1356 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 



Traveling-wave tubes really involve at least four waves: two space- 

 charge waves and two circuit waves. Usually, the backward circuit wave 

 is so far out of synchronism with the space-charge waves that we can 

 neglect its coupling with them. Further, if the space-charge waves are 

 well separated in velocity, that is, when cog is large enough, then when 

 one is coupled to the circuit wave the other isn't, and so we can get some 

 idea of traveling-wave tube operation by considering waves in pairs. 

 The simple mathematics of such coupling is given in Appendix D. 



In Fig. 10, the behavior of various phase constants, plotted as a func- 

 tion of oj/wo , is shoAvn qualitatively. Here co is radian frequency and Uo 

 is electron velocity. We may consider that co/uq is varied by changing 

 the electron velocity Vo and keeping the frequency w constant. The 

 horizontal line (3c is the phase constant of the forward circuit wave in 

 the absence of electrons, or when the coupling to the electrons is zero. 

 jSc does not change with electron velocity. /8« and ^f are the phase con- 

 stants of the slow and fast space-charge waves, respectively, with zero 

 coupling to the circuit wave. For the slow space-charge wave, the power 

 flow is negative, while for the circuit wave and the fast space-charge 

 wave the power flow is positive. Thus, for coupling between the slow 



0.5 



1.0 



1.5 



2.5 



Uo 



Fig. 10 — Suppose that at a constant radian frequency w we change the electron 

 velocity wo in a traveling-wave tube. If the waves of the electron stream were not 

 coupled to the waves of the helix, the phase constants, /3<- of the forward circuit 

 wave, (3s of the slow wave, and /?/ of the fast wave, would vary approximately as 

 shown. 



