496 BELL SYSTEM TECHNICAL JOURNAL 



metallic circuit. The analysis of this problem, while furnishing the 

 fundamental formulas and a good deal of information regarding what 

 takes place in a system of parallel wires, is not well adapted for 

 actual calculations, except for relatively simple systems; in particular 

 it is not adapted to deal with the important problems of crosstalk 

 and interference. 



In Section II the problem of propagation over a circuit or line 

 exposed throughout its length to an arbitrary impressed field of force 

 is taken up. The resulting solution is immediately applicable to 

 interference problems, where the field of the disturbing source is 

 supposed known, and to the theory of the wave antenna. Moreover, 

 as shown in Section I la, it is particularly well adapted to the problems 

 of 'crosstalk,' or interference between circuits of the parallel system. 



Sections I, II and I la furnish the formal analysis and the funda- 

 mental formulas. Sections III and IV, constituting the remainder 

 of the paper, "are devoted to the development and the application to 

 representative problems of a more physical or synthetic treatment, 

 in which the general theory and formulas are interpreted in terms of 

 'equivalent electromotive forces'; this concept permits of an intuitive 

 or physical grasp of the various problems, and has been found quite 

 useful in dealing with crosstalk and interference, and also with the 

 wave antenna. 



I 



Propagation of Periodic Currents over a System of Parallel 

 Wires, with Impressed Field Concentrated at Terminals 



The physical system under consideration is supposed to consist of 

 n parallel wires, numbered from 1 to n, which may either be a system 

 of overhead wires parallel to the surface of the earth, or a multi-wire 

 cable enclosed in a sheath. The formal analysis applies equally well 

 to both cases ; but the calculation of the circuit parameters is a matter 

 of considerably greater difficulty in the case of the cable, due to the 

 close juxtaposition of the wires. Even in this case, however, the 

 circuit constants are rather easily calculable to a first approximation 

 from the dimensions of the system; and they are, in any case, experi- 

 mentally determinable. 



Let /], ^2 • • • In be the currents in the n wires, which are taken as 

 parallel to the x'-axis, which is itself parallel to the surface of the 

 earth or to the sheath (in the cable case). A steady state is assumed; 

 that is to say, the currents are sinusoidal and involve the time / only 

 through the common factor exp (icot), where co/lir is the frequency and 

 i denotes V— 1 ; consequently the differential operator d/dt is replace- 

 able by ico in accordance with the usual methods of alternating current 

 theory. 



