VARIABLE FREQUENCY ELECTRIC CIRCUIT THEORY 527 

 sufficient accuracy in the neighborhood of co = coc by the expression 



y{ic,) = yiic,;) ( 1 + ^^^^^ \ . (53a) 



This approximation should be vaHd over the frequency range from 



W = OJc ~~ ^ to CO = COc + X- 



Supposing that this condition is satisfied, the wave, after passing 

 over the transducer and through the terminal frequency detector, is 

 (omitting the constant y • Y) 



I = (l +^s{t)\ - ^^p(i f'^di)- (54) 



If y is a pure reactance, wi is a pure real; due to unavoidable dissi- 

 pation it will actually be complex. To take this into account we 

 replace wi in (54) by wig""'" where now wi is real; (54) then becomes 



/ = I 1 -I cosa-5(/) + i — sin a-s{t) 1 exp ( * j Udn - (55) 



The amplitude A of this wave is then 



^ = I / 1 +Acosa-5(/) y+ ('^sina-5(/) y V''. (56) 



Now let X/coi be less than unity and let the wave (55) be impressed on 

 a straight-line rectifier. Then the rectified or detected output is 



/i , ^ /,^^ f 1 , / Xsina-5(/) \2 1i/2 ^_. 



1+— cosa-5(/) 1-f , . \a ) \ > (57) 



\ wi /[ \coi -\- \ cos a -sit) / J ' 



or, to a first order, 



1+ — cos a- sit) -\- l^^sin' a- s'{t). (58) 



coi Z coi 



The second term is the recovered signal and the third term is the first 

 order non-linear distortion. 



Inspection of the foregoing formulas shows at once that, for detection 

 by straight rectification, the following conditions should be satisfied: 



(1) X/wi must be less than unity. 



(2) The terminal network should be as nearly as possible a pure 



reactance to make the phase angle a as nearly zero as possible. 



