Time-domain low frequency approximation for the off-diagonal terms of the ground impedance matrix

The ground impedance is one of the parameters needed for field-to-transmission line coupling calculations. In the case of multiconductor lines this ground impedance is a full matrix with diagonal (self-impedance) and off-diagonal (mutual impedance) terms, The expression for the mutual ground impedance between two conductors i and j has been derived by Sunde (1968) and a low frequency approximation by Carson (1926). The general expression for the ground impedance matrix terms in the frequency-domain does nor have an analytical inverse Fourier transform. Therefore, the elements of the "transient ground resistance" matrix in time-domain should be, in general, determined using an inverse Fourier transform algorithm. In low-frequency approximation, however, Timotin (1967) and Mok and Costache (1992) has found an analytical inverse Fourier transform for the ground self-impedance. We extend the Timotin formula to the off-diagonal terms of the ground resistance matrix.