NETWORKS FOR IMl'KDANCE FUNCTIONS 



391 



parallel network of Fig. 6 and the series network of Fig. 7 is the dual of 

 ilu^ parallel network of Fig. 9. These dual i-elationships are of course a 

 result of the fact that the impedance of an open-circuited line is the dual 

 of the impedance of the same line when short-circuited. 



Example 2: Short-circuited Concentric Line (or Toroidal Cavity with 

 E Radial). The preceding example considered a fictitious transmission 

 line of invariable parameters, R, L, G, C, having a perfect short circuit 

 at one end. The present example has to do essentially with the same 

 problem but considers it from a more practical point of view. The vari- 

 ation of R and L with frequency is taken into account and the impedance 

 of the "short-circuit" is no longer neglected. 



Let the line be the piece of coaxial cable plugged at both ends with 

 conducting material as illustrated in Fig. 1 1 . Considered from an alter- 

 native point of view, our line is now a toroidal cavity oscillating in the 



(Ro=o) 



2. 

 G 



-VAr 



2. 

 G 



Fig. 10- 

 Fig. 8. 



-L ^R_ 2L_ 



-Network of the second kind eciuivalent to the open-circuited line of 



2R 2L 



2 



mode where the electric force E is directed radially and the magnetic 

 force H lies in planes at right angles to the axis. If w^e assume the cavity 

 to be excited, or "driven," from one end,* the impedance that is effective 

 in defining the selective characteristic of the cavity with respect to fre- 

 quency is the total impedance at that end, that is, the sum of the im- 

 pedance Zi , viewed into the cavity, and the impedance, Z2 , of the ad- 

 jacent end-plug. Therefore, w^e have to deal with the impedance, 



Z = Zi + Z2. (2-1) 



By "impedance" is here meant the same thing that one considers in look- 

 ing at the problem from the point of view of transmission line theory, 

 namely, the complex ratio, for exponential oscillations, of the voltage 

 between the inside and outside cylindrical surfaces to the total current 



* For determining the "natural frequencies" of oscillation of the cavity, it is 

 immaterial at what point along it the imi)edance is taken; the total impedance 

 at every point has the same roots. The impedance is, nevertheless, not the same 

 at all points so that the behavior of the cavity, when driven, will depend to some 

 extent on the driving point. 



