CHA RA CT ERISTICS OF TOLL TELEPHONE CA BLES 297 



R (resistance) = R'(l - f c/L'C) + • • • 



L (inductance) = L'i\ - i co'L'C • • •) + ^ R'^C + • • • 



G (conductance) = G'(l - f c/L'C • • •) - i ^'co'C' + | R'o^^L'C'^ ■ - ■ 



C (capacitance) = C'{\ - | c/VC + f R'G' • • •) (1) 



These formulas give accuracies within one per cent for reel-lengths of 500 

 feet or less and frequencies up to 100 kilocycles for 19-gauge cables having a 

 capacity of .062 mf per mile. All the curves of R, L, G, C herewith are 

 based on true values obtained from such computations. 



Resistance 



The quantity R, series resistance in ohms per mile, has a large variation 

 with frequency produced by the well-known phenomenon called skin effect 

 and another large increment, resulting from the closeness of the wires in 

 cables, known as the proximity effect. " . The magnitude of the proximity 

 effect varies with the diameter of the conductors as well as with their separa- 

 tion. The curves in Fig. 2 show the increment in resistance resulting from 

 skin effect and the total increase including proximity effect as computed 

 for a pair of wires separated by various multiples of their diameters. The 

 abscissa, B, in Fig. 2 is a sort of universal parameter used in data on skin 

 effect so that a single curve will sufl&ce for various gauges. If / is frequency 

 in cycles per second and Ro is the d-c. resistance for 1000 feet of the wire 

 (not a 1000-foot loop), the parameter B is given by the equation 



B = Vf/Ro = VjT^ (2) 



for 19-gauge wire so that ^ = 80 corresponds to 51,200 cycles. According 

 to the curves at 5 = 80 (51.2 kc), the skin effect increases the a-c. resistance 

 to about 12 per cent more than the d-c. resistance. 



For a separation of two diameters between centers of the wires of a pair 

 {k — .25) the proximity effect adds another 6 per cent to the resistance 

 ratio making the total a-c. resistance about 1.18 times the d-c. resistance 

 at 51.2 kc. If the wires are closer together {k = .4) the a-c. resistance is 

 computed to be about 1.30 times the d-c, which is about the ratio actually 

 measured. The effects caused by the presence of the adjacent pair in a 



* J. R. Carson, "Wave Propagation over Parallel Wires — The Proximity Effect," 

 Phil. Mag., Vol. 41, April 1921, pp. 607-633. 



^ A. E. Kennelly, F. A. Laws and P. H. Pierce, "Experimental Researches on Skin 

 Effect in Conductors," A.I.E.E. Trans., Vol. 34, Part 2, 1915, pp. 1953-2021. 



* A. E. Kennelly and H. A. Affel, "Skin Effect Resistance Measurements of Conductors 

 at Radio Frequencies," I.R.E. Proc, Vol. 4, No. 6, Dec. 1916, pp. 523-574. 



^Giinter Wuckel, "Physics of Telephone Cables at High Frequencies," EFD 47, 

 (Nov. 1937) pp. 209-224. 



