LAMINATED TRANSMISSION LINES. II 1137 



()ii((M- stack at pi \vc have 



z ='^ ^^'o(r^P2)/i(r,6) + /vi(r,6)/o(r,p2) .^^^. 



^' g /vi(r,p2)/i(r,6) - /vi(r,/>)/i(r,p.>) " 



'i'he coiKlitiou that the radial impedances shall be matched at the sur- 

 faces of the main dielectric is given by equation (38) of Section II, which 

 lakes the form 



KoKo(koPi) + iweoZiKiJKoPl) _ KoKo{koP2) — io}€oZ2Kl{KoP2) CiOn) 



KoIo{koPi) — iCiieoZiIi(KoPl) KohiKopo) + ZCO€oZ2/i(koP2) 



where kq is related to T( by equation (330). If we substitute the expres- 

 sions (337) and (338) for Zi and Zo into (339), we have a single equation 

 whose roots in T( correspond to all the circular transverse magnetic 

 modes on the coaxial Clogston line. The propagation constant y of 

 each mode is given in terms of Tf by equation (327). 



Once the boundary conditions have been satisfied for a particular 

 mode by a suitable determination of F^ , it is a routine matter to obtain 

 the field components for this mode. In the main dielectric the fields are 

 of the form given by equations (33) of Section II. Hence for pi ^ p ^ P2 

 we have 



H^ = U/i(kop) + BKriKopW, 



E, = -X- [A/i(kop) + BK.iK.pW', (340) 



E.= -!^ U/o(kop) - BIU{K,p)]e-'% 



where one of the constants A and B is arbitrary, but the ratio A/B 

 must be taken equal to either side of equation (339). The fields in the 

 stacks are of the form of equations (277) of Section VIII, where the 

 constants are to be determined so that H^ = at p = a and p = h, and 

 so that the tangential field components are continuous at pi and p2 . 

 Imposing these conditions, we find that in the inner stack, for a ^ p ^ Pi , 



H^ = C[Ki(r,a)7i(r,p) - /i(^,a)/^l(^,p)]e-^^ 



E, = ^, C[Ki(r,a)/x(r,p) - /i(r,a)/^i(r,p)]e-^ (341) 



icoe 



E. = ^ C[K,{Tca)UTtp) + /l(^,a)/Co(^,p)]e-^^ 



g 



