LAMINATED TRANSMISSION LINES. II 1125 



where .1 and 11 arc arbitrary constants and Ff and A' ai'c defined as 

 boforo. The lioiiiidary conditions at p = a and p ^ h take the foi-m 



AIo{l\a) - BIu{Y,a) ^ y . . 

 "^ Ahir^a) + Blu{l\a) ''"^^^' ^^^^^^ 



AUTtb) - BKojTcb) _ 

 ^ AI,{r,h) + BK,{Tcb) ''^^^' 



and these e([nati()ns can be satisfied by \alues of ,1 and />' that are 

 not botii zero if and only if 



/avo(r,a) +Za{y)IuiTca) ^ KKoJT^b) - ZMK,{Tth) , 

 KhiTca) - ZMhiV^a) KhiVcb) + Z,(y)I,{T,b) ' ^*^^^^ 



Now K is given in terms of T( by equation (273), while frf)m equa- 

 tion (272) we have 



y- - -co'/Ze - {io)e/g)T]. (280) 



Hence if the dependence of the boundaiy impedances on y is known, 

 equations (274) and (276) for the plane line and equation (279) for the 

 coaxial line are transcendental relations from which in principle we may 

 determine T( , and therefore y, for each mode of the type that we are 

 considering. If the value of T( for a particular mode satisfies the in- 

 equality 



r^, 



osfig 



« 1 , (281) 



then on taking the square root of the right side of equation (280) by 

 the binomial theorem, we find that the attenuation and phase constants 

 of the given mode are approximately 



a = Re 7 = -Re ^' , (282) 



/3 = Im 7 = CO V^ - Im — ^= . (283) 



2^V)"/e 



Throughout the rest of this section we shall consider only the lowest 

 or principal mode. In a parallel-plane line the principal mode corre- 

 sponds to the lowest root in Tt (that is, the root having the smallest 

 modulus) of equation (274), which may be written in the form 



^TcataiDh^Tta = -^gaZniy). (284) 



