## 3 Finite Element Model
### Global FEM Model, Boundary Conditions and Solver Setup  
The global FEM model of the three-phase cable system is depicted in Figure 5a, where the computational domain is a square of side \(a=20\) m, sufficiently large to determine the behavior of the magnetic field well outside from the cable central axis. A discretized section around the power cables is given in Figure 5b. The power cables are buried--according to the geometrical dimensions in Figure 4a (for trefoil formation) and Figure 4b (for flat formation)--in a ground with the electrical conductivity \(\sigma\) = 0.01 S/m, the relative magnetic permeability \(\mu_{r}\) = 1 and the relative electric permittivity \(\varepsilon_{r}\) = 10. The half top layer in Figure 5a models the air. The model depth (cable length) is 1 m.  
All power cables are modeled as presented in Figure 5c, using a simplified four-layer cable model consisting of Al core conductor (\(\sigma_{20}\) = 35.38 \(\times\) 10\({}^{6}\) S/m, \(\mu_{r}\) = 1 and \(\varepsilon_{r}\) = 1), XLPE insulation (\(\sigma\) = 1 \(\times\) 10\({}^{-15}\) S/m, \(\mu_{r}\) = 1 and \(\varepsilon_{r}\) = 2.5), Cu shield (\(\sigma_{20}\) = 58 \(\times\) 10\({}^{6}\) S/m, \(\mu_{r}\) = 1 and \(\varepsilon_{r}\) = 1) and HDPE oversheath (\(\sigma\) = 1 \(\times\) 10\({}^{-14}\) S/m, \(\mu_{r}\) = 1 and \(\varepsilon_{r}\) = 2.3). The equivalent thickness (\(th\)) of the Cu shield layer is chosen so that it closely matches the nominal 25 mm\({}^{2}\) cross-sectional area of the real wire shield, while its mean radius is exactly  
Figure 5: **The 2D FEM model for computing magnetic fields from a three-phase underground cable system with solid bonding: (a) global geometric model; (b) discretized section around the power cables; (c) simplified cable model; (d) coupled circuit for shield bonding. **  
the same, 14.8 mm. For cables with metallic sheets, the sheet layer in the cable model will simply reflect the cross section of the real sheet. In simulation, the aforementioned conductivities (at \(T_{0}=20\)\({}^{\circ}\)C) of the metallic layers will be adjusted to maximum operating temperatures (90 \({}^{\circ}\)C for Al core conductors; approximately 80 \({}^{\circ}\)C for Cu shields), by taking into account the following temperature coefficients (of resistivity, \(\rho=1/\sigma\)) per K at 20 \({}^{\circ}\)C [26]: 4.03 \(\times\) 10\({}^{-3}\) for Al and, respectively, 3.93 \(\times\) 10\({}^{-3}\) for Cu. Since the analyzed cable structure is assumed to be completely isolated from other magnetic fields or sources of current, Balloon boundary conditions are assigned to all four edges of the defined computational domain. In this case, the magnetic vector potential, \(\mathbf{A}_{z}\), goes to zero at infinity; the magnetic flux lines are neither tangential nor normal to the Balloon boundary. At the interfaces between objects, natural boundary conditions are automatically assigned by the eddy current module. The values of the wire-core currents (amplitude and phase) are assigned using the software functionality \"Current Excitation\", while the shield bonding is encoded in the model by the coupled circuit in Figure 5d, where external windings are used for controlling the induced shield currents. This electrical circuit was defined with Maxwell Circuit Editor. As mentioned above, only perfectly balanced currents are considered for model implementation and further validation, but unbalanced loading conditions can also be managed. By simulation, the effect of grounding resistance was proved to be insignificant. A fine mesh was defined for analysis, totalizing a number of triangle elements in the order of 1,300,000. The mesh refinement was achieved by restricting the maximum length of the elements in all model blocks, ensuring that high-resolution magnetic field profiles are generated over a large distance of interest with respect to the cable central axis (a number of preliminary tests with mesh size were performed).