## 2 Theoretical Study and Simulation Models  
Three-phase electrical power is a commonly used method to generate, transmit, and distribute alternating-current electrical power. This method is similar to the polyphase system, which is a common method of transmitting energy from electrical networks around the world. In this study, we assumed that four alternating currents \(I_{R}\), \(I_{S}\), \(I_{T}\), and \(I_{N}\) flow in four cables of length 2 L had different phase angles, with the phase shift of \(2\pi/3\) between each phase, with a relative phase angle between the phases of \(-120\) degrees. The magnetic field B at any point was calculated according to the Biot-Savart law:  
\[R_{i}=r_{i}\hat{r}-z^{\prime}\hat{z} \tag{1}\]\[dl^{\prime}\times R_{i}=dz^{\prime}\hat{z}\times(r_{i}\hat{r}-z^{\prime}\hat{z})=r_ {i}dz^{\prime}\hat{\phi} \tag{2}\]  
where index \(i\) = S, R, T, and N.  
Substitution of (2) into the Biot-Savart law gives:  
\[B_{i}=\int dB_{i}=\frac{\mu_{0}I_{i}\hat{\phi}_{i}}{4\pi}\int_{-L}^{L}\frac{r_ {i}dz^{\prime}}{\left(z^{\prime 2}+r_{i}^{2}\right)^{\frac{3}{2}}}=\frac{\mu_{0}I_{i }L\hat{\phi}_{i}}{2\pi r_{i}\sqrt{L^{2}+r_{i}^{2}}} \tag{3}\]  
Then the total magnetic field is:  
\[B=\sum_{i=A}^{C}B_{i}=\sum_{i=A}^{C}\frac{\mu_{0}I_{i}L\hat{\phi}_{i}}{2\pi r_{ i}\sqrt{L^{2}+r_{i}^{2}}} \tag{4}\]  
In general, three-phase systems are more economical, prompting many nano fabs to use three-phase power systems [38]. A three-phase power system provides greater power density than a single-phase system at the same voltage, making it a less-expensive process with a smaller wiring size. It is also easier to reduce harmonic currents and load balancing in three-phase systems. These systems are used for electric motors and high-power tools because they optimize the use of electrical capacity to increase power efficiency. The phase angle between each phase is not always zero, and depends on the load type [39]. One-dimensional finite element method (1-D FEM) simulations were performed for the examination of the three-phase, four-cable framework. This method is widely used for solving numerical differential equations arising in engineering and mathematical modeling. Conventional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential are among the typical problem areas in 1-D FEM simulations [40; 41; 42]. The cable assemblies were arranged into two types in our reproduction: a counterclockwise framework and a clockwise framework. These arrangements are shown in Figure 1, where R, S, T, and N represent the four alternating current cables with different phase angles, with a phase shift of 2\(\pi\)/3 between each phase, and a relative phase angle between phases of \(-\)120 degrees. The reason for this phase change is spatial displacement. The clockwise and counterclockwise framework arrangements help to increase understanding of the paper since the ideal permutation consists of the combination of these frameworks. Figure 1 shows how the optimal ideal permutation system is applied in a nano fab with a three-phase, four-cable system. Three-phase electric force is the basic process used to age, transmit, and disperse the rotary flux electric force. This strategy is similar to the polyphase framework, which is the basic technology for transmitting energy over a global power network [33]. Figure 2 shows a list of the most optimal and ideal permutation cases applicable in the nano fab according to our simulation results. In this study, the three-phase electric power cables (R, S, T, and N) are bound in a square shape as one group. A twelve-group system was simulated in this study, with different phase angles, with the phase shift of 2\(\pi\)/3 between each phase. The relative phase angle between the phases was \(-\)120 degrees for the optimization of electromagnetic influence in a high-technology nano fab. Figure 1: Diagram showing clockwise and counterclockwise cable system arrangement.