Hemmatian et al. / J Zhejiang Univ Sci A 2009 10(8):1191-1198 1191 Earth return path impedances of underground cable for three-layer earth B. HEMMATIAN, B. VAHIDI †‡ , S. H. HOSSEINIAN (Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran) † E-mail: vahidi@aut.ac.ir Received Aug. 12, 2008; Revision accepted Oct. 20, 2008; Crosschecked June 10, 2009 Abstract: One of the factors that affect the parameters of an underground cable is earth return path impedance. Pollaczek de- veloped a formula for the case of one-layer (homogenous) earth. But in practice the earth is composed of several layers. In this study we develop a new formula for earth return path impedance in the case of a three-layer earth. To check the accuracy of the obtained results, a comparison has been made with the finite element method (FEM). A comparison between the results of the Pollaczek formula and results of the obtained formula for a three-layer earth has been made, showing that the use of the Pollac- zek formula instead of the actual formula can cause serious errors. Key words: Underground cables, Earth return path impedance, Earth stratification, Finite element method (FEM) doi:10.1631/jzus.A0820598 Document code: A CLC number: TM75 INTRODUCTION Nowadays, there are such a variety of cable de- signs that developing a general formula for calculat- ing the electrical parameters of all cables is difficult, if not impossible (Dommel, 1986). One of the most important factors that affect the impedance of the cable is the earth return path. The earth-return path self impedance of underground cables can reach up to 90% of the cable impedance, and the mutual im- pedance of underground cables, which in the case of underground cables is very significant, is contributed totally by the earth return path (Nguyen, 1998a). A variety of studies have been made to obtain the earth return path contribution to transmission line impedances. The first attempts in this field were made in 1926. Carson (1926) obtained a correction term to consider the earth return path. But the Carson formula is true just for overhead transmission lines. Pollaczek (1926) developed a new formula for earth return path impedance, which is applicable for both overhead transmission lines and underground cables. The Pollaczek formula is used only for ho- mogenous earth (Pollaczek, 1926). But the earth is composed of several layers (Lagace et al., 1996; Zhang et al., 2005). Sunde (1968) extended a ho- mogenous earth solution and proposed formulas in the case of a two-layer earth. Other formulas are proposed for two- and three-layer earth cases, but all of them are used for overhead lines (Ametani and Schinzinger, 1976; Nakagawa and Iwamoto, 1976; Papagiannis et al., 2005b). Tsiamitros et al.(2005) proposed a formula for earth return path impedance of underground cables in the case of a two-layer earth and developed a formula for the case of a three- layer earth. All of the formulas developed for earth return path impedance include expressions with complex infinite integrals (Nguyen, 1998a). Some algebraic series were proposed for calculation of the Carson formula (Carson, 1926) and there are also some ap- proximations for this formula (Alvarado and Betan- court, 1983; Davies et al., 1995; Noda, 2006). But integration of the Pollaczek formula is not so simple and there is no closed solution for Pollaczek’s inte- gral (Papagiannis et al., 2005a). Some closed-form Journal of Zhejiang University SCIENCE A ISSN 1673-565X (Print); ISSN 1862-1775 (Online) www.zju.edu.cn/jzus; www.springerlink.com E-mail: jzus@zju.edu.cn ‡ Corresponding author