[1] In this paper, the developed formulation, which we shall call the ''reference'' one, is used to assess the validity of the most popular simplified approach for the calculation of the lightning electromagnetic field over a conducting earth, namely, the Cooray-Rubinstein (CR) approximation. This formula provides a simple method to evaluate the radial component of the electric field which is the component most affected by the finite ground conductivity and which plays an important role within the Agrawal et al. (1980) field-to-transmission line-coupling model. Several configurations are examined, with different values for the ground conductivity and different field observation points. A thorough analysis of all the simulated field components is carried out and comparisons are also made with the ''ideal'' field, namely, the field that would be present under the assumption of perfectly conducting ground. It is shown that for channel base current typical of subsequent strokes and for very low conductivities, the CR formula exhibits some deviations from the reference one but it still represents a conservative estimation of the radial field component, since it behaves as un upper bound for the exact curve. The developed algorithm is characterized by fast performances in terms of CPU time, lending itself to be used for several applications, including a coupling code for lightning induced overvoltages calculations.Citation: Delfino, F., R. Procopio, M. Rossi, F. Rachidi, and C. A. Nucci (2008), Lightning return stroke current radiation in presence of a conducting ground: 2. Validity assessment of simplified approaches,

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[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.

The number of power installations lying underground has been increasing in the last few years, and such devices are very sensitive to the effect of the lightning electromagnetic fields, due to the massive presence of power electronics. As a consequence, the scientific community has devoted much effort in the direction of a more accurate modeling of underground lightning fields and their coupling to cables. The exact expressions of the underground lightning fields have been derived by Sommerfeld decades ago. However, their numerical evaluation has always been a hard task because of the presence of slowly converging improper integrals. In the past, some approximate formulas have been derived, which have been included in field-to-transmission line coupling models to estimate the effect of lightning on buried cables. In this paper, an efficient algorithm for the evaluation of the Sommerfeld expression for underground fields is presented, and its mathematical features are discussed. The numerical treatment of the Sommerfeld integrals is based on a proper subdivision of the integration domain, the application of the Romberg technique, and the definition of a suitable upper bound for the error due to the integral truncation. The remarkable efficiency in terms of CPU time of the developed algorithm makes it possible to use it directly in field-to-buried cable coupling simulation codes. Finally, the developed algorithm is used to test the validity of the Cooray's simplified formula for the computation of underground horizontal electric field. It is shown that predictions of the Cooray's formula are in good agreement with exact solutions for large values of ground conductivity (0.01 S/m). However, for poor conductivities (0.001 S/m or so), Cooray's expression yields less satisfactory results, especially for the late time response.