A closed-form approximation for ground return impedance of underground cables

Calculation of the ground return impedance generally requires the evaluation of a complex infinite integral. Converging infinite series that usually converge to the exact value have been known for many years, and most handbook formulas were derived from these series, with only the first one or two terms retained. Several approximations with closed-form solutions have been proposed recently for overhead lines, but nothing seems to have been done in the case of underground cables. This paper describes a complete, simplified closed-form approximation for the self and mutual impedances of undergound cables. The simplified equations are first presented and then related to Pollaczek's and Wedephol's equations. Finally, the errors, which in most cases are very satisfactory, are numerically evaluated.