432 BELL SYSTEM TECHNICAL JOURNAL 



Here /3i is 7" times the transit angle in radians from cathode to anode. For 

 Va we use a velocity fluctuation with the mean-square value given by (10.9). 

 Suppose now that there is a constant-potential drift space following the 

 diode anode, of length fii/j in radians. If we apply (10.6)-(10.8) and assume 

 that the space-charge is small and the transit angle long, we find that qb , 

 the value of qb at the end of this drift space, is given in terms of qa and Va , 

 the values at the beginning of this drift space, by 



q'l. = iq'a + {h/ubW'a)e~^' (10.19) 



The case of Vb , the velocity at the end of this drift space, is a little dif- 

 ferent. The first term on the right of (10.8) can be shown to be negligible 

 for long transit angles and small space charge. The last term on the right 

 represents the purely kinematic bunching. For the assumption of small 

 space charge the middle term gives not zero but a first approximation of a 

 space-charge effect, assuming that all the space-charge field acts longitu- 

 dinally. Thus, this middle term gives an overestimate of the effect of space- 

 charge in a narrow, high-velocity beam. If we include both terms, we ob- 

 tain 



Vb = Htq'a + e~%!, (10.20) 



Here the term on the right is the purely kinematic term.* 



Now, the current from the gun is assumed to go into the drift space, 

 so that qa is qb from (10.17) and Va is Va from (10.18). The d-c velocity at 

 the gun anode and throughout the drift space are both given by Ub . If 

 we make these substitutions in (10.19) and (10.20) we obtain 



q'b = {lo/ubm - 0,)e~'^'^^'\'a (10.21) 



,',= -(2|;+ l)e-'^'+^^%„ (10.22) 



The term l/Si/lS^ in (10.22) is the "space-charge" term. We will in the fol- 

 lowing analysis omit this, making the same sort of error we do in neglecting 

 space charge in the traveling-wave section of the tube. If space charge in 

 the drift space is to be taken into account, it is much better to proceed as 

 in 9.7. 



From the drift-space the current goes into the helix. U is now necessary 

 to change to the notation we have used in connection with the traveling- 

 wave tube. The chief difference is that we have taken currents as positive 

 to the right, but allowed h to be the d-c current to the left. If i and v are 



* The first term has been written as shown because it is easiest to use the small space- 

 charge value of //* for the drift region (//*) in connection with the space-charge limited 

 value of /•"* for the cathode-anode region rather than in connection with (10.17). 



