ELECTRON BALLISTICS IN HIGH-FREQUENCY FIELDS 331 



where the K^s are a function of the transit time, of the field distribution and 

 of the entrance phase, and we will proceed to evaluate these coefficients. 

 The average energy per unit of change as expressed in volts is then simply 



— at the end of the field while the gain is: 



F.v = mw - Zo) = ^-^^ + ^^ + ... 



where the bar means that we are averaging over all values of the entrance 

 phase. 



It is of interest to evaluate the value of velocity y- which individual elec- 

 trons receive as a function of the entrance phase. For small signals it is 

 usually sufficient to evaluate y~ maximized with respect to the starting 

 phase, then 



F_ = m)iK - KoU. = ^-^' + 1|^^ + . . .] 



max. 



We can further define the ratio of Fmax to the largest value it can have as a 

 coefficient (3, sometimes called the modulation coefficient. 



But now to evaluate the K's. There are many ways of doing this as I 

 have intimated. We will proceed by writing 



y = yo{t) + vyiit) + v'^y2{t) + mysiO + • • • 



where the y's are coefficients depending upon the transit time / which in 

 itself is a function of the applied field thus 



t = to + vli + fh + V% + .... 

 We can then expand each function of time into a series remembering that 



/(. + „) =;(,)+w+mi:... 



or for our particular case 



yo(to)[vh + V' h -^ V^ h + • • •] 



>(/) = yo(/o) + 



1! 



, yo(to)[nh + 77^/2 + r?'/3 +•• •]' , 

 2! 



Now we can expand yi(t), y-iit) etc. in exactly the same way. Finally we 

 get a collection of terms which can be grouped in like powers of ij thus 



y = yo(to) + V [terms in y, y, ti , h , etc.] -\- r [ ] . . . 



The coefficient of the y] is in fact y^{h) ti + yi{k). We will not bother to 

 write the rest. This expression can then be differentiated to get y and then 



