852 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 



may be seen to give a resultant increasing wave traveling somewhat 

 slower than the initial wave, and thus slower than the electron velocity. 

 Returning to Fig. 7 we see that electrons in the decelerating field [from 

 +30 to +210° in Fig. 7(a)] have been slowed down, and because of their 

 initial velocity being faster than the wave velocity, have moved forward 

 in the wave giving a region of minimum velocity somewhat in advance 

 of the point of maximum retarding field (greatest negative slope in the 

 wave potential). Also, bunching due to acceleration and deceleration of 

 electrons has produced a maximum of electron current density which, 

 because of the initial excess electron velocity, is somewhat to the right 

 of the potential maximum (downward). 



As the level is increased the modulation increases and at 17 db below 

 saturation drive, Fig. 7(b), some nonlinearity is evident. The velocity 

 and current are no longer sinusoidal, but show the beginnings of a cusp 

 in the velocity curve and a definite non-sinusoidal bunching of the 

 electrons in the retarding field region (between +30 and 210°). 



In the next pattern, Fig. 7(c), at 14 db below saturation a definite cusp 

 has formed with a very sharp concentration of electrons extending sig- 

 nificantly below the velocities of the other electrons. We already have a 

 wide range of velocities in the vicinity of the cusp, and at this level the 

 single valued velocity picture of the traveling wave tube breaks down. 

 Although it cannot be distinctly resolved, the study of many such pic- 

 tures leaves little doubt that the cusp and its later development is really 

 a folding of the velocity line. 



The next pattern at 12 db below saturation drive. Fig. 7(d), shows a 

 greater development of the spur and a somewhat greater consolidation 

 of current in the main bunch between +60° and +180°. It is interesting 

 that the velocity in this region has not changed significantly. In order 

 for this to be true the space charge field must just compensate for the 

 circuit field. In the vicinity of the 60° point the space charge field ob- 

 viously must reverse, accounting for the very sharp deceleration evident 

 in the very rapid development of the low velocity spur. The decelerating 

 field must be far from that of the wave, inasmuch as the electrons just 

 behind the cusp are much more sharply decelerated than those preced- 

 ing the cusp. We conclude that there are very sharply defined space 

 charge fields much stronger than the helix field. At this relatively low 

 drive, the velocity spread has already achieved its maximum peak value. 



The succeeding three patterns show a continuing growth of the spur, 

 a continued bleeding of electrons from the higher velocity regions, and 

 a consolidation of the main bunch just in advance of the spur. Presum- 

 ably the increased concentration of space charge in the bimch has kept 



