Lightning Electromagnetic Field Calculations in Presence of a Conducting Ground: the Numerical Treatment of Sommerfeld Integrals

This chapter deals with lightning electromagnetic field calculations in presence of a lossy ground, tackling the problem of the efficient and accurate numerical computation of the so-called Sommerfeld’s integrals appearing in the electromagnetic field expressions. From a mathematical standpoint, the Sommerfeld’s integrals are improper integrals (i.e. over a semi-infinite domain) of complex functions characterized by the presence of branch points. The coexistence in the integrand of highly oscillating Bessel functions and of integrable singularities are the main causes of numerical troubles and slow convergence. Here, a method is discussed to treat them without resorting to any kind of approximated formulas. The method is based on a proper subdivision of the integration domain and on the analytical determination of a suitable upper bound for the error due to the integral truncation. The developed analysis refers to the following main configurations: lightning event on 1) homogeneous lossy ground with constant conductivity and permittivity; 2) homogeneous lossy ground with frequency-dependent soil electrical parameters; 3) stratified lossy ground. In all the cases, the exact field expressions are derived starting from the Maxwell equations both in air and in the ground.