510 BELL SYSTEM TECHNICAL JOURNAL 



of the wires. ^ When this is done it is found that the mutual impedance 

 comprises not only a reactance component, which is no longer propor- 

 tional to the frequency, but also a resistance component, which does 

 not vary in any simple way with the frequency. Both of these com- 

 ponent departures of the mutual impedance from its simple fila- 

 mentary value increase the difficulties of balancing out crosstalk, and 

 the resistance component has also an important effect on the attenua- 

 tion at carrier frequencies. These matters have recently assumed con- 

 siderable importance on account of the rapidly increasing interest in the 

 possibilities of communication transmission over non-loaded cable 

 circuits with the aid of carrier currents having frequencies high com- 

 pared with those of speech. As an approximate guide to the behavior 

 of twisted circuits in cables the theory and formulas for straight wires, 

 as developed in this paper, have proved to be of considerable service. 

 The present paper deals with the mutual impedances of two or more 

 straight parallel wires from two aspects: In Part I the physical theory 

 is developed and expounded. The current in a wire is there regarded 

 as made up of an indefinitely large number of parallel filamentary 

 current elements. On this basis it is shown (among other things) that 

 the current distribution over the cross-section of each conductor is 

 necessarily non-uniform, and that this non-uniformity gives rise to a 

 mutual resistance term in the mutual impedance, besides a change in 

 the mutual reactance term. In Part II electromagnetic wave theory 

 is applied to develop formulas for the mutual and self impedances of a 

 pair of long straight parallel transmission circuits in close juxtaposition. 

 Calculations of the mutual impedance made with these formulas over 

 a very wide range of frequencies (1 to 1000 kilocycles per second) are 

 found to be in very satisfactory agreement with available experi- 

 mental results.^ In both parts of the paper an endeavor has been 

 made to bring engineering concepts and formulas into closer relation- 

 ship with electromagnetic theory. 



'The convenient term "proximity effect" when applied to the distribution of 

 the current over the cross-section of a given conductor means the deviation of this 

 distribution from the "intrinsic distribution," the latter meaning the distribution 

 when the given conductor is far enough from all other conductors so that the distri- 

 bution in it is sensibly unaffected by them. 



When the given conductor is a straight uniform wire of circular cross-section, its 

 "intrinsic distribution" is of course axially symnietrie^al. 



Not every axially symmetrical distribution is the same as the corresponding 

 intrinsic distribution, as is evidenced by the case of two coaxial conductors, where the 

 proximity effect in the outer conductor may be large although the current is axially 

 symmetrical in each conductor. 



2 See the paper by R. N. Hunter and R. P. Booth, in the April issue of this 

 Journal, entitled "Cable Crosstalk — Effect of Non-Uniform Current Distribution in 

 the Wires," which includes the results of some rather extensive sets of measurements 

 of the mutual impedance of straight wire circuits, and also of twisted circuits in 

 cables, and a brief physical discussion with particular regard to the effect of non- 

 uniform current distribution. 



