## 2 Physical Model Selected for Analysis  
The single-core power cables in a three-phase circuit can be laid out in a number of formations. Typical ones are trefoil and flat. In the first case, the three cables are placed in the corners of an equilateral triangle, as presented in Figure 1a; in the second case, the three cables are placed in the same horizontal plane, at equal distances between adjacent cables, as presented in Figure 1b. The choice of use depends on several factors such as shield bonding method, conductor area and available space for installation [21; 22].  
In solid bonding systems (suitable for cable lengths over 500 m), the cable shields are grounded at both ends of the cables, as depicted in Figure 2, where \(R_{g}\) represents the grounding resistance. This bonding scheme will reduce the induced shield voltages, but there will be circulating shield currents proportional to the wire-core currents. This will cause losses in the shields, which reduce the cable current carrying capacity. At the same time, the circulating shield current generates a magnetic field that is significantly out of phase with respect to the wire-core magnetic field. In the following, only the total magnetic field will be investigated.  
Since solid bonding is more common for medium-voltage applications, a 12/20 kV NA2 XS(F)2Y single-core cable (Figure 3) has been selected for finite element analysis. The cable is constructed from aluminum (Al) core conductor with the cross-sectional area of 150 mm\({}^{2}\), semi-conductive layer over conductor, core insulation of cross-linked polyethylene (XLPE), semi-conductive layer over insulation, swelling tape, copper (Cu) wire shield with the cross-sectional area of 25 mm\({}^{2}\), waterproofing tape and outer sheath of high-density polyethylene (HDPE). The current carrying capacity when buried in the ground is 319 A for trefoil formation and 352 A for flat formation, respectively. Other cable characteristics are given in Table 1.  
Figure 1: Typical cable formations: (**a**) trefoil formation; (**b**) flat formation.  
Figure 2: Solid bonding.  
The physical layout of the analyzed three-phase cable system is presented in Figure 4, where two cases are considered in conformity with the national regulations regarding the design and execution of the electrical cable networks [23]:  
* The three phases are buried in the ground, at a depth of 0.8 m, in trefoil formation (the spacing between the centers of any two cables is 36 mm, as dictated by the cable outer diameter), Figure 4a;
* The three phases are buried in the ground, at a depth of 0.8 m, in flat formation with clearance between cables of 70 mm (the spacing between the centers of the adjacent cables is 106 mm), Figure 4b.  
In both cases, the power cables are located in a rectangular trench 0.9 m deep and 0.4 m wide, between two sand layers of 0.1 m thickness. Both ends of all cable shields are connected to ground, as indicated in Figure 2.  
For model implementation and its validation, it is assumed that the three-phase cable system has exactly balanced currents, as follows: \(\mathbf{I_{1}}=I\angle-120^{\circ}\), \(\mathbf{I_{2}}=I\angle 0^{\circ}\) and \(\mathbf{I_{3}}=I\angle 120^{\circ}\). Computations will be performed at maximum rated current (319 A and 352 A, respectively), assuming that the actual laying depth has no significant influence on the maximum admissible load current, which is otherwise compensated by the reduction in the ambient temperature and the more favorable specific thermal resistances of the ground at bigger laying depths (according to [23], the usual laying depth range is approximately 0.7 m \(\div\) 1.2 m). Additionally, the surrounding ground is considered electrically homogenous and non-magnetic.