The hysteresis of magnetization in the generalized critical-state model Low-field approximation

The Bean Model in Superconductivity: Variational Formulation and Numerical Solution

“…If, as was assumed by Bean, the critical current density curl ϭ , div ϭ 0, does not depend on the magnetic field, the set of admissible functions is fixed: K (h) ϵ K. The inequality (15) becomes ͉͉ Ǟ 0 as ͉x͉ Ǟ ȍ. a variational inequality:…”

“…The same numerical algorithm can be applied everywhere, e.g., in the critical and subcritical regions h ʦ K (h), (15) of the superconductor, and the front-tracking becomes (Ѩ͕h ϩ H e ͖/Ѩt, Ϫ h) Ն 0 for any ʦ K (h), unnecessary (see [23]). However, the solution of (16) must be calculated in an infinite domain, and this is an h͉ tϭ0 ϭ h 0 , additional complication.…”

“…current density depends on the magnetic field and various Maxwell equations with the displacement current omitrelations of the type J c ϭ J c (͉H͉) have been proposed (see, ted read e.g., [14,15]). The constraint on the current density may be written as…”

Exponential critical state model applied to ac susceptibility data for the superconductor YBa2Cu3O7−δ

“…The concept of critical state model was introduced by Bean 10,11 and London 12 in a successful effort to derive magnetic properties of hard type-II superconductors. Different critical state models, 8,[13][14][15] which assume that supercurrents flow inside the sample with a critical density J c (H i ), where H i is the internal magnetic field, have been used to study the response of type-II superconductors to an external field H. [16][17][18] In particular Chen et al 17 have shown that the exponential law, J c (H i ,T)ϭJ c (T)exp(ϪH i /H 0 ), originally introduced by Fietz et al, 14 is very useful to interpret ac susceptibility data of high-T c superconductors. Following, we show the main steps of the calculation which is allowed to express the complex susceptibility, ϭЈϩiЉ, in terms of the exponential critical state model.…”