Electromagnetic Modeling of Superconducting Bulks in Applied Time-Varying Magnetic Field

Abstract

An integrodifferential model formulated in terms of the electric vector potential is developed for the 3D numerical modeling of the electromagnetic field in superconducting bulks, for AC losses evaluation. The Newton Raphson method is applied to accelerate the convergence. The model is validated on a benchmark. The comparison results show the accuracy of the model and its performances in terms of computation time compared to classical approaches.

1. Introduction

The evaluation of critical quantities and AC losses are among the main steps for the sizing of high temperature superconductor (HTS)-based systems and the associated cryogenics. It is mainly founded on the evaluation of local quantities due to the strong, anisotropic, and nonlinear dependence of the electrical properties of such materials to the magnetic field.

Analytical modeling is of very limited use due to the strong assumptions on the geometry and the physical properties [1,2]. In fact, 3D time domain numerical modeling is required in most cases. The finite element method (FEM) is the most popular in the community of HTS modeling [3,4]. However, the reported computation times are often considerable and not compatible with computer-aided design approaches. On the other hand, the evolution of computers seems in favor of the development of integral methods. Indeed, memory has been several thousand times multiplied these last decades, while the CPU only experienced little improvement. With the possibility of large data storage and treatment, integral methods become very competitive due to the numerous advantages they offer. Indeed, they allow one to limit the discretization to only the active parts of the modeled systems, and do not require boundary conditions. They involve a direct interaction between the modeled system parts, which is a real advantage in optimization problems. They are particularly suitable for distributed multisource systems, or systems presenting multiscale dimensions such as HTS coils [5,6,7,8,9]. In 3D modeling, the conservation of the electric current is, however, not ensured using only integral equations, in particular in anisotropic and nonlinear materials. In this case, hybrid (integrodifferential) models are more suitable [9,10]. In previous works, we highlighted the efficiency of such methods for the electromagnetic field modeling in HTS tapes and coils, even with the presence of ferromagnetic materials [8]. In this work, an integrodifferential model based on the electric vector potential is developed for the modeling of the electromagnetic field in HTS bulks for AC losses evaluation.

HTS bulks are used either for the magnetic field trapping or screening, for applications in power systems, in particular in electrical machines [11]. They are not predestined to operate with time-varying magnetic fields; however, they may be subjected to variable magnetic fields created by the armatures or due to their function of magnetic field modulation in electrical machines. Given that the AC losses, and in particular the hysteresis losses, increase with the size of the HTS, their evaluation and reduction in HTS bulks is crucial for the development of HTS electrical machines and their cryogenic systems.

The electromagnetic field modeling in HTS bulks is found to be more delicate than in HTS tapes, which incorporate matrices of finite electrical conductivity, ensuring a certain electrical stability when the resistivity of the HTS tends to zero. Thus, in complement to the previous works on the AC losses modeling in HTS tapes [9], a damped Newton Raphson method is applied to accelerate the convergence. A benchmark model of a superconducting cube submitted to an external magnetic field is considered, and the results, as well as the model performances, are compared to those reported in the literature [4].

The modeled system and the model formulation are presented in the next section. The model implementation is presented in the third section, followed by the results and discussions.

2. The Modeled System

The modeled system consists of an HTS cube of side “a”, submitted to an external sinusoidal time-varying magnetic flux density ${\overrightarrow{B}}^s$, with an amplitude $B_m^s$ and a frequency $f$, uniformly distributed in the space and oriented in the z direction, as shown in Figure 1. Thus, we have: ${\overrightarrow{B}}^s=B_m^s\sin (2\pi ft){\overrightarrow{e}}_z$.

The equivalent electrical resistivity ρ of the HTS, given by Equation (1), is derived from the power law, where J is the electric current density, $E_c={10}^{-4} \mathrm{V}/\mathrm{m}$ is the critical electric field, which is a criterion value for the insurgence of the normal state, n is the creep exponent, and $J_c$ is the critical electric current density depending on the magnitude of the total magnetic flux density (B) as given in Equation (1), where $J_{c0}$ is the zero field critical current density and $B_0$ is a constant depending on the considered material. The resistivity ${\rho }_0={10}^{-14} \mathsf{\Omega }\cdot \mathrm{m}$ is added to avoid instabilities when the conductivity of the HTS tends to infinity. Notice that the total magnetic flux density B in Equation (1) is the sum of the source and magnetic reaction of the superconductor.

$$\left\{\begin{array}{l}\rho (J,B)=E_c{J}_c^{-1}(B){[J_c^{-1}(B)J]}^{n-1}+{\rho }_0\\ J_c(B)=J_{c0}{(1+B_0^{-1}B)}^{-1}\end{array}\right.$$

