Efficient numerical models are required for the design of systems with high temperature superconductor (HTS) coils, as fully resolved finite element simulations of individual coated conductors become computationally prohibitive. This work applies the foil conductor model (FCM) to insulated HTS coils using magnetic field conforming h-(full), h-ϕ, and t-ω formulations. The approach replaces individual turns by a homogenized bulk and ensures physically consistent current density distributions in the coils by using additional voltage basis functions in the finite element formulations. The models are verified in 2D axisymmetric and 3D geometries with a pancake coil simulation under AC transport current excitation. All FCM formulations show excellent agreement with reference detailed simulations, with coefficients of determination above 0.99 for instantaneous AC losses. In 3D, the h-ϕ and especially the t-ω formulation substantially reduce the number of degrees of freedom by using the magnetic scalar potential in non-conducting regions. Scalability is demonstrated with a 3D stack of racetrack coils model with a field- and angle-dependent critical current density. For the stack of racetrack coils, while maintaining accurate loss prediction, the t-ω FCM achieves a speedup factor of 22 and reduces degrees of freedom by 78

No-insulation (NI) and metal-insulation (MI) high-temperature superconducting (HTS) magnets require three-dimensional (3D) models to describe the current distribution around critical current defects. In this work, we design and validate the EXTRA homogenisation method, standing for explicit turn resolution with anisotropic homogenisation method. It allows 3D magneto-thermal finite-element (FE) simulations of large-scale magnets to be performed with high accuracy at a reasonable computational cost. The method combines the anisotropic homogenisation of turn-to-turn contact layers (T2TCLs) and their neighbouring winding turns with the explicit resolution of specific T2TCLs. In particular, the inner- and outermost winding turns and adjacent contact layers are explicitly resolved to properly describe the current distribution near current leads. In addition, the method is able to simulate local J_c defects for a broad range of turn-to-turn contact resistances, provided the winding turns and T2TCLs next to the defect are explicitly resolved. For efficiency, the resolved T2TCLs are modelled using the surface contact approximation. The consistency of the proposed method is first verified on a 50-turn single pancake benchmark. It is shown to reproduce AC losses and temperature distributions obtained with a turn-resolved FE reference model, for both nominal operation and during thermal runaway. The computational efficiency of the EXTRA method is demonstrated with the simulation of a stack of three 150-turn pancake coils, for which computation time is reduced by a factor of up to 13 with respect to a turn-resolved FE reference model. Finally, the results of a large-scale 3D FE simulation, currently out of reach of turn-resolved models, are provided for an insert HTS magnet with 10,000 turns. The EXTRA method is open-source and input files to reproduce all results are made available.

Conductor-on-round-tube (CORT) cables are a potential solution for carrying AC power in a small cross-section. Due to the geometry of the cable and the helical arrangement of the coated conductors (CC), the current follows a non-trivial pattern inside each CC. For instance, for the case of a single-layer cable, the current flow is mostly axial along the outer face of the CCs and mostly azimuthal along their inner face. Such a current distribution, known as the Garber current pattern, affects the transport AC losses. In numerical models, commonly adopted simplifications are either based on straight conductors or infinitely thin CCs. Such approaches neglect the Garber current pattern and thus misrepresent both the detailed current flow within the CC and the resulting 3D distribution of the fields. In this work, the detailed 3D current distribution in the CCs is investigated in a one-layer CORT cable, as a function of the cable geometrical parameters such as the conductor thickness, the pitch angle, and the gap between adjacent CCs. In particular, the impact of the Garber current pattern is studied on the two largest contributions to the AC losses, namely the surface losses (associated with the penetration of the component of the magnetic field parallel to the wide faces of the superconducting layer) and the edge losses (associated with the penetration of the perpendicular component of the magnetic field occurring in the vicinity of the gaps between the CCs). The detailed distribution of the currents in the CCs is examined and its relationship with the different AC loss mechanisms is established. This study is carried out by means of an effective 2D model that uses a system of coordinates conforming with the helical structure of the cable.

