138 BELL SYSTEM TECHNICAL JOURNAL 



the authors are indebted to E. B. Payne and H. R. Kimball for filters, 

 and to H. Whittle and A. G. Ganz for transformers and retard coils. 



Appendix 

 Grid Current Modulator, Large Grid Potentials 

 Making use of the observation that the positive lobes of the input 

 wave are effectively suppressed with a sufificiently large external grid 

 resistance, we first define a function equal to zero when the independent 

 variable is positive, and equal to the variable when the variable is 

 negative. This is evidently a representation of the potential effective 

 on the grid in terms of the applied potential. If we denote the grid 

 potential by — f{y) where y is the impressed potential, it may be 

 expressed as a Fourier series 



— f{y) = bo/2 + 2&„, cos miry I Y + o,„ sin mwy/Y, (14) 



in which the coefficients are determined by the usual relations 



1 r^ niTTV Y , s 



b,n = — \ y cos -^^dy — -^ir:-^ (cos ;H7r — 1), 



m-TT- 



(15) 



1 / ^ . WTTV , ( .s,„_i Y 



a,n = -T7 I V sm — =j- flv = (— 1) '■ — cos m-rr, 

 Y Jq ' Y ' niT 



&o = yJ^^<v= Y/2, 



In these equations y represents the generator e.m.f. and Y is its 

 maximum value. If we put (15) in (14), we have 



_/(y) = F/4 - 2ZY™' '^"' ~ '^'^'/^ 



'""-' • ' (16) 



the last term of which may be summed to v/2 and (16) may be re- 

 written as 



„,, , ., 2F'«="cos (2w - l)7rv/F .... 



- /(v) = F/4 + yll -~, H 77- T^ . (1 /) 



which represents the desired solution. It will be observed that the 

 first two terms of the right member contribute only a d.c. term and 

 the fundamentals, so that the other modulation products must come 

 from the summation. This expression is a perfectly general one as far 

 as the form of y is concerned. In the case of a sinusoidal grid e.m.f. it 

 is possible by more customar>- methods to find the grid potential, but 



