WAVE PICTURE OF MICROWAVE TUBES 



1347 



Here Vo is the accelerating voltage specifying the electron velocity Uo 

 and /o is the total beam current. 



We see that the characteristic impedance 7vi of the .slow wave is 

 negative. This means that the power flow in the +2 direction is negative. 

 We could also say that positive power flows in the —z direction, but this 

 may carry an unfortunate implication as to the direction in which 

 causality acts. An example may be helpful. 



Fig. 1 shows an electron beam acted on by the fields of two devices 

 A and B. The fields in A are such as to set up the slow wave only. This 

 travels between A and B. The fields of B are such as to just remove the 

 slow wave entirely, so that the electron beam leaves B with no ac dis- 

 turbance on it. The electron velocity Wo , phase velocity v, group velocity 

 Vg and negative power flow —P are all directed in the -\-z direction, that 

 is, to the right. 



We must remove a power P from A to set up the slow w^ave. A power 

 — P flows from A to B. We must add a power P to Bto remove the slow 

 wave from the electron beam. Causality acts from A to 5. To change the 

 ampUtude of the slow wave betw^een A and B we must change the fields 

 in A, not the fields in B. 



The power flow is the group velocity times the stored energy per unit 

 length. As the group velocity for the slow wave is positive and the power 

 flow is negative, we see that the stored energy must be negative. 



If we moved with the electrons and observed the weaves, we would 

 find that the average kinetic energy associated with the ac electron 

 velocity was equal to the average potential energy of the electric field, 

 and that both were positive ; this is characteristic of waves in a stationary 

 medium. The kinetic energy of the electrons relative to a fixed observer 

 is proportional to the square of their total velocity, that is, the ac velocity 

 plus the average velocity. The average velocity is larger than the ac 

 velocity, so that energy terms involving the product of the average 



UNMODULATED 

 BEAM 



ELECTRON VELOCITY 



Uo- 



PHASE VELOCITY 



GROUP VELOCITY 



POWER FLOW 



P I TOUT 



UNMODULATED 

 BEAM 



P lIlN 



Fig. 1 — Device A sets up the slow space-charge wave only, and device B 

 removes it. uo , v, Vg and —P are respectively the electron velocity, the phase 

 velocity, the group velocity and the power flow between A and B. 



