This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1 Accurate and Approximate Evaluation of Power-Line Earth Impedances Through the Carson Integral Octavio Ramos-Lea˜ nos, J. L. Naredo, Senior Member, IEEE, F. A. Uribe, Senior Member, IEEE, and J. L. Guardado, Senior Member, IEEE Abstract—Earth impedances of aerial power-lines usually are calculated through methods and approximate formulas based on the Carson integral. This paper presents a methodology to eval- uate their accuracy and to serve as aid to select an appropriate method or formula in specific applications. Reference values for the Carson integral are first obtained through a numerical algo- rithm guaranteeing relative errors below 10 –9 . These values are first used to validate the general solution of Carson integral em- ploying an algorithm proposed by Theodoulidis. The reference val- ues are then used to produce maps of errors and of differences for the following methods: Carson series, Gary–Dubanton formulas, Alvarado–Betancourt formulas, and double symmetry-plane for- mulas. Improvements are further proposed for the Carson-series method and the Alvarado–Betancourt formulas. Finally, the ef- fectiveness of the proposed methodology is illustrated through an application example. Index Terms—Aerial power lines, Carson integral, Carson se- ries, Earth impedances, electromagnetic transients, exponential integral functions, Struve functions. I. INTRODUCTION T HE calculation of Earth impedances of aerial power-lines is made through methods and formulas that are largely based on the Carson integral. In his 1926 paper [1], Carson in- troduced this integral and provided its general solution in terms of Bessel and Struve functions [2]–[5]. The difficulties at that time to handle these special functions led Carson to produce two sets of complementary series intended for practical calcu- lations and to prescribe their application ranges. The Carson series have become the standard method for calculating aerial- line Earth impedances [6]. In 1969, Dubanton produced a simple formula for the self- Earth impedance of an aerial conductor by introducing the con- cept of complex depth within the method of electromagnetic images [7]. Dubanton’s formula is equivalent to a previous one obtained empirically by Sunde in 1949 [8]. In 1976, Gary Manuscript received December 4, 2016; revised February 14, 2017; accepted March 2, 2017. O. Ramos-Lea˜ nos and J. L. Naredo are with the Department of Electri- cal Engineering and Computer Sciences, Cinvestav, Zapopan 45019, Mexico (e-mail: octavio.ramos@polymtl.ca; jlnaredo@gdl.cinvestav.mx). F. A. Uribe is with the Electrical Engineering Department, Universidad de Guadalajara, Guadalajara 44100, Mexico (e-mail: uribe_felipe@yahoo.com). J. L. Guardado is with the Instituto Tecnologico de Morelia, Morelia 58120, Mexico (e-mail: lguarda@prodigy.net.mx). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2017.2679213 extended Dubanton’s method and proposed a simple formula for the mutual Earth-impedance of two aerial conductors [9]. The agreement between the Carson series and the Gary–Dubanton formulas was found to be very close in [9] and Gary suggested the possibility of these being the actual solution to the Carson integral. Nevertheless, in 1981, Deri et al. showed that these formulas are mere good approximations to the actual solution [10]. In passing, Deri et al. also extended the applicability of the Gary–Dubanton formulas to stratified soils. An improvement to the Gary–Dubanton formulas was pro- vided in 1983 by Alvarado and Betancourt [11] by adding extra terms to increase their accuracy. Another improvement in this direction was proposed by Pizarro and Erikson [12] and refined later by Noda [5]. It consists in the synthesis of Earth-impedance formulas by introducing two complex-depth symmetry planes into the method of electromagnetic images. Although, multilayer approximations to the Earth return impedance exists [13], most popular EMTP-type programs still use Carson’s uniform soil approximations to compute the Earth impedance of aerial transmission lines [6], [14]. Ametani [15] compared switching surges for three-layer soils with different resistivity combinations concluding that: 1) in all the cases, the largest surge value is only affected by the deepest soil layer; and 2) uniform soil approximations present similar re- sults to those obtained with multilayer soils when soil resistivity ρ 1 ≈ ρ 3 <ρ 2 or ρ 1 <ρ 2 <ρ 3 or ρ 1 >ρ 2 >ρ 3 . Showing that, uniform soil approximations are still valuable and widely used tools. Approximate methods and formulas for the uniform soil Earth-impedance calculation often are evaluated through their comparison with the Carson series available from the Line Pa- rameter Program of EMTP [6]. One problem here is that the ac- curacy of these series had not been well determined. Mainly for this reason, various power-system specialists resort to solving the Carson integral through numerical methods [5], [16], [17]. Alternatively, attempts to evaluate Earth impedances directly by the general solution given by Carson in [1] have often failed. The main reason for this is that it involves differences between Struve and Bessel functions that can attain very large and similar val- ues; in consequence, their differences are prone to magnifying round-off errors. Nevertheless, as it is discussed in Section III-C, this problem has been circumvented recently in an effective manner with an algorithm proposed by Theodoulidis [18]. The first and main objective of this paper is to establish ac- curacy ranges for the methods and formulas of most common 0018-9375 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.