*2.4. IF Dosimetry*

It is well known that numerical dosimetry up to intermediate frequencies (10 MHz) can exploit the fact that the induced currents in the human body do not perturb the external

magnetic field [16,17]. For this reason, the simulations of the external magnetic field source can be separated by the evaluation of the electric field induced inside the human bodies. This division makes it possible to select the most suitable formulations for the two steps.

In this paper, the simulation of the WPT system and the car body is handled by a numerical hybrid formulation based on BEM and SIBC methods [18]. This formulation is particularly suitable to handle the multi-scale problem as the car body has a significant surface with a very small thickness.

The numerical dosimetry computations are instead performed using the Scalar Potential Finite Element (SPFE) method, which is implemented in the commercial software tool Sim4Life. Based on the magneto-quasi-static (M-QS) approximation and the conductioncurrent-dominant characteristics of biological tissues in the IF region, a simplified scalar potential equation is given by

$$
\nabla \cdot \sigma \nabla \phi\_{\varepsilon} = -j\omega \nabla \cdot \sigma \mathbf{A} \tag{1}
$$

where **A** is the magnetic vector potential, *φ<sup>e</sup>* is the scalar electric potential, *ω* is the angular frequency, and *σ* is the conductivity. Due to the fact that the magnetic field source is handled by a hybrid formulation based on BEM/SIBC, we cannot directly compute the necessary magnetic vector potential **A** on the right hand side of Equation (1). Therefore, the magnetic flux density **B** is computed via step (1), and a compatible magnetic vector potential **A** is then evaluated by using one of the curl-inversion procedures described in [22–24]. Specifically, Sim4Life implements the curl-inversion procedure based on Laakso et al. [22], though different schemes can be exploited by providing an external text file.

Once the magnetic vector potential **A** is provided, Equation (1) is discretized using the Galerkin Finite Element Method and linear nodal basis functions on a rectilinear grid. The resulting linear equation system is then solved using a conjugate gradient solver with a stopping criterion of 10 orders of magnitude reduction for the initial residual. Upon solving the unknown scalar potential *φ<sup>e</sup>* , the induced electric field **E** can be computed from

$$\mathbf{E} = -\nabla \phi\_{\ell} - j\omega \mathbf{A}.\tag{2}$$
