268 BELL SYSTEM TECHNICAL JOURNAL 



where P is the proximity effect factor, given in a paper by J. R. 

 Carson."' At high frequencies this factor reduces to the asymptotic 

 value 



P = -=L= • (38) 



\V2 - 1 



For a given high frequency and given wire separation, assuming the 

 dielectric constant and conductivity to be fixed, equation (37) becomes 



^''' (39) 



yju^ — 1 cosh ' V 



where K3 is a constant. 



For a given wire separation this expression is minimized when 



u = ^i= 2.27. (49) 



b 



For open-wire pairs, which may be considered as approaching pairs in 

 space, it is ordinarily cheaper to obtain any desired attenuation at a 

 given frequency by using a wide separation and relatively small 

 conductors rather than a narrow^ separation and conductors of such 

 size as to satisfy (40). This relation is of considerable utility, how- 

 ever, in that it is a reasonably close approximation to the optimum 

 for many kinds of shielded pairs. The corresponding ratio for the 

 shielded solid pair, as given by {33). and (34), is approximately 2.5. 



Shielded Stranded Pair 



The preceding discussion of shielded pairs has been limited to types 

 of enclosed conductors such that high-frequency currents are crowded 

 toward the conductor surfaces. There will now be found the optimum 

 proportioning when the enclosed conductors are stranded.'^ 



The capacitance and inductance between two shielded stranded 

 wires when surrounded by a cylindrical shield are approximately 



C = p z 5-=T abfarads per cm., (41) 



^ 1 - c^ 

 2v 



4 log. 

 L = 4 log. 



2v 



1 +C72 



1 - (T- 

 1 + (T^ 



+ 2L, abhenries per cm.. (42) 



where Li is the internal inductance of each conductor. 



If it be assumed that the current distribution is uniform over the 

 cross-section of the enclosed conductors, the resistance of each is the 



