Propagation of Periodic Currents over a System of 

 Parallel Wires 



By JOHN R. CARSON and RAY S. HOYT 



Synopsis: The first section of this paper is devoted to the formal mathe- 

 matical theory of the propagation of periodic currents over a system of 

 parallel wires energized at its physical terminals only. The theory de- 

 veloped is essentially a generalization of the classical theory of transmission 

 over a single wire (with ground return) or over a balanced metallic circuit. 

 The solution here given furnishes the fundamental formulas and a good 

 deal of information regarding what takes place in a system of parallel 

 wires; for actual calculations, however, the method of treatment is not 

 so well adapted as that developed in the remaining sections of the paper. 



The second section deals analytically with the problem of propagation 

 over a line or a circuit exposed throughout its length to an arbitrary im- 

 pressed field of force. The resulting solution is immediately applicable to 

 problems of crosstalk and interference, and to the theory of the wave 

 antenna. 



The last two sections are devoted to the development and application 1 

 of a more physical or synthetic method of treatment, based on the substitu- 

 tion of 'equivalent electromotive forces' for the arbitrary impressed field. 

 This synthetic treatment, which permits of an intuitive or physical grasp 

 of the various problems, has been found quite useful in dealing with cross- 

 talk and interference, and also with the wave antenna. The method is 

 illustrated (in the last section) by application to two representative problems 

 of a diverse nature. 



IN the modern telephone system, transmission takes place over a 

 circuit which is usually in close juxtaposition to a number of 

 parallel circuits, and which may be, and frequently is, exposed to inter- 

 ference from power circuits or other disturbing sources. The mathe- 

 matical theory of wave propagation over such a circuit involves two 

 problems: (1) propagation over a system of parallel wires, and (2) 

 propagation over a wire or metallic circuit in an arbitrary impressed 

 field of force. 



The first Section of this paper is devoted to the formal mathematical 

 theory of the propagation of periodic currents over a system of parallel 

 wires, energized at its physical terminals only.^ This problem is 

 essentially a generalization of the problem of transmission over a 

 line of uniformly distributed resistance, inductance, capacity and leak- 

 age; and involves the formulation and solution of a dififerential equa- 

 tion which may be termed the generalized telegraph equation in con- 

 tradistinction to the well-known telegraph equation which characterizes 

 transmission over a single wire (with ground return) or a balanced 



1 This is the assumption underlying ordinary transmission theory. 



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