(1)

Considering the electric current conservation $\overrightarrow{\nabla }\cdot \overrightarrow{J}=0$, the current density $\overrightarrow{J}$ is derived from the electric vector potential $\overrightarrow{T}$:

$$\left\{\begin{array}{l}\overrightarrow{J}=\overrightarrow{\nabla }\times \overrightarrow{T}\\ \overrightarrow{n}\times \overrightarrow{T}{(\overrightarrow{r})}_{\overrightarrow{r}\in \Gamma }=\overrightarrow{0}\end{array}\right.$$

(2)

where Γ is the frontier of the HTS cube, and $\overrightarrow{n}$ is its normal vector.

From the Maxwell–Faraday equation $\overrightarrow{\nabla }\times \overrightarrow{E}=-{\partial }_t\overrightarrow{B}$, where $\overrightarrow{E}$ is the electric field, and by using the Biot–Savart law to evaluate the magnetic field produced by the current density $\overrightarrow{J}=\overrightarrow{\nabla }\times \overrightarrow{T}$, we obtain the following equation:

$$\overrightarrow{\nabla }\times \rho (‖\overrightarrow{\nabla }\times \overrightarrow{T}‖,B)\overrightarrow{\nabla }\times \overrightarrow{T}(\overrightarrow{r})=-{\displaystyle \frac{\partial }{\partial t}}\left\{{\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \underset{\Omega }{\int }\frac{\overrightarrow{\nabla }\prime \times \overrightarrow{T}(\overrightarrow{r}\prime )\times (\overrightarrow{r}-\overrightarrow{r}\prime )}{{\left|\overrightarrow{r}-\overrightarrow{r}\prime \right|}^3}}dv+{\overrightarrow{B}}^s(\overrightarrow{r})\right\}$$

(3)

For an anisotropic material, the resistivity is introduced in Equation (3) as a nonlinear tensor [9].

3. Discretization and Implementation

The cube is discretized into $Ne=n_x\times n_y\times n_z$ to rectangular elements, where n_x, n_y, and n_z are the number of elements, respectively, in the x, y and z directions. A simple collocation method is used for the discretization of the integral part in Equation (3):

$$\overrightarrow{T}(\overrightarrow{r})={\displaystyle {\sum }_{i=1}^{Ne}\overrightarrow{T}\delta (\overrightarrow{r}-{\overrightarrow{r}}_i)}$$

(4)

The differential part is discretized by the finite difference method. An implicit scheme is used for the time derivative, with the time step $\Delta t$, leading to the discrete form given by Equation (5), where $\overline{\overline{C}}$, $\overline{\overline{R}}$, and $\overline{\overline{M}}$ are the curl, the resistivity, and the integral matrices, respectively. The matrix ${\overline{\overline{\mathcal{l}}}}_m$ and the vector ${\overline{\mathcal{l}}}_s$ account for the boundary conditions, cancelling the tangential component of the electric vector potential, ensuring the cancellation of the normal component of the current density on the frontier Γ of the HTS cube.

$${\overline{\overline{\mathcal{l}}}}_m\left[\overline{\overline{C}}{\overline{\overline{R}}}^{(t)}\overline{\overline{C}}+{(\Delta t)}^{-1}\overline{\overline{M}}\overline{\overline{C}})\right]{\overline{T}}^{(t)}={(\Delta t)}^{-1}\left[{\overline{B}}^{s(t-1)}-{\overline{B}}^{s(t)}+\overline{\overline{M}}\overline{\overline{C}}{\overline{T}}^{(t-1)}\right]{\overline{\mathcal{l}}}_s$$

(5)

The matrix Equation (5) can be written: ${\overline{\overline{G}}}^{(t)}{\overline{T}}^{(t)}={\overline{S}}^{(t)}$. The damped Newton Raphson (NR) method is used to speed up the convergence. We introduce the residual vector ${\overline{r}}^{(t)}={\overline{\overline{G}}}^{(t)}{\overline{T}}^{(t)}-{\overline{S}}^{(t)}$, and at each time (t), we execute iteratively the algorithm in Equation (6), where $\overline{\overline{D}}({\overline{r}}_i^{(t)})$ is the Jacobian of ${\overline{r}}_i^{(t)}$ and $\beta$ is a damping factor chosen in the way to satisfy $‖{\overline{r}}_i^{(t)}‖<‖{\overline{r}}_{i-1}^{(t)}‖$ [12].

