842 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 



same basis that under high space charge and elevated voltage conditions, 

 efficiencies might be as high as 



77 = 8C (3) 



J. C. Slater^ on the other hand considered the motion of electrons in a 

 traveling wave and concluded that the maximum possible reduction in 

 beam velocity would also indicate a limiting efficiency of 2C. Taking a 

 more realistic account of the electron velocity, Pierce showed that these 

 considerations lead to a value of 



V = -^yiC (4) 



which, since t/i ranges between —}'2 and —2, leads to the same range of 

 values as the other predictions. 



None of these papers purport to give a physical picture of the over- 

 loading phenomenon, but only specify clear limitations to the linear 

 theory. L. Brillouin on the other hand found a stable solution for the 

 flow of electrons bunched in the troughs of a traveling wave. This he 

 supposed to represent the limiting high level condition of traveling wave 

 tube operation. His results give an efficiency of 



V = 2hC (5) 



In the first numerical computations of the actual electron motion in a 

 traveling wave tube in the nonlinear region of operation, Nordsieck pre- 

 dicted efficiencies ranging between 2.5 and 7 times C and showed that 

 there would be a considerable reduction in efficiency for large diameter 

 beams, due to the non-uniformity of circuit field across the beam diame- 

 ter. He also gave some indication of the electron dynamics involved. 

 Improving on this line of attack, Poulter calculated some cases includ- 

 ing the effect of space charge and large values of C. 



Tien, Walker and Wolontis carried computations still further for small 

 values of C by including the effect of small beam radii upon the space 

 charge terms, and showed that space charge and finite (small) beam radii 

 result in much smaller efficiencies than were previously predicted. J. E. 

 Rowe^ got similar results and gave more information on the effects of 

 finite values of C. Computations for large values of C by Tien showed 

 that a serious departure from the small C conditions takes place above 

 values of C = 0.1 if space charge is small (i.e., below QC = 0.1) and 

 above C = 0.05 for larger values of space charge. They indicated that a 

 maximum value of efficiency as high as 40 per cent should be possible 

 using C = 0.15, QC = 0.1 and elevated beam voltages. 



These five papers give some insight into the electron dynamics of power 



