LAMINATED TRANSMISSION LINES. II 



1183 



e(|iuiti()u (520), define the values of 7 wliicli are the propagation constants 

 of t!i(^ \-ariou.s modes of the Hue. 



While it is possible to find special foruis of th(> fuiu^tious €('//), jl(ij), 

 a:ui (jiu) such that (520) can be solved exactly in terms of known func- 

 tions, it is easier to make certain approximations in the beginning 

 which retain only the important terms. For this purpose we shall write 



e = Co + At, 

 M = juo + Afi, 



g = g^ + Ag, 



(522) 



where eo , mo , and yo are constants representing the average values of 

 e, jl, and g across the stack, so that the average values of At, Ajl, and A^ 

 across the stack are zero.* Furthermore the fractional variations in the 

 stack parameters will be assumed small compared to unity; in practical 

 cases they will never be larger than a few per cent and will usually be 

 only a fraction of one per cent. 



Referring now to equation (520), we see that the coefficient of i/x 

 contains the large factor co/xg, which is of the order of l/5i, as compared 

 with the term d''HJdy~, which is presumably of the order of (l/a)Hx . 

 Hence small changes in e and p. will make relatively large changes in 

 the coefficient of H^ , since 7 is a constant. On the other hand, the 

 coefficient of dHx/dy will be small for any reasonable variations in the 

 small quantity Ag/g^ . Hence we shall neglect this term entirely and 

 deal with the approximate equation 



d-H. 

 dy- 



■loijlg 





Hx = 0. 



(523) 



If we substitute (522) into (523) and drop second order terms in 

 Ae eii , Afl/fio , and Ag/go , we find that the coefficient of Hx becomes 



ig r 2_. , 2n 



■ — [w ^e + 7 J 

 coe 



tgo 



1 + ^ - ^^ 

 ^0 Co , 



and if 



7 + w"/xoeo + w fioeo 



r^ = ^- lo) Moeo + 7 J, 



Am Ae 



Mo €o/_ 



(524) 



(525) 



* The j)reseut use of zero subscripts on *o , Mo , and go has of course nothing to 

 do with the earlier convention that associated zero subscripts with the main 

 dielectric in Clogston 1 lines. 



