CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1227 



where 8' is real, then (52) may be written 



e = €o[l + 5' - i{l + 8') tan <p] = eo(l + 8), (54) 



where 



8 = 8'- i{l + 8') tan <p. (55) 



No changes in the formal analysis result from the fact that 8 now has a 

 complex value. 



If the compensator is designed (assuming a lossless dielectric) so that 

 the total power coupled from TEoi into all spurious modes is small at 

 all points, it is reasonable to assume that the principal effect of dielectric 

 loss on the TEoi mode will be seen in the modified phase constant |S[oi] 

 of this mode in the presence of the compensator. If the compensator is 

 made by filling a certain part Si of the guide cross section with a medium 

 of constant (complex) permittivity, and the rest of the cross section with 

 air, then from (32) and (40) the modified phase constant of the TEoi 

 mode is 



^[01] = hm +^ f (grad T(oi])' dS, (56) 



-^"[01] •'Si 



where 8 is given by (55). Since 5 is complex, the attenuation constant is 



T _ j8^(l + 8') tan (p f , , r,^ \2 70 /-„x 

 oc[m = -ImiS[oi] = ^ • / (gradT[oi]) dS, (o7) 



where the integration may be carried out as soon as the area Si is speci- 

 fied. 



The approximation (57) for the attenuation constant due to dielec- 

 tric losses has a simple physical interpretation. It corresponds to the 

 power which would be dissipated in a medium of conductivity coe" if 

 I the electric field existing in the medium were the same as the field of the 

 TEoi mode in a straight, empty guide. This is probably a very good 

 approximation if 8' is small, as it will be for the foam dielectrics from 

 which compensators are most likely to be made. 



It is doubtful that (57) furnishes a good approximation to the dielec- 

 itric loss when the permittivity of the compensator is high (5 not small 

 [compared to unity). If the permittivity is high the cross section of the 

 dielectric member will be small, but the field perturbation may be large 

 jin the immediate neighborhood of the dielectric. The series which rep- 

 resent the fields in terms of the normal modes of the empty guide may 

 converge slowly; in other words, when using the telegraphist's eciuations 

 one must consider the coupling between TEoi and a large number of 



