) describes the coupling between the second stage and the third stage. This is kind of a partitioned first neighbour coupling: on one hand, the second stage is coupled to the first stage in system A and on the other hand it is coupled to the third stage in system B. In order to get a complete first neighbour coupling for the second stage, we couple (by redefining the system equation) the two differential equations describing the second stage in both systems (i.e. (77) and (78) ) in the following equation (79):

Instead of having two systems of dimension 2 (system A and B), we can gather both systems in a larger one of dimension 3x3 (equation ( 80)), replacing both the second equation of system (75) and the first equation of system (76) by the unique equation ( 79), representing a first neighbour coupled three-stage system (see Figure 40): with the following notation to simplify the handling:

[
[ 𝐼 1 𝐼 2 𝐼 3 ] + [𝜏] 𝐴∪𝐵 [ 𝐼 ̇1 𝐼 ̇2 𝐼 ̇3] = [𝑌]𝐵 ̇𝑎 (81)
We can see in equation ( 80) that the first line is the same as the first line of the system equation of system A (equation ( 75)) and the last line in equation ( 80) is the last line of system equation of system B (equation ( 76)). The equation of the second stage in (80) is thus a new equation replacing both the last equation of system A and the first equation of system B. The new system, depicted in Figure 40, represents the three interacting stages: the system 𝐴 ∪ 𝐵. As earlier at the two stage level, the current 𝐼 1 , 𝐼 2 and 𝐼 3 are amplitudes of currents used to compute the current circulating on the red superconducting shell of basic elements of COLISEUM. The current of an element 𝑘 1 of the second stage number 𝑘 2 of the third stage number 𝑘 3 is expressed following formulation and assumptions from [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] as:

𝐼 𝑘 1 𝑘 2 𝑘 3 = 𝐼 1 cos( 𝛼 1 𝑧 + 𝜙 𝑘 1 ) + 𝐼 2 𝑁 1 cos( 𝛼 2 𝑧 + 𝜙 𝑘 2 ) + 𝐼 3 𝑁 1 𝑁 2 cos( 𝛼 3 𝑧 + 𝜙 𝑘 3 )
We now have the second stage, in interaction with its first neighbours through the same matrix equation (i.e. the first and the third stage).

In addition, first and third stages are only coupled to the second stage as we only consider here a first neighbour coupling three-stage model. We choose in this modelling to neglect the interaction of nonconsecutive stages. As already said earlier, non-diagonal terms of the time matrix have to do with notions of mutual inductances between two stages. The first line above and below the diagonal (in equation 80) stands for first neighbour interaction between the three stages. As an example, 𝜏 12 and 𝜏 21 , and, 𝜏 23 and 𝜏 32 , respectively reflect the mutual inductance between sub-stages 1 and 2 and between sub-stages 2 and 3.

In some way, we could have included the interactions between all the stages by redefining one of the equation of the system (see equation 79) and thus we could have added the interaction between stages 1 and 3 in equation ( 80). In order to add these extra terms to the time matrix (in other words, the time constants 𝜏 13 and 𝜏 31 ), we would have to redefine the magnetic vector potential in order to be able to consider non-consecutive elements from different cabling stages. In addition, this analytical process should be carried out for each new cabling stage considered in the model (i.e. for 3 stages, for 4 stages, etc.), and therefore this way does not seem to be the best approach as the model we are building here for the community also aims at being practical and not too heavy analytically speaking.

Therefore, we only consider here first neighbour interactions in order for the already existing two-stage model to be iterated and to form a system capable of describing several consecutive stages of a cable in an effective way.

Once the new time matrix of equation ( 81) is diagonalized, we find the three coupled time constant of the three considered stages.

[𝜏] 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 = 𝑉[𝜏] 𝐴∪𝐵 𝑉 -1 ( 82
)
where 𝑉 is the matrix composed of the eigenvectors related to each time constant. This eigenvector matrix [𝑉] is used to go from the initial basis (tridiagonal time matrix), where the current flows can be identified as due to the shielding of one another cabling stages, to the coupled basis (diagonal basis) where currents are intertwined and can thus no longer be identified to one specific cabling stage shielding.

We also have

𝑌 𝑏 = 𝑉 -1 𝑌 (83)
Compared to the initial two-stage COLISEUM, this iteration now includes the coupling between neighbouring stages. This is due to the fact that the second stage reaction to its surrounding stages (second stage of system A and first stage of system B, which describe the same object), is now explicit in only one differential equation (79). Thus, at this point, stage 1 is only coupled to stage 2, stage 3 is only coupled to stage 2, and stage 2 is coupled to stage 1 and 3 (see Figure 41).

Scale change is the heart of this iteration. The two first stages (system A) are coupled/united with the second and third stages (system B). We can also argue that in order for both subsystems to be coupled, at some point before their union, we have to compute the value of the uncoupled shielding coefficient of the super-stage of system A in order to define the radius 𝑅 𝑓 of the superconducting tubes we have to use in the substage of system B for both systems (A and B) to be consistent. The coupling of these two subsystems can be approximately illustrated by the case of masses related with springs in first neighbour interaction. At first, we have two systems of equations: one to describe the interaction of mass 1 and 2 and one to describe the interaction of mass 2 and 3. If we now want to couple the three masses in first neighbour interaction, the system equation will be changed as we have shown earlier. Using the methodology developed in section II.3.2.5, we found that the dissipated power per unit volume in a three-stage system (1-2-3) is: It follows:

𝑃 𝑣𝑜𝑙 123 = 𝛼 1 2 𝑁 1 𝑁 2 𝑁
𝑛𝜅 𝑖 = 2𝜇 0 ∑ ∑ 𝜆 𝑘 𝑉 𝑘𝑖 𝑌 𝑏 𝑖 𝑉 𝑘𝑙 𝑌 𝑏 𝑙 𝜏 𝑖 + 𝜏 𝑙 3 𝑙=1 3 𝑘=1 (86)
with, as already defined in section II.3.2, 𝜏 𝑖 being the time constants extracted by diagonalization of the above time matrix of dimension three. 𝑉 𝑖𝑗 are the components of the eigenvector matrix associated to the coupled time constant of the system. 𝑌 𝑏 = 𝑉 -1 𝑌 with 𝑌 being the vector of the second member of the above equation (80).

The coupling losses per cycle of magnetic excitation 𝐵 𝑎 = 𝐵 𝑚 sin(𝜔𝑡) + 𝐵 𝑜𝑓𝑓 per unit of circumscribed area for such a system writes:

𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 (𝑓) = ∑ 𝑛𝜅 𝑖 𝐵 𝑚 2 𝜇 0 𝜋𝜔𝜏 𝑖 1 + (𝜔𝜏 𝑖 ) 2 3 𝑖=1 (87)
In order to check if the uncoupled behaviour of this three-stage system is still in agreement with the independent calculation led using the one-stage COLISEUM we set to zero the terms reflecting the mutual inductances (see section II.3.2.5 for definition) of the system equation, which becomes:

[ 𝐼 1 𝐼 2 𝐼 3 ] + ( 𝜏 11 0 0 0 𝜏 22 0 0 0 𝜏 33 ) 𝐴∪𝐵 [ 𝐼 ̇1 𝐼 ̇2 𝐼 ̇3] = [ 4𝜎 1 𝑅 𝑐 1 sin 2 ( 𝜋 𝑁 1 ) ( 𝑙 𝑝 1 2𝜋 ) 2 4𝜎 2 𝑅 𝑐 2 sin 2 ( 𝜋 𝑁 2 ) ( 𝑙 𝑝 2 2𝜋 ) 2 4𝜎 3 𝑅 𝑐 3 sin 2 ( 𝜋 𝑁 3 ) ( 𝑙 𝑝 3 2𝜋 ) 2 ]
𝐵 ̇𝑎 (88)

As the time matrix is already diagonal it follows:

𝑉 = ( 1 0 0 0 1 0 0 0 1 ) 𝑎𝑛𝑑 𝑌 𝑏 = 𝑌. ( 89
)
Hence, we have:

{ 𝑛𝜅 1 = 𝜇 0 𝜆 1 𝑌 𝑏 1 2 𝜏 11 𝑛𝜅 2 = 𝜇 0 𝜆 2 𝑌 𝑏 2 2 𝜏 22 𝑛𝜅 3 = 𝜇 0 𝜆 3 𝑌 𝑏 3 2 𝜏 33
Simplifying their expression using the above given terms for 𝜆 𝑖 , 𝑌 𝑏 𝑖 and using the expression of 𝜏 𝑖𝑖 given in section II.1.3 the uncoupled time constant of the system is equivalent to:

{ 𝑛𝜅 1 = 𝜇 0 ∏ (𝑁 𝑘 ) 3 𝑘=1 𝑅 𝑐 1 2 2𝛾 1 𝜋𝑅 𝑐𝑖𝑟𝑐 2 𝑛𝜅 2 = 𝜇 0 ∏ (𝑁 𝑘 ) 3 𝑘=2 𝑅 𝑐 2 2 2𝛾 2 𝜋𝑅 𝑐𝑖𝑟𝑐 2 𝑛𝜅 3 = 𝜇 0 ∏ (𝑁 𝑘 ) 3 𝑘=3 𝑅 𝑐 3 2 2𝛾 3 𝜋𝑅 𝑐𝑖𝑟𝑐 2 (90)
These expressions are perfectly consistent with the formula of the partial shielding coefficient 𝑛𝜅 of the one-stage COLISEUM model, equation [START_REF] Bruzzone | Upgrade of operating range for the SULTAN test facility[END_REF] which is recalled here

𝑛𝜅 = 𝜇 0 𝑁 2𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 𝜋 ( 𝑅 𝑐 𝑅 𝑐𝑖𝑟𝑐 )
Although the complexity of the diagonalization of the time matrix of the system depends on the dimension of the three-stage COLISEUM, which itself depends on the number of interacting stages, we are able to analytically compute the uncoupled magnetic parameters of an 𝑛-stage COLISEUM. We will show in the section III.1.3 that it is true for every 𝑛.

The methodology to recover the uncoupled behaviour of each independent stage being part of the threestage cable is identical to the one presented in section II.3.2.5 for the two-stage model. This is done by setting the terms reflecting the mutual inductances to zero in the time matrix of the system in order to cancel the coupling effect of the several stages composing the cable. In dimension 2 (modelling only two-stage of a cable), to set the terms reflecting the mutual inductances to zero is strictly equivalent to taking the limit of both 𝜎 to zero. Regarding the new three-stage system of equation ( 80) we verified we also have the same results.

For example, taking 𝜎 1 → 0 and 𝜎 2 → 0 , we have 𝐼 1 = 𝐼 ̇1 = 𝐼 2 = 𝐼 ̇2 = 0. This implies that the only equation remaining from the initial system of equation ( 80) is:

𝐼 3 + 𝜏 33 𝐼 ̇3 = 4𝜎 3 𝑅 𝑐 3 sin 2 ( 𝜋 𝑁 3 ) 𝐵 ̇𝑎/𝛼 3 2
The one of the uncoupled third stage, equivalent to the one given for this third stage as modelled with the one-stage COLISEUM. The same methodology can be applied with both (𝜎 1 → 0 𝑎𝑛𝑑 𝜎 3 → 0) to recover the uncoupled behaviour of the second stage and with both (𝜎 2 → 0 𝑎𝑛𝑑 𝜎 3 → 0) to recover the uncoupled behaviour of the first stage.

Also if we set to zero only one 𝜎, we can obtain several already observed configurations. For example if 𝜎 1 → 0 thus 𝐼 1 = 𝐼 ̇1 = 0 and we are left with a system describing the coupling between the second and the third stage only as depicted below: Which is exactly the system of equation of system B. Of course, similar behaviour is observed if we set only 𝜎 3 → 0 the remaining system will describe the coupling between stages 1 and 2 without the third stage, as described by system A.

[
If we set 𝜎 2 → 0, we recover both the uncoupled first stage alone and the uncoupled third stage alone as the first neighbour of both stages has been electrically insulated.

It is also interesting to note that setting the terms reflecting the mutual inductances to zero is more efficient to compute the uncoupled magnetic parameters of the subsystems as it allows us to obtain them in one computation instead of three as presented in the above lines. It also allows us to reach analytical expression of the time constants and shielding coefficients for an uncoupled cable. This method ("mutual inductance terms" to zero, see section II.3.2.5), in dimension 2 and in the iterative version of COLISEUM, is in agreement with the limit of the modelling while setting 𝜎 to zero. It is coherent to think that the coupling between several stages, as described by COLISEUM through these "mutual inductance" terms 𝜏 𝑖𝑗 , is strongly related to the transverse conductance between them as this conductance drives the current loops taking place in these stages.

We can see that, iterating the two-stage COLISEUM to a three-stage model, we stay in line with the formulation of the initial formulation : a formulation of coupling losses as defined in MPAS with one set of magnetic parameters (𝜏, 𝑛𝜅).per stage. Initially starting from a four dimensions two-stage system, we have succeeded in reducing the model to two dimensions; this step has allowed us to iterate this twostage model to create a three-stage COLISEUM that can be iterated to an 𝑛-stage model.

We can also comment the fact the assuming a first neighbour interaction is due to the formulation of the initial development of the model. The interaction between stage one and stage three for example is considered here as a higher order correction on the modification of time constants and shielding coefficients. This interaction of second neighbour in this example should be driven by a terms 𝜏 13 𝐼 ̇3 in the equation of the first stage and by a terms 𝜏 31 𝐼 ̇1 in the equation of the third stage in the above system.

We have also shown here above that the uncoupled behaviour obtained by setting to zero the "mutual inductance" terms of the time matrix is, as for the two-stage model, perfectly equal to the uncoupled behaviour given by the one-stage COLISEUM when each stage is simulated alone.

III.1.2 Numerical application

III.1.2.1 Two-stage COLISEUM: two independent modellings.

As an example, we will take the geometrical parameters of the JT-60SA TF strand (see [START_REF] Zani | Starting EU production of strand and conductor for JT-60SA TF coils[END_REF]) to demonstrate the process and show the difference between magnetic parameters from two distinct computations using on one-hand two systems of dimension two and on the other hand one system of dimension three.

First we start simulating the three first stages of JT-60SA TF cable (see [START_REF] Zani | Starting EU production of strand and conductor for JT-60SA TF coils[END_REF]) with two runs of the two-stage COLISEUM. The first run will simulate the triplet of triplets of strands (two first stages) and the second simulation will model a triplet of triplets representing the second and the third stages of the considered cable.

For the first simulation, the geometrical parameters are defined using tangency condition for cabling radius and taken from cable specifications for others such as the twist pitch length, the radius of superconducting elements, and the number of superconducting elements in each cabling stage. The electric parameters are chosen, as in the previous section, to be in a realistic range regarding the previous measurements and work done with the University of Twente (see [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]). Simulating system A with a two-stage COLISEUM gives us the following magnetic parameters (already exposed in Table 3).

Table 16: magnetic parameters of the two first stage of the JT-60SA TF

For the second simulation, 𝑅 𝑒𝑙𝑒𝑚 𝐵 used in the second system is equal to 𝑅 𝑒𝑙𝑒𝑚 𝐴 + 𝑅 𝑐 1 = 𝑅 𝑐𝑖𝑟𝑐 𝐴 as shown in Figure 39 (scale change).

In order to determine the 𝑅 𝑓 𝐵 /𝑅 𝑒𝑙𝑒𝑚 𝐵 ratio in the second simulation (for system B), we have to determine the shielding coefficient of the uncoupled second stage of the system A. The method to obtain its value is already presented in section II.3.3. Using the same methodology, we compute the uncoupled shielding coefficient of the second stage of system A. Then, we determine the equivalent 𝑅 𝑓 𝐵 /𝑅 𝑒𝑙𝑒𝑚 𝐵 ratio corresponding to this uncoupled shielding coefficient. In this way, the uncoupled first stage of system B and the uncoupled second stage of system A are defined the same way. So, using the analytical formulae (equation ( 70) and ( 73)), we gather in Table 17 the magnetic parameters of the uncoupled second stage of system A:

Using the same methodology as in section II.3.3, we found that (

𝑅 𝑓 𝑅 𝑒𝑙𝑒𝑚 ) 𝐵 = 0.68.
The definition of this value is necessary to define the system B and to be consistent with system A. In this way, the second stage, both in system A or in system B, will be represented and described the same way.

Using the above value of 0.68 for the ratio

𝑅 𝑓 𝐵 𝑅 𝑒𝑙𝑒𝑚 𝐵
in the second and third stage of JT-60SA TF cable using the two-stage COLISEUM, we have: Simulating the second and third stages with a two-stage COLISEUM gives us the following magnetic parameters.

Table 19: magnetic parameters of the second and third stage of the JT-60SA TF If we take a look at the uncoupled limit of system B for verification (second and third stage simulated with two-stage COLISEUM with 𝜎 3 → 0), we found that the remaining time constant and shielding coefficient at the first stage of system B are these of the uncoupled second stage, equal by construction to those of the uncoupled second stage of system A as seen in Figure 42 and in Table 20. The uncoupled behaviour given by the one-stage COLISEUM can be reach by taking the limit of the coupled model where the "mutual inductance" terms are set to zero.

In the table below, we display the time constants and partial shielding coefficients values obtained for each cabling stage with the one-stage COLISEUM model and the electrical and geometrical parameters presented above. We checked that, either calculated with the one-stage COLISEUM (Table 16) or with the uncoupled version of the two-stage COLISEUM (Table 15), the uncoupled second stage is well defined as both versions of COLISEUM gives the same magnetic description, i.e. the same time constant and partial shielding coefficient This is also verified for the first and the third stages in this three-stage system.

III.1.2.2 Iterative three-stage COLISEUM: Unique modelling for three consecutive stages.

Uncoupled case:

Using the methodology developed in section III.1.1, we compute the magnetic parameters of the three independent stages of the cable. The results are gathered in Table 22 below:

Table 22: Uncoupled magnetic parameters of the three stage system using the limit of the iterative three stage COLISEUM without mutual inductances. Shielding coefficients are referred to the circumscribed area of the system.

Coupled Case:

We gather the coupled magnetic parameters obtained from the three-stage model in Table 23 below:

Table 23: Coupled magnetic parameters of the three stages system using the iterative COLISEUM. Shielding coefficients are referred to the circumscribed area of the system.

We can see in Table 22 that the uncoupled time constants and shielding coefficients of each independent stage of the three-stage model are equal to the description given by the three independent one-stage COLISEUM presented in Table 21. We can thus be confident in the fact that the development of an 𝑛stage COLISEUM is trustworthy as it is consistent with the previously developed models (as one-stage and two-stage COLISEUM models).

As shown earlier for the two-stage COLISEUM, we depict the time constants evolution dynamic from coupled to uncoupled cases in Figure 43 below. Globally, coupling modifies time constants, accentuating the spreading between them, and also modifies the shielding coefficients. We can note that between a two-stage and a three-stage system, the time constant variation is slightly different. Here, it seems that the difference between the biggest time constants and all others is bigger in the coupled case than in the uncoupled case. The modification of the shielding coefficients is due to the fact that now, the second stage of the cable shields the external magnetic excitation with the first and the third stages at the same time.

Stage

IV.1.2.3 Discussion

The iterative COLISEUM is well defined, at the three-stage level, as it provides us both a coupled and uncoupled description of three consecutive stages. The uncoupling of the three-stage COLISEUM is as described for the two-stage COLISEUM in section II.3.2.5. "Mutual inductance" terms are set to zero, which also corresponds to taking the limit of 𝜎 to zero. However, taking the limit of 𝜎 to zero is heavier to manage analytically and numerically; this is why we have chosen to carry out the uncoupling of the three systems through the extinction of the terms reflecting the mutual inductances. We have to keep in mind that this uncoupling is based on physical hypothesis because setting an interstage conductance to zero can be considered as representative of an insulated stage (petal wrapping).

