VARIABLE FREQUENCY ELECTRIC CIRCUIT THEORY 535 



Appendix 1 



Formula (40) et sequa establish the fact that the actual frequency of 

 the wave (29) varies between the limits 



provided s(t) is a pure sinusoid X sin co/ and X:»co. This agrees with the 

 concept of instantaneous frequency. 



When s(t) is a complex function — say a Fourier series — the frequency 

 range of Wean be determined qualitatively under certain restrictions, as 

 follows: 



We write 



W = exp ( iwct + iX i sdtj ■ (la) 



F{iw)e'"^doi. (2a) 



00 



The Fourier formulation is supposed to be valid in the epoch — t — T 

 and T can be made as great as desired. Then 



F(io)) = T I exp ( i\ I sdt — icot ) dt. {Za) 



We now suppose that, in the epoch — ^—2", 



X I 5(// " (4a) 



becomes very large compared with lir. On this assumption, it follows 

 from the Principle of Stationary Phase, that, for a fixed value of co, the 

 important contributions to the integral {2)a) occur for those values of 

 the integration variable t for which 



^^{x £ sdt - .i) = 0, 



or 



o) = \s{i). 



Consequently the important part of the spectrum F{iui) corresponds " 

 to those values of co in the range 



Therefore the frequency spread of W lies in the range from coc + X^min 



to Wc + X^max or COc ± X If 5njax = — ^min = 1- 



