616 



BELL SYSTEM TECHNICAL JOURNAL 



The initial average velocity is small so that the low-frequency 

 plate impedance may be written 



r„ = 



el, 



OT' 



llhme^ 



(67) 



Thus for any transit angle, the mean square noise generator voltage 

 is given' by 



£/ = 12 M - ^ j krpTdfl -, [2 + 0^ - 2(cos 6 -\- 6 sm <?)] 



= 4Ski0.64:4T)rpdf 

 where 



4 



S = 



2 + 0"- - 2(cos e -\- sin 6) 



(68) 



For low transit angles, this expression reduces to 



Ef' =12 1 



5 ) kr.Tdf, 



(69) 



which is precisely the limiting value obtained by the much longer, 

 but more rigorous analysis. 



It must be understood that (68) is an approximation since the 

 transit time effect in the region between cathode and potential minimum 

 was entirely neglected, and because the validity of the average velocity 

 concept does fail at the very high frequencies. 



Some knowledge of the extent of the operating conditions for which 

 the above equations are good approximations may be obtained from 

 the d-c. current-voltage relation. For the boundary conditions 

 assumed, the low frequency current equation derived from the general 

 solution given by Llewellyn reduces to 



I - 



2.33(F- Vy- 

 10«(x - x')- 



1 -f 2.66 



\e[ r - V) J 



(70) 



This equation was shown by Langmuir to be a very good approxi- 

 mation for the plate current for most operating conditions and fails 

 only for very low values of plate voltages. Thus, it may be concluded 

 that (68) is a good approximation for all operating conditions except 

 for very low plate voltages and a small space charge. 



The plot of (68) given in Fig. 4 shows that the magnitude of the 

 mean square noise generator voltage decreases by five per cent only 

 for transit angles as large as one radian. 



