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BELL SYSTEM TECHNICAL JOURNAL 



while from (7.16) we have 



h 



V 



2Fo 



(12.3) 



We expect non-linear effects to become important when an a-c quantity is 

 no longer small compared with a d-c quantity. We see that because (1/5C) 

 is large, | i/h \ will be larger than | v/uo \ . 



The important non-linearity is a sort of over-bunching or limit to bunch- 

 ing. For instance, suppose we were successful in bunching the electron flow 

 into very short pulses of electrons, as shown in Fig. 12.2 As the pulses ap- 

 proach zero length, the ratio of the peak value of the fundamental com- 

 ponent of convection current to the average or d-c current /o approaches 2. 

 We may, then, get some hint as to the variation of power output as various 

 parameters are varied by letting \i\ = 21 o and finding the variation of power 

 in the circuit for an a-c convection current as we vary various parameters. 



TIME *" 



Fig. 12.2 — If the electron beam were bunched into pulses short compared with a cycle, 

 the peak value of the component of fundamental frequency would be twice the d-c cur- 

 rent /o . 



Deductions made in this way cannot be more than educated guesses, but in 

 the absence of non-linear calculations they are all we have. 



From (7.1) we have for the circuit field associated with the active mode 

 (neglecting the field due to space charge) 



E = 



T^T,(FJ/I3'-P) 



2{rl - r') 



(12.4) 



This relation is, of course, valid only for an electron convection current i 

 which varies with distance as exp(— Fs). For the power to be large for a 

 given magnitude of current, E should be large. For a given value of i, E will 

 be large if F is very nearly equal to Fi . This is natural. If F were equal to 

 Fi , the natural propagation constant of the circuit, the contribution to the 

 field by the current i in every elementary distance would have such phase 

 as to add in phase with every other contribution. 



Actually, Fi and F cannot be quite equal. We have from (7.10) and (7.11) 



-l\ = ^^{-j - jCb - Cd) 



(12.5) 



•r ^ 0e(-j + jCyi + Cxi) 



(12.6) 



