Technical Digests - Currents and Potentials along Leaky Ground-Return Conductors

* Digest of a paper to be presented at the Winter Convention of the American Institute of Electrical Engineers, New York, January 25-29, 1937, and published in full in Electrical Engineering, Vol. 55, No. 12, pp. 1338-1346, December, 1936. 110 CURRENTS AND POTENTIALS ALONG CONDUCTORS 111 For conductors with large leakance, these simplifying assumptions are not justified. When the earth resistivity is uniform or varies with depth only, the electric force may be formulated in terms of the current along the entire length of the conductor, in which case the usual differential equation for the current is replaced by an integrodifferential equation. A general solution of this equation and for conductor and earth potentials has been obtained. For homogeneous earth, rigorous as well as approximate solutions of special cases of interest in connection with the railway electrification problems mentioned above have also been derived. One of these cases is of general interest since it may be regarded as fundamental to the solution of the general case of an arbitrary impressed electric force along the conductor. In this case a voltage 2V(0) is impressed across a break in the conductor at a certain point, which may be taken as the origin. The conductor current and potential are given by rather complicated integrals, which, in order to obtain practical formulas, may be expanded in series as: I(x) = h(x) + J (*); /,(*) = hi{x) + /«(*), (1) 2 V(x) = Vi(x) + Vi(x); V (x) = Vn{x) + V (x), 2 22 (2)