Magnetic forces of a cylinder shape bulk high-temperature superconductor (HTS) over a permanent magnet guideway (PMG) are studied mathematically. One cylindrical bulk HTS with a diameter of 30 mm and 15 mm in height is used. Two types of PMG are employed for external magnetic fields consideration. The relationship of magnetic forces of bulk HTS under different lateral offsets over PMG is studied with 3D-model finite element method (FEM). The calculation results show that the maximum magnetic levitation force of bulk HTS over PMG is tightly related to the applied magnetic field distribution. For the symmetrical PMG, the maximum magnetic levitation force decreases linearly with the increase of lateral offset of the bulk sample. For the Halbach PMG, when lateral offset changes from 0 mm to 25 mm, the maximum magnetic levitation force increases with the increase of lateral offset of the bulk HTS. When the lateral offset exceeds the center of the Halbach by 25 mm, the maximum levitation force decreases rapidly with the increase of the lateral offset of the bulk sample.

Since the discovery of bulk high temperature superconductor (HTS) which can stably levitate above permanent magnet (PM), the magnetic levitation transportation system has attracted many researchers focus on its potential application [

During the bulk HTS Maglev system optimization design, many researchers focus on the peak value of magnetic levitation and guide forces of bulk HTS. The maximum magnetic force is tightly related with bulk HTS critical current density, flux pinning ability, and the peak value of applied magnetic fields. Many practice prototype Maglev test vehicles involve bulk HTS arrays and PMG. How to arrange the bulk HTS arrays above the PMG and the magnetic field distribution induced by the PMG also needs to be investigated. For cylindrical bulk HTS arrays, two main arrangements are often used, as Figures

The two cylindrical bulk HTS arrays above PMG and the schematic diagram of two types PMG (a) bulk HTS side-by-side array (b) bulk HTS stagger array (c) symmetrical PMG (d) Halbach PMG.

In the paper, the influence of magnetic field distribution of PMG on the magnetic levitation force of bulk HTS is investigated. The investigation is carried out by the method that magnetic levitation forces of bulk HTS with different lateral offsets over PMG are simulated with 3D-model numerical method. The lateral offset of the bulk HTS over PMG increases from 0 mm to 30 mm with the step size equal to 5 mm. The proposed 3D-model method uses two virtual bulk samples for mathematical descriptions of high-

For the proposed numerical method introduced in this section, one cylindrical shape bulk HTS levitated above PMG is considered, as Figure _{1} and domain a_{2}. The two calculation regions are satisfied for Maxwell-Ampere’s law and Faraday’s law:

The cross section of bulk HTS-PMG levitation 3D-model.

For the quasi-approximation problem, we assume no displacement currents are considered. Here

The

In order to simplify the equation derivation, a virtual Ohm’s law is considered for all the calculation regions:

For the calculation region of domain a_{1} which corresponds to bulk HTS,

From (

The virtual conductivity of domain a_{1} can be derived by (

Critical current density in the bulk YBCO interior along a-b plane is about two times than that along _{1} (see Figure _{2} and _{1} (see Figure

Formulation (

The outer boundary of domain a_{2} is a dynamic boundary. Here we use a time dependence function to describe

The detailed numerical method and implementation procedure here are the same as those in our previous research (please see [

After the magnetic fields and current density distribution are resolved, the magnetic forces acting on the bulk HTS can be obtained by

Here

The prototype magnetic levitation system is simulated with the proposed methods. As Figure ^{2}. Figure

The magnetic flux density distribution of Halbach PMG symmetrical PMG.

The magnetic levitation force-gap loops are simulated with different lateral offsets. Figure

The magnetic guidance forces are also calculated. Different from the levitation forces calculation, the bulk HTS was cooled in field-cooling condition. The sample was firstly moved to the position which is about 3 mm to the top surface of the PMG. It is called a field-cooling height (CH). When the bulk HTS was cooled and changed into superconducting state, it started to move laterally. During the process, the gap between the bulk sample and the PMG equals to 3 mm. It is called working height (WH). The guidance force simulation was carried by moving the bulk HTS along

Because of flux pinning potential ability, in experiment, the bulk HTS could freeze some magnetic flux inner. Mathematically, each mesh node’s magnetic field of the sample was initialized with the applied magnetic fields. By the method, the guidance force was successfully calculated.

Based on the proposed numerical method, the magnetic levitation force-gap loops of bulk HTS over PMG with different lateral offsets are obtained. The calculation parameters are of ^{2}, ^{7} A/m^{2}, and ^{−4} V/m. Figures

Magnetic levitation force loops of bulk HTS over symmetrical PMG with different lateral offsets.

Magnetic levitation force loops of bulk HTS over Halbach PMG with different lateral offset.

Figure

Figure

Maximum levitation forces of bulk HTS above two types PMG with different lateral offsets.

Figure

Guidance force-displacement curves of bulk HTS over two types PMG along

The PMG is assumed infinite-long in

From the discussion above we can get the conclusion that, in a Maglev system applied of symmetrical PMG, the bulk should be located close to the center of the PMG, which means HTS stagger array is better than side-by-side array arrangement (see Figures

In the paper, levitation forces of cylindrical bulk HTS with different lateral offsets over two types of PMG are numerically presented. The simulation results show that the levitation force will decrease monotonously with the increase of lateral offset of the bulk HTS over symmetrical PMG.

For Halbach PMG, there is a value of the lateral offset of the bulk over the PMG, when the lateral offset changes from 0 mm to the value, the levitation force will increase monotonously; when the lateral offset exceeds the value, the levitation force decreases rapidly. From the simulation results we can get the conclusion that the optimization of magnetic levitation transportation system composed of bulk HTS and PMG not only consider the applied magnetic field enhancement, but also the applied magnetic field distribution. For symmetrical PMG, the bulk HTS stagger array arrangement may be better than others, for Halbach PMG, the bulk HTS side-by-side array is better than the stagger array arrangement.

This work is supported by the National Natural Science Foundation in China (no. 11205080).

_{c}superconducting Maglev vehicle practical application