598 BELL SYSTEM TECHNICAL JOURNAL 



Up to this point, the present analysis is very similar to that given 

 by Spenke.^ For the reasons stated above, the two methods digress 

 hereafter. 



Since the instantaneous position of the potential minimum depends 

 upon the operating conditions as indicated above, the elimination of 

 this variable by the addition of the two equations (15) and (16) 

 results in a simpler expression for the cathode-anode spacing, namely: 



e 



Nhm lir , V" „, ,, , .,- . 



-y—rfr? • (18) 

 . .... ^M I 



1 _ f r'^^ ^ p 



To separate the noise or fluctuation component of the potentials 

 from their average values, it is necessary to assume at any plane in 

 the diode that it is possible to express both the instantaneous voltage 

 and velocity as a steady state or average value plus a very small 

 superimposed fluctuation component. Since this assumption does not 

 result in any discontinuities, it is possible to express the instantaneous 

 values of the dimensionless variable as 



V ^ ria -\- Vi 

 and 



v' = Vo' + Vi', 



(19) 



where both 771 and rji are very small. From this assumption, it may 

 readily be shown that the d-c. solution and the first approximations 

 to the fluctuation component of the solution for (18) are respectively 



ex r Nhm Itt , 



'" = Firjo') + f(vo) (20) 



and 



ex [Nhm fir _ tl^'^rji' , 



dF(rjo') dfivo) 



~3 — 7 r li — 7- 



70 



i r r°'^ . , f ° 13(8) 



^[Nhm ' ^ ' ' ---^^'' ' ' '-^'^'' 



(21) 



In the last equation, the average or d-c. values of all quantities 

 except 771, t;/ and 8(uc) are to be used. 



To avoid the necessity of using long, awkward equations, it will be 

 of great convenience to define several new variables. 



