WAVE PICTURE OF MICROWAVE TUBES 1367 



The change AU in the "voltage" U is 



ATT '^'<^^ AT/ 



m 



The con^'ection current i flows against the voltage Al^, so that a power 

 AP is transferred from the beam to the resonator which is attached to 

 the grids. 



AP = -ReAVi* (B4) 



Thus, the change in the power in the beam in passing through the grids 

 must be — AP 



-AP = Re {- AVi*) = ReAUi* (B5) 



AP = -ReAUi* 



According to the expression we ha\'e used in calculating beam power, 

 if the "\-oltage" of the beam on reaching the grids is U, and the convec- 

 tion current is i, then the beam power Pi on reaching the grids is 



Pi = ReUi* (B6) 



After passing through the grids, U is increased by an amount AU while 

 the current is unchanged, so that the power P2 of the beam leaving the 

 grids is 



P2 = Pe(f/ + AU)i* (B7) 



The loss of power in the beam, AP, is 



AP - Pi - P2 = -RcAUi* (B8) 



This agrees with (B5), in which AP was calculated as the poAver lost from 

 the beam to the resonator. 



Appendix C 



the effect of the velocity distribution in the electron beam 

 on the effective plasma frequency 



Consider an electron beam in which electron motion is confined to the 

 ^-direction, and in which the electrons have a velocity spread with a 