In fact, all trends shown for the two-stage COLISEUM are conserved with this iteration. This is another point which strengthens the analytical basis to iterate the model to an 𝑛-stage COLISEUM.

The behaviour of the magnetic parameters is similar, and the relation between uncoupled and coupled magnetic parameters is conserved. Indeed, as we can see from the value presented in Table 22 and Table 23, we have: when referred to the circumscribed area of the system. We precise that the values of magnetic parameters exposed in Table 22 and in Table 23 are given with limited precision, i.e. a check with those values would artificially show a deviation from the strict equality. It is stressed that this was numerically verified and demonstrated in section II.3.3.

∑ 𝑛𝜅 𝑖 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝜏 𝑖 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 = ∑ 𝑛𝜅 𝑗 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝜏 𝑗 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑
In sections III.1.1 and III.1.2, we have demonstrated that we could iterate the two-stage COLISEUM to a three-stage model. Initially taking into account the coupling between two neighbouring stages, we enhance the modelling to take into account first neighbours coupling between several consecutive stages. Now we can model the three first stages in interaction instead of simulating on one side the first and the second stages and on the other side, the second and the third stages using two independent computations with the two-stage COLISEUM. In the next section, we will show how to iterate the modelling of COLISEUM to 𝑛-stage cable using the same methodology as presented in section III.1.1

III.1.3 Iterative process

III.1.3.1 Analytical development

Using the methodology developed in section III.1.1, COLISEUM is turned into an iterative model capable of describing 𝑛-stage cable.

At this point, we take the system equation of the three first stages modelled using the above three-stage COLISEUM:

[
In equation ( 93), we have a first neighbour coupling. The third stage being the actual last one, it only interacts with stage two. To this system, we will add the coupling between the third and the fourth stage which are initially modelled using a two-stage COLISEUM with the following system equation:

[
For the above equation (94) to be well defined, as done in section III.1.1, we have to use the uncoupled shielding coefficient of the last stage of equation ( 93) (here the third stage) to compute the value of the radius of the filamentary zone of the third stage we have to use in system equation (94), allowing to obtain the needed equality to iterate, i.e. 𝜏 33 𝐴∪𝐵 = 𝜏 33 𝐶 . We have to note that system equation (94) describes the coupling between stage three and stage four.

We can now combine system 𝐴 ∪ 𝐵 with system 𝐶, as follows:

[
Where we ensure by construction that the common terms between system 𝐴 ∪ 𝐵 and 𝐶 are all equal, i.e. 𝜏 33 in the time matrix, 𝐼 3 and 𝐼 ̇3 in the current vectors and 𝑌 3 the third component in the vector result. Iterating the same methodology for 𝑛-stage cable, we end with: 𝐵 ̇𝑎 (96)

[
The time matrix [𝜏] gives the coupled time constants after diagonalization:

[𝜏] = [𝑉] -1 [𝜏] 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 [𝑉] (97)
The uncoupled time constants are the coefficients 𝜏 𝑖𝑖 on the diagonal of the matrix [𝜏] as for the initial two-stage model, and we have shown in II.3.3 that they are strictly equal to the time constants computed for each isolated stage by using the one-stage COLISEUM.

It is important to note that in order to iterate to the 𝑛-stage description, we have to determine at every step of the process the uncoupled shielding coefficient of the current last stage to well define the next stage that will be added to the system.

The methodology to compute the dissipated power per unit circumscribed volume by coupling currents is not different from the one presented in section III.1.1. The same holds true as for the shielding coefficients.

𝑃 𝑣𝑜𝑙 = 𝛼 1 2 ∏
𝑃 𝑣𝑜𝑙 = 𝜆 1 𝐼 1 2 + 𝜆 2 𝐼 2 2 + ⋯ + 𝜆 𝑛 𝐼 𝑛 2 (99)
The lineic power dissipated in such a cable is:

𝑃 𝑙𝑖𝑛𝑒𝑖𝑐 = 𝑃 𝑣𝑜𝑙 𝜋𝑅 𝑐𝑖𝑟𝑐 2 (100)
with 𝑅 𝑐𝑖𝑟𝑐 the circumscribed radius of the last stage of the considered system and 𝑃 𝑙 the lineic dissipated power. 𝜆 coefficients are defined in section II.3.2.5. We recall their expression in dimension 𝑛 here:

𝜆 𝑘 = 𝛼 𝑘 2 ∏ (𝑁 𝑙 ) 𝑛 𝑙=𝑘 8𝜎 𝑘 sin 2 ( 𝜋 𝑁 𝑘 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2 (101)
It follows:

𝑛𝜅 𝑖 = 2𝜇 0 ∑ ∑ 𝜆 𝑘 𝑉 𝑘𝑖 𝑌 𝑏 𝑖 𝑉 𝑘𝑙 𝑌 𝑏 𝑙 𝜏 𝑖 + 𝜏 𝑙 𝑛 𝑙=1 𝑛 𝑘=1 (102)
where the 𝜏 𝑖 are the coefficients on the diagonal of the matrix [𝜏] 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 , i.e. the eigenvalues of the matrix [𝜏].

Coupling losses per cycle of external magnetic excitation 𝐵 𝑎 = 𝐵 𝑚 sin(𝜔𝑡) + 𝐵 𝑜𝑓𝑓 per unit circumscribed volume for such a system writes:

𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 (𝑓) = ∑ 𝑛𝜅 𝑘 𝐵 𝑚 2 𝜇 0 𝜋𝜔𝜏 𝑘 1 + (𝜔𝜏 𝑘 ) 2 𝑛 𝑘=1 (103)
We will apply the method to uncouple the 𝑛 stages. Taking the uncoupled limit of this 𝑛-stage COLISEUM, we set to zero the "mutual inductance" terms between consecutive stages:

[ 𝐼 1 𝐼 2 𝐼 3 𝐼 4 ⋮ 𝐼 𝑛 ] + ( 𝜏 11 0 0 0 0 0 0 𝜏 22 0 0 0 0 0 0 𝜏 33 0 0 0 0 0 0 𝜏 44 0 0 0 0 0 0 ⋱ 0 0 0 0 0 0 𝜏 𝑛𝑛 ) [ 𝐼 ̇1 𝐼 ̇2 𝐼 ̇3 𝐼 ̇4 ⋮ 𝐼 ̇𝑛] = [ 4𝜎 1 𝑅 𝑐 1 sin 2 ( 𝜋 𝑁 1 ) /𝛼
Thus we have:

𝑉 =
( 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 ⋱ 0 0 0 0 0 0 1 )

𝑎𝑛𝑑 𝑌 𝑏 = 𝑌. (105)
The shielding coefficients are then expressed as:

{ 𝑛𝜅 1 = 𝜇 0 𝜆 1 𝑌 𝑏 1 2 𝜏 11 𝑛𝜅 2 = 𝜇 0 𝜆 2 𝑌 𝑏 2 2 𝜏 22 ⋮ ⋮ 𝑛𝜅 𝑛 = 𝜇 0 𝜆 𝑛 𝑌 𝑏 𝑛 2 𝜏 𝑛𝑛 𝑛𝜅 𝑖 = 𝜇 0 ∏ (𝑁 𝑘 ) 𝑛 𝑘=𝑖 𝑅 𝑐 𝑖 2 2𝛾 𝑖 𝜋𝑅 𝑐𝑖𝑟𝑐 2 (106)
Here again, these expressions (106) are perfectly consistent with the formula of the partial shielding coefficient 𝑛𝜅 of the one-stage COLISEUM model for each isolated stages, equation [START_REF] Bruzzone | Upgrade of operating range for the SULTAN test facility[END_REF] which is recalled here

𝑛𝜅 = 𝜇 0 𝑁 2𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟𝑡 𝜋 ( 𝑅 𝑐 𝑅 𝑐𝑖𝑟𝑐 ) 2

Sum of 𝒏𝜿𝝉 :

In the case of 𝑛 uncoupled stages and in the low frequency regime, we can also demonstrated the equality between coupled and uncoupled sum of product of 𝑛𝜅𝜏. As already done in section II.3.3, we can neglect the magnetic coupling between currents induced in all elements. This implies that 𝐼 ̇ in the differential equation of the system (equation ( 52)) can be neglected, so we have

𝐼 1 = 4𝜎 1 𝑅 𝑐 1 𝐵 ̇𝑎 sin 2 ( 𝜋 𝑁 1 ) ( 𝑙 𝑝 1 2𝜋 ) 2 ; … ; 𝐼 𝑛 = 4𝜎 𝑛 𝑅 𝑐 𝑛 𝐵 ̇𝑎 sin 2 ( 𝜋 𝑁 𝑛 ) ( 𝑙 𝑝 𝑛 2𝜋 ) 2
Thus, we have the instant coupling power per unit volume of cable using the limit of equation ( 99):

𝑃 𝑣𝑜𝑙 (𝑧) = [ 𝑁 2 (2𝜎 1 𝑁 1 𝑅 𝑐 1 2 sin 2 ( 𝜋 𝑁 1 ) ( 𝑙 𝑝 1 2𝜋 ) 2 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2 + ⋯ + (2𝜎 𝑛 𝑁 𝑛 𝑅 𝑐 𝑛 2 sin 2 ( 𝜋 𝑁 𝑛 ) ( 𝑙 𝑝 𝑛 2𝜋 ) 2 ) 𝜋𝑅 𝑐𝑖𝑟𝑐 2 ] 𝐵 ̇𝑎 2
The sum of product 𝑛𝜅𝜏 is thus proven for any 𝑛-stage system. Strengthening our analytical development by proving that the already existing relations are conserved from the initial two-stage COLISEUM to this newly developed 𝑛-stage model.

We have here succeeded in defining and iterating the initial two-stage COLISEUM to describe an 𝑛stage cable. It models the coupling between several stages of a cable assuming that they are in first neighbour interaction. 𝑛-stage COLISEUM can be used to model five-stage cables (or more), for deriving the induced currents in each stage and also, the magnetic parameters (𝜏, 𝑛𝜅) needed to compute the coupling losses generated in such a cable. These magnetic parameters can be computed in what we have named the "coupled cases" where the time matrix of the system equation is tridiagonal: "selfinductance" terms on the diagonal and "mutual inductance" terms above and below the diagonal. In addition, they can also be computed in the "uncoupled cases" as done since section II.1.4. This uncoupled modelling corresponds to the modelling of each stage as if it were simulated alone with the one-stage COLISEUM. It must be recalled that, of course, in each one-stage the superconducting elements are coupled together. The coupling of several consecutive stages is well handled by the 𝑛-stage COLISEUM and is consistent with the results initially given by the one-stage COLISEUM for the uncoupled case.

A schematic view of the methodology to define a five-stage cable is given in the next sub-section and in Figure 44.

We will present an applicative case of the JT-60SA TF conductor. In section IV, this model will be benchmarked on AC losses data measured at CEA Cadarache on a sample relevant to of JT-60SA TF conductor.

III.1.3.2 Algorithmic scheme of the n-stage COLISEUM

In order to well define a 𝑛-stage system, we carefully have to go through several steps:

Step 1.

Define the basic two-stage system we start with (here 1 𝑠𝑡 and 2 𝑛𝑑 stages of JT-60SA TF)

Step 2.

Compute the uncoupled shielding coefficient of the current last stage (2 𝑛𝑑 stage)

Step 3.

Using the above shielding coefficient, deduce the 𝑅 𝑓 /𝑅 𝑒𝑙𝑒𝑚 ratio and thus the 𝑅 𝑓 value that stands as input for the next two-stage system (i.e. 2 𝑛𝑑 and 3 𝑟𝑑 stages)

Step 4.

Couple 1 𝑠𝑡 -2 𝑛𝑑 to 2 𝑛𝑑 -3 𝑟𝑑 to obtain a three-stage system

From here, we have to repeat steps 2, 3 and 4 adapting the actual last stage which is now the third stage:

Step 5.

Compute the uncoupled shielding coefficient of the current last stage (3 𝑟𝑑 stage).

Step 6.

Using the above shielding coefficient, deduce the 𝑅 𝑓 /𝑅 𝑒𝑙𝑒𝑚 ratio and thus the 𝑅 𝑓 value that stands as input for the next two-stage system (i.e. 3 𝑟𝑑 and 4 𝑡ℎ stages)

Step 7.

Couple 1 𝑠𝑡 -2 𝑛𝑑 -3 𝑟𝑑 to 3 𝑟𝑑 -4 𝑡ℎ to obtain a four-stage system It is also interesting to look at the "dynamics" of the magnetic parameters while going from the uncoupled to the coupled case as depicted in Figure 47. The dynamics of the magnetic parameters observed here for the iterative five-stage model is the same one as observed for the two-stage model (Figure 15) and for the iterative three-stage model (Figure 43).

We also verify

∑ 𝑛𝜅 𝑖 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝜏 𝑖 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 = ∑ 𝑛𝜅 𝑗 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝜏 𝑗 𝑢𝑛𝑐𝑜𝑢𝑝𝑙𝑒𝑑 5 𝑗=1 = 22.3843 … (𝑢𝑝 𝑡𝑜 17 𝑑𝑖𝑔𝑖𝑡𝑠) 𝑚𝑠 5 𝑖=1
The value is referred to the circumscribed area of the system. We precise that the values of magnetic parameters exposed in Table 26 and in Table 27 are given with limited precision, i.e. a check with those values would artificially show a deviation from the strict equality. It is stressed that this was demonstrated in section III.1.3 for the 𝑛-stage COLISEUM.

III.1.4.2 Analytical modelling of cables including composite strand

As shown in subsection II.3.4, we have succeeded in modelling multiplets of superconducting composite strands (see Figure 40). Following the same method, we have simulated a six-stage cable which is constituted of the composite strand at the first stage plus five cabling stages. Using this modelling, we are able to get both intra-strand coupling losses and inter-strand coupling losses simulated in the same system.

The geometrical and electrical parameters of such a system are shown in It is seen that the conductance of the "new" first stage is large. It is the one which determines the time constant of the stage (as explained in sect II.3.1 and in II.3.4)

We can give a quick check that the uncoupled behaviour of this six-stage system in Table 30 in order to further check the agreement with the coupled description as done for each simulated system.

III.1.5 Discussion

At first glance, it seems that coupled and uncoupled cases are equal at low frequency. A noticeable difference on the coupling losses per cycle is clearly identified at the higher frequencies as depicted in Figure 49 below:

The difference between the uncoupled and coupled approach can be explained looking at the contribution of each stage. It is clear that in the coupled case, the shielding coefficients are redistributed in favour of the last stage which plays the bigger role to shield the cable. For the case of JT-60SA TF, in the uncoupled case, shielding coefficients of the lowest stages are overestimated, which explains the higher losses (blue curve over red curve in Figure 49) at high frequency.

We have to keep in mind that this uncoupled version of the 𝑛-stage COLISEUM does not represent any physical system, we use it as a reference model and to verify that our development of the coupled 𝑛stage COLISEUM is well established (conservation of the sum of the product 𝑛𝜅𝜏).

Even though the system is well defined in dimension 𝑛, above dimension 3, the diagonalization of the time constants matrix is necessarily done by numerical tools. We would have to solve polynomial equations with a degree equal to or higher than four, which is not possible in all cases. We have initially succeeded to reduce the initial two-stage COLISEUM from dimension four to two, allowing full analytical calculations of the observables. But iterating the two-stage model to an 𝑛-stage model, for cables composed of three stages or more, brings back the problem of the analytical diagonalization of the time matrix. Nevertheless, assuming that the time matrix is diagonalizable, and using numerical tools to compute the eigenvalues and eigenvectors, we have been able to fully calculate the magnetic parameters of the system. As exposed above in section III.1.3.2, this iterative model has to be defined gradually. The scheme of the algorithmic process in Figure 44 well describes this process.

An interesting feature of the Iterative COLISEUM is that its only free parameters are the interstage conductances. Several configurations of interstage conductances can be used, for example:

-all 𝜎 equal.

-Increasing 𝜎 -Decreasing 𝜎

The three above configuration can be reach by using the following set of conductance:

𝑞[𝜎 1 𝑝𝜎 1 𝑝 2 𝜎 1 𝑝 3 𝜎 1 𝑝 4 𝜎 1 ]
These three types of distribution of conductances are chosen to limit the number of free parameters from five conductances (in a five stage cable) to only three: one conductance 𝜎 1 , the ratio between consecutive 𝜎 named 𝑝 and a multiplicative factor 𝑞. As done in MPAS, this simplification is done in order to facilitate the use of the model for future user. Also, it is more instructive to go from the more restricted cases to the one with no restriction.

Some coupling losses curves are depicted in Figure 50 below with several configurations of interstages conductances, they are used to exhibit the impact of the interstage conductances on the model.

We can see that, taking an initial distribution of sigma (see legend of Figure 50), we can modify the value of each 𝜎 with the factor 𝑞 keeping the ratio in between 𝜎 ′ 𝑠 constant. Another trend that can be evidenced is the one where the ratio in-between 𝜎 ′ 𝑠 varies while the conductance of the first stage of the cable is kept constant, this scenario is presented below in Figure 51.

The trend of the curves are understandable because, increasing the value of 𝜎 ′ 𝑠 by a constant factor 𝑞, increases all the time constant of the system. As conductances are linearly related to time constants, a global increase in the set of interstage conductances will increase the value of the set of time constants obtained with the iterative COLISEUM. This increase of the time constants values is clearly seen in Figure 43 above as an increase in the initial slope of the 𝑄(𝑓) is visible. The maximum of the curve is also moving towards low frequencies since the values of the time constants have increased. The fact that the maximum amplitude of the curve is kept while changing all sigma by the same factor is remarkable.

The tendencies shown above help us to better understand the behaviour of the iterative COLISEUM with respect to its free parameters. It will also give us an insight on the methodology that will be used to fit experimental data using the iterative COLISEUM. IV.1 The JOSEFA Facility IV.1.1 Presentation of the facility JOSEFA facility (Figure 54 and see [START_REF] Decool | The CEA JOSEFA test facility for subsize conductors and joints[END_REF]) located at IRFM CEA Cadarache is, among other purposes, used to measure magnetization cycles on a conductor on which a transient transverse magnetic field is applied. This facility can host a superconducting dipole named MARIUS which is immersed in a liquid helium bath kept at 4.2 K. This superconducting dipole is a paired saddle coil (Figure 52) used to generate a uniform magnetic field in a 3D zone where the sample holder is located. MARIUS dipole has been manufactured at CEA/Saclay during the year 1979 as a coil model for UNK [START_REF] Ciazynski | Final report on task MJOI-1, Modification and upgrading of a test facility to allow testing of conductor joints[END_REF] project. It has been installed and tested during MJOI task [START_REF] Perot | Superconducting dipole models for UNK[END_REF]. It is a NbTi superconducting device, able to produce a 4 T magnetic field within a cylinder of 74 mm of diameter (corresponding to the diameter of the hole inside the dipole) and along 440 mm (approximatively the length of the dipole). Its inductance is 𝐿 𝑀 = 5 𝑚𝐻.

IV. Experimental AC losses measurement

𝐼 𝐵 𝑎

This produced magnetic field is homogeneous (>92%) on a length of 220 mm on both sides of the equatorial plan (440 mm in total) as we can see from the magnetic field profile produced by MARIUS (Figure 54).