$$\left|\begin{array}{l}{\overline{T}}_0^{(t)},\beta =1,i=1\\ {\overline{T}}_{i+1}^{(t)}={\overline{T}}_i^{(t)}-{\displaystyle \frac{\beta }{2^{(i-1)}}}{\left(\overline{\overline{D}}({\overline{r}}_i^{(t)})\right)}^{-1}{\overline{r}}_i^{(t)}\\ i=i+1(\mathrm{UNTIL}\mathrm{CONVERGENCE})\end{array}\right.$$

(6)

The Jacobian of ${\overline{r}}_i^{(t)}$ can be expressed as follows:

$$\overline{\overline{D}}(\overline{r})=\overline{\overline{G}}+\overline{T}\cdot {\partial }_{\overline{T}}\overline{\overline{G}}=\overline{\overline{G}}+\overline{T}\cdot ({\partial }_{\overline{J}}\overline{\overline{G}})\overline{\overline{C}}$$

(7)

In Equation (7), the term ${\left\{\overline{T}\cdot ({\partial }_{\overline{J}}\overline{\overline{G}})\overline{\overline{C}}\right\}}_{i,j}=T_i{\left\{({\partial }_{\overline{J}}\overline{\overline{G}})\overline{\overline{C}}\right\}}_{i,j}$ leads to a nonlinear equation in Equation (6). It can be linearized by using the solution of the electric vector potential $\overline{T}$ at the previous iteration. In this work, this term is simply ignored, setting $\overline{\overline{D}}(\overline{r})\simeq \overline{\overline{G}}$, which is actualized at each iteration. To improve the convergence, a linearization of the constitutive power law (Equation (1)) can also be used [13].

4. Results and Discussions

A benchmark model of a superconducting cube submitted to an external magnetic field [4] is considered to test the performances of the proposed model. The system specifications are given in

Table 1. Used parameters.

Quantity	Symbol	Value
HTS properties	JC0 |B0 |n	2.54 106 A/m2|0.3 T|23
Cube side|Discretization	a|nx × ny × nz	10 mm|10 × 10 × 10
Source|Freq.|Time step	$B_m^s$|f|Δt	5 mT|50 Hz|5 10−5 s

.

Figure 2 presents the calculated instantaneous AC losses in the cube over a period. A common shape of the time evolution of the AC losses is obtained, where the steady state of the electromagnetic penetration in the material is obtained in the second half of the period. Figure 3 represents the eddy current repartition in the cube at t = T/4 (5 ms) and t = 3T/4 (15 ms). As the magnetic field is rather feeble, the penetration of the current in the cube, which depends on the critical current, is not complete. Indeed, the full penetration of the magnetic flux density into the cube is [4]: $B_p={\mu }_0{J}_{c}a/2\approx 16\mathrm{mT}$.

The model has been implemented and run in a 3.50 GHz, 32.0 Gb RAM computer. As shown in

Table 2. Results and performance comparison.

Quantity	Obtained	Published [4]
Average losses P (mW)	0.74	0.66 to 0.87
Computation time (min)	14	6.4 to 134

, the computation time is about 14 min, while it varies from 6.4 to 134 min for the models used in [4], depending on the model and the performances of the computer used.

Next, the cube is subdivided into two sub-cubes in the y direction at x = a/2 (0.5 mm). This is done to evaluate the reduction of the AC losses in the HTS cube, appreciating in the same time the performances of the model to deal with a multidomain geometry. This is achieved simply by cancelling the normal component of the current applying the boundary condition on both sides at the interface between the sub cubes. Eddy current and AC losses are evaluated with and without considering the electromagnetic coupling between the sub cubes. Figure 4 shows the eddy current repartition in the sub cubes for both cases. As expected, the penetration of the current is more important in the case where the coupling is considered (Figure 4b). Figure 5 shows a comparison of the instantaneous losses evaluated over a period in the entire cube and in the sub-cubes with and without electromagnetic coupling between them. As expected, the subdivision of the cube leads to a reduction of the mean AC losses, which are of 0.62 mW considering the magnetic coupling between the sub-cubes. The computation time is about 17 min. When the coupling between the sub-cubes is ignored, the mean value of the AC losses is 0.54 mW and the computation time is about 4.5 min.

5. Conclusions

In this work, an integrodifferential model formulated in terms of the electric vector potential is developed for the 3D numerical modeling of the electromagnetic field in HTS materials, for AC losses evaluation, in the aim of their integration in electrical power systems. The numerical issues are highlighted and the results show the performances of the model in terms accuracy and computation time, in comparison with classical approaches. The model, however, inherits the limits of the finite difference method in dealing with complex geometries.