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Mutual impedance calculation between buried conductors of finite length

Konstantinos Rallis (Department of Mechanical Engineering, University of Western Macedonia, Kozani, Greece)
Theodoros Theodoulidis (Department of Mechanical Engineering, University of Western Macedonia, Kozani, Greece)



The paper aims to develop expressions for calculating the mutual impedance between isolated conductors buried in homogeneous earth. The conductors have finite length and arbitrary position.


The conductors are represented by the use of elementary electric dipoles. Well‐known existing expressions are employed for the electric field of these dipoles. The induced voltages are evaluated and the final expressions for the mutual impedance are derived. The resulting expressions involve infinite double integrals, evaluated by using adaptive quadratures that are, however, time consuming. Therefore, an alternative approach is followed involving Sommerfeld integrals (SI) for representing the electric field of a dipole and a recently devised method for computing the SI, in the spatial domain, by using calculation of discrete complex images.


Final expressions for parallel and perpendicular conductors were derived and numerical results for several values of frequency, conductors' length and horizontal distance between them, were produced. Comparison to results produced with the well‐known Pollaczek formula showed excellent agreement.

Research limitations/implications

For future research, it is possible to use the developed expressions for earthing systems study, where the grounding grid is discretized and the moment method is invoked.


Currently, the formulas used for calculating mutual impedance are valid for parallel conductors of infinite length. With the present work, accurate expressions are given for finite length and arbitrary horizontal positioned conductors. In addition, the use of SI and the discrete complex image method results in a rapid and efficient tool for massive impedance calculations.



Rallis, K. and Theodoulidis, T. (2011), "Mutual impedance calculation between buried conductors of finite length", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 30 No. 4, pp. 1248-1259.



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