214 BELL SYSTEM TECHNICAL JOURNAL 



conditions restrict the phase shift between resonators to a set of specific 

 values. These correspond to the travehng waves which on reflection at 

 the terminations constructively interfere. 



The cylindrical magnetron anode structure is a series of N resonators 

 connected in a ring. It may be thought to be a section of a linear array of 

 resonators rolled into a cylinder. The boundary condition imposed is that 

 of connecting together the resonators at the ends of the string. Under 

 these circumstances only those modes of oscillation are possible for which 

 the total phase shift around the ring is 2x« radians, n being any integer 

 including zero. The oscillations in adjacent cavities then differ in phase 



by -—z~ radians. Again this means that only those waves traveling around 



the anode block which constructively interfere are possible solutions. These 

 are waves which, after leaving an assumed starting point and traversing the 

 anode once, arrive back in phase with the wave then leaving in the same 

 direction. The anode potential waves and the RF interaction fields in the 

 interaction space to which they correspond have already been discussed in 

 connection with equations (12) and (13). In these electromagnetic field 

 patterns, the electric and magnetic field components are displaced both in 

 space and time phase by 7r/2 radians relative to one another, similar to the 

 manner in which voltage and current on a terminated transmission line are 

 related. 



6.3 The Modes of the Resonator System: It has been seen that the modes 

 of oscillation of a magnetron resonator system are characterized by definite 



values of the phase shift between adjacent resonators specified by — , in 



which the parameter n may assume only integral values including zero. 

 Each such mode of oscillation has a frequency different from the frequency 

 of any other mode and from the frequency of one of the N resonators oscil- 

 lating freely and uncoupled from its neighbors. In the general case of N 

 coupled resonators, as in the case of two coupled resonators previously dis- 

 cussed, the modes of oscillation have different frequencies because of the 

 effect of the mutual coupling between the resonators. For N = 2, the 

 oscillations in the two resonators are either in phase or tt radians out of 

 phase, the induction in one circuit by the other either directly adding to or 

 subtracting from the self induction. In the case of the multiresonator sys- 

 tem the mutual induction effect may bear phase relations to the self induc- 

 tion other than and tt radians. Thus not only the magnitude of the 

 coupling but also this phase relationship determines the magnitude of the 

 effect of the mutual induction and hence the amount of deviation of the 



