436 



BELL SYSTEM TECHNICAL JOURNAL 



Now substitute the approximate values of the I' for small x. The result is 



Since Zo(r) is tinite, it follows that 



•qH' ' — ' 



1 



a; log 



('-3^ 



and it is easily shown that r)II' — > <» as r; -^ 0. On the other hand, rfH' -^ 

 as ?7 -^ 0. Hence, () - — > as 77 — ^ 0. On the other hand, for tj = 0, a 



A 



0.50 



0.45 



q: a. 



0.40 



0.30 



Q 0.25 



z 

 o 



'-' 0.20 



0.15 

 ^ 0.10 

 0.05 



0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 



_a_ 



L 



R=-r- 



Fig. 9 — Coaxial resonator. TE Oil mode Contour lines of ()- 



A 



perfect cylinder exists whose (J - is not zero. It is concluded that the ex- 



X 



pression for Q - does not apply for small 7/ for the TM modes with /" = 0. 



A 



s 



Thus it is seen that the expressions for the factor (() -) reduce to those 



A 



given for the cylinder, when t; = 0, except for TM modes with /* = 0. 

 For these latter cases, the factor approaches zero as 7/ approaches zero, 

 because 77//' increases without limit. This means that an assumption 



