HIGH-FREQUENCY AMPLIFIER 41 



3. Design Curves 



It is proposed to present in this section material for actually evaluating 

 the gain of the increasing wave for a particular geometry of electron How. 

 In this section there is some repetition from earlier sections, so that the 

 material presented can be used without referring unduly to section 2. In 

 order to avoid confusion, much of the mathematical work on which the 

 section is based has been put in the appendix. 



The flow considered is one in which electrons of two velocities, «i and u-i , 

 corresponding to accelerating voltages Vy and Vo , are intermingled, the 

 corresponding current densities /i and J2 being constant over the flow. 

 The flow occupies a cylindrical space of radius a. It is assumed that the 

 surrounding cylindrical conducting tube is so remote as to have negligible 

 effect on the a-c. fields. 



It will be assumed, according to (12), that the current densities and the 

 voltages Vi and Vo are specified in terms of a "mean" current Jo and a 

 "mean" voltage Vo corresponding to a velocity Uo , by 



_Ji_ 2l. 1± r.r. -s 



yzn - yzn - yzn (12aj 



The gain will depend on the beam radius, the free-space wavelength X, 

 and on Jo and Fo , and on the fractional velocity separation 



b = 2(^^L^J^) (6) 



wi + W2 



The wavelength in the beam, Xs , which is associated with the voltage 

 Fo is given by 



X. = X^ = X^2^ 



c c 



X, = 1.98 X KT'xVVo (26) 



Here c is the velocity of light. 



A dimensionless parameter IF is defined to be 



w = -; = -^ (27) 



TI'^ = 8.52 X 10^ 7—^ (28) 



Here We is the electron plasma frequency associated with the average space 

 charge density Jo/uo , and co is the radian frequency corresponding to the 

 wavelength X. In (28), the constant is adjusted so that Jo is expressed in 



