188 BELL SYSTEM TECHNICAL JOURNAL 



high magnetic field, provided V has the proper value, that the scallops be- 

 came relatively small variations in an otherwise smooth orbit (see Fig. 18). 



In the cylindrical magnetron, the radial variation of the DC electric field, 

 resulting in a decrease in the mean angular velocity of the electrons as they 

 approach the anode, would make it impossible for an electron to keep step 

 with the fields across the anode gaps were not a mechanism of phase focusing 

 operative. That such focusing is inherent in the interaction of electrons and 

 fields will be seen later. 



3.2 The Interaction Field: The electronic mechanism which has been 

 discussed in terms of electron motions through the fields at the gaps in the 

 multisegment anode, may also be discussed in terms of the traveling waves 

 of which the RF interaction field may be considered to be composed. The 

 RF interaction field corresponds to anode potential waves like those plotted 

 in Figs. 9, 10, and 11. The interaction fields for the several modes of oscil- 

 lation of the resonator system are thus to be distinguished by the number, 

 n, of repeats of the field pattern around the interaction space. Since the 

 potential at the anode radius is nearly constant across the faces of the anode 

 segments and varies primarily across the slots, the azimuthal variation of 

 the field cannot be purely sinusoidal but must involve higher order 

 harmonics. 



For a mode of angular frequency co = lirf, corresponding to a phase diflfer- 



Ztt 

 ence between adjacent resonators of n — , the anode potential wave is of 



periodicity n around the anode and may be written as a Fourier series of 

 component waves traveling in opposite directions around the interaction 

 space: 



k = n + pN, p = Q, ± 1, ±2, ■■■ . 



Note that the summations are taken over all integral values of k given by 

 equation (10). 



The interaction field for any mode of periodicity n is thus represented by 

 two oppositely traveling waves, whose fundamentals are moving with 



p f 

 angular velocities - = — ~, and whose component amplitudes, Ak and Bk, 

 n n 



in general are not equal. 7 and 5 are arbitrary phase constants. The ex- 

 pression (12) may be reduced to the form: 



Vhf = T.i^k- Bk) cos (co/ -kd -\-y) 



+ E 25.cos(c./ 4- l±j)cos(>fee - -"-^y 

 k = n -\- pN, p = 0, ± 1, ± 2, 



(13) 



