Series expansions for line series impedances considering different specific resistances, magnetic permeabilities, and dielectric permittivities of conductors, air, and ground

A power series and for very high frequencies, an asymptotic series expansion for the calculation of the self and mutual line impedances are derived and verified. The different specific resistances, magnetic permeabilities, and dielectric permittivities of the three relevant materials, namely, conductor, air, and ground, are completely taken into account, which is not completely considered in the available literature. Breaking up the power series after the first terms results in simplified expressions near power frequency. The defining equations for the coefficients of the power series expansion are given by infinite series expansions. Their limiting values, as well as the frequency curves of the self and mutual resistances and inductances, are calculated for different magnetic ground permeabilities. Due to the current displacement in the lossy ground, the resistances increase with frequency whereas the inductances decrease. With increasing permeability of the ground, the resistances and inductances increase due to the higher magnetic flux in the ground.