Sample holder is inserted in-between the two parts of the paired saddle coil as shown in Figure 53. Hence, the sample (30 cm long) and its replica are fully immersed in the uniform magnetic field produced by the superconducting dipole MARIUS and in the same liquid helium bath. We can note that only the ends on both side of the sample are subject to 98.75% of the applied field which is fairly close to nominal field.

In our configuration of AC losses measurement, both the MARIUS dipole and the tested sample will be immersed in the same bath with a constant temperature of 4.2 K. The used cryostat is a cylindrical one (around 50 cm of diameter and a 2 m height), as seen in Figure 53 this cryostat is layered at its external surface (as Russian doll). A liquid helium bath fills the inside of the cryostat surrounded by a liquid nitrogen shield. Between the liquid nitrogen shield and the internal liquid helium, a layer of void is maintained to limit the conducting part of heat transfer. Also, three aluminium shields are used along the sample holder in order to limit the liquid helium vaporisation (as depicted in Figure 53).

We monitor the whole facility using a control command unit (Panorama) while the acquisition of the data is done on another computer using LabVIEW 7.1. The system (cryostat + sample) is mounted at room temperature and it is cooled through two phases. Sample holder is placed in the cryostat, the system is thermalized to 80 K using the liquid nitrogen shield (as shown in Figure 54). Then the cryostat is filled with liquid helium to reach 4.2 K in order for the sample and MARIUS to be in superconducting state.

In order to decrease the temperature from 80 K to 4.2 K, with the sample holder in place, the average level of helium needed is around 280 l. An automat is present to check and maintain the level of liquid nitrogen and liquid helium between set levels (above 95% for the liquid nitrogen and between 55% and 65% of the cryostat height for the liquid helium as depicted in Figure 54). All manipulations were done with the help of the Cryogenic group from CEA Cadarache.

This ensures that:

-The superconducting magnet MARIUS and the sample are well isolated from external heat flux through nitrogen shield. -The internal liquid helium level stays above the superconducting dipole and sample top.

Nitrogen supply is taken from a direct line of liquid nitrogen tokamak whereas helium is stored and transferred from a 1000 l tank. Helium is taken from a stock used for WEST, helium availability is thus dependent on the exploitation of the tokamak. Using one tank we are able to perform at least two cool down of the complete installation and several measurements rounds.

Current leads for MARIUS are made of copper and can be actively cooled with the helium flowing out from cryostat. This outflow is mainly driven by the heat produced by MARIUS and the sample during operation. This heating is due to the current applied on MARIUS to generate the magnetic field and to the induced currents inside the CICC sample. It is interesting to note that it is possible to measure AC losses using calorimetric measurement of the deposited power inside helium as done in SULTAN (see [START_REF] Nijhuis | Optimization of ITER Nb3Sn CICCs for coupling loss, transverse electromagnetic load and axial thermal contraction[END_REF], [START_REF] Ilyn | Effect of cyclic loading and conductor layout on contact resistance of full-size ITER PF CICC[END_REF], [START_REF] Breschi | Analysis of AC losses in a CS conductor sample for ITER project[END_REF] and [START_REF] Breschi | Analysis of AC losses in the ITER central Soleinoid insert coil[END_REF]). Power supply can deliver up to 2000 A positive current corresponding to a 0 to 2 T magnetic field amplitude for MARIUS dipole. This current supply is originally set to generate ramps or steady state signal, we had to couple it with a remote handler in order to additionally generate sinusoidal current. This is the first time that sinusoidal applied field is implemented and tested in JOSEFA.

Several electric fans had to be added on the current supply in order to cool it down efficiently to avoid several security shutdowns. Before adding the fans to the power supply front wall we faced numbers of shutdowns (about a dozen over our 6 campaigns) which caused wasting time, helium, and human resources during the experimental campaign. Many little implementations had to be done in order to ensure the efficiency of the facility exploitation especially on the current supply which was not developed initially to generate sinusoidal current.

IV.1.2 Presentation of the method

The methodology to generate and measure the voltage related to the magnetization cycles from a CICC sample is the magnetization method (also called compensated pick up coils method) and is described in the following:

-A pick up coil is longitudinally wound on the conductor, another pick up coil is wound on the replica of the tested conductor. -The replica is a copy of the tested sample with the same external dimension. Its role is to measure the flux due to the applied field through its surface in order for us to remove this component from the pick-up wound on the conductor. Thus we could compute the component of the induced current in the pick-up of the conductor due to the magnetization of the sample. We could have chosen a replica with different external dimension we just would have to rescale the data during analysis. This method to measure AC losses is largely used in the fusion community as seen in [START_REF] Muzzi | Magnetic method for AC losses measurement of coil wound CICCs in pulsed regimes[END_REF].

Both are maintained on a sample holder, immersed in the cryostat once fully instrumented. Sample holder can be seen in Figure 55. CICC sample and its replica are placed onto a bakelite support (Figure 55 a) ), this support is enclosed with plastic strip and metal screw and attached to a steel tube (Figure 55 b) ). Cables coming from both pick ups coil are passing through the steel tube until its top. To avoid heat transfer (convection or radiation), three aluminium shield are also placed along the steel tube in order to prevent heat transfer coming from the top of the cryostat which is its hottest part).

The length of the tested samples is chosen at one last stage twist pitch (e.g. 290 mm for JT-60SA TF conductor) to get the full contribution of each cabling stages (especially the last one) to the cable coupling losses. The applied external magnetic field 𝐵 𝑎 is sinusoidal:

𝐵 𝑎 = 𝐵 𝑚 sin(𝜔𝑡) + 𝐵 𝑜𝑓𝑓 . (107)
We can note that the implementation of an applied sinusoidal field is new for this facility which has been upgraded during the three years of the thesis. Here, the background field 𝐵 𝑜𝑓𝑓 and the amplitude of the applied field 𝐵 𝑚 are parallel (see Figure 52) conversely to e.g. SULTAN facility, where they are orthogonal (see Figure 8).

The replica is used to ensure that the flux from MARIUS dipole is properly measured to be removed from the flux measurement of the sample containing the embraced flux of the applied field plus the flux coming from the magnetization response of the sample. Another enhancement done on the facility was to strengthen the whole sample holder and to weld some part of it which was originally screwed (top tap to the steel bar, see b) and c) in Figure 55). We noticed that while applying a magnetic field above 1 𝑇 on the sample holder, this screwed part of the system shows weakness in the clamping of the steel bar as it allowed the steel tube to rotate inside the cryostat. The magnetization signal measured from the pick-up coils was degraded and did not allow to extract any reliable data. This welding of the superior tap to a thicker steel bar prevents from torsion and thus displacement in question.

Each turn of the pick-up coil (on the replica) will see the flux created in one turn by the applied field and on the sample it will see the flux of the applied field plus the magnetization (𝑀) produced in reaction by the sample.

In order to get a voltage signal related to the only magnetization generated by the sample, pickup coils voltages are balanced trough a Wheatstone bridge. We describe the electric scheme of the experimental setup in Figure 56.

𝑅 Ω c (conductor related resistance) is a standard resistance (978 Ω) and 𝑅 Ω r (replica related resistance) is variable and can be adjusted with a ± 1 Ω precision (due to decade counter precision). Number of turns of the wound coils 𝑛 𝑡 is the same on the conductor and on the replica with 𝑛 𝑡 = 21. Dimensions of the conductor and of the replica are chosen to be equal in order to simplify the balancing of our Wheatstone bridge configuration. We have

𝑛 𝑟 𝑆 𝑟 2 𝑛 𝑐 𝑆 𝑐 2 = 𝑅 Ω r 𝑅 Ω c ( 108
)
with 𝑆 being the surface of a pick-up turn for both the replica and the conductor. 𝑟 and 𝑐 indexes are used for respectively replica related variable and cable/conductor related ones. Thus, using equal dimension on the conductor and the replica and equal number of turns in both pick up coils, we should set 𝑅 Ω r = 𝑅 Ω c into our Wheatstone bridge. Nonetheless, pick-ups coils are not perfectly similar and small differences in the winding and in the surface of a turn (dimension of the sample) can unbalance our prepared Wheatstone bridge in a way that the ratio between both resistances is not exactly equal to one. So at the end we have to manually adjust 𝑅 Ω 𝑟 during the firsts runs in order to get a good balance. For all these samples, the strands are composed with NbTi filaments.

Using the voltage related to the magnetization of the CICC, we reconstruct the magnetization cycle (𝑀(𝐵)). From this, integrating each 𝑀(𝐵) cycle over time with respect to the frequency of the applied field, we obtain the total losses vs frequency curves.

𝑄 𝑡𝑜𝑡 (𝑓) = ∫ 𝑀𝑑𝐵 𝑎 = 𝑓 𝑔𝑒𝑜 ∫ (∫ 𝑉 𝑚 𝑑𝑡) 𝑑𝐵 𝑎 (109)
𝑓 is the frequency of the applied magnetic field and 𝑓 𝑔𝑒𝑜 a geometrical factor taking into account the shape and dimension of the conductor (round or square cable, strand radius, number of strand, length of the considered sample). This geometrical factor is taken from a technical CEA note [START_REF] Schild | Mesures de pertes par aimantations: Principe et méthode d'analyse[END_REF].

𝑓
𝑛 𝑠 and 𝑛 𝑡 are respectively the strand number and the pick-up coil turn number. 𝑥 and 𝑦 are coordinates of the pick-up position in our experiment referential, see Figure 59. 𝐿 is the length of the tested sample and 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 the radius of the strand used inside the sample. 123

Using JOSEFA facility, we performed the measurement of total losses 𝑄 𝑡𝑜𝑡 for all samples. Two components contribute to the total losses 𝑄 𝑡𝑜𝑡 : Hysteresis losses 𝑄 ℎ and Coupling losses 𝑄 𝑐𝑜𝑢𝑝 . We have to separate the two contributions in order to be accurate on the magnetic parameters (time constant and shielding coefficient) linked to 𝑄 𝑐𝑜𝑢𝑝 . These magnetic parameters, as stated above in II.1, are related to coupling currents circulating inside the CICC. Thus we have to be careful in the determination of the hysteresis losses we remove from the total measured losses.

In the next sub-section, we will describe both type of losses (Hysteresis and Coupling). We will explain our methodology to extract the hysteresis losses from the total measured losses.

IV.2 Hysteresis Losses

Our samples are subject to a transverse magnetic field generating hysteresis losses due to the current circulating inside the superconducting filaments. Hysteresis losses are not related to applied field derivative function (𝐵 ̇𝑎) but to its amplitude (𝐵 𝑚 or 𝐵 𝑖 depending on the selected consideration) as we will see below. To subtract the right amount of hysteresis losses from total losses 𝑄 𝑡𝑜𝑡 , two options are available. We could experimentally evaluate them by extrapolation of the ordinate at origin of the 𝑄 𝑡𝑜𝑡 (𝑓) curves. We will compare both approaches in the following. An alternative way to proceed is to compute their analytical value.

Using the analytical calculation of the hysteresis losses will allow us to evaluate for each curve 𝑄 𝑐𝑜𝑢𝑝 (𝑓), the value of the effective filamentary diameter 𝑑 𝑒𝑓𝑓 and establish a statistic of these values.

We will extract a mean value and its standard deviation to see whether the measured effective filamentary diameter agrees with the value given by the manufacturer of the strand 𝑑 𝑓𝑖𝑙 .

From now, two cases are identified and named partial penetration and full penetration. If the amplitude of the applied field 𝐵 𝑚 is sufficient, the superconducting filament inside the strand will be filled with current in its whole section whereas if the amplitude is not sufficient, the current will circulate on the edge of the filamentary zone. In order to know if the amplitude of the applied field will be sufficient to penetrate filaments until their centre, we look at the factor 𝜷 = 𝑩 𝒎 𝑩 𝒑

(see [START_REF] Wilson | Introduction" in Superconducting Magnets[END_REF]) where we call 𝐵 𝑝 the penetration field supposing that the filament is not virgin but has been penetrated in a first cycle (first magnetization).

𝐵 𝑝 = 𝜇 0 𝐽 𝑐 * (𝐵, 𝑇)𝑑 𝑓𝑖𝑙 𝜋 ( 111
)
𝐵 𝑝 is the threshold where filaments start to carry current on their whole section with induced currents, with the average current density 𝐽 𝑐 * (𝐵, 𝑇):

𝐽 𝑐 * (𝐵, 𝑇) = 1 2𝐵 𝑚 ∫ 𝐽 𝑐 (𝐵, 𝑇) 𝑑𝐵 𝐵 𝑜𝑓𝑓 +𝐵 𝑚 𝐵 𝑜𝑓𝑓 -𝐵 𝑚 (112)
Rather than simply take the value of 𝐽 𝑐 (𝐵 𝑜𝑓𝑓 ) for the evaluation of the current density inside the strands 𝐽 𝑐 * is computed to get a better agreement with the real value of current density flowing inside superconducting strands. We recall here that 𝐵 𝑚 and 𝐵 𝑜𝑓𝑓 are defined is equation (107). The value of 𝐽 𝑐 * is calculated using past experimental measurement on the K006-01C strand using VSM technique [START_REF] Zani | Starting EU production of strand and conductor for JT-60SA TF coils[END_REF]. We can see in Figure 34, in the appropriate field range we felt into, 𝐽 𝑐 (𝐵) from [START_REF] Zani | Starting EU production of strand and conductor for JT-60SA TF coils[END_REF]. Interpolation is used to determine the value of 𝐽 𝑐 in between two measured points.

Based on 𝛽, we clearly have two cases that can be discriminated. If the external field amplitude is great enough to be above the penetration field or if it is below the penetration field. We will respectively talk about full penetration (𝛽 > 1) and partial penetration (𝛽 < 1).

We use a Bean model ( [START_REF] Bean | Magnetization of Hard Superconductors[END_REF], [START_REF] Bean | Magnetization of High Field Superconductors[END_REF]) to estimate hysteresis inside filaments, we can see in the following scheme both approaches (full and partial penetration) depicted with respect to the applied magnetic field region they correspond (see a), b), c) and d) in Figure 61).

𝑡 (𝑠)

B (T)

The field penetrating in the superconducting filament follows the same variation as the external field (see Figure 62 below).

Where we use Δ𝐵 = 2𝐵 𝑚 .

As said earlier, depending on the amplitude of the applied field, we can determine if we will be above or below the penetration field in order to subtract the right amount of hysteresis losses over our integration of the magnetization of the sample. The case of the full penetration is presented in Figure 63 below. We use an approach developed in CEA to compute the hysteresis losses: extracted from a CEA internal note written by B. Turck [START_REF] Turck | CEA Internal Service Note TS41[END_REF]. We have confronted to those used by M.N. Wilson [START_REF] Wilson | Time-varying fields and A.C. losses[END_REF] with success as seen in Figure 64 below. The considered cycle contains two variations ΔB.

In partial penetration 𝛽 <

both expressed in 𝐽. 𝑚 -3 . 𝑐𝑦𝑐𝑙𝑒 -1 of composite strand with 𝑥 being the ratio of non-superconducting material over superconducting ones inside a strand.

Hysteresis losses cycle can be used to check the balance of the Wheatstone bridge, in order to do so we can check several points:

-We try to minimize the amplitude of 𝑉 𝑚 .

-We look at the orientation of the 𝑀(𝐵) (see Figure 65). 𝛤 is a coefficient given in [START_REF] Wilson | Time-varying fields and A.C. losses[END_REF] in order to compute hysteresis losses. As we can see in Figure 65, filaments start to carry current during the first magnetization (from 0.3 𝑇 to 0.48 𝑇) until they reach the critical current density 𝐽 𝑐 . Once the current density inside filaments reaches 𝐽 𝑐 , it only depends on the internal field 𝐵 𝑖 and the temperature of the filaments 𝑇. Same logic can be applied to the lower branch.

For practical use, we depict again in Figure 65 one magnetization cycle ( 𝑀(𝐵) ) at a very low frequency ( 𝑓 = 5. 10 -3 𝐻𝑧 ). We can also note that, using equation (112) we found that 𝐽 𝑐 * = 1.4. 10 10 𝐴. 𝑚 -2 and taking 𝑑 𝑓𝑖𝑙 = 18 𝜇𝑚 from the manufacturer value, we found 2𝐵 𝑝 𝑡ℎ𝑒𝑜 = 0.19 T for the tested sample (from equation ( 111)).

Computing the hysteresis losses for the case of Figure 65, we obtain 11.3 𝑚𝐽. 𝑐𝑚 -3 . 𝑐𝑦𝑐𝑙𝑒 -1 using equation ( 114) which is in agreement with the value found using the formula extracted from [START_REF] Wilson | Time-varying fields and A.C. losses[END_REF].

The experimental 𝐵 𝑝 𝑒𝑥𝑝 is measured on this magnetization cycle, 𝐵 𝑝 𝑡ℎ𝑒𝑜 is in good agreement with 𝐵 𝑝 𝑒𝑥𝑝 . 𝐵 𝑝 𝑒𝑥𝑝 is evaluated by taking the field range from the lowest to the field point where the magnetization variation is null using the upper branch. It is important to note that measuring 𝐵 𝑝 for a field variation below 𝐵 𝑜𝑓𝑓 is different from the 𝐵 𝑝 measured for a field variation above 𝐵 𝑜𝑓𝑓 . This is due to the fact the critical current density changes between 0.3 𝑇 and 0.7 𝑇.

In fact, once filaments have experienced a first magnetization they are filled with current and these currents are driven by the 𝐽 𝑐 (𝐵 𝑚 , 𝐵 𝑜𝑓𝑓 , 𝑇) curves. That's why on the upper branch we have two behaviours, first it is increasing then it decreases and on the lower branch, it is monotonic and only decreasing.

Determining if 𝛽 > 1 or 𝛽 < 1 is important in order to remove the right amount of hysteresis losses regarding the total measured losses using JOSEFA. The more accurate we will be on the hysteresis removal the more reliable our coupling losses measurements will be in the following work.

IV.3 Coupling Losses

As the total losses during a cycle is a sum of hysteresis and coupling losses, we can write:

𝑄 𝑡𝑜𝑡 = ∫ 𝑃 𝑐𝑜𝑢𝑝 (𝑡)𝑑𝑡 + 𝑄 ℎ𝑦𝑠𝑡 𝑇 0 (115)
with 𝑇 being the period of a magnetic cycle. Once hysteresis losses have been removed, we are left with coupling losses generated by the time variation of the applied field.

In the low frequency regime, we assume that 𝐵 𝑖 ≈ 𝐵 𝑎 , we have:

𝑃 𝑐𝑜𝑢𝑝 = 𝑛𝜏𝐵 ̇𝑎 2 𝜇 0 (116)
with 𝑛𝜏 being the effective time constant of the system. For a sinusoidal applied field

𝐵 𝑎 = 𝐵 𝑚 sin(𝜔𝑡) + 𝐵 𝑜𝑓𝑓 (117)
we recall that 𝐵 𝑚 is parallel to 𝐵 𝑜𝑓𝑓 . We have

∫ 𝑃 𝑐𝑜𝑢𝑝 (𝑡)𝑑𝑡 = ∫ 𝑛𝜏𝐵 𝑚 2 𝜔 2 cos 2 (𝜔𝑡) 𝜇 0 𝑑𝑡 𝑇 0 𝑇 0 = 𝑛𝜏𝐵 𝑚 2 𝜔 2 𝜇 0 ∫ 1 + cos(2𝜔𝑡) 2 𝑑𝑡 𝑇 0 = 𝑛𝜏𝐵 𝑚 2 𝜔 2 𝜇 0 1 2 (∫ 𝑑𝑡 + 𝑇 0 ∫ cos(2𝜔𝑡) 𝑑𝑡 𝑇 0 ) = 𝑛𝜏𝐵 𝑚 2 𝜔 2 𝜇 0 1 2 (𝑇 + [ 1 2𝜔 sin(2𝜔𝑡) + 𝐶𝑠𝑡𝑒] 0 𝑇 ) = 𝑛𝜏𝐵 𝑚 2 𝜔 2 𝜇 0 1 2 (𝑇 + 1 2𝜔 sin(2𝜔𝑇))
using sin(2𝜔𝑇) = sin(4𝜋) = 0 and 𝜔 2 𝑇 = 4𝜋 2 𝑓, we obtain:

∫ 𝑃 𝑐𝑜𝑢𝑝 (𝑡)𝑑𝑡 𝑇 0 = 𝑛𝜏𝐵 𝑚 2 𝜔 2 𝜇 0 1 2 (𝑇 + 1 2𝜔 sin(4𝜋)) = 𝑛𝜏𝐵 𝑚 2 2𝜋 2 𝑓 𝜇 0 . (118)

IV.4.1 MAG 42 studies

After the presentation of the method to perform our post-processing, we present a measurement done on MAG 42-3 sample (see Figure 67). It is relevant of the JT-60SA TF conductor in terms of geometry, type of strand, void fraction. Total losses are extracted from our measurement.

