NEGATIVE FEEDBACK 



435 



Then all derivatives of Y with respect to co above the first are zero, and 

 the steady-state response is equal to its response at the instantaneous 

 frequency of the applied wave. Hence after conversion we shall have 



aAB[ao + ax{piS — p2^o")] cos coo^ + | {piS — p2k(r)dt 



-f aBQl^ao + ai(wn — p2^o-)] cos (wo + a)„)/ — I p2k(Tdt + 0„ 



(48) 



Application of (48) to a linear amplitude detector will yield a low-fre- 

 quency output proportional to its amplitude. The amplitude factor is 

 readily calculated for the case where AB ^ BQ. For if 



then 



X cos X -\- Y cos y = Z cos z 

 Z = VX2 + 72 -f 2XFCOS (x - y) 



and when X '^ Y 



Z = X -\- Y cos (x — y) . 

 Hence the output of the linear detector will be 



7 ( aAB\^aQ -f ai{piS — p2^o-)] + aBQ\^aQ -f ai(w„ — P2^cr)] 



X cos ccj — pi I 5'(f/ + 0n 



(49) 



(50) 



The term ajABao represents direct current. Assuming that this is not 

 fed back to the local oscillator we can then write 



where 



0- = A'\^ai(piS — P2^(t)] + Q'[,ao + ai(co„ — P2^o-)] cos ^ (51) 



A' = ayAB 

 Q' = ayBQ 



Solving for a- 

 1 



^ = (wj — I plSdt -f (j)n \ 



1 + ai^'^p2 



aiQ'kpi cos ^ "1 -' 

 ^ 1 + a,A'kp2\ 



■ X [^'aipi5 + Q'{a, -f Oicon) cos ^]. (52) 



