CONSTANT FREQUENCY OSCILLATORS 99 



the current. It is convenient to write the series in the following form: 



/ = Z ^t.*«"' (3) 



?{ = — 00 ^ 



or, for brevity, in the symbolic form 



/ = E/(«a;) (4) 



where the summation is understood to extend from minus infinity to 

 plus infinity. Substitution of (4) and (2) into (1) gives: 



E = RZI(no,) + LZino:I{nw) + y^Z^^ 



C tnoj 



+ Fo + ^iE/(«co) + ^2EE/(ww)/(mco) 



+ A3ZZJ:iinc)I(mo:)I{lco) + • • •. (5) 



For the component of fundamental frequency, co, we get 



£(a)) = i?/(co) + Lfco/(a;) + ^ + ^i/(co) 



Cico 



+ ^2 E /(wC0)/(»/C0) +^43 E /(wC0)/(WC0)/(/c0) + • • • (6) 



where the summation terms involve the products of all frequency com- 

 ponents which beat together to give the fundamental, as indicated. 

 In order to put the last expression in symmetrical form, it is convenient 

 to multiply and divide each of the summation terms by /(a;) so that 

 we may write 



E{oj) = I{c 



i? + Lico + -i- + ^1 + ^ E/(«co)/(wa;) 

 + 4\E/(M^(wco)/(/a;) + 



(7) 



This expression exhibits the terms in square brackets in the form of an 

 impedance, and shows that the vacuum tube may be treated as an 

 ordinary linear circuit element if it is considered as having the im- 

 pedance 



Z = ^i+-^^/(„co)/(mco) +4^yi(nc,)I{mo:)I(lo:) + .••. (8) 



Of course, the numerical value of such an impedance cannot be 

 found from this expression alone, but in oscillator analysis there is no 

 necessity for its numerical evaluation. The very important fact that 

 the nonlinear elements in a circuit network may be replaced by equiva- 