As mentioned in section III.1.1, the amplitude of our input signal decreases with the frequency as shown in Figure 68. . 𝑐𝑦𝑐𝑙𝑒 -1

)

Measured losses Vs Frequency

We can see in 𝐵 𝑚 (𝑓) above that the first points are performed at the nominal 𝐵 𝑚 and then the power supply does not keep up the nominal amplitude with respect to the increasing frequency. As a consequence, we are confident in the first points of 𝑄 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 (𝑓) (rough data from measurement, see Figure 67) allowing us to extract an approximate value of 𝑛𝜏 𝑒𝑓𝑓 named 𝑛𝜏 𝑒𝑓𝑓 𝑎𝑝𝑝𝑟𝑜𝑥 from the slope at origin of 𝑄 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 (𝑓). Our measurement at very low frequency shows us a very good linearity of 𝑄 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 (𝑓) which provides a trustworthy evaluation of 𝑛𝜏 𝑒𝑓𝑓 𝑎𝑝𝑝𝑟𝑜𝑥 using only two points in the extrapolation as shown below in Figure 69.

In the example given, by extrapolation, we found that 𝑄 ℎ𝑦𝑠𝑡 𝑒𝑥𝑡𝑟𝑎𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛 = 4.05 𝑚𝐽. 𝑐𝑚 -3 . 𝑐𝑦𝑐𝑙𝑒 -1 . This value will be compared to the one computed using the analytical formulae of hysteresis losses of section III.2.

We can extract what we defined earlier as 𝑛𝜏 𝑒𝑓𝑓 𝑎𝑝𝑝𝑟𝑜𝑥 from 𝑄 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 . For this measurement we found:

𝑛𝜏 𝑒𝑓𝑓 𝑎𝑝𝑝𝑟𝑜𝑥 = 580 𝑚𝑠 Using 𝐵 𝑚 and 𝑛𝜏 𝑒𝑓𝑓 𝑎𝑝𝑝𝑟𝑜𝑥 (assuming that 𝑛 = 2) we can compute 𝐵 𝑖 (see equation ( 4)) as shown in Figure 70 below. We point out the fact that the use of 𝐵 𝑖 is considered more accurate as the filaments inside the strands composing a cable experience 𝐵 𝑖 while subject to a transverse applied magnetic field.

In order to determine the 𝑑 𝑒𝑓𝑓 to use in the above equations ( 113) and (114), we have to invert the equality and solve the equation. The nominal value of 𝑑 𝑒𝑓𝑓 can be calculated from [START_REF] Zani | Starting EU production of strand and conductor for JT-60SA TF coils[END_REF] using the copper to non-copper ratio in the composites and the number of filaments in the filamentary zone of the considered strand, which is an information provided by the strand manufacturer. Once 𝑄 ℎ𝑦𝑠𝑡 is removed from 𝑄 𝑡𝑜𝑡 (𝑓) we obtain the coupling losses 𝑄 𝑐𝑜𝑢𝑝 (𝑓) for the considered sample as shown in Figure 40.

𝑄 𝑐𝑜𝑢𝑝 (𝑓) is biased by our power supply which cannot deliver the nominal field amplitude and due to the their nature (analytical expression see equation ( 118)), they can be rescaled with 𝐵 𝑚 2 . Coupling losses are rescaled for each measured point, so to recover outputs as if they were all measured at nominal 𝐵 𝑚 . Hysteresis losses are analytically computed and removed in Figure 71 using partial penetration formula in this case (𝐵 𝑜𝑓𝑓 = 0.5 𝑇 and 𝐵 𝑚 = ±0.1 𝑇).

In order to extract the right amount of hysteresis losses using the analytical approach we have to adjust our free parameter: the diameter of the superconducting filament in the formula. This modification can be clearly justified as the superconducting filaments are considered cylindrical in the analytical model whereas in reality their shape is hexagonal in assembly phase. Furthermore in the manufacture process, this hexagonal shape is deformed during fabrication of the strand, especially near the external frontier of the filamentary zone where filaments are the first to carry current and thus generating hysteresis losses. That is why in the following we will consider the notion of effective filamentary diameter 𝑑 𝑒𝑓𝑓 . We start by using a range close to the manufacturer value (computed from [START_REF] Zani | Starting EU production of strand and conductor for JT-60SA TF coils[END_REF]) for this effective filament diameter which is for this Nb-Ti strand 𝑑 𝑓𝑖𝑙 = 18 𝜇𝑚 .

We can assume that once the right amount of hysteresis losses has been removed, we are left with only coupling losses. As coupling losses do depend linearly on frequency (in the low frequency regime), we should have 𝑄 𝑐𝑜𝑢𝑝 (𝑓 = 0) = 0 (see Figure 72). We will use this fact to adjust the 𝑑 𝑒𝑓𝑓 in the analytical formula of hysteresis losses used for each measurement done on each sample of type MAG42 (as strands are the same in all samples, them being fabricated with the same cable). Using this method, we are able to establish a statistical database of 𝑑 𝑒𝑓𝑓 in order to compare it with the manufacturer nominal value. The use of 𝐵 𝑚 or 𝐵 𝑖 for the computation of hysteresis losses does not matter too much at low frequency as we can assume that 𝐵 𝑎 ≈ 𝐵 𝑖 but at higher frequency this affirmation is not true anymore. Thus we are confident that this methodology ensures that we remove the right amount of hysteresis losses at each frequency according to the internal field of the conductor.

Once we have our rough coupling losses (see red curve in Figure 71), we have to rescale each point of the curve to the right 𝐵 𝑚 2 . This is done using the value of 𝐵 𝑚 measured for each point (Figure 68) of the red curve shown in Figure 71. Result is shown in Figure 73. From this curve of coupling losses, we can now compute the effective time constant of the system which is, as described above, related to its slope at origin. We found:

𝑛𝜏 𝑒𝑓𝑓 = 596 𝑚𝑠
which is a more reliable value, actually not so far from 𝑛𝜏 𝑎𝑝𝑝𝑟𝑜𝑥 (relative difference of 2.7 %). This difference is due to the way we remove hysteresis losses from the measured data. 𝑛𝜏 𝑒𝑓𝑓 is by definition more accurate than 𝑛𝜏 𝑎𝑝𝑝𝑟𝑜𝑥 because its calculation process take into account the effective internal field produce by the cable rather than extrapolation a constant value of hysteresis losses.

During the experimental campaign, various combinations of 𝐵 𝑜𝑓𝑓 and 𝐵 𝑚 were used. Rescaling of the coupling losses was done using Bi 2 as we have:

𝐵 𝑖 1 𝐵 𝑖 2 = 𝐵 𝑚 1 𝐵 𝑚 2 . ( 121
)
As we have shown in section III.3 that 𝑄 𝑐𝑜𝑢𝑝 linearly depends on 𝐵 𝑚 2 .

Thus the rescaling is done as follows: We can see that once rescaled to the same 𝐵 𝑚 (here ±0.1 𝑇) our different measurements agree very well with each other. This rapid analysis comforts us regarding the reproducibility aspect of our measurement approach and post processing methodology.

𝑄 𝑐𝑜𝑢𝑝 (𝐵 2 ) = 𝑄 𝑐𝑜𝑢𝑝 (𝐵 1 ) ( 𝐵 2 𝐵 1 ) 2 (122)
We also check the influence of the change of 𝐵 𝑜𝑓𝑓 on the coupling losses. It appears that the range of variation allowed by our power supply regarding 𝐵 𝑜𝑓𝑓 (0 -1.5 T) is too small to see any effect (such as copper magnetoresistance) on coupling losses curves as depicted in Figure 75. It is added that higher 𝐵 𝑜𝑓𝑓 (see [START_REF] Zani | Starting EU production of strand and conductor for JT-60SA TF coils[END_REF] and [START_REF] Zani | Completion of TF strand production and progress of TF conductor manufacture for JT-60SA project[END_REF]) globally decreases the coupling losses as shown in SULTAN facility but here the change in 𝐵 𝑜𝑓𝑓 is too small. At the end, this will not drastically modify the outcomes of our experimental work, as this aspect is not the main focus of our work. Moreover, the smaller the background field is, the greater the coupling losses generated are. Thus we are in the domain of great interest for operation of tokamak.

The cable being composed of several stages, we assume (as in MPAS and COLISEUM) that one time constant is related to each stage of the cable, thus each stage 𝑖 is described by a 𝑛𝜅 𝑖 𝜏 𝑖 .

But using the experimental data, we can only extract the effective 𝑛𝜏 from each 𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 (𝑓) curves with

𝑛𝜏 𝑒𝑓𝑓 = ∑ 𝑛𝜅 𝑖 𝜏 𝑖 𝑖 . (123)
Several 𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 (𝑓) are measured for the samples of Table 33 and used to compute 𝑛𝜅𝜏 𝑒𝑓𝑓 (see Table 12) and effective filamentary diameter 𝑑 𝑒𝑓𝑓 (gathered in Figure 77).

The number of series for each samples is the number of various 𝐵 𝑜𝑓𝑓 + 𝐵 𝑚 tested during this experimental campaign to construct a consolidated database of effective 𝑛𝜅𝜏 𝑒𝑓𝑓 which are computed using the average of each 𝑛𝜅𝜏 𝑒𝑓𝑓 for each sample. We can see in Figure 76, the increase of the 𝑛𝜏 𝑒𝑓𝑓 with respect to the decrease of the void fraction. DP4-UP and MAG42-3 are samples sharing the same geometrical parameters, except that DP4 is strictly part of the JT-60SA TF coil production whereas the MAG42-3 was manufactured separately at a different time. This consistency between these two samples comforts us in our global methodology and in the reproducibility of our experimental process (measurement and post processing). Decreasing the void fraction from 36% to 26% increases the 𝑛𝜏 𝑒𝑓𝑓 by a factor 3. We can note that the 𝑛𝜏 𝑒𝑓𝑓 of MAG42-4 sample is not in agreement with the linear fit used as guide for the eyes.

The variation of the initial slope is related to the intuitive hypothesis that interstage conductances increase with compaction rate [START_REF] Nijhuis | Impact of void fraction on mechanical properties and evolution of coupling loss in ITER Nb3Sn conductors under loading applied superconductivity[END_REF], as a consequence of inter-strands surface growth. Inter-strands surface is the contact surface between two neighboring strands. In order to obtain comparable 𝑛𝜏 𝑒𝑓𝑓 with COLISEUM, 𝜎 ′ 𝑠 ranges are in the range of 10 8 S. m -1 , which seems to be in a realistic range with evaluation from Twente University (see [START_REF] Nijhuis | Change of interstrand contact resistance and coupling losses in various prototype ITER NbTi conductors with transverse loading in the Twente Cryogenic Cable Press up to 40000 cycles[END_REF] and [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]).

As mentionned above, a statistic of effective filament diameter (see Figure 77) has been made along the different measurement conducted. We can take a closer look to the results given by adapting 𝑑 𝑒𝑓𝑓 to perfectly fit the requirement 𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 (𝑓 = 0) = 0 while removing hysteresis losses. 33). From this quasi normal distribution, we can extract the mean value 𝑑 𝑒𝑓𝑓 𝑒𝑥𝑝 and the standard deviation given by our experimental setup and family of samples:

𝑑 𝑒𝑓𝑓 𝑒𝑥𝑝 = 17.6 ± 0.9 𝜇𝑚

We can notice that this statistical study has been led for varying 𝐵 𝑜𝑓𝑓 , 𝐵 𝑚 and void fractions and however shows an outstanding agreement (within 2%) with the manufacturer data 𝑑 𝑓𝑖𝑙 = 18 𝜇𝑚 (we can note that varying 𝐵 𝑜𝑓𝑓 as said above also modifies the value of 𝐽 𝑐 * as seen in equation ( 112)). Our hysteresis losses value being found strongly consistent with measurement we can then consider that our extraction process of coupling losses (namely post processing from raw data) is this way reinforced too.

The shape (normal distribution) and the mean value are quite expected as the fabrication process of the strand used same dimension rods at the beginning of the compaction process. At the end, the filaments on the edge of the filamentary zone are quite deformed from their original hexagonal shape. It is well known that filaments at the external edge of the filamentary zone are more squished than the ones in the centre which remain mainly hexagonal (depending on their initial shape). As the strand shields itself from external fields using its outer edge filaments, the filaments carrying current are more susceptible to be the ones deformed from their original shape. Thus, the filamentary diameter we measure is affected by this deformation.

Apart from increasing the 𝑛𝜏 𝑒𝑓𝑓 , we can also notice that increasing the compaction of MAG42 samples does modify the position of the maximum of 𝑄(𝑓) curve (in terms of frequency), we can see it in Figure 78 and values are reported in the Table 34 determined by hand. Amplitudes and global curves shape remain unchanged with void fraction variation. This shift in frequency of the maximum of the 𝑄 𝑐𝑜𝑢𝑝 (𝑓) curves could be explained by a specific modification of the interstage transverse conductances.

It is possible to demonstrate that a given coupling losses curve, characterized by a maximum named 𝑄 𝑚𝑎𝑥 at the frequency 𝑓 𝑚𝑎𝑥 , which is the result of the combination of several stages contribution (as stated in COLISEUM), can be shifted towards left or right of the graphics with regard to its initial position.

Assuming the analytical expression of the coupling losses is:

𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 (𝑓) = ∑ 𝑛𝜅 𝑖 𝐵 𝑚 2 𝜇 0 2𝜋 2 𝑓𝜏 𝑐 𝑖 1 + (2𝜋𝑓𝜏 𝑐 𝑖 ) 2 𝑗 𝑖=1
With 𝑗 the number of considered stages in the cable. We recall that in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF], the computation of the components of the time constant matrix are linearly dependant on the conductances 𝜎 as depicted in equation [START_REF] Duchateau | Stability of a cable in conduit conductor under fast magnetic field variations[END_REF]. Therefore, if we consider a transformation of the 𝜎 distribution where they are all multiplied by the same value (thus keeping the ratio of consecutive 𝜎 constant), it is strictly equivalent for 𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 expression above to change variable.

If we take 𝜏 𝑐 𝑖 = 𝑧𝜏̅ 𝑐 𝑖 as homothetic transformation of all 𝜎, it will be seen as the change of variable into 𝑧𝑓 = 𝐹, giving:

𝑄 ̅ 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 (𝑓) = ∑ 𝑛𝜅 𝑖 𝐵 𝑚 2 𝜇 0 2𝜋 2 (𝑓𝑧)𝜏̅ 𝑐 𝑖 1 + (2𝜋(𝑓𝑧)𝜏̅ 𝑐 𝑖 ) 2 𝑗 𝑖=1 = ∑ 𝑛𝜅 𝑖 𝐵 𝑚 2 𝜇 0 2𝜋 2 𝐹𝜏̅ 𝑐 𝑖 1 + (2𝜋𝐹𝜏̅ 𝑐 𝑖 ) 2 𝑗 𝑖=1 .
This kind of homothetic transformation conserves the maximum amplitude of the initial curves and only shifts the position (in frequency) of its maximum (and other points). Between the two above expressions of 𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 , amplitude is not changed but position of each point is moved according to the change of variable applied, i.e. 𝑓𝑧 = 𝐹. This statement shows that the behaviour of the coupling losses curves we measured for the different compaction rate of MAG 42 can be analytically explained and predicted (position of the maximum, amplitude, shape, etc.) in spite of the complexity of a system composed of several contributions. We will see in a next section that this can be fully explained by a global increase in the interstage conductances in COLISEUM as demonstrated above in Figure 50. We can show on the data of Figure 78 that a homothetic transformation can make all five coupling losses curves coincide as depicted in below. The best agreement is found for the initial slope and the position of the maximum whereas the high frequency losses are not so well adjusted.

MAG42

The coefficients of the homothetic transformation to make the curve coincide given in Table 35 are not in agreement with the one strictly given by the ratio of the position of the maximum of Table 34. The error made on the measurement is estimated (see Figure 80) linked to the uncertainty associated with the calibre range we used to perform each measurement. From the characteristics of our acquisition card, we can extract the error made on the measurement of 𝑉 𝑚 . This error can be integrated as 𝑉 𝑚 in the computation of the total losses and thus of coupling losses. As hysteresis losses are analytically computed we consider the error on this calculation as negligible compared to the error made on the measurement. Thus, the error computed from 𝑉 𝑚 is fully reported on 𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 as shown in Figure 80 and Figure 81.Uncertainty range on the measurement is not always depicted in the previous and following figures for clarity reason. They are always evaluated in order to support the trustworthy aspect of our measurements.

In Figure 81, we crosscheck our measurement using magnetization method with the calorimetric measurement led at SULTAN facility [START_REF] Bruzzone | Upgrade of operating range for the SULTAN test facility[END_REF].

They are in good agreement in terms of slope at origin, shape and position of the maximum of the curves. We can also see that each curve (from JOSEFA) is smooth as stated before. These facts consolidate our experimental process of AC losses measurement, especially the rescaling of AC losses with 𝐵 𝑚 2 of the different measurements done on each sample. However, at 3 Hz some discrepancy seems to appear (within 14%). This discrepancy could be generated by the difference in measurements methods.

We can also take into account the fact that our sample is 30 cm long while in SULTAN the length of the leg is about 2 meters. Considerations on influence of the conductor length on AC losses (see Ries and Takacs in [START_REF] Ries | Coupling losses in finite length of superconducting cables and in long cables partially in magnetic field[END_REF]) showed that such differences could appear. A longer sample could generate more losses because of the current loops closing outside of the applied field zone, adding coupling losses to the helium mass flow treated. We consider consequently that our results on JOSEFA are trustworthy for our studies. Zooming on the low frequency range of the Figure 81, we obtain the Figure 82 where the dashed line is added as a guide for the eye. Hysteresis removal process seems to be well done as we effectively obtain 𝑄 𝑐𝑜𝑢𝑝 (𝑓) = 0 when 𝑓 = 0 (see Figure 82). The experimental methodology to extract AC losses from measurement done on the JOSEFA facility is described in details. The methodology, based on magnetization measurements, proved to be robust considering the consistency from several measurement rounds. Seven samples from the JT-60SA TF production had been tested all along the thesis work. Some part of the work done in this experimental part have to be continued, as for the characterization of the MAG42-6 sample which coupling losses are not measured at high frequency due to unexpected bad quality measurements.

In the next sub-section IV.4.2, side-work issued from a collaboration with INFLPR (Ro) will be presented.