The simulation of large-scale high-temperature superconducting (HTS) magnets is a computational challenge due to the multiple spatial scales involved, from the magnet to the detailed turn-to-turn geometry. To reduce the computational cost associated with finite-element (FE) simulations of insulated HTS coils, the simultaneous multi-scale homogeneous (SMSH) method can be considered. It combines a macroscopic-scale homogenized magnet model with multiple single-tape models and solves both scales monolithically. In this work, the SMSH method is reformulated using the h-ϕ thin-shell (TS) approximation, where analyzed tapes are collapsed into thin surfaces, simplifying mesh generation. Moreover, the magnetic field is expressed as the gradient of the magnetic scalar potential outside the analyzed tapes. The discretized field is then described with nodal functions, further reducing the size of the FE problem compared to standard h formulations. The proposed h-ϕ SMSH-TS method is verified against state-of-the-art homogenization methods on a 2-D benchmark problem of stacks of HTS tapes. The results show good agreement in terms of AC losses, turn voltage and local current density, with a significant reduction in simulation time compared to reference models. All models are open-source.

While the prediction of AC losses during transients is critical for designing large-scale low-temperature superconducting (LTS) magnets, brute-force finite-element (FE) simulation of their detailed geometry down to the length scale of the conductors is a computational challenge. Multi-scale methods, balancing between a coarse approximation of the fields at the scale of the magnet and a detailed description at the scale of the conductors, are promising approaches to reduce the computational load while keeping a sufficient accuracy. In this work, we introduce a neural network approach to accelerate multi-scale magneto-thermal simulations of LTS magnets by replacing costly single-turn FE models with neural network surrogates. The neural network architecture is presented and discussed, together with an automated procedure for generating simulation data for its training. The resulting accelerated multi-scale model is used to simulate current ramp-up procedures for the IBA S2C2 magnet. The surrogate-based multi-scale model is compared with a conventional multi-scale model based on a composite wire-in-channel FE model. The surrogate model is shown to reproduce single-turn filament hysteresis, inter-filament coupling, and eddy losses, while the computational time of the multi-scale method is reduced by a factor of 800.

Many nanoscale magnetic imaging techniques are constrained by the maximum magnetic field that can be applied during measurements, due to geometrical limitations or interactions with the probe or the detected signal (e.g., electrons). Here, it is demonstrated that sample-integrated metamaterial-inspired magnetic flux concentrators (MFCs) locally amplify magnetic fields, allowing observation of magnetization processes beyond instrumental limits. Micrometer-sized MFCs fabricated directly on the samples are tested in photoemission electron microscopy experiments employing X-ray magnetic circular dichroism as magnetic contrast mechanism. At low applied fields, substantial amplification factors enable observation of magnetization reversal in a chain of magnetite nanoparticles synthesized by magnetotactic bacteria at an applied field of 8 mT, substantially smaller than the ∼50 mT predicted by simulations in the absence of MFCs. At higher fields, the field enhancement extends the accessible field range by a factor of five, enabling for the first time, imaging of the field-dependent magnetic domain structure evolution of an isolated giant magnetofossil. Finally, we show how MFC geometry and material parameters can be tuned to optimize performance considering sample and experimental constraints, providing a tunable and broadly applicable strategy for extending the accessible field range in a wide variety of nanoscale magnetic imaging techniques.

We extend the foil winding homogenization method to magnetic field conforming formulations. We first propose a full magnetic field foil winding formulation by analogy with magnetic flux density conforming formulations. We then introduce the magnetic scalar potential in non-conducting regions to improve the efficiency of the model. This leads to a significant reduction in the number of degrees of freedom, particularly in 3-D applications. The proposed models are verified on two frequency-domain benchmark problems: a 2-D axisymmetric problem and a 3-D problem. They reproduce results obtained with magnetic flux density conforming formulations and with resolved conductor models that explicitly discretize all turns. Moreover, the models are applied in the transient simulation of a high-temperature superconducting coil. In all investigated configurations, the proposed models provide reliable results while considerably reducing the size of the numerical problem to be solved.

Defects in superconducting systems are ubiquitous and nearly unavoidable. They can vary in nature, geometry, and size, ranging from microscopic-size defects such as dislocations, grain boundaries, twin planes, and oxygen vacancies, to macroscopic-size defects such as segregations, indentations, contamination, cracks, or voids. Irrespective of their type, defects perturb the otherwise laminar flow of electric current, forcing it to deviate from its path. In the best-case scenario, the associated perturbation can be damped within a distance of the order of the size of the defect if the rigidity of the superconducting state, characterized by the creep exponent n, is low. In most cases, however, this perturbation spans macroscopic distances covering the entire superconducting sample and thus dramatically influences the response of the system. In this work, we review the current state of theoretical understanding and experimental evidence on the modification of magnetic flux patterns in superconductors by border defects, including the influence of their geometry, temperature, and applied magnetic field. We scrutinize and contrast the picture emerging from a continuous media standpoint, i.e. ignoring the granularity imposed by the vortex quantization, with that provided by a phenomenological approach dictated by the vortex dynamics. In addition, we discuss the influence of border indentations on the nucleation of thermomagnetic instabilities. Assessing the impact of surface and border defects is of utmost importance for all superconducting technologies, including superconducting resonators, superconducting single-photon detectors, superconducting radio-frequency cavities and accelerators, superconducting cables, superconducting metamaterials, superconducting diodes, and many others.

