On the Validity of Approximate Formulas for the Evaluation of the Lightning Electromagnetic Fields in the Presence of a Lossy Ground

The accuracy of approximate formulas for the evaluation of lightning electromagnetic fields above and inside a lossy ground is investigated taking as reference full-wave numerical results obtained using a parallel implementation of the finite-difference time-domain (FDTD) technique. The return stroke channel is modeled using the antenna theory model with fixed inductive loading which is appropriately incorporated into the FDTD algorithm. First, the validity of the Cooray-Rubinstein (CR) formula for the evaluation of above-ground horizontal electric field is evaluated. The obtained results confirm the recent findings of Cooray according to which it is important to include propagation effects over a lossy ground in the magnetic field used in the CR formula, except at very close distance ranges for which propagation effects can be neglected. Second, we test the validity of the Cooray formula for the evaluation of underground horizontal electric fields. The Cooray formula allows for the evaluation of the underground horizontal electric field starting from the horizontal electric field at the air-soil interface. This latter can be calculated using either 1) the original CR formula in which the magnetic field at ground level is evaluated assuming the ground as a perfect conductor or 2) taking into account propagation effects in the computation of the magnetic field. We show that the assumption of a perfectly conducting ground for the evaluation of the magnetic field at ground level might lead to inaccuracies in the Cooray formula. However, the use of the Cooray formula taking into account propagation effects in the computation of the magnetic field results in acceptable predictions of the underground horizontal electric fields.