In the next section V, both the 𝑛-stage COLISEUM and the MPAS model will be applied to the experimental data exposed in Figure 81 trying to propose a set of magnetic parameters that best fit the data of sample MAG42-3. We choose to benchmark our models on these data, as we have performed the measurements from JOSEFA and from SULTAN on this conductor.

IV.4.2 Complementary study: tomographic analysis

This work was led with INFLPR (Romania) and is part of a collaboration.

With the intuition that compaction and interstage conductances are related, we started an analysis on tomographic images (see Figure 83). In the previous thesis work, an algorithm capable to reconstruct 3D trajectories of strands inside a CICC from tomographic images was developed. It is called IVORA and can be used on distant slice of tomographic images (300 voxels between two images) where strand position cannot be identified from one slide to another.

Contrariwise, to the program developed during this thesis which is used for very close tomographic slices (30 voxel between two slices). As the tomographic quality of images improves, we are not constrained to use the IVORA code, which is more complex than the program developed here.

The recent program is a simply first neighbour coupling of closest strand from one slice to another. As the voxel between two slices is small enough, we can easily follow a strand from a slice to another (Figure 84) as its displacement in the section between two consecutive slices is very small. In addition to the reconstruction work, we work on contact surface between strands in the cable. We started to apply mask on strand of each tomographic images (see Figure 85) and to measure the contact in pixel between two neighbouring strands. care to remove the overlap of two neighbouring circles but not for the overlap of three circle (which represent a very small part of the total contact).

In Table 32, we can find the value of void fraction extracted from [START_REF] Zani | Experimental and analytical approaches on JT-60SA TF strand and TF conductor quality control during qualification and production manufacture stages[END_REF]. We can see that the computed void fraction for MAG42-5 sample (over compacted JT-60SA TF) is about 28,1% and with our methodology we measure a mean value of 28,6% of void fraction along the hundred slices we measure (see Figure 88).

The relative error between the given void fraction and the one extracted from tomographic analysis is small (1.74% of relative error). Using this methodology, we reduce the number of free parameters from five to two. Equation ( 124) and (125) relate consecutive conductances with a linear relation as the ratio 𝑝 is equal for all stages. We can either simulate linearly increasing or decreasing interstage conductances. Constant distribution of 𝜎 is also possible with 𝑝 = 1.

The initial slope given by the 𝑛-stage COLISEUM is 𝑛𝜅𝜏 𝑒𝑓𝑓 𝐶𝑂𝐿𝐼𝑆𝐸𝑈𝑀 = 617 𝑚𝑠.

The prediction of the 𝑛-stages COLISEUM is in good agreement with the experimental data. The shape of the curve (blue) is very similar to the one found with JOSEFA and SULTAN. The position of the maximum and the initial slope are also in good agreement. We stress that this fit is obtained by hand without the use of a specific method to minimize the discrepancy between prediction and data (as the least square method). )

Table 38: Magnetic parameters found for the best fit of Figure 89. Shielding coefficient are refered to the area of superconducting strand in the cable.

We therefore consider this fit as the best fit without going through the same statistical mechanism we will use for MPAS in section V.2.

The analytical basis of the model is developed and tested here above. Once again, the validity of a multitime constants model is proven to be useful in order to describe coupling losses in a wide range of frequency. The absence of statistics for the 𝑛-stage COLISEUM is due to the fact that each computation of a five stage cable as JT-60SA TF using the 𝑛-stage COLISEUM takes at least 4 minutes whereas it is made in seconds when using MPAS. Here we encounter the limit of this modelling which is pretty costly in execution time if we want to do statistical studies. In order to do so, another strategy should be adopted. We can start by hand, finding some fit close to the experimental data. This will drastically reduce the domain of free parameters to explore. Knowing that with the 𝑛-stage COLISEUM applied to a five-stage cable we have five free parameters (five conductances), the number of points in our statistical study will increase therefore with a power five. V.2 MPAS application to experimental data

Stage

V.2.1 Statistical study

We recall that for the MPAS model (restricted or advanced), once the magnetic parameters of each stages had been determined, the coupling losses per cycle writes:

𝑄 𝑐𝑜𝑢𝑝 𝑀𝑃𝐴𝑆 = ∑ 𝜅 𝑗 5 𝑗=1 𝐵 𝑚 2  0 𝜏 𝑗 ( 2 𝜏 𝑗 2 + 1) . ( 126
)
For both MPAS models, we choose to probe the space of free parameters:

-(𝜏 5 , 𝜅 5 ) for the restricted MPAS (see section II.1.1.1).

-(𝜏 5 , 𝜅 𝑎 , 𝜅 𝑏 ) for the advanced MPAS (see section II.1.1.2).

First of all, this exploration will show if, in the chosen range of free parameters, we can find a set of magnetic parameters which well fit the experimental coupling losses, for a given experimental curve.

This exploration will also help to understand if the best fit that has been found using either the restricted MPAS or the advanced MPAS is unique or not. If it is not, several sets of distinct free parameters can be used to obtain a best fit. If it is unique, the best fit is thus described by using one set of free parameters.

Restricted MPAS:

Using the iteration rules from equation ( 9) and ( 12) (restricted MPAS), we can compute the magnetic parameters of a five-stage cable for a given couple (𝜏 5 , 𝜅 5 ). In order to get a global view of all the possible fits, we plot each corresponding 𝑄(𝑓) curves on top of each other as we can see in Figure 91. On top of these curves generated with the restricted MPAS, we plot the experimental data to fit. The curves panel in plain line is obtained with the restricted MPAS using 𝜏 5 ∈ [0.2,0.7] and 𝜅 5 ∈ [0.8,1.8]. These ranges of free parameters are chosen to be in agreement with the magnetic parameters usually encountered in the description of the coupling losses of a cable. A constrain we consider is that the product of the last stage shielding coefficient and time constant (𝑛𝜅 5 𝜏 5 ) cannot be superior to the initial slope of the experimental data. If it were the case, last stage magnetic parameter would be sufficient to describe the initial slope and all addition of another contribution from another stage would be already too much to describe losses at the low frequency regime. As a first comment, we can see that all curves are above the experimental data at high frequency. This means that whatever the fit we choose, we will have few or no cases where both high and low frequency regimes will describes the experimental data at the same time.

Nevertheless, our experimental data fell into the curve envelope shaped by the curves panel in plain lines (i.e. fit of the restricted MPAS).The interval for last stage time constant and shielding coefficient are chosen to encompass the experimental data.

As the number of fit is huge, we cannot see them on the same plot or their will cover each other as in the above Figure 91. The two dimensional plans of Figure 91 can be turned into a two dimensional grid where in each small area, the number of fit curves is represented. These curves are thus gathered inside some distribution over the experimental data as seen in Figure 92. These figures gives a first insight on the ability of the restricted MPAS to well fit the experimental data.

In fact, we can take a look at the spreading of the fit curves over the coupling losses axis. The larger the distribution will be, the greater the variability of free parameters is important whereas if all fit curves are close to each other, their distribution won't be as spread as before because the sensitivity in the free parameters is not as important as before. Using a least square approach to choose the best fit among all, we are able to extract a set of magnetic parameters. As suspected using the restricted MPAS, we cannot fit both high and low frequency zones of the 𝑄(𝑓) curve at the same time (for this peculiar sample from JT-60SA TF) as seen in Figure 93.

The corresponding magnetic parameters are given in the In Figure 94 below, we can see a global view of the normalized mean square obtained for each 𝑄(𝑓) with respect to the free parameters of the restricted MPAS (see equation ( 9) and ( 12)). We can clearly see that the minimum we found is unique in the whole explored domain of free parameters. We can clearly see that the minimum we found is unique (red dot) in the explored range of free parameters. However, we can see visually that the best fit found is close to the experimental data. Conversely, we could try to shrink the domain of interest of our fit (to target initial slope only for example) in order to see if the restricted MPAS is able to give us a good description of losses in a certain range of frequency.

We choose thus to best fit the initial slope and to show the discrepancy obtained at high frequency using this restricted MPAS in Figure 95 below.

The best fit we found is thus:

With the following magnetic parameters: It has been shown that, even if the restricted MPAS is not able to fit the experimental data over the whole frequency domain, it can be reduced to a domain of interest (here low frequency regime) and be used to assess the magnetic parameters generated by such a cable. The implementation of the new iteration rules for the shielding coefficients appears to be efficient as it allows MPAS to fit the data by adapting its basic hypothesis using the assumption given by the onestage COLISEUM that consecutive stage with a different number of elements should have a different iteration rules (as equation ( 22) over equation ( 9) when applied to the case of JT-60SA TF). This allows MPAS to be more flexible and thus to better fit the experimental data.

Stages
The magnetic parameters related to the Q(f) simulated with the advanced MPAS, presented in Figure 98 above are given in Table 41 below. As done above for the restricted MPAS, we can now check the unicity and the local aspect of the best fit found. In Figure 99, a three dimensional cartography of the value obtained with the least square approach is given.

Using a defined threshold of 200% of the minimum value, we can segregate the values among all the simulations satisfying this constrain but we can also analyse the shape of the volume surrounding the minimum value found (see Figure 100). Here, we obtain a volume inside which all fits are close to the best fit found as we have three free parameters. We can understand that our best fit from Figure 98 is surrounded of quasi-equivalent fits with a least square value close to the best fit one (see Figure 100). Taking a set of magnetic parameters close to the one used for the best fit will also have a very low value of least square.

In order to check the sensitivity of the advanced MPAS we decided to plot all fit with a least square value equal to the value of the minimum mean square plus 10 % as depicted in Figure 101 below. This threshold is only chosen for convenience in order to reduce the number of plot in the Figure 101 below.

It is clear that all fits around the minimum are also fits of good quality, the variation in the magnetic parameters is small. The maximum of the coupling losses curves given by the advanced MPAS is varying in a window of 1 𝑚𝐽. 𝑐𝑚 -3 . 𝑐𝑦𝑐𝑙𝑒 -1 and the initial slope is constant within a millisecond.

V.3 Discussion between the two approaches

In its actual form, the best fit given by the advanced MPAS is visually as good as the one given by the 𝑛-stage COLISEUM. Both initial slopes (COLISEUM and MPAS) are in good agreement with the one found for MAG42-3 in section IV.4 (we remind the value: 637±34 𝑚𝑠). The position of the maximum is in best agreement with the experimental data for the fit given by COLISEUM. It is a little shifted on the left with the fit given by the advanced MPAS.

Despite these slight discrepancies between the two approaches, both models are able to fairly describe the experimental coupling losses measured in JOSEFA.

Starting from their initial state: two-stage model for COLISEUM and no flexibility in stage assembly for the restricted MPAS, we can see that we have improved both modelling. COLISEUM is now able to describe 𝑛 consecutive cabling stages and MPAS now better takes into account geometrical parameters of the considered cable. Despite the analytical extension of COLISEUM, we have strengthened its analytical basis with the reduction of the number of time constants describing the two-stage system in order to iterate the model to the description of 𝑛-stage system.

The modelling of 𝑛-stage cable requires light computing time, gives information on coupling currents in each stages and has a predictive capability for coupling losses of a CICC that has to be verified by using measurements of its electrical and geometrical parameters as input for the model. Moreover, we show that it well matches the experimental results gathered with JOSEFA on JT-60SA TF samples.

Looking at Table 41 and

Table 38, we can see that shielding coefficients and time constants of both models are close to each other. It is shown that the last stage plays a major role in the coupling losses at low frequency, both 𝑛stage COLISEUM and MPAS agree on this point. The assumption of first neighbour interaction considered in the 𝑛-stage COLISEUM seems to be realistic enough to be in good agreement both with the data and with the MPAS model. The JT-60SA TF cable enables to have a simple approach regarding the modelling as the cable do not possess peculiarity as a central channel for cooling.

Stage

VI. Conclusions and Prospects

In the work presented in this document, we have attempted to rationalize the use of the MPAS model with respect to its streamlined version. The given improvement has been quantified by using statistical tools on experimental data. We have also continued the development of the two-stage COLISEUM started in a previous thesis work at CEA and extended its comparison with other existing models such as MPAS or JackPot. The COLISEUM can now be extended to the description of cables with indefinite number of stages. This extension of the model tries to stay in the same spirit as in the initial development: encompassing the model expression under the most advanced possible analytical expression keeping its "generic" validity (i.e. for any transient magnetic field regime).

In order to benchmark our models, we have measured experimental coupling losses on samples with various void fractions in the JOSEFA facility at CEA Cadarache. All sides of the experimental work have been explored: facility improvements, sample preparation, measurement process, data extraction and post processing. These measurements allow to confront the descriptions of the experimental coupling losses by both models (MPAS and COLISEUM). It is shown that even if their development are very different, they are visible similarities regarding the description for coupling losses of the presented case (JT-60SA TF conductor).

Along this thesis, all the analytical developments (𝑛-stage COLISEUM, MPAS) and the methods (data processing from JOSEFA) were numerically implemented as integrated tools with Matlab. They are ready to be used in a wider framework like application to stability calculations. In the following paragraphs, we give a summary of the presented work with concluding remarks. They can be gathered above four main axes, the rationalization of the MPAS model, the extension of COLISEUM to an 𝑛stage model, the experimental measurements of coupling losses and finally the benchmark of the developed models between each other and on the experimental data.

VI.1 Rationalization of MPAS

In an effort to facilitate the use of MPAS for fusion community, the MPAS model is presented in its streamlined version with a minimal number of free parameters. We have also presented its extension to the "advanced" version of the model with one extra degree of freedom (justified by the influence of cabling pattern in the computation of the shielding coefficients). Both versions are numerically implemented and applied to the experimental data under a statistical approach to validate the model improvements: the best fit obtained with the advanced MPAS is a local minimum (i.e. a good fit found visually is close to the absolute best fit).

VI.2 Analytical extension of the model: 𝑛-stage COLISEUM

At the end of this thesis, we have in hand an analytical model capable to assess coupling losses for CICC with various types of geometry. This objective was challenging regarding the complexity of the analytical description of the initial two-stage COLISEUM developed in a previous thesis work. The first modelling using the two-stage COLISEUM showed that a description with only two stages is not sufficient to represent the coupling losses generated in a CICC and in particular in a cable such as the JT-60SA TF conductor. In order to describe a full cable geometry we have decided to extend the twostage model to an 𝑛-stage model. The first step was the appropriation of the two-stage COLISEUM developed in the previous thesis work. Its reduction to a system of dimension two and the demonstration of the conservation of the sum of the product 𝑛𝜅𝜏 allow COLISEUM to be in line with the description of coupling losses given by MPAS: one couple of magnetic parameters per stage. This appropriation of the two-stage COLISEUM was a success with the modelling of multiplets of composites. Regarding the analytical complexity of the modelling, sophisticated numerical tools have been developed on an analytical basis in order to carry out faster calculations and to obtain the magnetic parameters of a system in minutes.

These important steps allow us to simplify the already developed COLISEUM, by strengthening its analytical basis at the two-stage level. At this stage, it appears that a direct extension of the two-stage model with a purely explicit analytical description of a fully coupled cable is mathematically out of reach and that the only reasonable path to explore is the one of an extension by iteration. Using a first neighbour coupling assumption, we are able to iterate the modelling used by the initial two-stage COLISEUM to describe a 𝑛-stage cable where each stage is coupled to its first neighbouring stage.

We succeed to demonstrate that the modelling of six-stage cables, including strand plus five-stage modelling, can be achieve using the 𝑛-stage COLISEUM. In other word, the strand can be embedded in the modelling of the coupling losses of a cable as a stage coupled the first cabling stage of the cable.

A main outcome of the calculations is that, due to the coupling, the last stage plays a predominant role in the description of the coupling losses. This is particularly true for the case of a five-stage cable and of a six-stage cable (strand + five stages). Moreover, this is all the more visible in the case of the JT-60SA TF conductor which is a sextuplet for the last stage. This modelling is analytically well defined as we have shown that strong trends as the equality of the sum of the product 𝑛𝜅𝜏 between uncoupled and coupled descriptions, already identified at the two stage level, is conserved all along the iteration of the method.

VI.3 Experimental coupling losses: JOSEFA facility

Regarding the experimental side of the thesis work presented here, we have established a process for the extraction of coupling losses, using CEA analytical formulae for the computation of hysteresis losses with respect to the internal field 𝐵 𝑖 . This process of subtraction of analytical hysteresis losses allows gathering a statistics of effective filament diameter 𝑑 𝑒𝑓𝑓 of the strand constituting our samples. We have shown that the found value of the effective filament diameter 𝑑 𝑒𝑓𝑓 is in agreement with the value given by the strand manufacturer. Thus, we are confident that the proposed process of evaluation of the hysteresis losses is correct.

The measurements are carried out thanks to the method with magnetization pick up coils in the JOSEFA facility at CEA Cadarache. We have measured the coupling losses of seven samples that are close to the JT-60SA TF conductor but with various void fractions ranging from 26 % to 36 %. An important point that strengthens our confidence in the experimental process, is that our measurement has been compared with other measurements led at SULTAN. They are in good agreement although the measurement methods are completely different.

This study has allowed to quantify the intuitive hypothesis that increasing the void fraction (in a real cable) is related to decreasing conductances in between stages and to model this effect using the 𝑛-stage COLISEUM. This result is experimentally evidenced with the shift of the presented curves of

  𝐵 𝑐 : critical magnetic field (T) 𝐼 𝑐 ; 𝐽 𝑐 : critical current/current density (A) (A/m²) 𝑙 𝑝 : strand or stage twist pitch length (m) 𝜏 : time constant of the induced current (s) 𝐵 𝑎 : applied field (T) 𝐵 𝑖 : internal field (T) 𝑃 : deposited power per unit volume (W/m 3 ) 𝜇 0 : vacuum permeability (H/m) 𝑡 : time variable (s) 𝑓 𝑔𝑒𝑜 : geometrical factor related (adim) 𝐵 𝑚 : amplitude of the applied field (T) 𝑓 : frequency (Hz) 𝜔 : pulsation (rad/s) 𝑄 : average losses per unit volume and per cycle (J/m 3 /cycle) 𝑅 : radius (m) 𝑇 𝑜𝑝 : temperature of operation (K) 𝐵 𝑚𝑎𝑥 : maximum field (T) 𝐸 : electric field (V/m) 𝐼 : current inside a strand or a group of strand (A) 𝑅 𝑐𝑖𝑟𝑐 : circumscribed radius (m) 𝑅 𝑒𝑙𝑒𝑚 : radius of an element (m) 𝑅 𝑓 : radius of the filamentary zone (m) 𝑅 𝑐 : cabling radius of a group of strand (m) 𝜎 : transverse conductance (S.m -1 ) 𝑁 : number of elements in a stage (adim) 𝛾 𝐵𝑖𝑜𝑡𝑆𝑎𝑣𝑎𝑟 : geometrical factor (?) 𝑛𝜅 : shielding coefficient (adim) 𝑑 𝑒𝑓𝑓 : effective diameter of the filament (m) 𝜌 𝑡 : transverse resistivity (Ω. 𝑚) 𝐵 𝑜𝑓𝑓 : background field (T) 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 : radius of the superconducting strand (m) 𝜙 : magnetic flux (T/m²) 𝐿 𝑖 : inductance of system 𝑖 (H) 𝑀 𝑖𝑗 : mutual inductance between system 𝑖 and 𝑗 (H) 𝑅 Ω i : resistance of system 𝑖 (Ω) 𝑅 Ω r : replica related resistance (Ω) 𝑛 𝑟 : replica winding turn number (adim) 𝑆 𝑟 : surface of the winding of the replica (m²) 𝑉 𝑟 : voltage related to the replica (V) 𝑅 Ω c : conductor related resistance (Ω) 𝑛 𝑐 : conductor winding turn number (adim) 𝑆 𝑐 : surface area of the winding of the conductor (m²) 𝑉 𝑐 : voltage related to the conductor (V) 𝑉 𝑚 : voltage related to the magnetization of the conductor (V) 𝑛 𝑠 : number of superconducting strands (adim) 𝐿 : length of the sample (m) 𝑀 : magnetization (T) 𝑄 𝑡𝑜𝑡 : total losses per unit volume of cable and per cycle (or mentioned if otherwise) (J/m 3 /cycle) 𝑄 ℎ𝑦𝑠𝑡 : hysteresis losses (J/m 3 /cycle) 𝑄 𝑐𝑜𝑢𝑝 : coupling current losses (J/m 3 /cycle) 𝐵 𝑝 : penetration field (T) 𝑑 𝑓𝑖𝑙 : theoretical diameter of the filament (m) 𝑇 : time period (s)

Figure 1 :

 1 Figure 1: Critical temperature of several materials with respect to the year of discovery.

Figure 2 :

 2 Figure 2: Critical surface description for two superconducting materials: Nb3Sn and NbTi.

Figure 3 :

 3 Figure 3: Scheme of a tokamak.

Figure 4 :

 4 Figure 4: Conductors of the two ITER model coils. On the left CS Model Coil (51mmx51mm, 40 kA), on the right an exploded view of the TF model Coil (40.7mm diameter, 80 kA)

Figure 5 :

 5 Figure 5: Scheme of a superconducting strands on the left. Detailed architecture of JT-60SA TF conductor strand (0.81 mm diameter) on the right.

