MUTUAL IMPEDANCES OF PARALLEL WIRES 511 



PART I 



Physical Theory 



The Physical System; Analysis of the Wire Currents into Filaments 

 Since the mutual impedance between any two parallel circuits can be 

 expressed wholly in terms of the mutual impedances between the 

 various wires composing the circuits,^ it will suffice in Part I to discuss 

 the mutual impedance between the two wires A and B in Fig. 1. 

 These are each of uniform cross-section, but they need not be alike in 

 cross-sectional shape and area nor in material. 



The wires in Fig. 1 will be assumed very long compared to the dis- 



POSITIVE DIRECTION 



__P__ ^ Jl_^_ ^__^ ^-^_..,-, 



B 

 m^NUMBER OF "FILAMENTS" IN A n = NUMBER OF"FILAMENTS" IN B 



Fig. 1 — Two "thick" straight parallel wires, p, q designate any two "filaments" 

 of wire A; <l>, d any two of B. 



tance between them, so that the end-effects* in the current distribu- 

 tion will be negligible, yet short enough compared with the wave- 

 length so that the charging current will be a negligible fraction of the 

 total current and therefore the current in each wire of sensibly the 

 same value throughout its length.^ These circumstances enable the 

 current in each wire to be treated as an aggregate of filamentary cur- 

 rents which are purely longitudinal, and correspondingly enable the 

 mutual and self impedances of the wires to be described and formulated 

 in terms of the mutual and self impedances of such filaments, thus 

 correlating well with the familiar treatment of a system of fine parallel 

 wires. This treatment by analysis into filaments has been chosen 



^ For example, the mutual impedance Zab between two circuits a and b, of which 

 a comprises wires 1 and 2 and b comprises 3 and 4, is given by Zab = Zn — Zu — Zn 

 -f Z24. However, since the wires are in general "thick," the value of each nmtual 

 impedance (also each self impedance) must depend on the presence of all four of the 

 wires. 



* These consist in the currents not being purely longitudinal near the ends of the 

 wires. 



^ Negligibility of the charging current does not by any means imply that the 

 distributed charges on the surfaces of the wires are negligible as regards the voltages 

 which they produce, for extremely small charging currents suffice to establish charges 

 which can produce relatively large voltages. 



For a discussion of this very important fact and other underlying concepts of 

 circuit theory, the reader is referred to a paper by John R. Carson, "Electromagnetic 

 Theory and the Foundations of Electric Circuit Theory," published in this Journal 

 for January, 1927. 



