6.2.1 Boundary Conditions

The supplied total current in the inductor wire is considered by the following condition for $\vec{H}_1$ :

$$I=\oint_{\partial\mathcal{A}}\!\!\vec{H}_1{\cdot} \mathrm{d}\vec... ...mathcal{A}}\!\!\vec{H}{\cdot} \mathrm{d}\vec{r} \simeq H_{t}\sum_{i=1}^{p}l_i,$$ (6.24)

where $\partial\mathcal{A}$ is an arbitrary closed loop around the conducting wire, $p$ is the number of edges, which build this loop, and $l_i$ is the length of the $i$ -th loop edge. The Dirichlet boundary part from (5.27) is expressed as

$$\vec{v} = \sum_{j=n+1}^Mc_j\vec{N}_j \mathrm{with} c_j = H_t \mathrm{for} j\in[n{+}1;M] \mathrm{and} p = M - n.$$

Only the supply parts of the wire, which are used to force the electric current, lie directly on the outer bound of the simulation domain. The remaining parts of the wire are surrounded by dielectric material. The loop $\partial\mathcal{A}$ is chosen to lie on the outer face of the simulation domain. The Neumann boundary $\mathcal{A}_{N1}$ consists of all edges lying on the outer boundary of the simulation domain excluding the edges building $\partial\mathcal{A}$ . In this work the dielectric environment enclosing the wire is assumed to be sufficiently thick so that $\vec{E}$ can be neglected on the dielectric part of $\mathcal{A}_{N1}$ . On the other hand the electric current density is forced in a direction perpendicular to the conductor boundary faces. Thus, for isotropic materials with respect to $\sigma$ , $\vec{E}$ will be also perpendicular to these faces and the homogeneous Neumann boundary condition (5.30) is used for the conductor surface parts.

For the calculations of $\psi$ it is sufficient that one node of the simulation domain is set to an arbitrary value. Thus, the Dirichlet boundary part of (5.27) is modified to read

$$v = c_{M+1}\lambda_{M+1} \mathrm{with} N = M + 1.$$ (6.25)

The Neumann boundary $\mathcal{A}_{N2}$ consists of all edges lying on the outer boundary of the simulation domain. The simulation domain is constructed sufficiently large to allow that the magnetic flux $\vec{B}$ can be neglected on the outer boundary $\mathcal{A}_{N2}$ . Thus the homogeneous Neumann boundary condition (5.32) is applied.