Figure 7 :

 7 Figure 7: Different type of strands. NbTi (1st line) and Nb3Sn (2nd line). Extracted from [24].

Figure 6 :

 6 Figure 6: Global current loops (in red) generated by applied field variation and generating coupling losses inside an example of composite strand.

Figure 8 :

 8 Figure 8: Respective orientation of pulsed field 𝐵 𝑚 (red) created by the dipoles and background field 𝐵 𝑜𝑓𝑓 (blue) in SULTAN test facility [25].

Figure 9 :

 9 Figure 9: Coupling losses for JT-60SA TF conductor in the frequency range (0-4 Hz) are represented with black dots. Resulting MPAS fit in blue and single time constant model in red (𝐵 𝑜𝑓𝑓 = 0 T left leg virgin 𝐵 𝑚 = 0.1 T).Data are Extracted from [59].

Figure 11 :

 11 Figure 11: Scheme of the generic cross-section geometry considered in CLASS. n layers of radius 𝑅 that can be either normal or superconducting.

Figure 10 :

 10 Figure 10: Detailed architecture of JT-60SA TF conductor strand (0.81mm diameter). Composed of NbTi superconducting filaments.

Figure 12 :

 12 Figure 12: Illustration of two systems modelled with the one stage COLISEUM composed of 8 elements (on the left) and of 3 elements (triplet on the right) with brown filamentary zone. All variables are defined in the text below.

Figure 13 :

 13 Figure 13: Scheme of a triplet of triplet, composed of three triplets at the sub stage level and 1 triplet at the super stage level. 𝑅 𝑐 1 and 𝑅 𝑐 2 are respectively the cabling radius of the substage and of the super-stage.

1 𝑁 1

 11 in the third line of the[𝑌]

2 𝑗=1= 4 .

 24 877 … (𝑢𝑝 𝑡𝑜 17 𝑑𝑖𝑔𝑖𝑡𝑠) 𝑚𝑠 𝒏𝜿 𝝉

Figure 14 :

 14 Figure 14: Scheme of the system simulated in Table 2,3 and 4. Coupled and Uncoupled systems are depicted only with their relevant magnetic parameters. a and b in the top part of the table are used to refer respectively to the first stage and to the second stage.

Figure 15 :

 15 Figure15: Time constants behaviour with respect to the coupling between both sub systems. Blue lines are used for coupled system, both red and green lines respectively correspond to uncoupled system.

8 ,

 8 but we had to use 𝑅 𝑓 𝑅 = 0.29 in the end to obtain the blue curve of Figure 16. It is shown in Figure 17 below how the parameters 𝑅 𝑓 𝑅 affect the coupling losses prediction of the onestage COLISEUM.

Figure 16 :

 16 Figure 16: Comparison of one stage COLISEUM and JackPot on the fourth stage of JT-60SA TF cable simulated as alone.

Figure 18 :

 18 Figure 18: Relative error between the prediction of the one-stage COLISEUM and the one-stage JackPot.

Figure 17 :Frequency

 17 Figure 17: Influence of the filamentary zone radius on the coupling losses predicted with the one stage COLISEUM for the case presented above of the fourth stage of JT-60SA TF conductor. Frequency (Hz)

Figure 19 :

 19 Figure 19: Schematic view of JT-60SA TF composite strand with its Normal Zone NZ1 and NZ2 and its Filamentary Zone FZ as simulated with CLASS on the right, picture of the real strand on the left. Radius of the strand is named 𝑅 𝑠𝑡𝑟𝑎𝑛𝑑 and radius of the filamentary zone (red external circle) named 𝑅 𝑓 𝑠𝑡𝑟𝑎𝑛𝑑 .

2 Figure 20 :Figure 21 :

 22021 Figure 20: Simplified composite simulated with CLASS from Figure 19, with simplified central zone (replaced by filamentary zone).

Figure 22 :

 22 Figure 22: Time consumption, for each N, to find 𝑅 𝑓 and σ while modelling the strand the strand.

  The

Figure 24 :

 24 Figure 24: Schematic view of half twist pitch of a doublet of doublet. With 𝑁 1 = 𝑁 2 = 2 and 𝑙 𝑝 2 𝑙 𝑝 1 = 2 . Red lines represent the current loop of the sub-stage and black lines represent one current loop of the super-stage. Hatched area defines the coupling between system 1 and system 2.

Figure 25 :

 25 Figure 25: Behaviour of the two time constants 𝜏 ± from the doublet of doublet. Asymptotical limit in blue. 𝜏 + in grey and 𝜏 -in black.

Figure 26 :

 26 Figure 26: Behaviour of the four time constants from COLISEUM for a triplet of triplet from Table2.

  1 where at some point the shielding coefficient of the first stage is negligible and we are left with one time constant.

Figure 27 :

 27 Figure 27: Variation of the shielding coeffients with respect to the conductances ratio. On the plot is depicted the conductance ratio we use in the above simulation for the two first stage of JT-60SA TF, 𝜎 2 𝜎 1 = 1.

Figure 29 :Figure 28 :

 2928 Figure 29: Variation of the shielding coefficients of the reduced two-stage COLISEUM with respect to the conductance ratio.

Figure 30 :𝜏 2 Figure 31 :

 30231 Figure 30: Variation of the value of the shielding coefficients for the analytical two stage COLISEUM with respect to 𝜉.

Figure 32 :

 32 Figure 32: Relative error on the value of 𝑛𝜅 between computation from the four dimension two stage COLISEUM and the two dimension two stage COLISEUM.. Computed using the geometrical parameters of the two first stages of JT-60SA TF from Table 2.

Figure 33 :

 33 Figure 33: Relative error on the value of 𝜏 between computation from the four dimension two stage COLISEUM and the two dimension two stage COLISEUM.. Computed using the geometrical parameters of the two first stages of JT-60SA TF from Table2.

Figure 34 :

 34 Figure 34: Overlap of two stages (uncoupled system) with one stage model. The dashed filamentary zone of the uncoupled superstage depict the fact that we do not have a filamentary zone at the superstage level but its shielding coefficient can gives us the equivalent filamentary zone radius to we have to put in the one stage COLISEUM in order to reproduce its magnetic behaviour.

Figure 36 :

 36 Figure 36: Time constants dynamic for the previously presented multiplet of composite with 𝜎 2 = 6.10 7 𝑆. 𝑚 -1 . Value of the times constants and shielding coefficient are given in the above Table11.

Figure 38 :

 38 Figure 38: Triplet of composite simulation using the two stages model on the left. Triplet of composite simulated with the one stage COLISEUM, strand details are absent from the sketch).

Figure 41 :

 41 Figure 41: Iteration scheme. Stages 1, 2 and 3 depicted in the centre. COLISEUM two stages (C2S) applied separately at first (two above brackets) and COLISEUM 3-stages (C3S) applied globally (bottom bracket).

Figure 40 :

 40 Figure 40: Schematic modelling of the three stage COLISEUM using the above iterative method. Cabling radius 𝑅 𝑐 are depicted in blue dashed lines and circumscribed radius 𝑅 𝑐𝑖𝑟𝑐 are depicted in black dashed line. Current 𝐼 𝑘 1 𝑘 2 𝑘 3 flows on red superconducting shell in each elements.

Figure 42 :

 42 Figure 42: Three-stage system modelled using two independent computations of the two-stage COLISEUM (top part of the figure). Uncoupled three-stage system modelled using three independent computations of the one stage COLISEUM (bottom part of the figure).

3 𝑗=1= 14 . 3 𝑖=1Figure 43 :

 314343 Figure 43:Magnetic parameters dynamic for a three-stage system. Coupled case in red. Uncoupled case in blue. All shielding coefficient are refered to the circumscribed area of the system.

Figure 44 :

 44 Figure 44: Schematic view of the establishment of a five-stage COLISEUM.

Figure 46 :

 46 Figure 46: Coupling losses in a five-stages cable simulated using the 𝑁-stages COLISEUM. In this configuration, each stage is in first neighbour interaction with surrounding stages. Using the shielding coefficient refered to the area of superconducting strands in the system.

Figure 47 :

 47 Figure 47: Magnetic parameters dynamic for a five-stage cable. Coupled case in red. Uncoupled case in blue. Shielding coefficients are refered to the area of superconducting strands in the system.

Figure 49 :

 49 Figure 49: Coupled (red) and Uncoupled (blue) approach simulated with the iterative five-stage COLISEUM. Using the shielding coefficient refered to the area of superconducting strands in the system.

Figure 50 :

 50 Figure 50: Q(f) curves modification with respect to the value of interstage conductances. All conductances are kept proportional. They all vary with the same ratio. Ratio between conductances are kept constants.

𝜎 1 = 7 .

 17 68.10 7 𝑆. 𝑚 -1 𝑎𝑛𝑑 𝑝 = 1.28

Figure 51 :

 51 Figure 51: Q(f) curves modification with respect to the ratio in-between interstage conductances. 𝜎 1 is kept constant. Solid lines show the modification of the 𝑄(𝑓) curves with respect to the ratio in-between the conductances, keeping 𝜎 1 constant. Dashed black line is here to remind the tendency shown in Figure 47 where the ratio in-between 𝜎 ′ 𝑠 is kept constant and the values of 𝜎 ′ 𝑠 are increased.

  , J. Duchateau, F. Topin, B. Turck and L. Zani, "Void fraction influence on CICCs coupling losses: Parametric measurements and analysis with MPAS model," IEEE on Transaction on Applied Superconductivity, 2019.[START_REF] Chiletti | Void fraction influence on CICCs coupling losses: Parametric measurements and analysis with MPAS model[END_REF]

Figure 52 :

 52 Figure 52: Sketch of a paired saddle coil on the left with the current 𝐼 and the applied field generated 𝐵 𝑎 . Photo of the dipole MARIUS on the right.

Figure 54 :Figure 53 :

 5453 Figure 54: Magnetic field profil generated by MARIUS. CICC samples we used are depicted in grey in the above sketch..

Figure 55 :

 55 Figure 55: Sample holder assembly views. a) describes the sample, its replica and the sample holder in brown bakelite,in b) and c) sample holder is attached to the superior tap with a welded stainless steel bar..

Figure 56 :

 56 Figure 56: Wheatstone bridge. V are voltages and 𝑅 𝛺 are resistances. c and r index are related respectively to the Cable and to the Replica, m index refers to the voltage related to the Magnetization of the sample. Arrow on the resistance of the replica is used to depict the decade box used here.

Figure 58 :

 58 Figure 58: Example of tested samples: MAG42 type,.

Figure 59 :

 59 Figure 59: Experiment referential. Magnetization of the sample 𝑀 ⃗⃗ and applied field variation 𝐵 ̇𝑎.

Figure 61 :

 61 Figure 61: One cycle of the applied field. Index using prime symbols are used to refer to the partial penetration case whereas other index are used for the case of full penetration.

Figure 60 :

 60 Figure 60: Measured value of critical current at T=4.2K for JT-60SA strand. Measured at ENEA, extracted from [38].

Figure 62 :

 62 Figure 62: Field partial penetration in a superconducting filament following the Bean model. First sketch number 1' depict the initial state after the first magnetization, then from 2' to 6' we go over a complete cycle of partiel penetration.

Figure 64 :

 64 Figure64: Comparison of hysteresis losses computed using Wilson or CEA aproach. 𝛤 is a coefficient given in[START_REF] Wilson | Time-varying fields and A.C. losses[END_REF] in order to compute hysteresis losses.

  lower part of the 𝑀(𝐵) curve (quasi-flat zones) are linked to the 𝐽 𝑐 (𝐵 𝑚 , 𝐵 𝑜𝑓𝑓 , 𝑇) curves.

Figure 65 :

 65 Figure 65: Magnetization cycle for MAG42-3 sample with 𝐵 𝑜𝑓𝑓 = 0.5 𝑇 and 𝐵 𝑚 = 0.2 𝑇. Numbers in black circle are here to depict the corresponding state inside filaments from Figure 63.

Figure 68 :

 68 Figure 68: Amplitude of the applied field with respect to the frequency.

Figure 67 :

 67 Figure 67: Sample MAG42-3 under 𝐵 𝑚 =±0.1 T and 𝐵 𝑜𝑓𝑓 =0.5 T . Measured losses (black)

Figure 69 :

 69 Figure 69: Low frequency zoom of the figure 39. Total losses in black. Dashed line used for the extrapolation at origin of the total losses.

Figure 70 :

 70 Figure 70: Internal field amplitude with respect to the frequency.

Figure 71 :

 71 Figure 71: Measured losses in black, hysteresis losses computed using 𝐵 𝑖 and 𝑛𝜏 𝑒𝑓𝑓 𝑎𝑝𝑝𝑟𝑜𝑥 in green, and coupling losses in red.Presented before rescaling of the coupling losses.

Figure 72 :

 72 Figure 72: Low frequency zoom of Figure 43. Total losses in black, hysteresis losses in green and coupling losses in black. Dashed line are used as linear extrapolation to the origin of each curves.

Figure 73 :

 73 Figure 73: Coupling losses after rescaling to 𝐵 𝑚 = ± 0.1 𝑇. For MAG42-3 sample with 𝐵 𝑜𝑓𝑓 = 0.5 𝑇, 𝐵 𝑚 = ±0.1 𝑇.

Figure 75 : 15 Figure 74 :

 751574 Figure 75: Variation of coupling losses regarding the change of 𝐵 𝑜𝑓𝑓 . The legend represents 𝐵 𝑜𝑓𝑓 ± 𝐵 𝑚 in Tesla. Curves are shown after rescale to 𝐵 𝑚 = 0.1 𝑇.

Figure 77 :

 77 Figure 77: Statistic of 𝑑 𝑒𝑓𝑓 . Extracted from all measurement made on MAG42 and DP4 samples (number of 𝑄 𝑐𝑜𝑢𝑝 (𝑓) given in Table33).

Figure 76 :

 76 Figure 76: Variation of 𝑛𝜏 𝑒𝑓𝑓 with respect of the void fraction of the sample. Dashed line used as guide for the eyes. Error bars are standard deviation and are taken from Table33.

Figure 78 :

 78 Figure 78: Comparison of 𝑄 𝑐𝑜𝑢𝑝 (𝑓) curves for various void fractions for 𝐵 𝑚 =±0.1 T. Without relative error to clarify the view.

Figure 80 :Figure 79 :

 8079 Figure 80: Effect of caliber change in the error made on the measurement at various frequencies. Sample MAG42-3 with 𝐵 𝑜𝑓𝑓 = 0.06 𝑇 𝑎𝑛𝑑 𝐵 𝑚 = 0.05 𝑇 , measurements are rescaled to 𝐵 𝑚 = 0.1 𝑇.

Figure 81 :

 81 Figure 81: Comparison of JOSEFA and SULTAN measurements on JT-60SA TF samples type.

Figure 82 :

 82 Figure 82: Zoom of Figure 81 at low frequency to show the precision of hysteresis removal. Dashed line used as guide for the eyes. We are in the low frequency regime, the frist measured point from SULTAN is above the maximum value of 0.03 Hz.

Figure 83 :

 83 Figure 83: JT-60SA TF conductor cross section typical images from tomography.

Figure 84 :

 84 Figure 84: Reconstructed strand tranjectories of a sample of JT-60SA TF conductor with the newly developed reconsctruction program. Z-axis depict the number of reconstructed slices while x and y-axis stands for the dimension in pixel.

Figure 85 :

 85 Figure 85:Mask applied on a tomographic slice.

Figure 88 :

 88 Figure 88: Void fraction in each slices. X-axis stands for the index of the considered slices and y-axis stands for the void fraction found in a given slice.

Figure 89 :

 89 Figure 89: Best fit obtain with n-stage COLISEUM with respect to JOSEFA and SULTAN data. Sample is MAG42-3 with 𝐵 𝑚 = ±0.1 𝑇. The contribution of each stage is depicted in dashed lines, plain blue line is the sum of the five contributions and represents the model of the n-stage COLISEUM.

Figure 90 :

 90 Figure 90: Zoom at low frequency of Figure 83.

Figure 91 :

 91 Figure 91: Q(f) curves from the restricted MPAS in plain line, experimental data depicted with red dot.

Figure 92 :

 92 Figure 92: View of the distribution of Q(f) curves generated with the restricted MPAS over the experimental data (red stars).

Figure 93 :

 93 Figure 93: Best fit obtained with the restricted MPAS using the mean square approach. Experimental data in red stars, best fit from the restricted MPAS in black plain line.

Figure 94 :

 94 Figure 94: Cartography of normalized mean square value obtained with each simulation using the restriceted MPAS. Continuous view on the top part and leveled view on the bottom part. Red dot represent the best fit parameters in the plan.

Figure 95 :

 95 Figure 95: Best fit obtained with the restricted MPAS using the mean square approach where we weighted the initial slope of the fit. Experimental data in red stars, best fit from the restricted MPAS in black plain line

Figure 98 :

 98 Figure 98: Best fit obtained with the advanced MPAS using the mean square approach. Experimental data in red star, best fit from the advanced MPAS in black plain line.

Figure 99 :Figure 100 :

 99100 Figure 99: 3D Cartography of the normalized mean square value with respect to the free parameters of the advanced MPAS. 𝜅 𝑎 (𝑎𝑑𝑖𝑚)

Figure 101 :

 101 Figure 101: Sensity of the advanced MPAS. 𝑄 𝑐𝑜𝑢𝑝𝑙𝑖𝑛𝑔 with a mean square value around the minimum are plot. Data from MAG42-3 in red stars, 50 curves with mean square around the minimum value plotted in plain lines.

