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Hybrid analytical and integral methods for simulating HTS materials under various applied magnetic field configurations

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Abstract

High Temperature Superconductors (HTS) are promising for applications requiring high power densities, such as superconducting electrical motors. Various approaches have been developed to model HTS, in particular for AC losses evaluation in thin wires and tapes. Indeed, AC losses are one of the key factors to size properly the cryogenic systems. In some applications, where the HTS materials are used as magnetic screens or as permanent magnets, such as in electrical motors, it is relevant to estimate properly the penetration of the magnetic field in order to optimize the magnetization processes and the integration of these materials in such systems. In previous works [1]–[3], analytical tools in 2D have been successfully developed for calculating the magnetic field distribution in different devices integrating HTS bulks by considering them as perfect diamagnetic materials, e.g. a superconducting electrical machine or an inductor with an iron core used for the pulsed field magnetization of a bulk HTS. These methods provide continuous derivatives and are useful tools for the design and optimization of such systems. They lead to meaningful solutions with helpful physical insights. However, they are limited to simple geometries and do not take into account either eddy currents or the variation of the critical current density with the magnetic field, which is crucial in HTS. On the other hand, rapid modeling approaches are needed, and the use of classical numerical tools is often ineffective in the design and optimization process due to considerable calculation time. Specific numerical approaches ensuring a better compromise between precision and calculation time are still required. In this context, the present work presents a hybrid model in which the HTS behavior is represented by the power law E(J, B) = Ec  (J /Jc (B)) n(B), taking into account the variation of the critical current density and the power exponent with respect to the magnetic flux density. The magnetic vector potential, the magnetic field and the current density distributions are computed by means of analytical and integral equations implemented on MATLAB. The resolution of the obtained stiff ordinary differential equations are performed using available solvers in MATLAB with adaptive time steps. The integral equations used to calculate the induced currents in the HTS materials are based on the well-known “Brandt method” developed in 1996 for strips and slabs, and later for disks and cylinders in an axial magnetic field. One of the main advantage of the hybrid method proposed here is that only the active parts are discretized. Distributions of the magnetic field over time and other quantities such as losses are presented. Calculation times for the studied problems are given and compared with those obtained from classical FEM software. [1]K. Berger, R.-A. Linares-Lamus, T. Lubin, M. Hinaje, and J. Lévêque, ‘Expression analytique de l’inductance mutuelle entre deux bobines circulaires possédant un noyau supraconducteur ou ferromagnétique’, in 8ème Conférence Européenne sur les Méthodes Numériques en Electromagnétisme (NUMELEC 2015), Saint-Nazaire, France, 2015, p. 25, id. 79. https://hal.archives-ouvertes.fr/hal-01159268 [2]G. Malé, T. Lubin, S. Mezani, and J. Lévêque, ‘Analytical calculation of the flux density distribution in a superconducting reluctance machine with HTS bulks rotor’, Math. Comput. Simul., vol. 90, pp. 230–243, Apr. 2013. https://doi.org/10.1016/j.matcom.2013.01.003 [3]G. Malé, T. Lubin, S. Mezani, and J. Lévêque, ‘2D analytical modeling of a wholly superconducting synchronous reluctance motor’, Supercond. Sci. Technol., vol. 24, no. 3, p. 035014, 2011. http://dx.doi.org/10.1088/0953-2048/24/3/035014
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... As a result, for problems involving HTS magnetization as in [9], fully analytical solutions may not be adequate and hybrid models may be required [16]. In [17] a hybrid combining analytical and integral methods were developed for the simulation of the magnetization of HTS bulks inside magnetic circuits, where the HTS was solved numerically and the remaining problem using analytical solutions. Results show that it is possible to obtain a good approximation of the final solution with lower times than using a fully numerical approach. ...
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