Wave Propagation Over Parallel Tubular Conductors: The Alternating Current Resistance

On t h e basis of Maxwell's laws a n d t h e conditions of cont i n u i t y of electric and magnetic forces at t h e surfaces of t h e conductor, t h e f u n d a m e n t a l equations are established for t h e axial electric force a n d t h e tangential magnetic force in a non-magnetic t u b u l a r c o n d u c t o r with parallel r e t u r n . T h e a l t e r n a t i n g c u r r e n t resistance per unit length is then derived as the mean dissipation per unit length divided by the mean square current. T h e general formula is expressed as t h e product of t h e a l t e r n a t i n g current resistance of t h e conductor with concentric return a n d a factor, termed t h e " p r o x i m i t y effect correction f a c t o r , " which formulates t h e effect of t h e proximity of t h e parallel return conductor. T h e auxiliary functions which a p p e a r in t h e general formula are each given by t h e product of t h e corresponding function for t h e case of a solid wire and a factor involving t h e variable inner b o u n d a r y of t h e conductor. In general, t h e resistance may be calculated from this formula, using tables of P :ssel functions. T h e most i m p o r t a n t practical cases, however, usually involve only t h e limiting forms of t h e Bessel functions. Special formulae of this kind are given for t h e case of relatively large conductors, with high impressed frequencies, a n d for thin tubes. A set of curves illustrates the application of the formulae. I. INTRODUCTION