In this article, we introduce two complementary approaches for the prediction of ac losses in large-scale low-temperature superconducting (LTS) magnets subjected to slow ramp rates. These methods account for the temperature rise within the LTS coil and its impact on ac losses. The first approach is multiscale and relies on the coupling between a macroscopic homogenized model of the LTS coil and a mesoscopic model of a single filament for loss prediction. The second approach is semianalytical and is based on analytical approximations for the hysteresis losses, which are validated against the single filament model. The second approach offers a faster computation suitable for initial design considerations, while the multiscale method is shown to take into account more complex phenomena for the ac loss evaluation at the filament scale. We apply both methods to the prediction of ac losses generated in the LTS coil inside the IBA S2C2 synchrocyclotron during its ramp-up procedure. Additionally, we discuss the convergence properties of the multiscale approach and demonstrate the good agreement between the numerical results and experimental data.

Magnetic shielding is a crucial aspect of many electronic devices and sensors. In this article, we investigate the magnetic shielding properties of ferromagnetic washers in a planar geometry in view of integration with on-chip devices, focusing on the transverse configuration, i.e., with the applied field parallel to the plane of the washer. We show that, in this configuration, the shielding factor of a washer can exceed the predictions of previous studies. We argue that the magnetic shielding results from the capture of the external magnetic flux by the upper and lower faces of the washer, and its subsequent channeling through the washer material and around the central hole. We present experimental results and numerical simulations and discuss the influence of the geometry of the washer, the permeability, and the saturation of the ferromagnetic material. We propose an explanation for the observed behavior and provide empirical formulas to estimate different quantities characterizing the shielding properties of washers in the transverse configuration.

Stacks of high-temperature superconducting tape annuli can be used as magnetic shields operating efficiently for both axial and transverse fields. However, due to their layered geometry and hybrid electrical and magnetic properties, implementing models of such structures is not straightforward. In this work, we propose two different modelling approaches with the finite element method: layered and homogenized. We compare their accuracy and numerical efficiency for three different formulations (h-phi, h-phi-b, and a-j), in both axial (2D-axisymmetric) and transverse (3D) configurations. We show that both approaches lead to comparable performance in the axial case, but that the homogenized model is considerably harder to use in the transverse case.

This paper deals with the modelling of superconducting and resistive wires with a helicoidal symmetry, subjected to an external field and a transport current. Helicoidal structures are three-dimensional, and therefore yield computationally intensive simulations in a Cartesian coordinate system. We show in this paper that by working instead with a helicoidal system of coordinates, the problem to solve can be made two-dimensional, drastically reducing the computational cost. We first introduce the state-of-the-art approach and apply it on the h-phi-formulation with helicoidally symmetric boundary conditions (e.g., axial external magnetic field, with or without transport current), with an emphasis on the function space discretization. Then, we extend the approach to general boundary conditions (e.g., transverse external magnetic field) and present numerical results with linear materials. In particular, we discuss the frequency-dependent losses in composite wires made of superconducting filaments embedded in a resistive matrix. Finally, we provide outlook to the application of the generalized model with nonlinear materials.

High-temperature superconducting coils are used in various large-scale applications, like rotating machines and high-field magnets. However, modeling these coils is a complicated and time-consuming process, especially due to the non-linearity of the current-voltage characteristics of the superconductors and the complex multiphysics involved. In this work, we used a fast homogenized method to model the coupled electromagnetic and electrothermal properties of racetrack and pancake coils for different applications. For this purpose, various formulations wielding homogenization methods are used and benchmarked with each other, as well as with models considering the detailed structure of the HTS tapes. We observe a very good agreement between different models (homogenized and detailed), and we discuss the pros and cons of the inclusion of insulating layers between the turns in homogenization. This work was performed under the collaboration of the COST action modeling teams and can be used as a review of the state-of-the-art superconductor modeling techniques, and a source for the development and benchmark of future numerical methods.

