IIIGII-IKEQUENCY AMPLIFIER 35 



Tlic following additional nomenclature will be used: 

 eo dielectric constant of vacuum 



€0 = 8.85 X 10-'- farad/meter 

 7} charge-to-mass ratio of the electron 



■q = 1.76 X lO'i coulomb/kilogram 

 d-c. current densities 

 d-c. velocities 

 d-c. charge densities 



Poi = —Ji/ui , po2 = —Jilut 

 a-c. charge densities 

 a-c. velocities 



d-c. voltages with respect to the cathode 

 a-c. potential 



;Sl = wlU]_ , ^2 = C0/U2 



Although the small-signal equations relating charge density to voltage V 

 have been derived many times, it seems well to present them for the sake 

 of completeness. For one stream of electrons the first-order force equation is 



di'i dvi , dii dV 



■-^ — + «l = 7? 



dt dt dx dx 



(co — I3ui)vi = —r]^V 



_ -r?/3K 



^'1 = -7^ ^. (2) 



Wi(|Si - ^) 



From the conservation of charge we obtain to the first order 



dpi ^ r \ \ 



^ = — — IPor^'i + piih) 

 dt dx 



wpi = PoiiSl'l + Uilipi 



From (2) and (3) we obtain 

 We would find similarly 







^ - U\{^2 - ^f ^'^ 



It will be convenient to call the fractional velocity separation b, so that 



2(wi - U2) , . 



