OPERATION OF THERMIONIC VACUUM TUBE CIRCUITS 439 



This illustrates strikingly the importance of the variation of /i in 

 modulators and detectors. 



Equation (14) is expressed in terms of e^, the voltage directly on 

 the grid of the tube. We may derive the expression for ip in terms 

 of e, a voltage impressed in series with a resistance, Q, in the external 

 grid circuit by noting that 



Hence, from (12), 



e, = {l-b,Q)e-b.Qe-+ . . . (17) 



Therefore 



ip = a,{\-b,Q)e-[aib2Q-a2{l-biQy]e'+ . . . (18) 



and, as in (13), ep= —ipZ. 



Substituting (17) and (18) into (5) and equating coefficients of like 

 powers of e, we get 



h = ^i~ T^aiZ 

 '~ l-fTiQ-T^a.ZQ' ^^^^ 



^[-a2ZT2-\-^Ts-arZT, + WZ-T,](l-b,Qr 

 ' l-^TrQ-T2aiZQ ^"^"^ 



The T's may be expressed in terms of r^ and v with the aid of (10). 

 The complete solution of this special case for first and second order 

 effects is then given by (18) above, in which we have now evaluated 

 the a's and &'s. 



Mathematical Digression 



Before the detailed steps in the complete development of the 

 general case, with general impedances instead of resistances, are at- 

 tempted, the following digression on the use of complex quantities in 

 non-linear equations is apposite. Included at this point, it serves a 

 two-fold purpose; first, the notation to be used is illustrated by means 

 of simple applications; second, it calls to mind the fundamental ideas 

 involved in the representation of impedances by complex quantities. 



Consider a current, /. If periodic, this current may be represented 

 by a Fourier series and expressed as the sum of a number of cosine 

 terms. Thus 



'=''{ ~2 )M T )+■■■ (21) 



where the symbol, j, represents the imaginary, V^- For brevity 

 this may be written 



I = (iih + iih) + (ilk + Ilk) + (22) 



