302 BELL SYSTEM TECHNICAL JOURNAL 



permissible variation in diameter for ordinary toll cable is ± 1 per cent 

 which means a d-c. resistance variation of about ± 2 per cent. 



Still another cause of resistance variation is the presence of small quan- 

 tities of impurities in the copper which show up as a reduction of as much as 

 2 per cent in the conductivity. This causes trouble in calibrating tem- 

 perature-resistance curves in the laboratory setup. 



Finally, in a single reel length the outside pairs are longer than the in- 

 side pairs. The total pair-to-pair variation in resistance from the average 

 of the reel caused by all these factors amounts to about ± 3 per cent with 

 a standard deviation of about 1.5 per cent. 



Inductance 



The inductance of a circuit formed by two parallel wires closely spaced 

 relative to their length is 



r 2D 



L = 0.64374 2.3026 logio -j" + M^ 



X 10 ^ henrys per loop mile (5) 



where d, the wire diameter , and D, the separation of the wires, are measured 

 in the same unit; /x is the permeability, and 5 is a frequency factor. 



As is well known, the tendency of alternating currents to concentrate 

 on the surface of a wire reduces the magnetic flux within the wire and 

 decreases the internal inductance of the wire. This internal inductance 

 is given by the term jib in Equation (5). In like manner, the "proximity 

 effect" produces a concentration of current density in the adjacent portions 

 of the two wires of a pair. 



Another term might well be added to formula (5) to represent this 

 proximity effect. The procedure outlined by J. R. Carson on pages 625 

 and 626 of the Philosophical Magazine paper"* of 1921 has been carried out 

 with the results given in an Appendix to this paper. Formula (11a) of 

 the Appendix gives the ratio, K, of the a-c. inductance of the pair (less the 

 "geometric inductance") to the a-c. inductance of a wire with concentric 

 return, which is given by a well-known formula (7a in the Appendix). 

 It will be seen that the factor introduced by proximity effect decreases with 

 frequency but is asymptotic to a definite value, depending upon the separa- 

 tion of the two wires, as the frequency increases indefinitely. Similar curves 

 are given in an extensive study of the mutual inductance of four parallel 

 wires of a quad by R. S. Hoyt and Sallie Pero Mead . Their theoretical 

 studies agree closely with experimental values given by R. N. Hunter and 



^ Ray S. Hoyt and Sallie Pero Mead, "Mutual Impedances of Parallel Wires," B.S.T.J., 

 XIV (1935), pp. 509-533. 



