## 3 Conductor Resistance under AC Supply
### Bimetallic Stranded Conductor with Magnetic Core under Power Frequency Supply  
For magnetic core conductors, \(r_{AC}\) depends not only on temperature and power frequency, but also on the magnitude of the electric current [18]. Due to the much lower electrical conductivity of the steel core, most of the AC current flow in ACSR conductors is through the aluminum strands [19]. The aluminum strands are wound helically around the magnetic core, creating an axial AC magnetic flux within the steel core, similar to a solenoid, which increases with the magnitude of the electric current [18]. The adjacent layers of helically stranded conductors are stranded in opposite directions so that their respective currents produce magnetic fields whose axial components are in opposite directions and tend to cancel each other out. Under power frequency supply, the axial component of the total magnetic field induces eddy currents and hysteresis losses in the magnetic core, while modifying the current distribution of the different aluminum layers. The magnetic coupling of the current through the steel core is known as the transformer effect [1], and its intensity is determined by the magnitude of the axial magnetic flux. The amplitude of the axial magnetic flux depends on the magnitude of the electric current, the supply frequency, the magnetic permeability of the magnetic core, and the lay length [20]. The magnetic permeability of the steel core depends on its temperature and tensile stress [21], which affect the AC resistance of steel core stranded conductors [1]. The cancellation of the axial component of the magnetic field generated by the different aluminum layers is not possible in a single-layer aluminum conductor, and it is maximum in two-layer aluminum conductors. Therefore, this cancellation is generally more important in conductors with an even number of layers than in conductors with an odd number of layers [22; 23; 24]. As a result, the core losses and the uneven distribution of current density among the different aluminum layers are more pronounced in ACSR conductors with an odd number of aluminum layers than in conductors with an even number of aluminum layers [1; 25]. Therefore, the effect of the core on \(r_{AC}\) depends on the topology of the stranded conductor. It is very small for two-layer ACSR conductors, maximum for single-layer conductors, where \(r_{AC}\) increases up to 20%, and moderate for three-layer ACSR conductors, where \(r_{AC}\) increases up to 5% [18]. However, these values depend on the current density through the aluminum strands, so if tabulated values of \(r_{AC}\) are used, they must be transformed using correction curves to obtain a multiplier factor that depends on the current density [18]. However, this method is impractical because it requires detailed conductor information, which is not always available. According to [20], the \(r_{AC}/r_{DC}\) resistance ratio of three-layer ACSR conductors can be found by correcting \(r_{DC}\) for temperature and skin effect at power frequency, but for conductors with an odd number of aluminum layers, a current-dependent resistance multiplier \(r_{AC,total}/r_{AC,skin,T}\) for core loss must also be applied, where \(r_{AC,total}\) is the total AC resistance and \(r_{AC,skin,T}\) is the AC resistance considering only temperature and skin effect. However, the resistance multiplier factor \(r_{AC,total}/r_{AC,skin,T}\) obtained from the experimental data of different three-layer conductors at different current densities shows a wide dispersion of results. In many papers, the conductor is considered as a homogeneous heat source, and the ohmic losses are calculated based on its DC resistance, as it can be found in manufacturers' datasheets and relevant standards. However, skin and proximity effects, the transformer effect, and the associated uneven current distribution between aluminum layers or radial temperature gradients cause temperature differences of about 10% between calculated and measured temperature values [1].