348 BELL SYSTEM TECHNICAL JOURNAL 



field oscillations occur with the magnetic field-strength very much above the 

 critical cutoff value and the efficiency on occasion reached as much as 70%. 

 While a careful reading of the literature will reveal that some of the earlier 

 experimenters were occasionally dealing with these oscillations, Posthumus' 

 observations represent a new departure in magnetron theory and practice 

 and one which we might do well to investigate. 



Posthumus' approach consisted in studying the electron paths in a mag- 

 netron in detail in order to find the conditions under which electrons may 

 reach the plate with considerably less energy than that corresponding to the 

 plate potential. He assumed a magnetron having k pairs of plates and 

 based his calculations on the supposition of a rotating electric field with k 

 pairs of poles. In reahty there exists a simple alternating field but this 

 can be resolved into two rotating fields rotating in opposite directions. 

 Power engineers will recognize this as identical with the procedure used in 

 analyzing single-phase rotating machinery. Posthumus neglected the field 

 opposite to the static angular velocity and considered only one component. 

 This is an approximation but a fairly plausible one which can be partially 

 justified. 



In the absence of oscillations there is a radial electric field independent 

 of the angular position and inversely proportional to radius (for the coaxial 

 cylindrical case). When oscillations are present there is an additional radial 

 field which varies as some periodic function of the angle and with a period 

 l-K, and a tangential component of the same general type. For simplicity 

 these functions are taken to be simple harmonic functions and can therefore 

 be split into two circular rotating fields. 



Posthumus writes the two simultaneous differential equations determining 

 the path of an electron, neglecting space charge, and inquires if a solution 

 is possible for an elecron path which travels at approximately the same 

 angular velocity as the rotating field but lags it by an angle a. An equally 

 satisfactory way of looking at this is to say that we transform our coordinates 

 from a fixed system to one rotating with the field and inquire if a solution 

 is possible where a the angular motion is always small. He finds that such a 

 solution is indeed possible and that for the electron motion to be stable the 

 value of a must be such that the electrons are somewhat behind the line for 

 which the field has its maximum retarding value. The electrons are thus 

 in a position to lose energy to the field and to spiral out toward the anode. 



Posthumus defined the value of the electron's radial velocity squared at 

 the anode as P and the total velocity squared at the anode as Q. Nor- 

 malized plots of these two parameters are shown in Fig. 15 as a function of 

 frequency. The upper plot shows the radial velocity. Obviously for elec- 

 trons to reach the plate at all they must have a positive velocity at the plate. 

 Electrons can therefore reach the plate with any given field value, say Z = 2, 



