258 BELL SYSTEM TECHNICAL JOURNAL 



Using the notation of Fig. 1, the resistances of the inner and outer 

 conductors, both with conductivity X, at frequencies where the walls 

 are sufficiently thin to avoid skin effect, are 



Ri = . ,,n ^abohms per cm. (17) 



Ra = . / .0 ^T-abohms per cm. (18) 



Tr\{a^ — c) 



Let the inner conductor have a fixed thickness b — a, the outer 

 conductor a thickness d — c, and let the ratio {b — a)l{d — c) be 

 represented by /. For small values of wall thickness 



b'- - a- ^ 2b{b - a) (19) 



and 



d^ - f2 = 2cid - f) = 2c ^^-^ . (20) 



Substituting these relations and the values of L and C from (4) and 

 (5) in equation (1), it is found that the attenuation for the circuit with 

 thin walls is 



a=-r^-7T (y-j nepers per cm. (21) 



47rXc {b — a)2 loge p 



Differentiation shows that minimum attenuation in the case of thin 

 walls is obtained when 



loge p = • (22) 



P 



The values of diameter ratio which satisfy this relation may be found 

 from the curve of Fig. 2, if the values of abscissa? on that curve are 

 interpreted as values of f^. 



If the conductor walls are thin, as above, and if in addition the 

 conductivities of the two conductors are not the same, that of the inner 

 conductor being n times that of the outer one, the condition for 

 minimum attenuation becomes 



loge p = . (23) 



P 



Figure 2 may be used to find the values of diameter ratio which satisfy 

 this relation also, the abscissae scale markings in this case being taken 

 as values of n^t-. 



