THE NATURE OF THE WAVES 

 8.3 Space-Charge Effects 



405 



J Suppose that we let d, the attenuation parameter, be zero, but consider 

 cases in which the space-charge parameter QC is not zero. We then obtain 



0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 



d 



Fig. 8.5 — For h = 0, that is, for electrons with a velocity equal to the circuit phase 

 velocity, the gain factor B falls as the attenuation parameter d is increased. For small 

 values of d, the gain is reduced by \ of the circuit loss. 



2.5 



1.0 



0.6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 



d 

 Fig. 8.6 — How the three x's vary for 6 = and for large losses. 



the equations 



{x^ - f-){h + );)-+- 2.^:23, -f \qc{h + v) + 1 = 



»[(^' - /) - 2>'Cy + ^) + <2C] = 



(8.26) 

 (8.27) 



Solutions of this have been found by numerical methods for QC = .25, 

 ,5 and 1; these are shown in Figs. 8.7-8.9. 



