REFLEX OSCILLATORS 467 



going from the cathode, it has a maximum retarding value for electrons 

 leturning through the gap. This is true also for If cycles, 2f cycles, n + f 

 cycles. Hence as Fig. 3 shows if the time electrons spend in the drift space 

 is 11 + f cycles, the electron bunches will return at such time as to give up 

 energy to the resonator most effectively. 



II. Electronic Admittance — Simple Theory 



In Appendix III an approximate calculation is made of the fundamental 

 component of the current in the electron stream returning through the gap 

 of a reflex oscillator when the current is caused by velocity modulation 

 and drift action in a uniform retarding field. The restrictive assumptions 

 are as follows: 



(1) The radio-frequency voltage across the gap is a small fraction of the 

 d-c accelerating voltage. 



(2) Space charge is neglected. Amongst other things this assumes no 

 interaction between incoming and outgoing streams and is probably the 

 most serious departure from the actual state of affairs. 



(3) Variations of modulation coefficient for various electron paths are 

 neglected. 



(4) All sidewise deflections are neglected. 



(5) Thermal velocities are neglected. 



(6) The electron flow is treated as a uniform distribution of charge. 



(7) Only two gap transits are allowed. 



An expression for the current induced in the circuit (/3 times the electron 

 convection current) is 



(0Vd\ j{ut-6) 



i = 2h^J,[^^Je^'^'-'\ (2.1) 



Here the current is taken as positive if the beam in its second transit across 

 the gap absorbs energy from the resonator. The voltage across the gap 

 at the time the stream returns referred to the same phase reference as the 

 current is v — Ve~^ "'"" ' . Hence the admittance appearing in shunt 

 with the gaps will be 



_ 21 (,13 (^Vd\ ,((W2)-9) (r. r.\ 



For small values of V approaching zero this becomes 



_ h^'O j((,r/2)-e) _ J((ir/2)-9) /^ ,, 



i es — r>-[T ^ Jef^ \^-Jj 



ZV 



