Starting from the electrical current conservation, we derive the latter as the curl of the electric vector potential (T):

$$\begin{array}{}{\displaystyle \overrightarrow{\mathrm{\nabla}}.\overrightarrow{J}=0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\iff \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{J}=\overrightarrow{\mathrm{\nabla}}\times \overrightarrow{T}}\end{array}$$(5)

Considering that the magnetic field varies sinusoidally with time, from the Maxwell-Faraday law in the frequency domain, with the angular frequency *ω*, we have:

$$\begin{array}{}{\displaystyle \overrightarrow{\mathrm{\nabla}}\times \overline{\overline{{\sigma}^{-1}}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\overrightarrow{\mathrm{\nabla}}\times \overrightarrow{T}=-i\omega \phantom{\rule{thinmathspace}{0ex}}\overrightarrow{B}}\end{array}$$(6)

We separate the magnetic flux density into its source and reaction terms, evaluated by integral equations (7) involving the source (*J⃗*_{S}) and the eddy currents. In (7) *B⃗*^{a} is an externally applied magnetic flux density satisfying ∇⃗ .*B⃗*^{a} = 0.

$$\begin{array}{}{\displaystyle \overrightarrow{B}(\overrightarrow{r})=\frac{{\mu}_{0}}{4\pi}\underset{\mathit{\Omega}}{\int}\frac{[{\overrightarrow{\mathrm{\nabla}}}^{\prime}\times \overrightarrow{T}({r}^{\prime})+{\overrightarrow{J}}_{S}]\times (\overrightarrow{r}-{\overrightarrow{r}}^{\prime})}{{\left|\overrightarrow{r}-{\overrightarrow{r}}^{\prime}\right|}^{3}}\phantom{\rule{thinmathspace}{0ex}}dv+{\overrightarrow{B}}^{a}(\overrightarrow{r})}\end{array}$$(7)

In the following, to compact the expressions of the formulas, the matrix writing is adopted, where *C̿*, *R̿* and *M̿* are respectively the curl, the resistivity and the integral matrices. Equation (7) can thus be rewritten as follows:

$$\begin{array}{}{\displaystyle \overline{B}={\overline{B}}^{a}+{\mu}_{0}\overline{\overline{M}}{\overline{J}}_{S}+{\mu}_{0}\overline{\overline{M}}\phantom{\rule{thinmathspace}{0ex}}\overline{\overline{C}}\phantom{\rule{thinmathspace}{0ex}}\overline{T}}\end{array}$$(8)

Introducing (7) in (6), we obtain the flowing integro-differential equation:

$$\begin{array}{}{\displaystyle \overline{\overline{C}}\phantom{\rule{thinmathspace}{0ex}}\overline{\overline{R}}\phantom{\rule{thinmathspace}{0ex}}\overline{\overline{C}}\phantom{\rule{thinmathspace}{0ex}}\overline{T}=-i\omega \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}({\overline{B}}^{s}+{\mu}_{0}\overline{\overline{M}}\phantom{\rule{thinmathspace}{0ex}}\overline{\overline{C}}\phantom{\rule{thinmathspace}{0ex}}\overline{T})\phantom{\rule{thinmathspace}{0ex}}}\end{array}$$(9)

The integral part of the formulation allows a direct interaction between the system elements, and thus only the active parts of the system are discretized. The finite difference method is used to discretize the curl operator. The matrix equation (9) is solved iteratively which allows to take into account the nonlinearity of the resistivity matrix as shown in the flowchart described in Figure 2. We start by solving (9) by considering only the source term of the magnetic flux density, then we evaluate the eddy currents, after that we evaluate the reaction term of the magnetic flux density; at this step, we actualize the value of the critical current density, then we actualize the conductivity tensor and solve again the equation (9) with the actualized values of the magnetic flux density and the conductivity tensor. The operation is repeated until convergence. An iterative method (conjugate gradient method) is used to solve the matrix equation. As the source term is divergence free, no gage condition is needed as shown by Z. Ren in [7].

Figure 2 Iterative solving procedure

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