ELECTRON BALLISTICS IN HIGH-FREQUENCY FIELDS 333 



cos(^/ + c) = cos(^« + .) + 



^sin (^ + c\[7)yx + -q y^ + • • •] H 



and as before equating like powers of -q with y defined as 



y = Jo + ^7 ji + ry-i + . . . 

 we finally arrive at 



.. 1 ^ I 



yi = cos {wt + (^) COS f ^ + c\ 



yi = —ynr/b cos (w/ + ^) sin [~^ -\- c\. 



Now we need only integrate these expressions to obtain the values of the y's 

 and the y's needed to evaluate the K's,, 



If we average y"^ over all values of the starting phase we can write the 

 energy contributed by the field to the electron's velocity. When this is 

 done one finds that the odd powers of 77 are identically zero leaving only the 

 even powers to be considered and for small signal analysis purposes we need 

 only consider K^ . The energy per electron expressed in volts is 



V = 2.49 X IQ'^E'XJid) 



where f (6) = oS^Ki , and the power is obtained by multiplying this expression 

 by the beam current in amperes. 



The end results can be expressed as curves oi f{d) against d as shown in 

 Fig, 1. Three examples are shown: the uniform field case and two different 

 harmonic distributions as indicated by the smaller plot in the lower left- 

 hand corner. You will note that there exist regions of positive f(0) where 

 the net transfer of energy is from the field to the electron and regions in 

 which the transfer is in the other direction ; the former portions are of con- 

 siderable interest in connection with the input gaps in velocity modulation 

 tubes, and for that matter in the cathode grid region of the negative grid 

 tube although this is more compHcated than is here indicated, as this trans- 

 fer of energy constitutes a loss to the field which loads the input circuit. 

 The latter portions may be utilized as was done in the Muller and Llewellyn 

 diodes to obtain sustained oscillations. 



If, as I have indicated, we maximize y- as a function of the starting phase 

 we can evaluate the modulation coefi&cient. The value for the uniform field 



