NOISE FIGURE 427 



Formally, F can be minimized by choosing the proper value of P. In Fig. 

 10.3, the minimum value of L, Lm , is plotted vs. the velocity parameter b 

 for zero loss and zero space charge (d = QC = 0). The corresponding value 

 of P, Pm , is also shown. 



P is a function of the cathode-anode transit angle di , which cannot be 

 varied without changing the current density and hence C, and of anode- 

 circuit transit angle 6-i , which can be given any value. Thus, P can be made 

 very small if one wishes, but it cannot be made indefinitely large, and it is 

 not clear that P can always be made equal to P,„ . On the other hand, these 

 expressions have been worked out for a rather limited case: an anode po- 

 tential equal to circuit potential, and no a-c space charge. It is possible 

 that an optimization with respect to gun anode potential and space charge 

 parameter QC would predict even lower noise figures, and perhaps at attain- 

 able values of the parameters. 



In an actual tube there are, of course, sources of noise which have been 

 neglected. Experimental work indicates that partition noise is very im- 

 portant and must be taken into account. 



10.1 Shot Noise in the Injected Current 



A stream of electrons emitted from a temperature-limited cathode has a 

 mean square fluctuation in convection current i\ 



T\ = 2ehBo (10.1) 



Here e is the charge on an electron, /o is the average or d-c current and B i^ 

 the bandwidth in which the frequencies of the current components whose 

 mean square value is il lie. Suppose this fluctuation in the beam current of 

 a traveUng-wave tube were the sole cause of an increasing wave 

 (F = V = 0). Then, from (9.4) the mean square value of that increasing 

 wave,, V'ls, would be 



K = {8eBVlc'/Io) I 5253 H (1 - 52/5i)(l - 53/5i) \~' (10.2) 



Now, suppose we have an additional noise source: thermal noise voltage 

 applied to the circuit. If the helix is matched to a source of temperature T, 

 the thermal noise power Pt drawn from the source is 



Pt = kTB (10.3) 



Here k is Boltzman's constant, T is temperature in degrees Kelvin and, as 

 before, B is bandwidth in cycles. If A'^ is the longitudinal impedance of the 

 circuit the mean square noise voltage Vl associated with the circuit will be 



F? = kTBK( (10.4) 



