Evaluation of lightning electromagnetic fields over a lossy ground

We present a new method for evaluating the electromagnetic field created by a lightning discharge under the assumption of a finite conductivity ground. In this case the expressions for the vertical and radial components of the electric field and for the azimuthal component of the magnetic one involve the so-called Sommerfeld integrals, which require a huge computational effort due to their characteristics of slow convergence. From a mathematical standpoint the Sommerfeld's integrals are improper integrals (i.e. over a semi-infinite domain 0 < x <infinity) of complex functions characterized by the presence of two branch points, occurring respectively when x is equal to the air and the earth wavenumbers k and k(E). The presence in the integrand of the highly oscillating Bessel function and of an integrable singularity are the main causes of numerical problems. The proposed algorithm allows the extraction of the singularity and performs part of the integral analytically. Consequently, a standard quadrature formula can be applied to the remaining term. The method has been developed in the frequency domain and, for the evaluation of each field component, about 1000 Sommerfeld integrals are necessary, each of them requiring about six seconds of CPU time on a Pentium IV, 1.6 GHz PC. This method has been numerically tested in the simulation of a straight vertical lightning channel electromagnetic field, assuming typical values for the ground conductivity. In particular, the field has been evaluated at different distances from the channel in order to point out the distance ranges where conductivity effects cannot be neglected. All the simulations have been carried out assuming, for the channel base current, the well-known Heidler's function and for the spatial-temporal distribution of the lightning current along the return stroke channel (i.e. the current return-stroke model) the classical MTLE expression.