256 BELL SYSTEM TECHNICAL JOURNAL 



The capacitance of the coaxial circuit now becomes 



(€2 - €l)W 



''+ ~s • 



C = ?r-i abfarads per cm., (13) 



2 loge p 



while the leakage conductance is 



G — — ■ 7i-T abmhos per cm. (14) 



5 2 loge P 



On substituting these values in formula (1) the following expression 

 results: 



1 // Vei^ + (c2 - €i)wp + Vw 



" = 4Vx; 



^^s 2 loge P 



poie^ w 1 



2 ^!s^|elS + (e2 - ei) 



w 



nepers per cm. (15) 



Once more the second term is independent of c and b, and the con- 

 dition for minimum attenuation is, as before, that given by equation (7). 

 The high-frequency characteristic impedance in this case, however, is 



Zo = —^^J^k£=: abohms. ( 1 6) 



(62 - ei)w 



^61 



The quantity in the denominator of the above expression is evidently 

 the weighted average dielectric constant of the insulating medium. 



In the case just considered, the gaseous and non-gaseous dielectrics 

 were separated from each other by planes perpendicular to the axis of 

 the conductors. Consequently, each line of dielectric flux passed 

 through only one kind of material. It can be shown that, as long as 

 this latter condition holds, the condition for minimum high-frequency 

 attenuation as given by equation (7) is valid, or, in other words, the 

 optimum diameter ratio is that shown in Fig. 2. Cases arise, however, 

 in which a line of dielectric flux, in going from one conductor to the 

 other, may pass through more than one kind of dielectric material. 

 It is extremely difficult to obtain a mathematical solution for the 

 diameter ratio which results in minimum attenuation for such cases, 

 since this involves a three-dimensional field problem. Consideration 

 of the problem, however, indicates that the optimum diameter ratio 

 will not differ appreciably from that given by Fig. 2, especially if the 

 dielectric is mostly gaseous, which, of course, is highly desirable. 



