660 BELL SYSTEM TECHNICAL JOURNAL 



APPENDIX \'II 



Ideal Drift Field 



The behavior of reflex oscillators has been analyzed on the basis of a uni- 

 form field in the drift space. It can be shown that this is not the drift 

 field which gives maximum efficiency. The field which does give maximum 

 efficiency under certain assumptions is described in this appendix. 



Consider a reflex oscillator in which a voltage V appears across the gap. 

 This voltage causes an energy change of /3F cos di for the electron crossing 

 the gap. Here di is the phase at which a given electron crosses the gap for 

 the first time. The effect of the drift space is to cause the electron to re- 

 turn after an interval Ta where Ta is a function of this energy. 



Ta = fm cos e,) (gl) 



Thus, each value of Ta will occur twice every cycle (lir variation of ^i). We 

 will have 



01 = co/i (g2) 



da = w(/i 4- Ta) 



= 01 + <p{d,) 



(p(di) = OiTa (g4) 



(g3) 



Here h is the time at which an electron first crosses the gap and (/i -f Tq) 

 is the time of return to the gap. ^i and 6a are the phase angles of the voltage 

 at first crossing and return. 



The net work done by an electron in the two crossings is 



W = |SF4-cos e, + cos {d, + <p{e,))] (g5) 



If the beam current has a steady value /o , the power produced will be 



P = (^17o/27r) I [-cos e, + cos (01 + ^{d,))\ dd, . (g6) 



The integral of cos d\ is of course zero. Further, from (g4) we see that 



<^(0i) = -^(-^i) 

 Hence 



p = {fiVh/2-K) I Icos (-01 -f ip{e,) -f cos (0, + .p(0i))1 de^ 

 Jo 



(g7) 



= (/3K/o/27r) f cos <^(0i) cos 0i (/0i . 



^0 



