## 3 Conductor Resistance under AC Supply
### Monometallic and Non-Magnetic Stranded Conductor under Power Frequency Supply  
Under power frequency supply, the current density within a stranded conductor is not uniform due to skin and proximity effects as well as variable contact resistance between strands [1]. As a result, at the same temperature, the resistance of a monometallic and non-magnetic conductor is greater under alternating current than under direct current. The magnitude of the skin effect depends on the supply frequency, the conductivity of the material, and the geometry of the conductor. Therefore, the radial current density distribution in the aluminum section and the power losses in the magnetic core are frequency dependent. The effective transverse resistivity also has a profound effect on the uneven current distribution of stranded conductors. It is greatly affected by the inter-strand contact resistance, the contact resistance at the crossings of the strands, and the characteristics of the contact surface layer, such as the degree of oxidation, the presence of grease, or the crys \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Conductor Type** & **Wires in Steel Layers** &
\begin{tabular}{c} **Strands in** \\ **Al Layers** \\ \end{tabular} & **Lay Ratio Steel Layers** & **Lay Ratio Al Layers** \\ \hline AAAC 4 layers & - & 61 (1/6/12/18/24) & - & 15/13.5/12.5/11 \\ ACSR 3 layers & 7 (1/6) & 54 (12/18/24) & 19 & 15/13/11.5 \\ ACSR 2 layers & 7 (1/6) & 26 (10/16) & 19 & 14/11.5 \\ ACSR 1 layer & 7 (1/6) & 10 & 19 & 14 \\ ACCC/TW & - & 36 (8/12/16) & - & 15/13.5/11.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Lay ratios of the analyzed cables according to the EN 50182 standard [14]. talline structure [9]. Therefore, the condition and the size of the contact area between the strands play an important role, along with the compression tension to which the conductor is subjected [15] and the angle of intersection between the strands [9]. There is an exact solution for the AC resistance of an infinitely long solid non-magnetic conductor of circular cross-section and also for an infinitely long tubular round conductor [16], so these solution take into account the skin effect. There is also an approximate analytical solution for two infinitely long parallel conductors of circular cross-section forming a single-phase system [16; 17], so this solution takes into account both skin and proximity effects. However, there are no exact methods for stranded conductors and for conductors with a magnetic core, so that the methods found in the technical literature [15] are based on many assumptions and the calculation process is complex. For stranded aluminum conductors with or without a magnetic core, in the 20-50 mm diameter range, the skin effect increases the resistance by approximately 1% and 10%, respectively [1]. In the case of having tabulated values of \(r_{AC}\) at two temperatures \(T_{2}>T_{1}\), the IEEE Std-738 for calculating the current-temperature of bare overhead conductors [18] suggests determining the conductor resistance at a given temperature \(T_{avg}\), where \(T_{1}\leq T_{avg}\leq T_{2}\), by applying the following linear fit:  
\[r_{AC}(T_{avg})=\frac{r_{AC}(T_{2})-r_{AC}(T_{1})}{T_{2}-T_{1}}.(T_{avg}-T_{1 })+r_{AC}(T_{1})\left[\Omega/\mathrm{m}\right] \tag{6}\]  
Note that when using Equation (1), the tabulated values of \(r_{AC}\) should include the skin effect, as well as the effects of the lay ratios and the magnetic core. However, some conductor manufacturers only provide the value of \(r_{DC}\) at 20 \({}^{\circ}\)C. It should be noted that the conductor temperature is not homogeneous within the cross-section, as it varies radially, with a maximum at the center and a minimum at the surface, so this effect must be considered as it affects the value of \(r_{AC}\).