WAVE PICTURE OF MICROWAVE TUBES 1359 



NOISE WAVES ON ELECTRON STREAMS 



Consider the electrons of the beam as they leave the cathode. If the 

 velocity distril)ution is MaxweJlian, and if the electrons leave inde- 

 pendently, there will be a mean-square fluctuation in convection current, 

 i", given by 



I = 2eIoB 



and an uncorrelated mean square fluctuation in ac velocity, v , given by 



^-'>(fj(~> 



Here /o is beam current, e and m are electron charge and electron mass 

 k is Boltzmann's constant, To is cathode temperature and B is bandwidth' 



Usually, space-charge-limited flow is used. In this case the beam cur- 

 rent is only a part of the emitted current ; the rest is turned back at the 

 potential minimum. In this case we may use the above relations, counting 

 Jo as the beam current, as some sort of approximation for the current 

 passing the potential minimum. 



The wave picture we have been discussing may be seriously inaccurate 

 near the cathode where the relative spread in electron velocities is large. 

 Suppose that we hope for the best and apply it. We find that in the most 

 general case our electron stream will have on it a noise standing-wave 

 pattern. If imin and Vax are the minimum and the maximum noise 

 currents, 



I ^'min I I ?max I 1 / CO \ f kT, 



I'm ax 



2ehB 2 \o}J \eVo 



Here a is a constant near to unity. 



The noise pattern is made up of two uncorrelated noise standing-wave 

 patterns, one from i at the cathode and the other from v at the cathode; 

 these patterns have amplitudes ?"i and 12 at their maxima; the minima 

 are of course zero. We have 



I imin I I tmax I = U'l I h'2 | slu ^ 



Here ^ is the relative phase angle of the standing- wave patterns associ- 

 ated with t'l and Z2 . That is, if the maximum of the 12 pattern is at that 

 of t'l ,■*• = 0, Avhile if the maximum of the 12 pattern is midway between 

 maxima of ii , then ^ = 7r/2. 



The first of these theorems says something about the noise current at 

 the maximum and that at the minimum, but it does not directly say how 



