Wave Propagation Over Parallel Tubular 



Conductors: 



The Alternating Current Resistance 



By SALLIE PERO MEAD 



Synopsis: On the b.isis of Maxwell's laws and the conditions of con- 

 tinuity of electric and magnetic forces at the surfaces of the conductor, the 

 fundamental equations are established for the axial electric force and the 

 tangential magnetic force in a non-magnetic tubular conductor with parallel 

 return. The alternating current resistance per unit length is then derived 

 as the mean dissipation per unit length divided by the mean square current. 

 The general formula is expresse<l as the product of the alternating current 

 resistance of the conductor with concentric return and a factor, termed 

 the "proximity effect correction factor," which formulates the efTert nf 

 the proximity of the parallel return conductor. The auxiliary functions which 

 appear in the general formula are each given by the product of the cor- 

 responding function for the case of a solid wire and a factor involving the 

 variable inner iKiundary of the conductor. 



In general, the resistance may be calculated from this formula, using 

 tables of P :ssel functions. The most important practical cases, however, 

 usually in.vjivc only the limiting forms of the Bessel functions. Special 

 formulae of this kintl are given for the case of relatively large conductors, 

 with high impressed frequencies, and for thin tubes. .A set of curves illus- 

 trates the application of the formulae. 



I. Introduction 



WHI-IRK circular conductors of relatively large diameter are 

 under consideration, the effect on the alternating current 

 risistance of the tubular as distinguished from the solid cylindrical 

 form becomes of practical importance. Mr. Herbert B. Dwight has 

 worked on a special case of this problem and developed a formula 

 for the ratio of alternating to direct current resistance in a circuit 

 a)mposed of two parallel tubes when the tubes are thin.' As infinite 

 sums of infinite series are involved, however, his result is not well 

 adapted to computation. 



Mr. John R. Carson has gi\en a complete solution for tlie alter- 

 nating current resistance of two parallel solid wires in his paper 

 "Wave Propagation Over Parallel Wires: The Proximity Effect," 

 Phil. Mag., April, 1921. The analysis of that paper may readily be 

 extended to the more general case of propagation over two tubular 

 conductors by a parallel method of development. This is done in 

 the present paper. As the underKing theory is identical in the two 

 problems, familiarity with the former pajier will be assumed and the 

 analysis will merely be sketched after the fundamental equations are 

 established. 



' "Proximity Effect in Wires and Thin Tubes," Trans. .1. /. E. £., Vol. XLll 

 (.1923), p. 850. 



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