726 BELL SYSTEM TECHNICAL JOURNAL 



voltage of each rectifier is the envelope of the voltage existing across its 

 associated impedance. Therefore, to find the total output of the balanced 

 frequency detector, we have to find the difference between the envelopes of 

 these voltages. 



It is necessary to specify the two impedances more precisely. It appears 

 that the best operation is obtained if the frequency of the carrier is midway 

 between the resonant frequencies of the two impedances. That is 



A = coo — coi = C02 — coo 



where coi , C02 are the resonant frequencies of Zi , Z2 , (previously written as 

 coc, for any impedance). Furthermore it appears that the two impedances 

 should have identical values of C and very nearly the same damping con- 

 stants. The design of the detector circuit is accordingly specified by 



Ci = C2 ^^ C 



R\ = Ro = R 



and then 



All the quantities are assumed to be substantially constant over the signifi- 

 cant frequency range. 



With the circuit constants so proportioned, it can be seen from (36) that 

 the envelope of the voltage across Zi differs from that across Z2 only in the 

 sign of A. Therefore, the output voltage of the balanced frequency detector, 

 when the instantaneous frequency variation is an impulse of moment 9 at 

 / = 0, is 



Vo{t) = €2(1) - c,{t) 

 h 



ic 



= t («^ + A^)"'^ ([1 + 2;r. sin (a/ - |) + y^J^' 



(38) 

 i > 



1 - 2w sin (m + I ) -r w j 



-at • 9 



= 0, / < 0, where m = 2e " sin - . 



