REFLEX OSCILLATORS 



661 



As cos ^(^i) cannot be greater than unity, it is obvious that this will have its 

 greatest value if the following holds 



< ^1 < 7r/2, .^(^i) = Inir, cos ^{6^) = + 1 (g8) 



7r/2 < ^1 < TT, ip{ei) = (2m + l)ir, cos (^(^i) = -1 (g9) 



These conditions are such that for a positive value of cos <p the gap voltage 

 is accelerating giving a longer drift time than obtains for a negative value 

 of cos ^(^i) for which a retarding gap voltage is required. Thus, physically 

 we must have 



2n > 2w + 1 



(glO) 



The simplest case is that for m = 1 and m = 0, so that in terms of the gap 

 voltage 



V <0, <p{di) = TT 



V > 0, ifidi) = Itt 

 This sort of drift action is illustrated by the curve shown in Fig. 136 



(gll) 



2 TT 



GAP VOLTAGE, V 



Fig. 136. — Ideal variation of drift time in the repeller region with resonator gap voltage. 



The problem of finding the variation with distance which would give this 

 result was referred to Dr. L. A. MacCoU who gave the following solution: 



Suppose Vo is the voltage of the gap with respect to the cathode and $ is 

 the potential in the drift space. Let 



Xo = \/2r]Vo/u (gl2) 



Here co is the operating radian frequency. Let x be a measure of distance 

 in the drift field. 



4> = Foil - (x/.vo)'], < X < .To 

 $= Foil - [(x/.Tor+ l]74(.v/.vo)'|, 

 This potential distribution is plotted in Fig. 137. 



X > Xo 



(gl3^ 



