430 BELL SYSTEM TECHNICAL JOURNAL 



We will notice that / and the g's are current densities and that, contrary 

 to the convention we have used, they are taken as positive to the left. Thus, 

 if the area is a, we would write the output convection current; as 



i — —aqh 



where qb is the convection current density used in (10.6) -(10.8). 



Peterson has used (10.6)-(10.8) in calculating noise figure by replacing 

 the actual multi-velocity flow from the cathode by a single-velocity flow 

 with the same mean square fluctuation in velocity, namely,^ 



t,2 ^ (4 - 7r)77 {kTjh)B (10.9) 



Here Tc is the cathode temperature in degrees Kelvin and h is the cathode 

 current. 



Whatever the justification for such a procedure. Rack has shown that it 

 gives a satisfactory result at low frequencies, and unpublished work by 

 Cutler and Quate indicates surprisingly good quantitative agreement under 

 conditions of long transit angle at 4,000 mc. 



We must remember, however, that the available values of the coeffi- 

 cients of (10.6)-(10.8) are for a broad electron beam in which there are 

 a-c fields in the z direction only. Now, the electron beam in the gun of a 

 traveling-wave tube is ordinarily rather narrow. While the a-c fields may 

 be substantially in the 2-direction near the cathode, this is certainly not 

 true throughout the whole cathode-anode space. Thus, the coefficients 

 used in (10.6)-(10.8) are certainly somewhat in error when applied to 

 traveling-wave tube guns. 



Various plausible efforts can be made to amend this situation, as, by 

 saying that the latter part of the beam in the gun acts as a drift region in 

 which the electron velocities are not changed by space-charge fields. How- 

 ever, when one starts such patching, he does not know where to stop. In 

 the light of available knowledge, it seems best to use the coefficients as they 

 stand for the cathode-anode region of the gun. 



Let us then consider the electron gun of the traveling-wave tube to form 

 a space-charge limited diode which is short-circuited at high frequencies. 



If we assume com.j)lete space charge (space-charge limited emission) 

 and take the electron velocity at the cathode to be zero, we find that the 

 quantities multiplying q^ in (10.6)-(10.8) are zero. 



/?* = £*=. //* = 0* (10.10) 



^ L. C. Peterson, "Sjiace-Chargc and Transit-Time KtTecls on Sifjnal and Noise in Mi- 

 crowave Tetrodes," Proc. l.R.E., Vol. 35, pp. 1264-1272, Novcmt)er, 1947. 



^ A. J. Rack, "Effect of Space Charge and Transit Time on the Shot Noise in Diodes," 

 Bell System Technical Journal, Vol. 17, pp. 592-619, October, 1938. 



