Contents I. Introduction
A reliable analysis of the tower-footing grounding (composed by counterpoises) transient behavior is of fundamental importance in evaluations of lightning performance of transmission lines [1], [2]. It has recently been shown that for such evaluations, the grounding transient behavior can be adequately represented by its impulse impedance () [3], [4]. is defined as the ratio between the peak values of the grounding potential rise (GPR), , and the current injected into the grounding (). This concept was introduced by Gupta and Thapar [5]. To determine , it is necessary to implement a consistent electromagnetic modeling that quantifies the wideband response of the grounding system. The modeling can be elaborated based on field [6]–[14], transmission line [15]–[20] or electric circuit [21]–[24] theories. The models’ solution is performed in the time [15]–[18], [22], [24] or frequency [6], [7], [9], [10], [12] domains, or both [13], [20]. It is not scope of this article to discuss advantages and disadvantages, limits of validity, etc., of such modeling and solution domains. In [25], for instance, these aspects are properly discussed. In this article, the well-known hybrid electromagnetic model (HEM) [10], [12] is implemented. The HEM is based on the field and circuit theories, with solution in the frequency domain [10], [12]. The integral equations that quantify the electromagnetic couplings are solved via method of moments (MoM) [26]. A crucial aspect in models of this nature corresponds to the segmentation process of the grounding electrode. In hybrid models, the size of the segment is ten times the radius of the electrode [6], [10], [12]. Other works adopt the segment size as a fraction of the minimum wavelength in the ground, (associated with the maximum frequency, fmax, typical of the current signal requesting grounding), such as [7], [8], [11] and from to [29]. Regardless of the segmentation adopted, the computational time to obtain the parameters that describe the transient grounding behavior may be significant [29]. Computational efficiency can be compromised even for simpler geometric configurations, decreasing: 1) with soil resistivity (), since the effective length ()
Consideration of is important in the discussion of computational efficiency. That is why his concept is presented here. The current dispersed to the ground along the grounding electrode shows nonuniform distribution. This nonuniformity is more pronounced at high frequencies. The high-frequency region corresponds to the lightning current wave front. Thus, in the first microseconds of the electromagnetic transient that is established in the electrode (due to the injection of the lightning current) the attenuation effects are intense. Therefore, the lightning current is greatly attenuated in this time interval. Associated with this attenuation there is a critical electrode length, such that in addition to this length there will be practically no current dispersion. This critical length is given the name of effective length, which depends on the soil resistivity and current waveform.
of the grounding electrode increases as increases [30], [31], and 2) in models whose segmentation is more restrictive (in the sense of a greater number of segments), as in the case of HEM. It is worth mentioning that accuracies of the computations MoM/HEM do not depend only on the conductor segmentation, but also on the choice of basis or expansion function of current along with a conductor segment [26]. Several functions can be used, among them the following stand out [27], [28]: piecewise constant, linear and sinusoid, and truncated cosine. However, in order to maintain all the characteristics of the HEM, with the exception of the segmentation process, the piecewise constant function is still used.