Table 1 :

 1 Superconducting fusion machines.

  Duchateau, F. Topin, B. Turck, L. Zani, "Analytical Modelling of CICCs Coupling Losses: Broad Investigation of Two-Stage Model", IEEE Trans. Appl. Superconductivity, Vol. 29, March 2019, Art. No. 4703005. [28]

Table 2 :

 2 Geometrical parameters of two first stage of JT-60SA TFParameters in Table2are illustrated in Figure

Table 4 :

 4 𝑵 𝟏𝑵 𝟐 𝝈 𝟏 (𝑺. 𝒎 -𝟏 ) 𝝈 𝟐 (𝑺. 𝒎 -𝟏 ) 𝒍 𝒑 𝟏 (𝒎) 𝒍 𝒑 𝟐 (𝒎)

	𝑹 𝒆𝒍𝒆𝒎 (𝒎𝒎)	𝑹 𝒄 𝟏 (𝒎𝒎)	𝑹 𝒄 𝟐 (𝒎𝒎)	𝑹 𝒇 𝑹 𝒆𝒍𝒆𝒎

Table 4 :

 4 Limit of the magnetic parameters obtained using the two-stage COLISEUM. In green is decpicted the only couple of magnetic parameters for which none of the components are either null or close to zero.

Table 5 :

 5 Magnetic parameters for JT-60SA TF strand generated using CLASS.

Table 6 :

 6

	𝑵	𝑹 𝒆𝒍𝒆𝒎 (𝒎𝒎) 𝑹 𝒄𝒊𝒓𝒄 (𝒎𝒎)	𝑹 𝒇 𝑹 𝒆𝒍𝒆𝒎	𝝈 (𝟏𝟎 𝟗 𝐒. 𝐦 -𝟏 )
	3	0.283	0.610	0.996	7.71
	4	0.231	0.558	0.938	7.25
	5	0.192	0.519	0.884	7.27
	6	0.164	0.491	0.814	7.47
	7	0.142	0.469	0.724	7.77
	8	0.125	0.452	0.627	8.13
	9	0.112	0.439	0.534	8.52
	10	0.101	0.428	0.445	8.93
	11	0.0921	0.419	0.363	9.37
	12	0.0846	0.412	0.297	9.82
	13	0.0783	0.405	0.235	10.3
	24	0.0427	0.369	0.0527	15.6

Geometric and electric parameters of the one-stage COLISEUM simulating the composite strand of JT-60SA TF. Parameters of the green line are the one used to model the JT-60SA TF strand.We recall that for each line of the above table, 𝑅 𝑐 = 𝑅 𝑓 𝑠𝑡𝑟𝑎𝑛𝑑 = 0.327 𝑚𝑚 and each simulation with the one-stage COLISEUM gives by construction 𝜏 = 18.2 𝑚𝑠 and 𝑛𝜅 = 1.24 .

Table 7 :

 7 Contribution to coupling losses of each magnetic parameters fromTable 3.

	𝒋	𝒏𝜿	𝝉(𝒎𝒔) 𝒏𝜿𝝉 (𝒎𝒔) 𝒂 𝒋 (%)
	0 5.10 -5	0.89	3.75.10 -5	8.10 -4
	1 4.3.10 -6 20.4	1.21.10 -6	3.10 -3
	2	0.09	1.30	0.121	2.56
	3 0.923	5.16	4.59	97.4

  Where 𝐼 2 and 𝐼 1 are now both describing amplitudes of currents without artificial weight. Once written again under matrix form, we obtain:

		𝑁 1 𝜏 21 𝐼 ̇1 + 𝑁 1 𝜏 22 𝐼 ̃2 = 4	𝑁 1 𝜎 2 𝑁 1	𝑅 𝑐 2 sin 2 (	𝜋 𝑁 2	) 𝐵 ̇𝑎/𝛼 2 2
	Using 𝐼 2 = 𝑁 1 𝐼 ̃2, we obtain the following equation
	𝐼 2 + 𝑁 1 𝜏 21 𝐼 ̇1 + 𝜏 22 𝐼 ̇2 = 4𝜎 2 𝑅 𝑐 2 sin 2 (	𝜋 𝑁 2	) 𝐵 ̇𝑎/𝛼 2 2
	[ 𝐼 2 𝐼 1	] + ( 𝑁 1 𝜏 21 (𝜎 2 ) 𝜏 11 (𝜎 1 )	𝜏 12 (𝜎 1 )/𝑁 1 𝜏 22 (𝜎 2 )	) [	𝐼 1 İ2 ̇] = [	4𝜎 1 𝑅 𝑐 1 sin 2 ( 4𝜎 2 𝑅 𝑐 2 sin 2 (	𝜋 𝑁 1 𝜋 𝑁 2	2 ) /𝛼 1 ) /𝛼 2 2	] 𝐵 ̇𝑎	(52)
	With
		[𝜏] = ( 𝑁 1 𝜏 21 (𝜎 2 ) 𝜏 11 (𝜎 1 )	𝜏 12 (𝜎 1 )/𝑁 1 𝜏 22 (𝜎 2 )	)

  𝑇𝑟[𝜏] = 𝜏 11 (𝜎 1 ) + 𝜏 22 (𝜎 2 ) and det[𝜏] = 𝜏 11 𝜏 22 -𝜏 21 𝜏 12 . 𝝉 𝟏 = 𝝉 𝟏𝟏 + 𝝉 𝟐𝟐 -√(𝝉 𝟏𝟏 -𝝉 𝟐𝟐 ) 𝟐 + 𝟒𝝉 𝟐𝟏 𝝉 𝟏𝟐 𝟐 𝒂𝒏𝒅 𝝉 𝟐 = 𝝉 𝟏𝟏 + 𝝉 𝟐𝟐 + √(𝝉 𝟏𝟏 -𝝉 𝟐𝟐 ) 𝟐 + 𝟒𝝉 𝟐𝟏 𝝉 𝟏𝟐 𝟐

	Written explicitly using initial uncoupled time constants we get:

  11 -𝜏 2 )𝑥 2 + 𝜏 12 𝑦 2 /𝑁 1 = 0 (𝜏 22 -𝜏 2 )𝑦 2 + 𝑁 1 𝜏 21 𝑥 2 = 0

			and {	(𝜏 11 -𝜏 1 )𝑥 1 + 𝜏 12 𝑦 1 /𝑁 1 = 0 (𝜏 22 -𝜏 1 )𝑦 1 + 𝑁 1 𝜏 21 𝑥 1 = 0	(57)
	After combining both equations in each system, we obtain:
	𝑥 2 𝑦 2	=	(𝜏 22 -𝜏 2 -𝜏 12 /𝑁 1 ) (𝜏 11 -𝜏 2 -𝑁 1 𝜏 21 ) = 𝐶 2 and	𝑥 1 𝑦 1	=

(𝜏

[START_REF] Gibson | The JET project[END_REF]

-𝜏 1 -𝜏 12 /𝑁 1 ) (𝜏 11 -𝜏 1 -𝑁 1 𝜏 21 ) = 𝐶 1

  𝝀 𝒌 𝑽 𝒌𝟐 𝒀 𝒃 𝟐 𝑽 𝒌𝒍 𝒀 𝒃 𝒍 𝝉 𝟏 + 𝝉 𝒍

	𝟐	𝟐
	𝒌=𝟏	𝒍=𝟏
	𝟐 𝒏𝜿 𝟐 = 𝟐𝝁 𝟎 ∑ ∑ 𝝀 𝒌 𝟐 𝒍=𝟏 𝒌=𝟏	𝑽 𝒌𝟏 𝒀 𝒃 𝟏 𝑽 𝒌𝒍 𝒀 𝒃 𝒍 𝝉 𝟐 + 𝝉 𝒍

  We are left with only one equation which is the one of the independent substage:

	) turn into: {	𝐼 1 + 𝜏 11 𝐼 ̇1 +	𝜏 12 𝑁 1	𝐼 ̇2 = (4𝜎 1 𝑅 𝑐 1 sin 2 ( 𝑁 1 𝜋 𝐼 2 = 𝐼 ̇2 = 0	) /𝛼 1 2 ) 𝐵 ̇𝑎
	𝐼 1 + 𝜏 11 𝐼 ̇1 = (4𝜎 1 𝑅 𝑐 1 sin 2 (	𝜋 𝑁 1	) /𝛼 1 2 ) 𝐵 ̇𝑎 𝑑𝑢𝑒 𝑡𝑜 𝐼 ̇2 = 0
	-[𝜏] = ( 𝜏 11 (𝜎 1 ) 𝜏 12 (𝜎 1 )/𝑁 1 0 0	det[𝜏] = 0 𝑇𝑟[𝜏] = 𝜏 11 ) and {
	-𝑌 = ( 4𝜎 1 𝑅 𝑐 1 sin 2 ( 𝑁 1 𝜋 0	) /𝛼 1 2	) and {	𝐶 1 = +∞ 𝜏 12 𝐶 2 = -𝑁 1 𝜏 11

Table 2 .

 2

	𝜏 1 /𝜏 11
	𝜏 2 /𝜏 11
	𝜏/𝜏 11
	𝜎 2 /𝜎 1

  𝜏 11 and 𝜏 2 = 𝜏[START_REF] Gibson | The JET project[END_REF]

				(70)
	𝐶 1 → +∞ and 𝐶 2 = 0	(71)
	𝑉 = ( 1 0 0 1	) and thus 𝑌 𝑏 = 𝑌	(72)
	Hence we have 𝑛𝜅 1 =	𝜇 0 𝜆 1 𝑌 𝑏 1 2 𝜏 11	and 𝑛𝜅 2 =	2 𝜇 0 𝜆 2 𝑌 𝑏 2 𝜏 22

  This calculation in dimension two is done without the extra terms (𝜏 00 , 𝜏 01 , 𝜏 10 , 𝜏 43 , 𝜏 44 𝑎𝑛𝑑 𝜏 34 ) in the time matrix [𝜏] as we have removed them from the description. Initially these extra terms (𝜏 00 , 𝜏 01 , 𝜏 10 , 𝜏 43 , 𝜏 44 𝑎𝑛𝑑 𝜏 34

Table 8 :

 8 and Table2) are recalled here below and segregated per stage: Geometric and electric parameters chosen for the first stage of JT-60SA.

	System 1:
	𝑵 𝟏 𝝈 𝟏 (𝑺. 𝒎 -𝟏 ) 𝒍 𝒑 𝟏 (𝒎)	𝑹 𝒆𝒍𝒆𝒎 (𝒎𝒎)	𝑹 𝒄 𝟏 (𝒎𝒎)	(	𝑹 𝒇 𝑹 𝒆𝒍𝒆𝒎	) 𝟏
	3	6.10 7	0.045	0.405	0.467		0.8

Table 9 :

 9

Geometric and electric parameters choson fro the second stage of JT-60SA TF.

  𝟐 𝝈 𝟐 (𝑺. 𝒎 -𝟏 ) 𝒍 𝒑 𝟐 (𝒎)

				𝑹 𝒆𝒍𝒆𝒎 (𝒎𝒎)	𝑹 𝒄 𝟐 (𝒎𝒎)	(	𝑹 𝒇 𝑹 𝒆𝒍𝒆𝒎	) 𝟐
	3	6.10 7	0.070	0.872	1.008		0.68

Table 10 :

 10 𝟏 𝑵 𝟐 𝝈 𝟏 (𝑺. 𝒎 -𝟏 ) 𝝈 𝟐 (𝑺. 𝒎 -𝟏 ) 𝒍 𝒑 𝟏 (𝒎) 𝒍 𝒑 𝟐 (𝒎)

	𝑹 𝒆𝒍𝒆𝒎 (𝒎𝒎)	𝑹 𝒄 𝟏 (𝒎𝒎)	𝑹 𝒄 𝟐 (𝒎𝒎)	𝑹 𝒇 𝑹 𝒆𝒍𝒆𝒎

Geometric parameters of the triplet of strands.

Figure 35: Sketch of the simulation of a triplet of strands of JT-60SA , strand is modelled with 𝑁 1 = 13 from Table 6. Again, black dashed lines represent circumscribed area of first and second stages. Blue dashed lines are used for cabling radii 𝑅 𝑐 1 and 𝑅 𝑐 2 .

Table 11 :

 11 Magnetic parameters of the triplet of strands fromTable 10 modelled with the two stage COLISEUM using 𝜎 2 = 6.10 7 𝑆. 𝑚 -1 .

can model One-stage COLISEUM can model First stage of the Two- stage COLISEUM can model

	Strand	 (previous work)	 (this work)	 (this work)
	Multiplet		 (previous work)	 (previous work)

Table 14 :

 14 View of the acomplished work.

Table 15 :

 15 Reminder of the table 2, geometrical and electrical parameters of the two first stage of JT-60SA TF cable.

  𝑵 𝟏 𝑵 𝟐 𝝈 𝟏 (𝑺. 𝒎 -𝟏 ) 𝝈 𝟐 (𝑺. 𝒎 -𝟏 ) 𝒍 𝒑 𝟏 (𝒎) 𝒍 𝒑 𝟐 (𝒎)

							𝑹 𝒆𝒍𝒆𝒎 𝑨 (𝒎𝒎)	𝑹 𝒄 𝟏 (𝒎𝒎)	𝑹 𝒄 𝟐 (𝒎𝒎)	𝑹 𝒇 𝑨 𝑹 𝒆𝒍𝒆𝒎 𝑨
	3	3	6.10 7	6.10 7	0.045	0.07	0.405	0.467	1.008	0.8
			Stage		𝒏𝜿			𝝉(𝒎𝒔)
			1 st		9.03.10 -2			1.30
			2 nd		0.923			5.16

Table 18 :

 18

geometrical and electrical parameters of the second and third stage of JT-60SA TF cable.

Table 20 :

 20 magnetic parameter of the uncoupled sub-stage of system B.Shielding coefficient is refered to the circumscribed area of the substage of system B.

	𝒏𝜿	𝝉(𝒎𝒔)
	0.807 4.77

Table 17 :

 17 Magnetic parameters of the uncoupled second stage of system A.𝑵 𝟐 𝑵 𝟑 𝝈 𝟐 (𝑺. 𝒎 -𝟏 ) 𝝈 𝟑 (𝑺. 𝒎 -𝟏 ) 𝒍 𝒑 𝟐 (𝒎) 𝒍 𝒑 𝟑 (𝒎)

	𝑹 𝒆𝒍𝒆𝒎 𝑩 (𝒎𝒎)	𝑹 𝒄 𝟐 (𝒎𝒎)	𝑹 𝒄 𝟑 (𝒎𝒎)	𝑹 𝒇 𝑩 𝑹 𝒆𝒍𝒆𝒎 𝑩

Table 21 :

 21 Uncoupled magnetic parameters of the first stage on the left, second stage in the middle and third stage on the right.

	𝒋	1	2	3
	𝒏𝜿 𝒋	0.395	0.521	0.770
	𝝉 𝒋 (𝒎𝒔)	1.68	4.77	14.7

Each one simulated using the one-stage COLISEUM. The considered case is the one of the three first stages of JT-60SA TF. Shielding coefficients are referred to the circumscribed area of the thrid stage (i.e. the circumscribed area of the system).

  Table 28 below.

	Stage i	Strand	1	2	3	4	5
	𝑵 𝒊	13	3	3	3	3	6
	15 𝝈 𝒊 (𝑺. 𝒎 -𝟏 ) 1.03.10 10 𝒍 𝒑 𝒊 (𝒎𝒎)	45 6.10 7	70 6.10 7	120 6.10 7	170 6.10 7	290 6.10 7

Table 28 :

 28 Geometric and electric parameters of a strand+five-stage cable.

	𝑹 𝒔𝒕𝒓𝒂𝒏𝒅 (𝒎𝒎)	𝑹 𝒇 𝒊𝒏𝒊𝒕	𝑩 𝒎 (T)
		𝑹 𝒔𝒕𝒓𝒂𝒏𝒅
	0.0783	0.235	0.1

Table 29 :

 29 Geometric parameters of JT-60SA TF (strand+ five-stage) plus 𝐵 𝑚 the amplitude of the applied field.

  1 (𝑖. 𝑒. 𝐵 𝑝 > 𝐵 𝑚 )

	𝑄 ℎ𝑦𝑠𝑡 =	2𝜋(2𝐵 𝑚 ) 3 * (𝐵 𝑚 , 𝐵 𝑜𝑓𝑓 )𝑑 𝑓𝑖𝑙 (1 + 𝑥) 2 𝐽 𝑐 6𝜇 0	(1 -	𝜋2𝐵 𝑚 * (𝐵 𝑚 , 𝐵 𝑜𝑓𝑓 )𝑑 𝑓𝑖𝑙 4𝜇 0 𝐽 𝑐	)	(113)
	and in full penetration 𝛽 > 1 (𝑖. 𝑒. 𝐵 𝑝 < 𝐵 𝑚 )
	𝑄 ℎ𝑦𝑠𝑡 =	4𝐽 𝑐 * (𝐵 𝑚 , 𝐵 𝑜𝑓𝑓 )𝑑 𝑓𝑖𝑙 2𝐵 𝑚 3𝜋(1 + 𝑥)	(1 -	𝜇 0 𝐽 𝑐 * (𝐵 𝑚 , 𝐵 𝑜𝑓𝑓 )𝑑 𝑓𝑖𝑙 𝜋2𝐵 𝑚	)

Table 33 :

 33 Effective 𝑛𝜅𝜏 for MAG42 samples, samples are presented inTable 32.

Table 34 :

 34 Frequency of the maximum of the Q_coup (f)curves for each MAG42 samples and DP4-UP.

Table 35 :

 35

Shift ratio found to rescale all curve with respect to the interstages conductances.

Table 37 :

 37 Interstages conductances used to best fit the experimental data using n-stage COLISEUM for JT-60SA TF.

  Table 39 below.

	Stages # 𝑖	𝒏𝜿 𝒊	𝝉 𝒊 (𝒎𝒔)
	1	0.381	13.7
	2	0.500	33.2
	3	0.656	97.6
	4	0.861	196
	5	1.13	570
	𝚺𝒏𝜿 𝒊 𝝉 𝒊	899 ms

Table 39 :

 39 Magnetic parameters for the above best fit of Figure 93 generated with the restricted MPAS.

Table 40 :

 40 Magnetic parameters of the weighted fit of the restricted MPAS.

Table 41 :

 41 Magnetic parameters for the above best fit of Figure98generated with the advanced MPAS. 𝜅 𝑏 = 0.63 for the best fit presented here.

	Stages # 𝒊	𝒏𝜿 𝒊	𝝉 𝒊 (𝒎𝒔)
	1	0.220	6.02
	2	0.254	14.6
	3	0.293	42.8
	4	0.340	85.9
	5	2.23	250
	𝚺𝒏𝜿 𝒊 𝝉 𝒊	604 ms

Acknowledgements

First, I would like to sincerely thank the Commissariat à l'Energie Atomique et aux énergies alternatives (CEA) and ASSYSTEM for their financial support which allowed me to be a PhD candidate within the IRFM facilities during three years and two month. Acknowledgements ................................................................................................................................. I. Introduction ...................................................................................................................................

Figure 23: Cartographies of the number of contribution to the coupling losses. Three different thresholds are presented to show the continuous vanishing of the contribution while we increase the threshold. From top to bottom we have the following three differents thresholds: s= 1 %, s= 4 % and s= 10 %. Dark blue is used for zone with 1 contribution, sky blue is used for domain with 2 contributions and green is used for domain with 3 contributions. Numbers of contributions for each zone are depicted at the limit between two domains.

𝑙 𝑝 2 𝑙 𝑝 1

Step 8.

…

Iterating this methodology, we can simulate a 𝑛-stage cable with (3𝑛 -5) steps:

Step 3𝑁 -7.

Compute the uncoupled shielding coefficient of the current last stage (𝑛 -1 𝑡ℎ stage).

Step 3𝑁 -6.

