# FEM Analysis and Sensor-Based Measurement Scheme of Current Distribution for Grounding Electrode
## 3 Calculation and Analysis of Current Distribution
### Calculation of Electric Field  
According to the distribution of soil electrical conductivity in the frozen soil environment in Inner Mongolia, two typical frozen soil models were established. In this article, the measurement cross-section of current density and potential distribution is located at 0.05 m below the ground surface and the parameters used in the simulation are shown in Tables 2 and 3.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Grounding** & **Radius (m)** & \begin{tabular}{c} **Length/** \\ **Thickness (m)** \\ \end{tabular} & **Buried Depth (m)** & \
\begin{tabular}{c} **Model** \\ **Size (m)** \\ \end{tabular} &
\begin{tabular}{c} **Injection** \\ **Current (A)** \\ \end{tabular} \\ \hline Linear & 0.002 & 0.2 & 0.05 & \(0.6\times 0.6\times 0.4\) & 10 \\ Cross-shaped & 0.002 & 0.2 & 0.05 & \(0.6\times 0.6\times 0.4\) & 10 \\ Ring-shaped & 0.1 & 0.002 & 0.05 & \(0.6\times 0.6\times 0.4\) & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Lab parameters for two typical frozen soil models.  
Figure 3: Frozen soil model with X and Y coordinate directions.  
Figure 2: Two typical frozen soil models and three typical grounding electrode sizes. (**a**) Three-layer soil model with three different grounding electrodes, the thickness of each layer is the same; (**b**) mixed-layer soil model with three different grounding electrodes, the soil width of the vertical layered soil is 0.2m; (**c**) the dimensions of linear, cross-shaped, and ring-shaped grounding electrodes are marked in detail.  
The current density and potential distribution in the three-layer soil with linear, cross-shaped, and ring-shaped grounding electrodes are shown in Figure 4.  
The current density and potential distribution of the linear grounding electrode are the highest, reaching 69.1 A/m\({}^{2}\) and 492 V, respectively. Meanwhile, the current and potential of the ring-shaped grounding electrode are the lowest, reaching 43.4 A/m\({}^{2}\) and 238 V, respectively.  
Furthermore, the current density and potential distribution under the mixed-layer soil model for linear, cross-shaped, and ring-shaped grounding electrodes are shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Relative Dielectric Constant** & **Conductivity (S/m)** \\ \hline \(\rho_{1}\) & 15 & 0.04 \\ \(\rho_{2}\) & 15 & 0.016 \\ \(\rho_{3}\) & 15 & 0.000043 \\ Grounding electrode & 1 & 59,900,000 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters for the three-layer soil model.  
Figure 4: Current density and potential distribution with different grounding electrodes. (**a**) Current density for linear grounding electrode; (**b**) current density for cross-shaped grounding electrode; (**c**) current density for ring-shaped grounding electrode; (**d**) potential distribution for linear grounding electrode; (**e**) potential distribution for cross-shaped grounding electrode; (**f**) potential distribution for ring-shaped grounding electrode.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Relative Dielectric Constant** & **Conductivity (S/m)** \\ \hline \(\rho_{1}\) & 15 & 0.04 \\ \(\rho_{2}\) & 15 & 0.016 \\ \(\rho_{3}\) & 15 & 0.000043 \\ Grounding electrode & 1 & 59,900,000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters for the mixed-layer soil model.  
Similarly, under the mixed-layer soil model, the current density and potential distribution of the linear grounding electrode are still the highest, which are 63.1 A/m\({}^{2}\) and 423 V, respectively. Meanwhile, the values of the cross-shaped grounding electrode are 49.5 A/m\({}^{2}\) and 291 V and the values of the ring-shaped grounding electrode are 37.2 A/m\({}^{2}\) and 199 V, respectively.  
We conclude that the ring-shaped grounding electrode has the best dispersion characteristics, the cross-shaped grounding electrode has the second best dispersion characteristics, and the linear grounding electrode has the worst dispersion characteristics.