Critical State Theory for the Magnetic Coupling between Soft Ferromagnetic Materials and Type-II Superconductors

Abstract

Improving our understanding of the physical coupling between type-II superconductors (SC) and soft ferromagnetic materials (SFM) is the root for progressing to the application of SC-SFM metastructures in scenarios such as magnetic cloaking, magnetic shielding, and power transmission systems. However, in the latter, some intriguing and yet unexplained phenomena occurred, such as a noticeable rise in the SC energy losses, and a local but not isotropic deformation of its magnetic flux density. These phenomena, which are in apparent contradiction with the most fundamental theory of electromagnetism for superconductivity, that is, the critical state theory (CST), have remained unexplained for about 20 years, given the acceptance of the controversial and yet paradigmatic existence of the so-called overcritical current densities. Therefore, aiming to resolve these long-standing problems, we extended the CST by incorporating a semi-analytical model for cylindrical monocore SC-SFM heterostructures, setting the standards for its validation with a variational approach of multipole functionals for the magnetic coupling between Sc and SFM materials. It is accompanied by a comprehensive numerical study for SFM sheaths of arbitrary dimensions and magnetic relative permeabilities μr, ranging from μr=5 (NiZn ferrites) to μr = 350,000 (pure Iron), showing how the AC-losses of the SC-SFM metastructure radically changes as a function of the SC and the SFM radius for μr≥100. Our numerical technique and simulations also revealed a good qualitative agreement with the magneto optical imaging observations that were questioning the CST validness, proving therefore that the reported phenomena for self-field SC-SFM heterostructures can be understood without including the ansatz of overcritical currents.

Maximum reduction of energy losses in multicore MgB2 wires by metastructured soft-ferromagnetic coatings

“…Based on our recent findings on the actual magnetostatic coupling between type-II SC wires and SFM materials 37 , in this section we explain not only the physical reasons behind the reported increment on the AC losses above mentioned, but also provide a solution pathway for achieving the maximum reduction of energy losses feasible in multifilamentary rounded MgB wires. For doing so, in Fig.…”

“…This is no other than the impact of the magnetostatic coupling between the SC and the SFM sheath shown by Fareed and Ruiz in monocore-wires 37 , whom have demonstrated that no matter the SFM used nor its dimensions, for , the physical coupling between the SC and the SFM leads to a large increment on the electric field outside the SC\SFM metastructure 35 , but not to a further increment in the density of power losses inside the SC. Moreover, this coupling gets dominated by the factor, where , being the SFM-sheath outer radius and its inner radius (outer radius of the Nb(Ti)/MgB filament in our case).…”

“…Furthermore, it is worth reminding that in conventional power systems encountering excitation frequencies greater than 50–60 Hz is uncommon, but more importantly is to remind that at least within the so-called quasi-steady low frequency regime (below radio frequencies 20 kHz), the superconductor and the ferromagnetic material can be modelled within the framework of the critical state theory with relatively no differences in he hysteretic losses 37 , 92 – 94 . Therefore, not only the standard H-formulation can still be used for determining the distribution of current density inside the superconducting filaments, but the eddy currents to be produced by the SFM and RE layers are not expected to overshadow the hysteretic losses produced by the superconductor, as long as this study is considered within the low frequency regime.…”

“…The trends found with the Nb or the NbTi barrier cases are the same, reason why we call it categorically as Nb(Ti). Also, the selection of this is not arbitrary, as a has serious repercussions on the total AC-losses of the MgB wire 37 . These aspects will be revised later, but for now it suffices to mention that a SFM with has been chosen, as it has been previously used for fabricating MgB monocore wires 95 – 98 .…”

“…Therefore, to enable MgB wires tap into the market of AC applications, it results imperative to find meaningful ways to reduce their AC-losses without a detriment of its critical current density, . Thus, in this paper we present a novel optimization method applied to realistic designs for multifilamentary MgB wires, which aided by the magnetic diverter principle of soft ferromagnetic coatings over superconducting wires 26 – 37 , can render to the maximum reduction of energy losses than can be achieved by first-physical principles. In this sense, we provide a detailed route for the adequate selection of materials, physical, and engineering dimensions, which in an optimal combination can lead to the fabrication of cost-effective metastructured multifilamentary MgB wires for high power density AC applications, that are nowadays accessible only by the use of much more expensive rare-earth coated conductors.…”