446 BELL SYSTEM TECHNICAL JOURNAL 



Under normal conditions of amplifier operation Ep and Ec are 

 fixed, and the alternating grid and plate potentials vary about these 

 potentials. It then becomes convenient to express the coefficients as 

 derivatives referred to the specific point Ep^, £,„. If we indicate the 

 variable components of the grid and plate potentials by e and v, 

 respectively, the tube equation may be put in the form 



h = f{Ep, + V, £,„ + e) = ^b,nnv"'e'^ 



= f{Ep„ £,„) + bi(,v + boie (3) 



+ ^20^'- + bnve + ^02^- 

 + h-MV' + 621^'"^ + bi-jX'e"^ + boiC^ • ■ ■, 



where 



1 (?"'+"/(£po, EJ 

 mini d"'Epd"Eo 



The b coefficients are functions of the operating point — specified by 

 £;,„, Eco — and change with the operating point, in general. The a 

 coefficients are definitely referred to the origin, however, so that 

 each b coefficient may be expressed in terms of the a coefficients which 

 correspond to it. It is possible, by determining this relation, to 

 follow the variation of the 6's in terms of Ep^ and Eco- To do this we 

 substitute for Ec and Ep in Eq. (1) the expressions Eco + e, Ep,, + e, 

 respectively, find the constant term, and obtain succeeding coefficients 

 by differentiation. Thus 



Ouo = a2oEpg" -\- UsoEpg' -\- OioEpg 



+ a-nEp^-Era + aziEp^EcQ + a-iiEp^Ec^, 

 and further 



, _ 5&on , _ db{m 



dEp, dE,; 



This concludes our consideration of the tube characteristics without 

 reference to the circuit to which the tube may be connected. Eq. (3) 

 rather than Eq. (1) will be used in the following. 



It should be noted in terminating this part of the discussion that 

 the treatment is capable of easy extension to characteristics depending 

 upon a larger number of variables. Thus a four element (double 

 grid) tube characteristic may be expressed by a triple power series, 

 and so on. When the potential of the second grid is maintained 

 constant it is evident that the tube characteristic is given by a double 

 power series in which the coefficients depend in addition upon the 

 potential of the second grid. To determine the dependence quantita- 

 tively, the triple series will serve. 



