NETWORKS FOR IMPEDANCE FUNCTIONS 397 



where 



The network is shown in Fig. 11. 



It will 1)(> foiuul that a "leakage" element, Gn , appears in the equiva- 

 lent network, although the air dielectric in the cavity was assumed to 

 have no leakage (G = 0). This element arises from the end-[)lugs and is 

 necessary to account for the dissipation in tlu^m. 



To obtain a network exactly equivalent to the cavity at all frequen- 

 cies, we should add a branch corresponding to n = 0, as was done in 

 example 1. This branch would make the equivalence hold do^vn to and 

 including zero frequency. But, inasmuch as the approximations that have 

 been made hold only for the high frequencies, where the resonances oc- 

 cur, it would be inconsistent to add this branch. What has been arrived 

 at, then, is a partial network representation that gives a close approxi- 

 mation to the impedance of the cavity at high frequencies, only. 



Example 3: Toroidal Cavity with E Axial. For further illustration, we 

 consider another mode of oscillation of the short-circuited concentric 

 transmission line investigated in the previous example. This time it is 

 assumed that the radial electric force vanishes while the axial electric 

 force between the end-plugs exists. The magnetic force is directed in 

 circles concentric mth the cylindrical central conductor, as before. This 

 situation is illustrated in Fig. 12, which is the same as Fig. 11, except for 

 the new disposition of the £"- vector. 



For the new mode of oscillation, where the wave is a cylindrical one 

 propagated back and forth between the inner and outer conducting 

 cylinders, the oscillatory space is naturally thought of as a "toroidal 

 cavity," while, in the previous example, where the wave was propagated 

 axially back and forth between the terminal discs, the space was called 

 a "concentric line." Actually, the cavity itself has the same geometric 

 form in the two cases. A pi'actical distinction may exist, however, in that 

 the axial mode of oscillation could be more easily excited in a cavity 

 whose axial length is large compared to its radius, while the cylindrical 

 mode would arise more easily in a flat "pillbox" cavity whose radius is 

 large compared to its axial dimension. 



The approach to the problem will be that of transmission line theory, 

 as before. This time, the "line" comprises two circular discs between 