Using the above shielding coefficient, deduce the 𝑅 𝑓 /𝑅 𝑒𝑙𝑒𝑚 ratio and thus the 𝑅 𝑓 value that stands as input for the next two-stage system (i.e. (𝑛 -1) 𝑡ℎ and 𝑛 𝑡ℎ stages)

Step 3𝑁 -5.

Couple 1 𝑠𝑡 -2 𝑛𝑑 -3 𝑟𝑑 -. . . -(𝑛 -1) 𝑡ℎ to (𝑛 -1) 𝑡ℎ -𝑛 𝑡ℎ to obtain an 𝑛-stage system.

The above methodology can be applied on a five-stage cable such as JT-60SA TF conductor which is quite academic as it does not possess any wrapping on petal or any central channel. The scheme of the algorithm to define an 𝑛-stage system is depicted in Figure 44 and is applied on a five-stage cable where all stages are triplets except the last one, which is a sextuplet. The scheme of Figure 44 is here to summarize all equations in the above development. It exposes in a synthetized way the principal steps to define a 𝑛-stage system. Despite the analytical development, it shows that the above method is, in fact, simple to apply and not time consuming with a fully explicit analytical calculation.

The system equation defined in step 4 in Figure 44 corresponds schematically to Figure 40. We do not depict the schematic view of the corresponding system of step 7 and step 10 for matter of clarity as it is not so useful and just depicts a complete fourth and five stage of JT-60 SA TF cable respectively, in tangency condition.

III.1.4 Numerical application

III.1.4.1 Iterative five-stage COLISEUM

We have chosen to model a JT-60SA TF conductor as the first modelling using the 𝑛-stage COLISEUM. The electric and geometric parameters are gathered in Table 24 andTable 25 The above parameters are taken from manufacturers specifications and are not extracted from measurements. Using these parameters we can model a five-stage cable in tangency condition. The cabling radii 𝑅 𝑐 𝑖 are thus all determined from the radius of the basic elements in the considered system.

The conductances 𝜎 𝑖 are taken all equal for the first simulation but could be modified in the following. Their values are taken in agreement with the work done by Twente University in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF] and in [START_REF] Nijhuis | Impact of void fraction on mechanical properties and evolution of coupling loss in ITER Nb3Sn conductors under loading applied superconductivity[END_REF].

is also selected such as it is in agreement with the description done of the strand modelling in section II.2.1.

Uncoupled case:

Stage 𝒏𝜿 𝝉(𝒎𝒔) As shown in section III.1.3, we can look at the behaviour of the system when we cut the coupling between surrounding stages in order to see how the coupling losses are affected by this modification. Results are gathered in the table above where shielding coefficients are referred to the circumscribed area of the system. Coupling losses associated to this set of magnetic parameters are depicted in Figure 45.

Coupled case:

Stage 𝒏𝜿 𝝉(𝒎𝒔) Using the 𝑛-stage COLISEUM to model our five-stage cable (JT-60SA TF), we have found the above magnetic parameters where shielding coefficients are referred to the circumscribed are of the system. Once these parameters are computed, we can readily give the coupling losses generated by such a cable as seen in Figure 46 below. And then compute the coupled magnetic parameters in Using the 𝑛-stage COLISEUM to model our five-stage cable (JT-60SA TF), we have found the above magnetic parameters where shielding coefficients are referred to the circumscribed area of the system. We can also depict the coupling losses in such a system in the usual 𝑄 𝑐𝑜𝑢𝑝 (𝑓) form as seen in Figure 48 below. We also verify that: The value is given here as referred to the circumscribed area of the system. We precise that the values of magnetic parameters exposed in Table 30 and in Table 31 are given with limited precision, i.e. a check with those values would artificially show a deviation from the strict equality. It is stressed that this was demonstrated in section III.1.3 for the 𝑛-stage COLISEUM.

Comparing this value of 23.04 ms with the value found in the above modelling of the five-stage cable (without the strand) we find a 3% additional coupling losses due to the presence of the strand. Added to the fact that the presence of the strand redistributed in a completely different way the shielding coefficient of the three first stages of the cable.

As the strand, simulated as the first stage of a six-stage cable, is in first neighbour interaction it only interacts with the second stage (first cabling stage of JT-60SA TF). However, its value is not so much affected by the coupling as we can see in Table 31 with respect to its uncoupled value in Table 30.

It appears that in the uncoupled case, the strand contributes more to the coupling losses (in terms of product 𝑛𝜅𝜏) than the first and second stage. In the coupled case, the strand contributes more to the coupling losses than the first, second and third stage. After coupling, the presence of the strand in the simulation has weakened the shielding coefficients of three first cabling stages. This was already seen for the triplet of composites in section II.3.4. It does not affect the shielding coefficients of the two last stages. The inclusion of the strand in the model seems to be well handled by the 𝑛-stage COLISEUM, as the result seems consistent (increase of initial slope). We can say that the contribution of the strand itself, is not insignificant in the modelling of CICC with respect to the contribution of each stage.

We have managed to integrate all shielding units (each stage plus the strand), and we have shown how they all play a role in the cable shielding.

We have implemented several improvements (material and post-processing) which have been carried out on this facility. New processes of winding (for pick-up coils) were implemented in order to shorten and simplify the exploitation of the facility between two tested samples (Figure 57). This is a great advantage for us as the exploitation of the facility is made easier and more flexible in order to be ready whenever helium is available for our tests.

The coils were wound with the same number of turns 𝑛 𝑡 and with the same orientation in order to be sure that the induced current inside the pick-ups coils will be similar in order to avoid useless postprocessing.

IV.1.3 Short presentation of the test of the MAG42 conductor sample

Information on all tested samples during the thesis work is gathered in the following Table 32 with their respective geometric parameters. Some of them are depicted in Figure 58. They were manufactured by ICAS in the framework of DEMO related R&D (see [START_REF] Della Corte | ITER and JT-60SA TF conductor production at ICAS[END_REF] and [START_REF] Zani | Starting EU production of strand and conductor for JT-60SA TF coils[END_REF]) and were tested at CEA in experimental campaigns of hydraulic pressure drop measurements (see [START_REF] Zani | Progresses at CEA on EU demo reactor cryomagnetic system design activities and associated R&D[END_REF]).

Six MAG-42 samples (plus the DP-4 UP sample, similar to MAG42-3) are used to study and quantify only the effect of compaction (void fraction) on coupling losses (see [START_REF] Zani | Progresses at CEA on EU demo reactor cryomagnetic system design activities and associated R&D[END_REF]). Combining equation ( 115) and (118) we can see that total losses writes

with 𝑎 𝑓 being the coupling losses under transverse sinusoidal field and 𝑏 the hysteresis losses. 𝑎 is the slope at origin of 𝑄 𝑡𝑜𝑡 (𝑓) and 𝑏 its ordinate at origin.

Thus, we have in seconds:

Where we consider a unique effective time constant 𝜏 𝑒𝑓𝑓 to characterize our measurements. Several measurements in a wide range of frequency allow us to construct the 𝑄 𝑡𝑜𝑡 (𝑓) curves.

The method applied to our experimental database will be to remove analytical hysteresis losses from 𝑄 𝑡𝑜𝑡 (𝑓) (rough measurements). This first step will give us an insight on the measured value of the effective filamentary diameter 𝑑 𝑒𝑓𝑓 . And then to look at the slope at origin of the coupling losses curves to extract 𝑛𝜏 𝑒𝑓𝑓 .

The post processing method can be decomposed as follows:

-Estimate an approximate 𝑛𝜏 𝑒𝑓𝑓 named 𝑛𝜏 𝑒𝑓𝑓 𝑎𝑝𝑝𝑟𝑜𝑥 from 𝑄 𝑡𝑜𝑡 (𝑓).

-Assuming 𝑛 = 2, we compute 𝐵 𝑖 using equation ( 4) and 𝜏 𝑒𝑓𝑓 𝑎𝑝𝑝𝑟𝑜𝑥 .

-Thus, we compute 𝑄 ℎ𝑦𝑠𝑡 (𝐵 𝑖 , 𝑑 𝑓𝑖𝑙 ), adjusting 𝑑 𝑓𝑖𝑙 to obtain 𝑄 𝑐𝑜𝑢𝑝 (𝑓 = 0) = 0.

-Rescale 𝑄 𝑐𝑜𝑢𝑝 (𝑓) to the nominal value of 𝐵 𝑚 .

-Extract 𝑛𝜏 𝑒𝑓𝑓 from 𝑄 𝑐𝑜𝑢𝑝 (𝑓).

This method is applied below on all samples from Table 32.

IV.4 Experimental studies

We have presented our methodology to extract coupling losses from our rough measurement. We depict in Figure 66 the coupling losses adding to the hysteresis losses in magnetization cycle while increasing the frequency of the applied field. Our goal in this section below is to properly assess the amount of hysteresis in the measured losses in order to get clean coupling losses measurements.

The study we are going to conduct on MAG42 samples (and DP4-UP) aims at comparing the coupling losses measured in JOSEFA to the one measured in SULTAN and at the end, to be confronted with both modelling from 𝑛-stage COLISEUM and the MPAS model. Magnetization

Hysteresis only

Coupling adding

We choose to measure the contact thickness as depicted in Figure 86:

Once done, we gather results under the form of a distribution of contact depending on their length. The chosen resolution is small enough to obtain a smooth distribution but not too small to avoid noise from the statistic.

The contact distribution we found in Figure 87 is in agreement with previous work led in [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF], and at Twente University.

Its shape is different from a monotonic decreasing distribution of contact. This can be explained by the fact that our cable is compacted and thus all the small contacts being crushed create greater contact. It can be easily expected as the contact area increase with the compaction of the cable.

Another part of this work was to extract the void fraction from each slice of tomographic images. In each slice, as we shown earlier, we applied masks on strands.

Considering the fact that the cross section of a strand is only slightly deformed, we can use our mask (which are calibrated on the real size of the strand) to assess the void fraction in each slices. We take V Benchmark of the two models

V.1 COLISEUM application to experimental data

The simulation of JT-60SA TF cable (see [START_REF] Zani | Starting EU production of strand and conductor for JT-60SA TF coils[END_REF]), will use the geometrical parameters of The only free parameter in this 𝑛-stage COLISEUM are the interstage conductances. As said earlier, we choose all conductances to be either equal, increasing or decreasing in order to see how the model behave in a restraint configuration. We have presented the case of equal conductances in Figure 45 and it is clear that this equal distribution of conductances will not be for an absolute optimum of the experimental data fit. Moreover, the maximum of the curves are not the same and we have shown in section III.1.5 that multiplying the distribution of conductances with a factor 𝑞 will only "translate" the curve horizontally, which will not be appropriate to optimally fit the experimental data.

Nevertheless, even if the constant distribution of conductances is not the best fit we can already say that the predicted coupling losses using the 𝑛-stage COLISEUM are in the good order of magnitude which is an important step for a first version of a predictive model close to an ab initio approach. Also, the shape of the curves is not so far from the experimental data measured with JOSEFA.

Using an increasing distribution of conductances given in Table 37 below, we succeed at fitting the experimental data from JOSEFA and SULTAN as depicted in Figure 89 and in Figure 90 below. The methodology to determine these conductances has been to consider the ratio 𝑝 between consecutive 𝜎 is constant. Here it is equal to 𝑝 = 1.28. We thus select 𝜎 1 and the ratio 𝑝 between consecutive 𝜎 to define the whole distribution of conductances.

All the 𝜎 are defined as followed with 𝑝 as first free parameter:

𝜎 1 is our second free parameter to initiate the iterative rules (124) giving:

Advanced MPAS:

We stress out that the restricted MPAS was already developed and used before the beginning of this thesis work whereas the advanced MPAS has been established in collaboration during this thesis work using results and discussions we had on the previous thesis work (see section II.1.1 and [START_REF] Louzguiti | Magnetic screening currents and coupling losses induced in superconducting magnets for thermonuclear fusion[END_REF]).Using the iteration rules from equation ( 12) and ( 22) (the advanced MPAS), we can compute the magnetic parameters of a five-stage cable for a given multiplet (𝜏 5 , 𝜅 𝑎 , 𝜅 𝑏 ). In order to get a global view of all the possible fits, we plot again each corresponding 𝑄(𝑓) curves on top of each other as we can see in Figure 96. On top of these curves generated with the advanced MPAS, we plot the experimental data to fit as done before in Figure 91.

Here, our free parameters are τ 5 ∈ [0.1,0.4] , κ a ∈ [2.0,2.4] and κ b ∈ [0.4,0.8]. We can see immediately that our experimental curve is in the middle of the curve envelope generated with the advanced MPAS. Thus, we are in a better situation to find a curve that best fits the whole experimental data as depicted in Figure 96 and Table 41.

In Figure 97 below, the distribution of curves generated with the advanced MPAS is given in order to see how the simulated curves are located regarding the experimental data. In both models, the last stage is the most contributing to the coupling losses. Magnetic parameters are in the same range. The shielding coefficients given by COLISEUM are not strictly decreasing from the last stage to the first one as in the MPAS model. We can clearly see that the magnetic parameters of the two last stages are similar in both modelling, it is the description of the three first stages which is slightly different from one model to another.

V.2.2 Discussion and conclusion

The statistical study of the two MPAS versions helps to better understand where the initial MPAS stands regarding its free parameters and the experimental data we try to fit. On the other side, we have checked how the cluster of curves given by MPAS is changed between the restricted version and the advanced version. Adding a strong physical consideration to MPAS allows to obtain a fit of the experimental data which is clearly better than the one given by the restricted MPAS.

In the fusion community, quite all the CICC's are composed with cabling stages of different types (JT-60 SA TF, CSJA6, etc). Making MPAS evolve in this direction is thus a substantial outcome, considering that with only three free parameters we obtain satisfactory fit quality across the whole measured experimental data (both high and low frequency) for the specific case of JT-60SA TF cable.

The 2D or 3D analysis of the mean square evaluation, respectively for the restricted MPAS and the advanced MPAS, are crucial to determine if the minimum found is unique or not in the explored range of free parameters. It appears that, in both cases, the found minimum is unique. In addition, we note that the explored range of free parameters in both cases is large enough to be realistic and close to the experimental data.

The numerical tools built and applied to obtain these statistical studies are efficient and not time consuming. A rather quick comparison between the two MPAS approaches is easily made. This methodology and these tools can be applied on an AC losses database in order to quickly get an insight on the MPAS prediction regarding the version to use.

A point that has not been checked out but that can be interesting to investigate is in case a fit close to the minimum is considered, how magnetic parameters will be affected, and e.g. how the Minimum Quench Energy (MQE) computed using the 𝑄(𝑓) curves might be affected?

We remind that the only modification we have done in MPAS has been to introduce a supplementary degree of freedom (𝜅 𝑏 ) and this modification only affects the shielding coefficients, not the time constants. In fact, in this case of JT-0SA TF, it appears that the iteration rule used for the shielding coefficients is determining in order to produce an acceptable fit. Interstage conductances (resistivity) are considered constant in this modelling whereas we showed that for COLISEUM, a constant conductance among all stages is not suitable to best fit the experimental data.

Some improvements could also be made on the advanced MPAS in order to:

-Take into account the difference of elements in each stage also in the time constant computation.

-Take into account the fact that interstage conductances/resistivities can be increasing or decreasing, and not only in a linear way.

In this way, we would add more free parameters to MPAS in order to relax the model. Nevertheless, the implementation of new free parameters have to be carefully made using physical considerations to relate all the free parameters between each other's.

An outstanding result is that with the newly developed 𝑛-stage COLISEUM, we are able to produce a satisfying fit of the experimental data without difficulties. It appears that this model seems to have a good predictive aspect as it allows us to fit with realistic distribution of conductances, which should be measures in order to properly assess the predictive character of this model.

It is noticeable that in the modelling given by the advanced MPAS and the 𝑛-stage COLISEUM, the majority of the contribution to the predicted coupling losses is due to the last stage because of its huge shielding coefficient whereas all other stages have a low contribution. From stage 4 to 1 shielding coefficients are below 0.425.

coupling losses with respect to the void fraction of the sample. This homothetic tendency is also evidenced when using the 𝑛-stage COLISEUM and it is proportional to the set of conductances.

VI.4 Models benchmark on data

Both advanced MPAS and 𝑛-stage COLISEUM are confronted to the experimental data. The fit presented for the 𝑛-stage COLISEUM is proposed using a restrained configuration of conductances (slight linear increase) and no statistical study of least square evaluation. If needed, there is some room for improvement of the quality of the fit, but it is already considered well enough for applications. We have put in light that despite the constrained set of conductances, which is a strong constraint, we are able to produce an acceptable fit of the experimental data, evidencing the robustness of the model.

In addition, the descriptions given by both models (stage contributions) are confronted to assess the correspondence between the intuitive hypothesis used in the MPAS model and the strong hypothesis of first neighbour coupling used in the 𝑛-stage COLISEUM. Both models showed that the last stage is the most contributing one. It appears that the first neighbour interaction used in the 𝑛-stage COLISEUM is fair enough to represent the balance in between stages.

A fit with the six-stage COLISEUM (strand + five stage of JT-60SA TF conductor) can be easily proposed but is not presented here because we wanted to compared both model on the same basis (five-stage modelling without the strand as it cannot be integrated in MPAS using the classical iteration rules [START_REF] Aymar | Tore Supra. Basic design Tokamak system[END_REF] or [START_REF] Gibson | The JET project[END_REF].

The numerical tools developed to evaluate the magnetic parameters and perform the statistical studies of the MPAS model can be used on a database of coupling losses in order to strengthen its validity and to be more confident in its basic hypotheses. They are used in this work to assess the quality of the enhancement of the MPAS model from its streamlined version to its advanced version.

The initial reduction of dimensions of the two-stage COLISEUM has allowed to be in line with the MPAS description and to perform fully analytical calculations of the magnetic parameters. Numerical tools have also been implemented to compute the magnetic parameters given by the 𝑛-stage COLISEUM as the dimension of the system equation increase as 𝑛 and as it is not analytically solvable for 𝑛 > 3. Nevertheless, we have shown that both enhanced models (MPAS and 𝑛-stage COLISEUM) are able to give descriptions of coupling losses that are in agreement even if the initial hypotheses and development methods are different.

VI.5 Prospects

As an overall synthesis, we can say that we have established a model capable to propose a description of coupling losses for CICC.

In particular with this cable modelling we have succeeded in introducing the strand description as a coupled stage.

The establishment of this 𝑛-stage COLISEUM opens towards many possible improvements:

➢ Conducting parametric study of the models to better understand their variations and limits. ➢ Conducting experimental measurements of coupling losses of various CICC geometries to confront the advanced MPAS and the 𝑛-stage COLISEUM. ➢ Confronting the advanced MPAS and the 𝑛-stage COLISEUM to an existing coupling losses database. ➢ Continuing the benchmark of the 𝑛-stage COLISEUM with JackPot for the model validations. ➢ Performing measurements of inter-stage conductances as entry for COLISEUM in order to evaluate its predictive capability. ➢ Continuing the tomographic study in order to find a way to get the inter-stages conductances from tomographic images and the contact network with direct electrical measurements. More accurate geometrical parameters could also be extracted from the tomography and used as input for the 𝑛-stage COLISEUM. ➢ Trying to relax the tangency condition hypothesis to reach cabling radii close to the ones of compacted cables. ➢ Starting conductor stability studies since the 𝑛-stage COLISEUM deals with all the needed elements: currents, losses, etc.

Finally, we can say that we have contributed, at our level, to enhance the description of coupling losses in superconducting cable in conduit conductors by using an analytical formulation. The 𝑛-stage COLISEUM is thus ready to be used on various cable geometries and in different experiments of coupling losses. We hope this work could help their design or their use in the future fusion reactors.