VARIABLE FREQUENCY ELECTRIC CIRCUIT THEORY 537 

 then 





(Sb) 



Replacing (46) by (5b) to take care of the distributed noise, the 

 noise term of {3b) becomes 



cos 



f X / 5C?M • — I (OJO + C0„ + IJLS)- cos {oint + 9n)d0}n 



+ sin ( X I 5C?/ j • — I (coo + con + iJLs) -sin (co„/ + dnjdcon. {%) 



Now this noise in the low frequency circuit is passed through a low 

 pass filter, which cuts off all frequencies above coa. Wo is the maximum 

 essential frequency in the signal s(t). 



It is therefore necessary to express (96) as a frequency function 

 before calculating the noise power. To this end we write the Fourier 

 integrals 



( X r sdn = - n Fc cos {oit + do)doi, (106) 



cos 



sm 



(\Csdt\ = -CF, sin (co/ + e,)doi. (116) 



We note also that 



''^ F, cos {oit + dM<^, (126) 



A 



fxs • cos ( X I ^'^^ ) ~ \'Tf sii^ ( ^ I sdt\ 



_ 1 r";. 



~ ""Jo 

 M5 • sin / X I ^^^] — ~ \^f cos ( ^ I sdt\ 



= - f" ^ F, sin (co/ + dc)dui. (136) 



Substituting (106), (116), (126) and (136) in (96) and carrying 

 through straightforward operations, we find that the noise is given by 



2-2 I Fpdw j I Wo + co„ + -CO j COS ((co — C0„)/ + Gp)d(X>n 



N r* /^-(c.-a,o) / ^ \ 



+ Tr^ I -^m^W I COo + W„ — -CO COS ((cO + OoJ/ + Qm)du>n, 



^■^ Jo J-(.+.,) V ^ / 



(146) 



^See "Transient Oscillations in Electric Wave Filters," Carson and Zobel, 

 B. S. T. J., July, 1923. 



