Lightning return stroke current radiation in presence of a conducting ground: 1. Theory and numerical evaluation of the electromagnetic fields

[1] The general theory describing the electromagnetic field radiated by a lightning stroke over a conducting ground is presented in this paper. The derivation of the Green functions necessary to solve the problem is discussed in detail, and the determination of the expressions for the electromagnetic field components is carried out in a form that minimizes the final computational costs. A method for the numerical evaluation of the electromagnetic field is then proposed, and it is shown that it can be used starting from any "engineering model'' representation for the lightning current distribution along the channel. Such method is based on a new efficient evaluation of the so-called Sommerfeld's integrals appearing in the electromagnetic field expressions, without resorting to any kind of approximated formulas for them. The numerical treatment of the Sommerfeld's integrals is characterized by a proper subdivision of the integration domain, the use of the Romberg technique and the determination of a suitable upper bound for the error due to the integral truncation. In the second part of this work it will be shown how the results provided by the developed theory can be used in order to assess the validity of the most common simplified approach for the calculation of the lightning radiation over a lossy ground plane.