*Equations*

To analyze the electromagnetic characteristics of the HTS bulk, the *H*-formulation and the well-known Maxwell equations (neglecting the displacement current term) [22] are as follows:

$$
\mu \frac{\partial H}{\partial t} + \nabla \times E = 0 \tag{2}
$$

$$E = \mathfrak{p} \int \tag{3}$$

$$J = \nabla \times H \tag{4}$$

where *H* is the magnetic field strength; ρ is the resistivity of the material, which is assumed to be isotropic in the two-dimensional (2D) model; *E* is the electric field; and *J* is the current density. Equations (2)–(4) are solved by the MFH interface in the AC/DC module.

The strong nonlinear relationship between *E* and *J* of the HTS material can be characterized by the experimental empirical *E*–*J* power law [23]:

$$E = E\_{\mathbb{C}} \left( \frac{J}{Jc} \right)^{\mathbb{M}} \tag{5}$$

where *E*c is the critical current criterion equal to 100 μV/m, and *m* is the power law exponent, which is usually 21. The parameter *J*c in the constitutive law depends on the temperature *T*. Thus, the thermal effect is coupled in the modeling using the heat transfer module. Four *J*c–*T* relationships are considered.

(1). The linear dependence [15]:

$$J\_{\mathbf{c}} = J\_{\mathbf{c0}} \frac{T\_{\mathbf{c}} - T}{T\_{\mathbf{c}} - T\_0} \tag{6}$$

where *J*c0 is the critical current density of the HTS bulk at *T* = *T*0 (77 K); *T*c is the critical temperature; and *T*0 is the coolant temperature.

(2). The nonlinear dependence with α = 1, 3/2, 2 turns into [10]:

$$J\_{\mathbf{c}} = J\_{\mathbf{c1}} \left( 1 - \left( \frac{T}{T\_{\mathbf{c}}} \right)^2 \right)^{\alpha} \tag{7}$$

where the critical current density *J*c1 is obtained by Equation (6) with *T* = 0 K. In addition, we calculated the case without considering the thermal effects. In that case, *J*c as a constant equals *J*c0.

The thermal equilibrium equation is expressed as:

$$C\_{\mathbb{P}} \frac{\partial T}{\partial t} - \nabla \cdot (\lambda \nabla T) = Ef \tag{8}$$

where *C*p is the heat capacity per unit volume of the superconductor; and λ is the thermal conductivity of the superconductor.

The levitation force is obtained by the Lorenz force formula at each time instant as:

$$F\_Y\left(t\right) = \int \mathbb{S}\left.B \times f\left[\frac{N}{m}\right]\right|\tag{9}$$

where *S* is the cross-section of the HTS bulk in the *x-y* plane. We assume four bulks placed along the *z*-direction to ge<sup>t</sup> the larger force; the total levitation force equals *Fy* (in the actual calculation, *Fy* is the force density along the length at the *z*-direction) times 128 mm, since each HTS bulk is 32-mm wide along the *z*-direction and there are four bulks in the experiments. Table 1 summarizes the simulation parameters. The material properties of the HTS bulk are based on the melt textured three-seeded rectangular YBaCuO bulk made by the ATZ GmbH (Torgau, Germany).

**Table 1.** Parameters for the modeling.

