CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1231 



Let US con.sider the case in which line 1 has a much higher attenuation 

 constant than Une 0; that is, 



(Xi » QIu 



(74) 



The second term on the right side of (73) is provided with a small co- 

 efficient, and also its exponential factor decays much faster than the 

 exponential in the first term. The second term, therefore, rapidly be- 

 comes negligible as z increases, and we may write, 



ao{z) 



1 + 





(75) 



(to — 7i)"_ 



If the attenuation constant of line is not modified* by the presence 

 of the coupled lossy line 1, then in the absence of line 1 the amplitude of 

 ao{z) would be e~"°% and the factor by which the amplitude is reduced 

 owing to the presence of line 1 is 



2 



1 + 



K^zKyQ-yi) 



(76) 



(to — Ti)' 



The first factor on the right is very nearly unity, but not less than unity 

 if K is real (lossless coupling mechanism) and 



(ar - a,)' ^ (^j - ^o)'. (77) 



Hence the factor by which the amplitude is multiplied is not less than 



(ao — aiJK'Z 



exp 



9 



K Z 



To 



Ti 



exp 



(ao - aiy + (/3o - ^iT 



(78) 



assuming that k is real. If the amplitude of the wave on fine is not to 

 be down by more than .V nepers, after a distance z, from what it would 

 have been in the absence of the coupled line, it suffices to have 



oJkz 



( 



"1 



oco) 



{ay - a^y + (/3i - |8o)^ 



= AT, 



or 



«i - ao = h[{Kz/N) + V{K'z/Ny - 4(^1 - ^o)-^J. 



(79) 



(80) 



2.2 TEoi-TMii Coupling in Plain and Compensated Bends 



In Jouguet's^ and Rice's^ analysis of propagation in a curved wave- 

 guide, the curvature is treated as a perturbation and the field com- 



* The value of ao rnay very well be modified by the coupling; but if it is this 

 can easily be taken into account when computing the over-all change in | ao(z) | 

 due to the presence of line 1. 



