GRID CURRENT MODULATION 139 



with a complex grid potential in which we are primarily interested no 

 simpler representation is known to the authors except where the two 

 frequencies involved are harmonically related. 



With the two frequency inputs considered, we have the grid potential 



3, = p cos /)/ + Q cos qt (18) 



and the summation term may be written 



2(P + Q) "g° cos [(2m - \)tt{P cos pt + Q cos g/)/(P + 0] 

 TT- m=i (2m - 1)- 



Upon expansion of (19) as the cosine of the sum of two angles we find 

 the terms cos {A cos 6) and sin {A cos 6) which require evaluation 

 before the solution can be put in significant form. This might 

 conceivably be done by direct expansion; for example, the first of the 

 expressions would then be 



f A nN 1 (^ cos ey- . {A COS ey 



cos {A COS e) = \ - ^ 2I — + ^ — 41 — 



Putting the terms of this series in terms of multiple angles gives us an 

 infinite series to be summed as the coefficient of each multiple angle 

 term. This may be done in terms of Bessel coefficients by Jacobi's 

 expansions ^- which are as follows 



cos {A cos 6) = E (- l)"e2n/2n(^) COS IflO, 

 n=0 



sin {A cos 6) = E (- l)"e2«+i/2n+i(^) cos (2« + \)d, 



in which Jk{A) is a Bessel coefficient of the ^th order and e^ is 

 Neumann's factor which is two for k not zero, and unity for k zero. 

 Carrying out the expansion, the sideband amplitude comes out as 



4(P + Q) 



J. ( ..^-^^ 1 / ' '^" 



P + Q \P + Q, 



(20) 



■^ y^^-pTo) ■^\t^) -^ 



which may be computed from tables to be found in Watson's book '^ 

 previously cited. Analogous expressions exist for the other com- 

 ponents. 



'^ Watson, "Theory of Bessel Functions," p. 22. 

 1' P. 666 et seq. 



