A double logarithmic approximation of Carson´s ground-return impedance

This paper proposes a simple closed-form formula of the ground-return impedance of horizontal parallel wires above a lossy ground plane. It is derived by a double logarithmic approximation of the integral term with a semi-infinite interval in Carson´s ground-return impedance formula. The proposed double logarithmic approximation gives better accuracy compared to a similar approximation proposed by Pizarro and Eriksson, and this is achieved by an advanced optimization technique used for determining the coefficients in the double logarithmic approximation and also by incorporating an additional variable which minimizes errors. In this paper, it is shown, using practical examples, that the derived formula gives more accurate results compared with other existing approximate formulas. It is also shown that a graphical interpretation of the double logarithmic approximation is to replace a homogeneous lossy ground with a pair of perfectly conducting return planes located at different complex depths.