## 4 Analytical Approach for FEM Model Validation  
To verify the numerical results obtained with the proposed FEM model, an interactive software tool based on the Biot-Savart law and the superposition principle has been developed. Assuming that the power cables are straight and infinitely long, the total magnetic flux density at any measuring point (\(x\), \(y\)) in the vicinity of a three-phase power cable with solid bonding can be calculated as (Figure 8):  
Figure 6: Example of simulation results for the three-phase power cable in trefoil formation: (**a**) instantaneous magnetic flux density profiles and calculated RMS magnetic flux density profile at the height \(h\) = 1 m above the ground; (**b**) a momentary magnetic field distribution in the cross section of the power cable (\(t\) = 16.94 ms). Figure 7: Example of simulation results for the three-phase power cable in flat formation: (**a**) instantaneous magnetic flux density profiles and calculated RMS magnetic flux density profile at the height \(h\) = 1 m above the ground; (**b**) a momentary magnetic field distribution in the cross section of the power cable (\(t\) = 15.56 ms). \[\textbf{{B}}_{\textbf{x}}=\sum_{i=1}^{3}\frac{-\mu_{0}}{2\pi}(\textbf{{I}}_{i}+ \textbf{{I}}_{\textbf{sh},i})\left[\frac{y-y_{i}}{r_{i}^{2}}\right]; \tag{4}\]  
\[\textbf{{B}}_{\textbf{y}}=\sum_{i=1}^{3}\frac{\mu_{0}}{2\pi}(\textbf{{I}}_{i}+ \textbf{{I}}_{\textbf{sh},i})\left[\frac{x-x_{i}}{r_{i}^{2}}\right]; \tag{5}\]  
\[\textbf{{B}}=\sqrt{\left|\textbf{{B}}_{\textbf{x}}\right|^{2}+\left|\textbf{{ B}}_{\textbf{y}}\right|^{2}}, \tag{6}\]  
where \(\textbf{{I}}_{i}\) is the phase current carried by the conductor located at \((x_{i},y_{i})\), \(\textbf{{I}}_{\textbf{sh},i}\) is the circulating current in the shield located at \((x_{i},y_{i})\), \(r_{i}=\sqrt{\left(x-x_{i}\right)^{2}+\left(y-y_{i}\right)^{2}}\) represents the distance between the conductor/shield and the measurement point \((x,y)\) and \(\mu_{0}=4\pi\times 10^{-7}\) H/m is the magnetic permeability of the free space. If considering the same phase sequence (\(\textbf{{I}}_{1}=I\angle-120^{\circ}\), \(\textbf{{I}}_{2}=I\angle 0^{\circ}\) and \(\textbf{{I}}_{3}=I\angle 120^{\circ}\)), it can be shown that the currents induced in the shields of the three-phase power cable in trefoil formation (Figure 1a) have the general form [21; 26]:  
\[\textbf{{I}}_{\textbf{sh},i}=-\textbf{{I}}_{i}\frac{jX}{R_{sh}+jX}, \tag{7}\]  
where \(X=2\ \times\ \omega\ \times\ 10^{-7}\ln(s/r_{sh})\) represents the shield reactance per unit length of cable, in \(\Omega\) /m, and \(R_{sh}\) is the shield DC resistance per unit length of cable, in \(\Omega\)/m. In the expression of \(X,s\) is the spacing between the centers of the (adjacent) conductors, in m, and \(r_{sh}\) is the mean of the outer and inner radii of the shield, in m.  
Similarly, for balanced phase currents, the currents induced in the shields of the three-phase power cable in flat formation (Figure 1b) have the following expressions [21; 26]:  
\[\textbf{{I}}_{\textbf{sh},i}=\frac{\textbf{{I}}_{2}}{2}\left[\frac{Q^{2}}{R_{ sh}^{2}+Q^{2}}-\frac{\sqrt{3}R_{sh}P}{R_{sh}^{2}+P^{2}}+j\left(\frac{R_{sh}Q}{R_{ sh}^{2}+Q^{2}}+\frac{\sqrt{3}P^{2}}{R_{sh}^{2}+P^{2}}\right)\right]; \tag{8}\]  
\[\textbf{{I}}_{\textbf{sh},2}=-\textbf{{I}}_{2}\left(\frac{Q^{2}}{R_{sh}^{2}+Q ^{2}}+j\frac{R_{sh}Q}{R_{sh}^{2}+Q^{2}}\right); \tag{9}\]  
\[\textbf{{I}}_{\textbf{sh},3}=\frac{\textbf{{I}}_{2}}{2}\left[\frac{Q^{2}}{R_{ sh}^{2}+Q^{2}}+\frac{\sqrt{3}R_{sh}P}{R_{sh}^{2}+P^{2}}+j\left(\frac{R_{sh}Q}{R_{ sh}^{2}+Q^{2}}-\frac{\sqrt{3}P^{2}}{R_{sh}^{2}+P^{2}}\right)\right], \tag{10}\]  
Figure 8: **The 2D analytical calculation of the magnetic flux density generated by a three-phase power cable with solid bonding.