870 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 



ation, the conditions for start of oscillation in a backward wave oscillator 

 are the same as for the output null in a traveling wave tube. Space charge 

 was first accounted for using the results of H. Heffner'^' " giving an 

 excellent check between predicted and computed helix impedance. Later 

 C. F. Quate showed that the same measurement could be used to de- 

 termine the space charge parameter QC as well as the helix impedance. 

 Since thermal velocity effects and the uncertainty of some of the assump- 

 tions used in evaluating the small signal effects of space charge cast 

 some doubt on the proper evaluation of this term, further measurements 

 were made on this factor, and a satisfactory correlation between the ob- 

 served value of QC and that computed from the Fletcher^'^ curves was 

 obtained. 



TOTAL ACCELERATING FIELDS 



From the velocity characteristics shown in Figs. 7 through 10, we can 

 deduce the electron accelerations, and thus the electric fields at any 

 point. While the curves are actually diagrams of velocity as a function 

 of phase, they closely correspond to the velocity- time or distance distri- 

 bution of the electrons in the traveling wave tube. Knowing these charac- 

 teristics we can deduce the motion of any element of charge, and thus the 

 force under which it moves. It is observed that over most of the curve 

 the shape of the velocity pattern does not change nearly so rapidly as 

 the redistribution of electrons within the pattern. Thus, we can approxi- 

 mate the situation at any amplitude by assuming the velocity pattern 

 to be constant, and that electrons move within the pattern according to 

 simple particle dynamics. This is a good approximation except where 

 the acceleration is high (i.e., vertical crossings of the wave velocity line). 



Consider then an element of the velocity pattern at phase $i and 

 velocity (wo + Aw). In an interval dt this element will move a distance 



(wo + Aw) dt (14) 



and will change velocity by 



du = E -dt (15) 



m 



At the same time the wave will have moved a distance v dt, resulting in 

 a relative change in phase between wave and current element of 



d^ = ^(uo - V -\- Aw) dt (16) 



In terms of equivalent differences the term in brackets can be written 



(«. - . + A„) = a/7T7. c ( ^° - ;: + "0 <i^> 



