## 2 Cable Model and Effective Length in 3D Twisted Conductors
### Definition of Total Power Loss \(P_{\text{loss}}\) and AC Resistance \(R_{\text{AC}}\)  
Under alternating-current conditions, the current density in a conductor is no longer uniform. The skin effect drives the current towards the conductor's surface, reducing its effective cross-sectional area and increasing its AC resistance. Additionally, when multiple conductors are close to each other, their magnetic fields interact and cause current redistribution in each conductor; this proximity effect can further concentrate current in some regions and intensify power losses relative to direct current conditions. When an AC current \(I\) flows through a conductor of volume \(V\) and electrical conductivity \(\sigma\), the total power loss \(P_{\text{loss}}\) arises from resistive (Joule) heating due to the current density distribution \(J\), and can be expressed as an integral over the entire volume in 3D, as shown in (3).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & \(\mu_{r}\) & \(\rho\) **(\(\Omega\cdot\text{m}\))** & \(\sigma\) **(S/m)** \\ \hline Copper & \(1\) & \(1.7241\times 10^{-8}\) & \(5.8001\times 10^{7}\) \\ Air & \(1\) & \(1.0\times 10^{10}\) & \(1.0\times 10^{-10}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties used in simulations.  
\[P_{\text{loss}}\ =\ I^{2}\ R_{\text{AC}}\ =\ \int_{V}\left[J\right]^{2}\frac{1}{ \sigma}\,\mathrm{d}V\quad(\text{W}), \tag{3}\]  
where:  
* \(P_{\text{loss}}\ (\text{W})\) is the total power loss due to resistive heating;
* \(I\ (\text{A})\) is the total current flowing through the conductor;
* \(R_{\text{AC}}\ (\Omega)\) is the conductor's AC resistance;
* \(J\ (\text{A}/\text{m}^{2})\) is the current density at each point of the volume;
* \(\sigma\ (\text{S}/\text{m})\) is the electrical conductivity of the conductor material;
* \(\mathrm{d}V\) represents the infinitesimal volume element.  
The total power loss within the conductors is extracted using APDL's POWER function, which performs a volume integral of the power loss across all finite elements in the model. This approach ensures that all contributions to power dissipation, including those arising from skin and proximity effects, are accurately accounted for.  
Rearranging (3) to solve for \(R_{\text{AC}}\) and given that \(R_{\text{DC-2D}}\) is expressed in Ohms per metre (\(\Omega/\text{m}\)) in this study, it is necessary to normalise \(R_{\text{AC}}\) similarly to facilitate comparative analysis across different models. This normalisation is achieved by dividing the total \(R_{\text{AC}}\) by the effective twisted length \(l_{\text{twist}}\) as shown in (4).  
\[R_{\text{AC-2D}}\ =\ \frac{P_{\text{loss}}}{I^{2}l_{\text{twist}}}\ =\ \frac{1}{I^{2}l_{\text{twist}}}\int_{V}\left[J\right]^{2}\frac{1}{\sigma}\, \mathrm{d}V\quad(\Omega/\text{m}) \tag{4}\]  
This yields the AC resistance per unit length \(R_{\text{AC-2D}}\) for each specific model considering the actual traveled length of the conductors under a twisting configuration. Finally, to assess the relative impact of AC effects compared to DC behavior, the ratio of AC resistance to DC resistance (\(R_{\text{AC-2D}}/R_{\text{DC-2D}}\)) is calculated for each model. This ratio provides insight into the additional resistive losses introduced under alternating current conditions, which are critical for applications where skin and proximity losses are present.