1134 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



111 Section X we shall give a similar but briefer treatment of the various 

 higher modes which can exist in partially or completely filled Clogston 

 lines. 



The notation for the parallel-plane line is established in Fig. 5 of 

 Part I. The stacks are bounded externally by high-impedance sheaths 

 at 2/ = ±l«, while the main dielectric is bounded by the planes 

 y = ±^5 = ±(^a — s). No restrictions are placed on the relative thick- 

 nesses h and s of the main dielectric and the stacks. The average electri- 

 cal constants of the stacks are e, Jx, and g, while the electrical constants 

 €o and Mo of the main dielectric are assumed to satisfy Clogston's con- 

 dition (102) but are otherwise arbitrary. 



As in Section II, the modes may be divided into two classes, according 

 to whether Hx is an even function or an odd function about the center 

 plane y ■= 0. The normal surface impedance Z{y) looking into either 

 stack may be obtained from equation (92) of Section III; if the imped- 

 ance of the outer sheath is effectively infinite we have 



Z{y) = K coth ViS = (Tt/g) coth Tts, (322) 



where 



T( = 



coe 



(w>e -f 7") 



(323) 



Substituting for Z(y) into equations (11) and (13) of Section II, we find 

 that the impedance-matching conditions become 



tanh iKoh tanh r,s = - — - , (324) 



g Kg 



ioieo r 



g 



for the even and odd modes respecti\-ely, where 

 From (323) Ave have 



coth i/cofc tanh Vts = _ !^« L' ^ (325) 



g Ko 



lio = (o-o — 7")' = ( — co'moCo — 7 )'• (326) 



7" = -coVe - (iw€/g)T] , (327) 



and so from (326), 



4 = -a)"(Mo€o - /ie) + (io}e/g)r] . (328) 



If Clogston's condition is satisfied, namely 



fjLaeo = p.e, (329) 



