SOME RESULTS Oi\ CYLINDRICAL CAVITY RESONATORS 431 



Corresponding expressions for the resonant wavelength are 

 ttD 0.805 D 



X = - 



a/\ + hnD\ ^1 + 0-1622 «2i22 



\2sL/ 

 0.820 D 

 Vl +0.1681 w2/?2- 



As an example, take n = 1, R = 1, then 



(Circular) Qc5 = 0.500 D X^ = 0.759 D 



(Elliptical) Q8 = 0.473 D X = 0.747 D 



Ratio = 0.946 Ratio - 0.984. 



Conclusions 



The mathematics of the elliptic cylinder have not yet been developed to the 

 point where the design of cavities of large ellipticity could be undertaken. 

 On the other hand, sufficient results have been obtained to indicate that the 

 ellipticity in a cavity intended to be circular, resulting from any reasonable 

 manufacturing deviations, would not have a noticeable effect on the reso- 

 nant frequencies or Q values, at least away from mode crossings. 



Full Cylindrical Coaxial Resonator 



The full coaxial resonator has been of some interest because of various 

 suggestions for the use of a central rod for moving the tuning piston in a 

 TE OUi cavity. 



The cylindrical coaxial resonator, with the central conductor extending 

 the full length of the resonator, has modes similar to the cylinder. In 

 fact, the cylinder may be considered as a special case of the coaxial. The 

 indices /, m, n have much the same meaning and the resonant frequencies 

 are determined by the same equation (1). However, now the value of r 

 depends in addition (see Fig. 1) upon 77, where 



_ diameter inner conductor _ ^ 

 diameter outer conductor a ' 



The problem now arises of how best to represent the relations between 

 /, a, b and L. The r's depend on tj; so one possibility is to determine their 

 values for a given 77 and then construct a series of mode charts, one for each 

 value of 77. 



A more flexible arrangement is to plot the values of r vs 77 and allow 

 the user to construct graphs suitable for the particular purpose in hand. 

 An equivalent scheme has been used by Borgnis.^^ 



It turns out that as 77 — ^ 1, r(l — 77) —> ftiir, for the TM modes and the 



