658 BELL SYSTEM TECHNICAL JOURNAL 



where a is the distance between centers of grid wires and c is the wire 

 radius. 



Comparing (90) and (91) we see that both are proportional to Xp, 

 but that (90) contains a correction term, h^jy which is normally very 

 small as h is usually of the order of one-third to one-tenth. The 

 presence of the correction term may be explained by remembering 

 that (90) was derived for conditions holding when electrons are flowing 

 while (91) applies strictly only to a cold tube in which no free electrons 

 are present. This being the case, it is to be expected that the two 

 equations would be equivalent if the correction term were omitted 

 from (90). The term being small, this is very nearly true in any case 

 and gives 



27r6 

 Cg = farads in cm.^ (92) 



The capacitance Cg which was introduced as existing between the 

 electron stream and the grid wires is thus shown to have a value which 

 can be calculated with fair exactness under the conditions when Max- 

 well's equation (91) holds, namely when the grid wires are small com- 

 pared with their separation. 



Attention is now directed to the low-frequency value of the plate 

 impedance, Zp. From (88a) together with (81), (83) and (84) it may 

 be shown that 



-V = r. [^Mo + 1 (1 + 3')(1 + /O - 3 (1 + hY~\. 



(93) 



This is thought to be the first instance of an expression for the plate 

 resistance derived on strictly theoretical grounds. The formula as 

 it stands contains Tc which is given by (65) and h which is given by 

 (61), and both involve Va- This latter may be found from (63) in 

 terms of the direct current. A convenient approximation for Va is 

 obtained by making the assumption that the presence of electrons 

 changes the d.-c. potential Va by a small amount only, so that its 

 value may be calculated from the static capacitances of a cold tube. 

 The appropriate diagram is shown on Fig. 2. 



From the figure, putting Fc = we obtain 



F..10 = 



Fpo + 7=r VgQ 





