Magnetization Distribution in a Superconducting Bulk
by means of an Equivalent Circuit Model
based on a Hysteretic Material Characteristic

Bulk-type YBCO superconductors prepared using melt processes show high critical current density value and are then promising for many kinds of applications, such as flywheels, motors, superconducting bearings and current limiters. However, a temporal decay of the magnetization is expected, and a periodic magnetization of the bulk is then needed. When superconducting bulks are magnetized using the Field Cooling (FC) procedure under a DC magnetic field, a large electromagnet system or a superconducting coil is needed. In contrast, the Pulsed Field Magnetization (PFM) method can be applied using small coils and would be the most suitable way for an in situ magnetization. Anyway, the experiments carried out have shown that the PFM process results in a low trapped flux in the core of the ring.

In [1] the current density inside the superconductor was evaluated utilizing a power-law model to relate the electric field to the current density. This model gave a good approximation of the trapped flux distribution in the central part of the ring bulk, while above the superconductive region the field distribution was not correctly reconstructed. On the other hand, the model approximated well the experimental results obtained with the FC process. The inconsistency between the experimental and the calculated results for the trapped flux profile in the PFM process was ascribed to the power-law characteristic used in [1], which may not be sufficient to characterize the behavior of the bulk superconductor if the transient involved is too fast.

PROBLEM FORMULATION

In order to improve the SC modelization a model based on an equivalent circuit network is derived [2, 3]. Starting from a 3D finite element discretization of the SC volume, the MQS form of the Maxwell equations is stated. By introducing the magnetic scalar potential y the following equation is derived:

(1)

where Hext is the external magnetic field. Equation (1) can be integrated between the centers of two neighboring elements leading to an equation that can be interpreted as the magnetic voltage balance equation of the branch of a (magnetic) circuit containing a known magnetomotive force generator, a

linear and a non-linear reluctance term and the difference of the scalar magnetic potentials in the nodes that are the centers of the elements. Moreover, assuming the magnetic flux density B to be a constant vector inside any element, its value can be linearly linked to the fluxes through the faces by means of a least sqare approach [3]. Therefore, the circuit variables are the magnetic fluxes through the element faces and the magnetic scalar potentials in the element centers.

The model takes into account the properties of the SC bulk by means of a hysteretic characteristic between magnetization and flux density field based on a vectorial Duhem model [4], expressed as follows:

(2)

This model gives a good agreement with experimental data [5] and allows the SC material to be characterized by few and simple-to-identify parameters. Fig. 1 shows the one-dimensional behavior when the external field is cycled with slowly increasing amplitude.

Fig.1 – Hysteresis curves of m0 M as functions of m0 H.

[1]  M. Fabbri, F. Negrini, P.G. Albano, M. Pretelli, H. Ohsaki, “Flux Trapping in a Ring-shaped YBCO Bulk by Pulsed Field Magnetization,” IEEE Trans. on Appl. Supercond., vol. 11, no. 4, pp. -, December 2001.

[2]  E. Cardelli, “3-D Circuital Approach for the Analysis of the Electromagnetic Transient Diffusion of Heat and Current in Conductive Bodies,” IEEE Trans. on Magn., vol. 32, no. 3, pp. 1034-1037, May 1996.

[3]  A. Morandi, A. Cristofolini, M. Fabbri, F. Negrini, P.L. Ribani, “Current distribution in a composite superconducting system by means of an equivalent circuit model based on a smooth E-J characteristic,” Physica C, Vol. 372-376, pp. 1771-1776, 2002.

[4]  A. Visintin, Differential Models of Hysteresis, Springer, Berlin, 1994

[5]  I.D. Mayergoyz, A.A. Adly, M.W. Huang, C. Kraft, “Experimental Testing of Vector Preisach Models for Superconducting Hysteresis,” IEEE Trans. on Magn., vol. 36, no. 5, pp. 2505-3507, September 2000.