The ability of bulk high-temperature superconductors to trap magnetic flux densities up to one order of magnitude larger than the saturation magnetization of conventional ferromagnetic materials offers the prospect of generating large magnetic flux density gradients. Combining multiple superconductors, akin to assembling a Halbach array of permanent magnets, may increase the generated gradient even further. The associated challenge is that superconductors are prone to demagnetization when exposed to field components perpendicular to their main magnetization direction. In the present work, we investigate the magnetic flux density gradient achieved with a pair of cubic, bulk, large-grain melt-textured superconductors in the presence of a background DC magnetic field at 77 K. We investigate the increase of the performance when decreasing the temperature down to 59 K. The studied configuration consists in two facing cubic YBa2Cu3O 7-x superconductors of 6 mm side with anti-parallel magnetization directions. It is obtained after the simultaneous magnetization of the samples followed by a rotation of 180 degrees of the top superconductor. Although the background field reduces the trapped field ability of individual samples, it is shown that this phenomenon is significantly mitigated at 65 K and at 59 K compared to 77 K. The results reveal that a sample-to-sample distance ( similar to 16 mm) of the order of their size is sufficient to avoid any mutual demagnetization effect during the rotational motion. Furthermore, it is shown that decreasing the temperature is not only beneficial in increasing the field and field gradient achieved but also in extending the range of background fields in which the superconductor can be rotated without demagnetization. This superconducting assembly yields a magnetic flux density gradient exceeding that of an isolated superconductor and has the potential to surpass the capabilities of permanent magnets.

For finite element (FE) analysis of no-insulation (NI) high-temperature superconducting (HTS) pancake coils, the high aspect ratio of the turn-to-turn contact layer (T2TCL) leads to meshing difficulties which result in either poor quality mesh elements resulting in a decrease of the solution accuracy or a high number of degrees of freedom. We proposed to mitigate this issue by collapsing the T2TCL volume into a surface and using a so-called thin shell approximation (TSA). Previously, two TSA have been introduced, one to solve the heat equation and the other for an $\vec{H}-\phi$ magnetodynamic formulation. In this work, we propose to combine the magnetodynamic and thermal TSA to create a coupled magneto-thermal TSA for three-dimensional FE analysis. Particular attention is paid to the detailed derivation of the coupling terms. In the context of NI HTS pancake coils, the TSA represents the electric and thermal contact resistance of the T2TCL. For the HTS coated conductor (CC) itself, an anisotropic homogenization is used which represents its multi-layered structure. In axial and azimuthal direction, it resolves the current sharing between the HTS and other layers of the CC. The coupled TSA formulation is verified against a reference model with volumetric T2TCL. The coupled TSA is shown to significantly reduce the solution time as well as the manual effort required for high-quality meshes of the T2TCL. The implementation is open-source and a reference implementation is made publicly available.

In this work we demonstrate the magnetic shielding ability of a stack of YBa2Cu3O7 tape annuli. The annuli are cut from a 46 mm wide second generation coated conductor deposited on a Ni-5at.%W alloy ferromagnetic (FM) substrate. The inner bore of the stacked tapes is 26 mm and the outer diameter is 45 mm. Three samples with different height (24 mm, 14.9 mm, 9.9 mm) are studied. All the experiments are carried out at both room temperature and liquid nitrogen temperature (77 K). The shielding efficiency is investigated when the magnetic field is applied either parallel to the axis of the stack (axial shielding) or perpendicular to it (transverse shielding). Under an axial field, magnetic shielding is found to be effective (SF > 3) up to magnetic flux densities of 0.67 T. The presence of the FM substrates is found to have two important consequences. First, the stack of annuli is able to shield transverse flux densities in spite of its layered structure. Second, a finite magnetic shielding effectiveness is demonstrated at room temperature. In order to understand the contribution of the FM substrates to the shielding mechanism, we use the experimental field dependence of the magnetic permeability as determined independently from hysteresis loop measurements on the same substrates. A finite-element homogenized model solved with an H-phi formulation is shown to successfully reproduce the shielding factor of the stack at room temperature and 77 K, both under axial and transverse applied fields. These models are also used to assess the influence of the critical current density and the magnetic permeability on the shielding efficiency. Finally, the results are used to predict the magnetic shielding properties of higher stacks, demonstrating their significant potential to shield axial fields of similar to 0.93 T (with SF > 10) at 77 K.

