Lightning induced disturbances in buried Cables-part I: theory

In this paper, we present a review of theoretical methods to compute lightning induced currents and voltages on buried cables. The evaluation of such induced disturbances requires the calculation of the electric field produced by lightning along the cable path. We show that the Cooray's simplified formula is capable of predicting accurately the horizontal electric field penetrating the ground, at distances as close as 100 m. Regarding the parameters of the buried cable, a comparison of several approximations of the ground impedance is presented. We show that the Pollaczek expression corresponds to the Sunde general expression, when the displacement current is neglected. The analysis shows also that all the proposed approximations provide very similar results for the considered range of frequencies (up to 30 MHz). Most of the approximate formulas neglect the contribution of the displacement current and, therefore, predict values for the ground impedance which tend to infinity at higher frequencies. This corresponds in the time domain to a singularity of the ground transient resistance at t=0. By analogy to the Sunde approximation for the ground impedance of overhead lines, we propose a logarithmic approximation for the ground impedance of a buried cable. In addition, unlike most of the considered approximations, the proposed formula has an asymptotic behavior at high frequencies; therefore, the corresponding transient ground resistance in the time domain has no singularity at t=0. It is also demonstrated that within the frequency range of interest, the wire impedance can be neglected, due to its small contribution to the overall longitudinal impedance of the line. The ground admittance, however, can play an important role at high frequencies (1 MHz or so) especially in the case of poor ground conductivity. The ground admittance needs to be taken into account in the calculation of lightning induced currents and voltages on buried cables. This is in contrast with the case of overhead lines in which its contribution is generally negligible, even in the MHz range. We also investigate the time-domain representation of field-to-transmission line coupling equations. The coupling model includes the effect of ground admittance which appears in terms of an additional convolution integral. An analytical expression for the ground transient resistance in the time domain is also proposed which is shown to be sufficiently accurate and nonsingular. Finally, we present a time domain solution of field-to-buried cable coupling equations using the point-centered finite difference time domain (FDTD) method, and a frequency domain solution using Green's functions. In our companion paper (Part II), we compare both solutions to experimental waveforms obtained using triggered lightning.