246 BELL SYSTEM TECHNICAL JOURNAL 



tion coefficient of amplitude, p, depending only on the termination [see 

 equation (30)], and of phase, <f), depending only on the frequency,/, and the 

 Hne length, f. Thus: 



from which 



/ = e. («) 



This equation expresses the linear relation between frequency and phase for 

 the load, specified by the reflection coefficient, into which the magnetron 

 operates. The heavy dashed lines in Fig. 35 represent this relation for 

 two particular line lengths, A and 4 , with 4 very much longer than A . 

 The difference in line length, 4 — A , corresponds to an input phase differ- 

 ence of many times tt radians. For the case in Fig. 35, however, this is 

 chosen as an integral multiple of t so that the curve, plotted for the funda- 

 mental period to r, will lie in the same range. 



The variation of operating frequency of the magnetron with variable 

 phase of the load reflection coefficient is a periodic function whose amplitude 

 increases with increasing amplitude, p, of the reflection coefficient. This 

 function may be determined graphically from a Rieke diagram of the mag- 

 netron, like that shown in Fig. 33, by traversing the appropriate circle 

 concentric with the center of the diagram and plotting the frequency of 

 operation against phase. In Fig. 35 are plotted such curves for two values 

 of p corresponding to different terminations at the end of the long line, 



A more detailed analysis of the condition of oscillation shows that it is 

 possible for the magnetron to oscillate stably at those intersections of the 

 magnetron and load frequency characteristics at which the slope of the load 

 line is greater than the slope of the magnetron characteristic. Thus, as 

 indicated in Fig. 35, oscillation may occur at only one frequency for the 

 line of length A if P = 0.2 but at two frequencies if p = 0.5. In the latter 

 case the middle intersection, indicated by an open circle, does not corre- 

 spond to stable oscillation. For a line of length A , on the other hand, two 

 oscillation frequencies are possible at both p = 0.2 and p = 0.5. If in 

 either case the line lengths A and A are increased by only an approximate 

 quarter wavelength, corresponding to a phase change of 7r/2 radians, the 

 light dashed lines labelled ([ and fi in Fig. 35 represent the load character- 

 istics, and oscillation can occur at only one frequency with p equal either 

 to 0.2 or 0.5. 



If one considers the relationships depicted in Fig. 35 it becomes clear that 

 there are two critical relationships between p and f. The first specifies the 



