MODULATION IN VACUUM TUBES 445 



of /x; one is a modification of Carson's analysis while the other involves 

 a reconsideration of the tube characteristic equation. The modifi- 

 cation of Carson's treatment may be carried out with the aid of an 

 expedient as follows: with a single input frequency the wave form of 

 the generator voltage acting in the plate circuit is distorted by the 

 variable amplification factor of the tube, and it is this distorted wave 

 which acts in the plate circuit, instead of the pure sine wave operating 

 with constant ^c. The method of procedure is then evident; if we 

 refer the actual distorted generator potential to the grid circuit — 

 that is, if we divide it by the average value of ^ — we have an input 

 wave which, when treated by Carson's well-known method ^ in which 

 ^i is assumed constant, will yield correct results inasmuch as the ix 

 variation has been taken into account — somewhat indirectly, it is 

 true. When complex waves are applied to the grid circuit the effective 

 grid potential is made up of numerous components and the treatment 

 becomes very cumbersome. 



Tube Characteristic Expressed by Double Power Series 



A more direct method which has been used with some success 

 consists in expressing the tube current-voltage relation by a double 

 power series, without invoking any special relations regarding the 

 connection between a grid potential change and the equivalent plate 

 potential. If we use the symbols /&, Ep, Ec to denote the plate 

 current, plate potential, and grid potential, respectively, we may 

 express the plate current as a double power series as follows : 



= floo + a-i^Ep + aQiEc. (1) 



-I- a^aEp^ + anEpEc + ao2E,r 

 -\- asoEp^ + a-nEj~Ec + a^EpEc^ + a^^zEJ^ 



+ ■•• 



where 



1 a'"+"/(o, 0) 



m\n\ dE'"dEc" 



(2) 



and in which it is understood that the development applies with the 

 operating point on the characteristic. The derivatives, it will be 

 noted, are evaluated at the point at which both Ep and Ec are zero. 

 Some of the coefficients of Eq. (1) may of course be eliminated by 

 reference to the evident properties of the tubes, but this need not 

 concern us here since it is more convenient to formulate the tube 

 equation in another way. 



'^Proc. I. R. E., 1919. 



