**1. Introduction**

Due to the inherent flux-pinning e ffect, high-temperature superconducting (HTS) bulk has been potentially used in HTS magnetic levitation (Maglev) transportation [1–4] and many other applications [5,6]. To investigate the electromagnetic characteristic of the HTS Maglev system, a proper HTS *E*-*J* constitutive law is necessary. For the HTS materials, the critical current density changes along the temperature. In the past, Matsushita et al. [7] investigated the single-grain YBaCuO specimen by measuring the critical current density *J*c–*T*, and proposed the current density as a function of *1*-( *T*/*T*c) 2. Yamasaki et al. [8] reported that *J*c is proportional to *(1-AT* +*BT*<sup>2</sup>*),* where *A* and *B* were constants for the HTS material of Bi-2223 thin films. Another research study [9] described *Jc* as a function of (*1-T*/*T*c) γ based on the exploration of Bi-2212/Ag wires. In addition, *J*c<sup>∝</sup>(*1-*(*T*/*T*c) 2) α for Bi-2212 tapes was mentioned in the book [10]. For 2G HTS YBCO bulks, Braeck [11] assumed a linear temperature dependence of the critical current as *J*c<sup>∝</sup>(( *T*c-*T*)/(*T*c-*T*0)); and then Tsuchimoto [12] presented an exact nonlinear relationship of *J*c<sup>∝</sup>(1-( *T*/*T*c) 2) 2.

Later, from the point of view of superconducting YBaCuO application, Tsukamoto et al. [13] employed the linear temperature dependence of the critical current density to calculate the temperature variation and the trapped magnetic field in YBCO bulks under an AC external field. Using this linear formula, Tixador et al. [14] calculated the current distribution and AC losses of the YBCO slab. In recent years, Ye [15] and Huang [16] studied the dynamic thermal e ffect of the HTS Maglev system using YBaCuO bulks by the linear *J*c*–T* relation.

In this paper, in order to elucidate which *Jc*–*T* relationship is more appropriate to model the HTS Maglev system, we used four di fferent *J*c–*T* functions to calculate the dynamic levitation force, the temperature distribution, and the current density distribution of the HTS bulk over the applied permanent magnetic guideway (PMG). The modeling characterized by the *H*-formulation and the *E*–*J* power law was implemented in the finite element software COMSOL Multiphysics 5.3a. With the magnetic field formulation (MFH) interface in the AC/DC module, the calculation subdomains of the HTS bulk and the PMG were built as the geometric entity. During the calculations, a vertical vibration with small amplitude was applied to the PMG to simulate the magnetic field fluctuation caused by the inevitable PMG irregularity. Di fferent from the previous modeling [17,18], the thermal e ffect was taken into account by coupling the heat transfer module. To study the levitation performance of the HTS Maglev system at high speed, a vibration with the frequency of 60 Hz was set to simulate the magnetic field inhomogeneity, which is equivalent to the linear velocity of about 1018 km/h of the circular PMG employed in the experiments. The converted linear velocity *v* of the circumferential PMG in the experiments can be expressed as:

$$w \, (\text{km/h}) = r(\text{m}) \times 2\pi n(\text{rpm}) \times \frac{60}{1000} \tag{1}$$

where *r* is the radius of the circular PMG, which is 0.75 m; the rotation rate of the PMG, *n* (rpm), is derived from the vibration frequency of the guideway set in the simulation. This occurs since one cycle of the magnetic field fluctuation during the dynamic modeling is approximately tantamount to the real PMG magnetic field, which the HTS bulk subjects to when the circular guideway rotates for one circle in the experiment [19,20]. Thus, the cycle of the magnetic field fluctuation is 1/60 s when the frequency is 60 Hz, and the PMG's rotation rate *n* is 3600 rpm. In that case, the calculation results by di fferent *J*c–*T* functions showed the levitation force attenuation during the vibration process. In comparison with the measurements in this paper, the *J*c–*T* relationship in which *J*c is proportional to (1-( *T*/*T*c) 2) 2 shows better agreemen<sup>t</sup> with the experiment.

## **2. Theoretical Model**

The HTS bulk and the opposite -polar-arranged PMG were built as the geometric entity in the COMSOL. The remanence of the permanent magne<sup>t</sup> *B*r was set at 0.8 T. With the MFH (magnetic field formulation) interface, we can easily simulate the magnetic field produced by any complex- shaped magnets. In this study, after the PMG reached the working height (WH), a small amplitude vertical vibration was applied to the PMG to simulate the magnetic field fluctuation. The first movement stage is regarded as the quasi-static process, since the speed is as low as 1 mm/s, while the following vibration process is the dynamic levitation stage.

Figure 1 shows the computation subdomains of the dynamic model and the movement process of the PMG. In the modeling, the PMG first moves from the field-cooling height (FCH = 30 mm) to the working height (WH = 15 mm) at 1 mm/s. This process takes 15 s. Afterwards, it relaxed from 15 to 120 s (Relaxation I) to let the flux fully redistribute. Then, a sine function with 1-mm amplitude is applied to the PMG for 20 s. After the dynamic condition, the second relaxation process (Relaxation II) takes place, which is from 140 s to the end. In the dynamic condition, the vibration frequency conditions of 2 and 60 Hz are calculated, respectively. According to the size of the circular PMG in the experimental equipment SCML-03 [21], the vibration frequencies of 2 and 60 Hz are corresponding to the linear velocity of 34 and 1018 km/h for the Maglev above the PMG, respectively.

**Figure 1.** Vertical displacement variation of the permanent magnetic guideway (PMG) and two-dimensional (2D) model for the high-temperature superconducting (HTS) Maglev system. Vibration amplitude A = 1 mm. Field-cooling height (FCH) = 30 mm, working height (WH) = 15 mm. The quasi-static condition from 0 s to 120 s includes the 0–15 s upward movement from FCH to WH and the Relaxation I process from 15 to 120 s. Then, after 20 s of vibration, the second relaxation process (Relaxation II) happened from 140 s to the end.
