Transpositions and the Calculation of Inductance from Geometric Mean Distances

The calculation of the inductance of power transmission lines by the method of geometric mean distances is discussed in most transmission line textbooks. The assumption is usually made therein that the single-phase case for which the return currents are established uniformly equal to -I/n in the return n conductors may be expanded automatically to encompass also the transposed 3-phase case even though the return currents of the other two phases obviously do not satisfy the -I/n condition. This paper demonstrates with mathematical rigour that the inductance per phase can be calculated by the geometric-mean-distance method, and further, it shows the necessary and sufficient conditions for the system of transpositions that must be satisfied for the general 3-phase case. For special 3-phase systems, possessing symmetry among certain spacing distances some of these transpositions, however, may then be unnecessary.