3-D Modeling of High-$T_{c}$ Superconductor for Magnetic Levitation/Suspension Application—Part I: Introduction to the Method

Abstract

A magnetic levitation technique with high- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Tc</i> superconductor (HTS) has received significant interest for a wide range of applications after its discovery due to its unique inherent stability, which gives a fundamental significance to evaluate the HTS magnetic levitation in both experiment and calculation. To numerically investigate the HTS magnetic levitation, a 3-D model describing the electromagnetic property of the HTS, including its anisotropic behavior, was established by incorporating the current vector potential and Helmholtz's theorem. In addition to the commonly considered nonlinear <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> - <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</i> characteristic in the reported calculation, we introduce an elliptical model to formulate the angular dependence of the critical current density <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Jc</i> resulting from the anisotropic behavior of the HTS. To numerically resolve the governing equations of the 3-D model, Galerkin's finite-element method and the Crank-Nicolson-θ method were employed to discretize the governing equations in space and time domains, respectively. The obtained algebraic equations were firstly linearized by the Newton-Raphson method, and then an extended format of the incomplete Cholesky-conjugate gradient method was applied to solve the linear algebraic equations. The 3-D model was implemented by a self-written numerical program based on a VC++ platform to calculate the magnetic force of a bulk HTS exposed to applied field generated by a permanent magnet guideway (PMG) assembled by the Nd-Fe-B magnets. In this paper, we present the numerical results of the levitation force of a moving bulk HTS above the PMG with different mesh densities and number of time steps. This presents a preliminary validation of the 3-D model proposed in this paper.