Assembling trapped-field superconducting magnets with mutually orthogonal magnetizations directions in a Halbach array configuration offers the prospect of generating both high fields and large field gradients. A major issue when assembling bulk superconductors in a Halbach array, however, consists in the alteration of the initial current density distribution during the assembly process. This reorganization of supercurrent loops limits the field generated by the system. We investigate two methods for reducing this demagnetization effect. The first method consists of using stacked tapes instead of bulk superconductors. For the second method, we propose a procedure leading to a re-magnetizing the superconductors of the array after the assembly. The procedure consists in putting two superconductors on top of one another, magnetizing them along the vertical direction, and then keeping the pair in place while two other superconductors, magnetized in an horizontal direction, are approached from left and right. The top central sample is then removed from the array, thereby providing the desired re-magnetization of the bottom one. The benefits of this procedure was investigated by finite element modelling and experiments carried out at 77 K both with bulk YBa2Cu3O 7-x superconductors ( similar to 14x14x14 mm3) and with stacks of second generation YBa2Cu3O 7-x tapes from Superpower ( similar to 12x12x12 mm3). The flux density measured above the array is compared to analytical results and finite element simulations. The results show that a re-magnetization of the central sample occurs, which allows the maximum field generated with Halbach arrays made of three bulk superconductors or three stacked tapes to be increased by 5% and 11% respectively. Numerical modelling shows that using a taller top sample with this method allows to recover almost the full potential of the array.

The development of coated conductors with a ferromagnetic substrate and their use in various applications require accurate knowledge of the magnetic properties of the substrate. In this work, we report measurements in the ac regime (30 Hz) of the magnetic hysteresis loops of the Ni-5at.%W alloy ferromagnetic substrate in a stack of second-generation YBa 2 Cu 3 O 7 (YBCO) tapes cut in the shape of annuli from 46-mm-wide-coated conductors. The stacked annuli form a closed magnetic circuit, which is the ideal configuration due to the absence of demagnetizing effect. The measurements are carried out at four temperatures between 77 K and 293 K. The peak intensity of the relative permeability is not significantly affected in the range of temperature considered. The coercive field dependence on temperature is also investigated. The hysteresis losses $Q$ as a function of the amplitude of the flux density $B_\text{m}$ exhibit a power law behavior ( $Q\sim B_\text{m}^{n}$ ) at the four temperatures. This power law behavior is also studied at low fields under the presence of a dc bias field. Finally, the hysteresis loops can be reproduced with the Jiles–Atherton model, which is used to estimate the hysteresis parameters and to discuss their temperature dependence.

We discuss the relevance of several finite-element formulations for nonlinear systems containing high-temperature superconductors (HTS) and ferromagnetic materials (FM), in the context of a 3D motor pole model. The formulations are evaluated in terms of their numerical robustness and efficiency. We propose a coupled h-phi-a-formulation as an optimal choice, modeling the problem with an a-formulation in the FM and an h-phi-formulation in the remaining domains. While maintaining a low number of degrees of freedom, the h-phi-a-formulation guarantees a robust resolution and strongly reduces the number of iterations required for handling the nonlinearities of HTS and FM compared to standard formulations.

In applications requiring a large magnetic force, permanent magnets with non-parallel magnetization directions can be assembled in a Halbach array to generate a large gradient of magnetic flux density. The saturation magnetization of permanent magnets, however, brings a fundamental limit on the performance of this configuration. In the present work, we investigate experimentally the assembly of cuboid bulk, large grain melt-textured YBa 2 Cu 3 O 7 − x superconductors ( ∼ 14 × 14 × 14 mm 3 ) with orthogonal c -axes so as to form a basic unit of Halbach array. The experiments are carried out at 77 K. The experimental distribution of the magnetic flux density above the array of trapped-field superconductors is compared to a similar array made of permanent magnets. A simple analytical model is developed and is shown to accurately reproduce the main experimental observations. The results suggest that a redistribution occurs in the current flowing in the central sample when the distance between the superconductors is reduced, whereas the neighbouring superconductors are unaffected. It is shown that this current redistribution yields a reduced contribution of the central sample to the magnetic flux density above the centre of the array and a new negative contribution associated with stray fields to the magnetic flux density at this location. This interpretation is confirmed by modelling of the distribution of transport currents in the superconductor using a 3D finite element model.