Theory of Curved Circular Waveguide 

 Containing an Inhomogeneous Dielectric 



By SAMUEL P. MORGAN 



(Manuscript received Febrviarj^ 25, 1957) 



Generalized telegraphist's equations are derived, following Schelkunoff, 

 for all modes in a curved circular waveguide containing an inhomogeneous 

 dielectric. Particidar attention is paid to the coupling between the TEoi 

 mode and other modes in the curved guide. The results are applied to the 

 problem of preventing the mode conversion from TEoi to TMn which rior- 

 mally occurs in a curved round waveguide, by partially filling the cross 

 section of the curved guide with a suitably shaped dielectric, such as poly- 

 styrene foam. Design equations are given for various compensators, and 

 criteria are set up for keeping the power levels of all spurious modes low in a 

 coynpensated bend. Dielectric losses, which may be important at millimeter 

 wavelengths, are briefly treated. The potentialities of different compensator 

 designs are illustrated by numerical examples. 



INTRODUCTION 



It has been recognized for several years that a major problem in the 

 transmission of circular electric waves through multimode round wave- 

 guides is the question of negotiating bends. Theoretical studies^' ^' ^ have 

 sho\Mi that a gentle bend couples the TEoi mode to the TEn , TEjo , 

 TEi3 , • • • modes and to the TMn mode. The TMn mode presents the 

 most serious problem, since it has the same phase velocity as TEoi in a 

 perfectly conducting straight guide. It follows that power introduced in 

 the TEoi mode at the beginning of a gradual bend will be essentially 

 completely transferred to the TMn mode at odd multiples of a certain 

 critical bending angle t?c . The angle §c is proportional to the ratio of 

 wavelength to guide radius but independent of bending radius; in other 

 words, power transfer cannot be avoided merely by using a sufficiently 

 gentle bend. 



S. E. Miller has discussed a number of methods for transmitting the 

 circular electric wave around bends with small net power loss to TMn • 

 These methods are of two general types. 



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