944 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



For the average power dissipated per unit length of line in the main 

 dielectric and the inner and outer stacks we have, respectively, 



APo = ^w/xo -z- log - (tan (^n + tan fn), (265) 



Ztt Pi 



APi 7^\uii' —- —- (tan 4>2 + tan f), 



llT Opi 



AP2 "^ ^co/i' — — -— (tan 4>2 + tan f). 

 zr op2 



(266) 



The part of the attenuation constant which is due to dielectric and 

 magnetic dissipation is therefore 



^ APo + APi + AP2 



"' 2(Po + Pi + P2) 



P2 1*"/^' / S\ S>\ 



log — (tan 00 + tan fo) + ^yi I ^ + ~ ) (tan ^2 + tan t) 

 _ 1 /-n Pi 3 jUQ \Pi P2/ 



log '-^ + i 4/?^? (^ + 'A 



Pi 3 K ^^e' \pi P2/ 



We need scarcely point out that if the loss tangents are not small 

 compared to unity, it may be impossible to satisfy Clogston's condition 

 (102) very closely with a real value of 6, and the resulting mismatch 

 may reduce the depth of penetration and increase the metal losses in 

 the stacks. In practice, however, the loss tangents will be of the order 

 of 0.001 or even 0.0001, and matching the imaginary parts of nata and 

 Jit will be much less of a practical problem than matching the real parts. 



Appendix I 



BESSEL FUNCTION EXPANSIONS 



Let Pi and po be the inner and outer radii of a cylindrical shell and 

 let the thickness t, given by 



t = p2 — pi , (Al) 



be less than pi . Then, following Schelkunoff, we may replace the Bessel 

 functions appearing in equation (68) of Section III by their Taylor ex- 

 pansions, namely 



1 S. A. Schelkunoff, Bell System Tech. J., 13, pp. 561-562 (1934). 



