624 BELL SYSTEM TECHNICAL JOVRNAL 



tor. There is no electric field outside of the resonator (except a very little 

 that leaks out near the grids). To take a charge from one grid to another 

 outside of the resonator require? no work. Thus the voltage across the gap 

 is #/(//, where \p is the magnetic flux around the axis. Actually, there 

 is a little magnetic flux between the grids, and hence the voltage near the 

 edges of the grids is a little less than that at the center. 



Current flows as a displacement current between the grids, and as con- 

 vection current radially out around, and back along the inside of the resona- 

 tor to the other grid. This current flow produces the magnetic field that 

 links the axis. 



Part of the magnetic flux links the coupling loop. If this part is \pi and 

 if the coupling loop is open-circuited, the voltage across the coupHng loop 

 will be d-^tjdt. 



Fig. 114. — Equivalent circuit for a resonator having 3 modes of resonance. 



In the resonator of Fig. 113, power leaks out through the iris into the wave 

 guide. Part of the wall current in the resonator flows out through the hole 

 into the guide; part of the magnetic flux in the resonator leaks out into the 

 guide. 



In dealing with resonators as resonant circuits of reflex oscillators, to 

 which the electron stream and the load are coupled, we are interested in 

 the gap and output impedances. For a clear and exact treatment, the 

 reader is referred to a paper by Schelkunoff. No exhaustive treatment 

 of the problem will be given here, but a few important general results will 

 be given. 



If the resonator is lossless, the impedance looking into the loop ma>- be 

 represented exactly by an equivalent circuit indicated in Fig. 114. As 

 coils used at low frequencies arc really not simply ideal "inductances," 

 (an idealized concept), but have many resonances (ascribed to distributed 

 capacitance), so the resonator has many, in fact, an intinity of resonances. 

 In the equivalent circuit shown in Fig. 114, only ^ of these are represented, 



'" S. A. Schelkunoff, Representation of InipccUuicc I'luutions in Tfrnis nf Resonant 

 Frequencies, Proc. I.R.E., 32, 2, pp. 83-90, Fcl)., 19-44. 



