LAMINATED TRANSMISSION LINES. II 



1131 



whoi'c ill the last step \v(> iia\'o made use of (804) and (306). The ratio 

 of voltafie aci'oss the stack to total one-way current is, from (303) and 

 (307), 



V 

 I 



M ./o(xic) 

 € 2ir 



1 



1 



Xi'>-/i(xi'>) Xifl^/i(xi«) 



(308) 



If there is no inner core, so tiiat a = 0, the e.xpressions which we have 

 just deri\ed become indeterminate forms, and it is simplest to make an 

 independent calculation of the fields for this special case. The Neumann 

 functions are now e.xcluded because of tlieir singularity at p = 0, and 

 the condition (293) is replaced b.y 



^i(x^) = 0, 



from winch 



XI = 3.8317/6. 

 The expi-essions for the fields are 



(309) 

 (310) 



(311) 



Xi 



E. = ^ HoMxip)e'"'% 

 Q 



where //o is now a different arbitrary amplitude factor. 



The total potential difference across the stack becomes, aftei- putting 

 in numerical values, 



y = -I E,dp = 0.3661 Vm/c Hobe~ 



Jo 



The conduction current density is 



Jz = gE, = xJhJo{x\p)e"^'; 

 and Jz changes sign at 



Jo(xi^) = 0, c = 2.4048/xi = 0.6276/>. 

 The total one-way current is 



/ = 27rf//„./i(xic)e~"-' = 2.047//obe"'', 

 and the ratio of total \oltage to t(jtal current is 



V/I = 0.1788Va7^ • 



(312) 



(313) 

 (314) 

 (315) 

 (316) 



