CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1249 



In view of (103) and (104) of Section 2.3.2, (123) is equivalent to 



and (122) becomes 



where 



^ 1.465X 



^ 8 I sin d/2 I ' ^^"^''^ 



= X/Xe = 3.8317 X/27ra (126) 



is the cutoff ratio for TEoi and TMu waves in a straight, empty guide. 



A TEoi to TEoi ± TMu" mode transducer may thus consist of a di- 

 electric sector, attached to the surface of the straight guide on the side 

 nearest the center of curvature of the bend, if the angle of the sector 

 satisfies (124) and the length satisfies (125). However, the condition (124) 

 can be satisfied by a real angle only if 



0.8483 ^ X/Xe ^ 1; (127) 



that is, only if the guide is very near cutoff; and the value of 6 which 

 satisfies (124) varies rapidly with X over the above range. This form of 

 normal mode transducer is therefore too narrow band to be of much 

 practical interest. 



A.3 A Broadband Compensator 



We shall now show how to design a dielectric compensator, in a straight 

 section of guide, which takes the mixture of TEoi and TMn" put out by 

 an adjacent bend and reconverts it to pure TEoi , independent of fre- 

 ; quency. 



First let us write the solution of (116) for a plain bend in terms of ar- 

 bitrary initial values ao(0) and ai(0). We have 



(128) 



ao{z) + a,{z) = [ao(0) + ai(0)]e-''""-'", 

 a,{z) - a,{z) = [ao(0) - ai(0)]e-''^+'", 

 and hence 



ao{z) = [ao(0) cos cz — zai(O) sin cz]e~''"', 

 «i(2) = [— mo(0) sin cz -{- ai(0) cos cz]e~'''^ 

 The bend may be compensated with a dielectric-loaded straight guide, 



(129) 



