Wave Propagation Over Continuously Loaded Fine Wires 



By M. K. ZINN 



The paper contains the resuhs of a theoretical investigation of wave prop- 

 agation along a pair of wires that are "loaded" by enclosing each wire 

 in a continuous sheath of magnetic material. The results of greatest 

 practical interest are certain approximate formulas that are sufficiently 

 simple to be adapted to engineering design studies, while having a high 

 degree of precision for all practical dimensions and frequencies. 



THE purpose of this investigation is to define the character of 

 wave transmission along a pair of wires each of which is loaded 

 with a continuous sheath of magnetic material. Exact expressions 

 for the propagation constants are developed from the general theory 

 that applies to such a system. Also, simple approximate formulas 

 are given for the sizes of wires that are generally used in paper-insu- 

 lated cables. 



Wave Propagation Along a Pair of Wires with Magnetic 



Sheaths 



For the benefit of those who are not interested in following the 

 theoretical work in detail, a general sketch of the method and a sum- 

 mary of the mathematical results will be given first, together with 

 a discussion of some numerical examples. Details of the theoretical 

 work have been placed in the Appendices. 



The analysis here given follows closely the methods developed by 

 John R. Carson ^ in a solution of the transmission of periodic currents 

 along a system of coaxial cylinders. The analysis for the case where 

 the outgoing and return conductors are coaxial is applied, with only 

 small modifications, to the case where the two conductors are parallel 

 and not coaxial. This application of the theory ignores the "proximity 

 effect." - That is to say, it assumes that the electric and magnetic 

 forces within each conductor are functions only of the distance from 

 its axis and of the coordinate in the direction of propagation, which 

 is strictly true where the cylindrical conductors are coaxial. 



1 "Transmission Characteristics of the Submarine Cable," John R. Carson and 

 J. J. Gilbert, Jour>ial of the Franklin Institute, December 1921. 



- This is the usual method of dealing with problems involving balanced parallel 

 conductors. The alternating-current resistance of the s\-stem may be expressed as 

 the product of the a.c. resistance, assuming a concentric return, and a "proximity 

 effect correction factor," which takes into account the influence of the parallel return 

 conductor. The "proximity effect" is in general negligible at voice frequencies for 

 conductors of sufficiently small cross-section, such as those of paper insulated cables. 

 References: "Wave Propagation over Parallel Wires: The Proximity Effect." John R. 

 Carson, Phil. Mag., April 1921, and "Wave Propagation over Parallel Tubular 

 Conductors," Sallie Pero Mead. Bell System Technical Journal, April 1925. 



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