TRANSMISSION LINES FOR RADIO SYSTEMS 415 



be obtained from an expression derived by Lord Rayleigh upon assum- 

 ing that the two tubes comprising the line are of negligible thickness. 

 This is permissible because at radio frequencies the conduction of cur- 

 rents is essentially a skin effect. Upon this basis the inductance per 

 unit length of a concentric-tube line becomes: 



L = 2 log^-e.m.u. (2c) 



a 



Upon substituting (2b) and (2c) into (2a) with proper regard of 

 units a simple expression for the characteristic impedance at radio 

 frequencies is obtained : 



Zo = 138 logio- ohms. (2d) 



The high-frequency resistance of concentric-tube lines has been 

 treated by a number of investigators, notably by A. Russell.'* The 

 asymptotic formula for resistance as the frequency is increased without 

 limit is: 



R = Af^ (^ -f ^ 10-« ohms/cm, (3) 



in which: 



p is the resistivity in e.m.u. (for pure copper p is about 1730 



e.m.u.), 

 /x is the magnetic permeability, 

 / is the frequency, c.p.s., 

 a is the outer radius of the inner conductor, 

 and 



b is the inner radius of the outer conductor, the two latter 

 being in centimeters. 



It is of interest to note that the wall thickness of the conductor is 

 not involved. At radio frequencies the current is confined to a very 

 thin layer on the outside of the inner conductor and on the inside of 

 the outer conductor.^ The skin effect is, of course, not so pronounced 

 at low frequencies and more complicated formulas involving wall thick- 

 ness must be employed. 



Some typical experimental data are submitted to show that for 

 frequencies higher than one megacycle and for several practical line 

 constructions the foregoing equation (3) holds with a very useful de- 

 gree of accuracy. The physical dimensions and construction details of 



^ A. Russell, Phil. Mag., April 1909; and "Alternating Currents," Vol. I, p. 222, 

 1914, Cambridge Press. 



^ Frequency is not the sole criterion, resistivity, wall thickness, and diameter also 

 being involved. 



