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

A magnetic levitation technique with high-
Tc
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
E
-
J
characteristic in the reported calculation, we introduce an elliptical model to formulate the angular dependence of the critical current density
Jc
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.