The alternating-current resistance of tubular conductors

Simple formulae are developed for the effective resistance to alternating currents of tubular conductors when used under the following conditions:¿ (a) Isolated conductor. (b) Single-phase system of 2 concentric conductors. (c) Single-phase system of 2 conductors of equal size laid parallel to each other. (d) Three-phase system of 3 conductors of equal size laid parallel to each other in a plane, with the middle conductor midway between the outer conductors. (e) Three-phase system of 3 conductors of equal size laid parallel to each other in triangular formation, with equal distances between conductors. In cases (a) and (b) rigid formulae are already available. The formulae developed in this paper are shown not to differ from the rigid formulae by more than a few parts in 1 000, while the reduction of labour in using the simple formulae is probably of the order of one hundredfold. In cases (c), (d), and (e), no rigid formulae are available, and the solutions put forward in this paper are the simplest and the most accurate which have yet been published. Experimental work has been undertaken to check the accuracy of the formulae and it has been found that the discrepancy between the experimental values and the calculated values is less than 3 per cent, provided that th´e ratio of diameter of conductor to spacing between axes of conductors is less than 0.85. This covers the great majority of cases occurring in practice. When the conductors are closer together than this, the errors in the formulae may be a little larger, but, even with the conductors almost touching, the greatest observed discrepancy between the experimental and calculated values of the ratio of a. c. resistance to d. c. resistance was only 6 per cent (11 per cent of the ratio of increase of resistance with a. c. to d. c. resistance).