*5.1. Validation Through an Academic Example*

In the first example, a thin magnetic conductive disk is considered with the circular conductor fed by a voltage of 1V and a frequency of 10Hz (Figure 3).

**Figure 3.** Thin magnetic conductive disk and conductor.

The device parameters are presented as below:

	- -Electrical conductivity <sup>σ</sup>*disk* <sup>=</sup> <sup>6</sup> <sup>×</sup> 107 <sup>S</sup>/<sup>m</sup>
	- -Permeability μ*<sup>r</sup>* = 200
	- -R = 1 m, *e* = 50 mm
	- -The skin depth δ = 1.45 mm is smaller than the disk's thickness
	- -Electrical conductivity σ conductor = 5.79 × 107 S/m
	- h = R/4 = 0.25 m

This academic example is solved by three numerical methods, where the first one is the axisymmetric FEM, the second one is a shell element formulation implemented in 3D FEM method [5] and the last one is the proposed integral method with the surface elements. In order to valid our method and to compare different approaches, we mainly focus on the computed current in the conductor and the magnetic field in the air region close to the device (calculated on the path AB, for A (0.25; 0; 0.1) and B (0.25; 0; 0.25)).

Let us note that the problem must be meshed very finely to have an accurate result with the FEM 3D (Table 1 and Figure 4) because of the high variations of the fields around the conductor. The current values greatly vary according to the number of the elements.

If the axisymmetric FEM is considered as our reference, the coupling integral method leads to an error of 0.1% (Table 1). Our method leads to more accurate results than the same shell element formulation but considering the air region treated with the 3D FEM.


**Table 1.** Current values in the conductor, where j is an imaginary unit.

**Figure 4.** Magnetic field on path AB.
