CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1241 



For future reference, the modified phase constants of the TEoi and 

 TMii modes may be calculated from (40) as: 



iS(ii) = /i(ii) + y4— [(^ - 0.29646 sin 6)^' 

 47r/l(ii) 



(0.70354 sin 9)xm], (103) 



/^[Ol] — ^[01] + 



fi'^Sd 



47r/l 



[01] 



The dielectric coupling coefficients for the single-sector compensator 

 are given by (50) and (51), provided that some of the Bessel functions 

 are integrated by Simpson's rule. In particular, the dielectric coupling 

 coefficient between TEoi and TMn is 



df^i^ = -0.12066 (38 sin 9/2. (104) 



Substituting (86) and (104) into (93), we obtain the decoupling condition 

 for a single-sector compensator, namely 



1.5295 a /,,^r^ 



5 = ^-jjK 7 . (105) 



sm 6/2 



No circular magnetic modes (TMom) and no higher circular electric 

 modes (TE02 , TE03 , etc.) are coupled to TEoi by the single-sector com- 

 pensator. We also observe that the coupling coefficients of the TEi^ 

 and TMim modes do not depend upon the sector angle 6 so long as 5 

 and 6 are related by (105). This is because these modes have the same 

 angular dependence as the TMn mode, which we are trying to compen- 

 sate. 



The most troublesome spurious modes are those whose phase velocities 



are closest to TEoi , namely TE21 and TE31 . The coupling coefficients and 



power transfer ratios for these two modes in |-inch and 2-inch guide are 



given in Table II. Either of these modes could be decoupled by proper 



' choice of the sector angle, provided we had a uniform, low-dissipation 



dielectric with the required value of 8, but it is impossible to decouple 



I both modes at once with a single sector. As a compromise, we might adopt 



I the sector angle which equalizes the maximum power transferred to TE21 



j and TE31 (the distances for maximum power transfer are of course not 



the same for the two modes). This angle is approximately 144°. 



Suppose we wish to employ a 144° sector in a |-inch guide at 5.4 mm. 

 If the criterion is that TEqi is to lose a maximum of 0.1 db each to TE21 

 and TE31 , so that the total mode conversion losses are of the order of 0.2 



