MAGNETRON AS GENERATOR OF CENTIMETER WAVES 



175 



IF 



eBdy 

 m dt ' 



d~y e 

 dr m 



dH 



( dx\ 



dt' 



= 0. 



(3) 



The particular case of most interest here is that for which the electron 

 starts from rest at the origin. The equations of motion then yield a cycloidal 

 orbit given by the parametric equations: 



X = vj — pc sin coj = Pc{(jict ~ sin cocO 

 y = pc{\ — cos cocO; 



in which: 



Vc = 



Pc 



E 

 B' 



mE 

 ~eB'' 



m 



B. 



(4) 



(5) 

 (6) 

 (7) 



This motion may be regarded as a combination of rectilinear motion of 

 velocity Vc in the direction of the x axis, perpendicular to both E and B, 

 and of motion in the xy plane about a circular path of radius Pc, at an 

 angular frequency Wc, the cyclotron frequency. Fig. 3 shows the resulting 

 cycloidal path and its generation by a point on the periphery of the rolling 

 circle. In the case of cylindrical geometry, it is often convenient to think 

 in terms of the plane case. 



In the case of cylindrical geometry with radial electric and axial magnetic 

 tields, the electron orbit, neglecting space charge, approximates an epicycloid 

 generated by rolling a circle around on the cyhndrical cathode. The orbit 

 is not exactly an epicycloid because the radial motion is not simple harmonic, 

 which state of affairs arises from the variation of the DC electric field with 

 radius. The approximation of the epicycloid to the actual path is a con- 

 venient one, however, because the radius of the rolling circle, its angular 

 frequency of rotation, and the velocity of its center, for the epicycloid, 

 all approximate those for the cycloid of the plane case. These approxima- 

 tions improve with increasing ratio of cathode to anode radii. Several 

 electron orbits in a DC cylindrical magnetron are shown in Fig. 4 for several 

 magnetic fields. 



