Author: Arnold A.H.M.
Title: The theory of sheath losses in single-conductor lead-covered cables
A theory of the losses in the lead covering of an insulated cable is developed which combines reasonable accuracy with simplicity. Formulae are obtained for the induced sheath voltages and for the losses with bonded sheaths. It is shown that when the cables of a 3-phase system are arranged in a plane the sheath losses occurring in the two outer cables are not equal, although previous writers have usually assumed this to be the case. A numerical example is given in Appendix 2 to show the possible differences which may exist. Experimental work, verifying the results obtained both for single-phase and for 3-phase currents, with two arrangements of cables, symmetrical and in a plane, is given in Appendix 1. The formulae obtained are given in a convenient form for reference in Section (5). The examples of the numerical values of the sheath voltages and sheath losses, given in Appendix 2, have been carefullychosen to show the method of applying the formulae developed in the paper, and also to give an indication of the magnitude of the effects involved. It is difficult, with formulae containing so many variables, to give a few numerical examples which will cover the whole range of conditions met with in practice, but the following examples will give an idea of the importance of the effects considered. The frequency assumed is 50 cycles per second. (A) Sheath voltages. (Sheaths bonded together at one end only.)?These voltages are directly proportional to the line current and to the length of the line for all arrangements of cables. The only other variable, at a given frequency, is the ratio of the spacing between axes of adjacent cables to the mean diameter of the lead sheath. The voltages are directly proportional to the logarithm of this ratio. As an illustration of the magnitude of the effects, values are given below for a line current of 1 000 amperes and a line length of 1 mile. Two spacings are considered, the first with the cables almost in contact, and the second wi- th their axes 5 diameters apart. The diameter referred to is the mean diameter of the sheath. (i) Single-phase.?The voltage between sheaths varies from about 150 volts, when the sheaths are almost touching, to 470 volts when the cables are 5 diameters apart. (ii) Three-phase?Cables arranged at the corners of an equilateral triangle.?The voltage between sheaths varies from about 130 volts, when the sheaths are almost touching, to 400 volts when the cables are 5 diameters apart. (iii) Three-phase?Cables arranged in a plane with the middle cable equidistant from the two outer cables.?The voltage between the sheaths of the two outer cables varies from about 250 volts, when adjacent sheaths are almost touching, to 530 volts when adjacent cables are 5 diameters apart. The corresponding figures for the voltage between the sheath of each outer cable and the sheath of the middle cable are 150 volts and 410 volts respectively. (B) Sheath losses. (Sheaths bonded together at both ends.)?The ratio of these losses to the copper losses does not directly depend on the line current, but it increases as the size of the cable is increased. In general, the losses do not become serious until the line current exceeds 500 amperes. The losses, also, do not increase quite in proportion to the spacing of the cables, that is, the curve of variation of sheath losses with spacing is slightly convex to the spacing axis in all cases, but for spacings up to 5 diameters the variation is approximately linear. The convexity of the curve increases as the size of the cables is increased and also as the spacing is increased. The figures given below are for a cable with a core resistance of 0-04 ohm per mile and with a sheath resistance of 0.3 ohm per mile. Such a cable would be capable of carrying a current of about 1 000 amperes. (i) Single-phase.?With the cables in contact the sheath losses are equal to 55 per cent of the copper losses, and with the axes of the cables spaced 5 diameters apart the sh