TRANSIENT RESPONSE OF AN FM RECEIVER 725 



From this it is evident that the first term of (30) contains the demodulation 

 sum product of frequency on the order of 2ajo . This frequency will be 

 suppressed by the diode load circuit and consequently the second term of 

 (30) is an adequate representation of the envelope function. Therefore we 

 write 



a{t) + ib{t) = hk ^- { . _ ^ (33) 



?ajo + 7 



and with the above approximations this is very nearly equal to 



a{t) + ib{t) = A (J ' )' + ' . (34) 



2C a -\- iA 



One deduction that can be made immediately from this formula is that the 

 frequency of the oscillation in the output of the rectifier caused by the phase 

 jump at the input is A, the radian frequency interval by which the applied 

 carrier frequency differs from the resonant frequency. The oscillation is 

 heavily damped, however, because a, while being very small compared to 

 coc , is comparable in magnitude with A in circuits commonly used. 



The angle of the complex envelope function (34) represents merely 

 a phase shift of the carrier frequency coo . We are interested only in the 

 magnitude of the function, viz.: 



ciO = [a'it) + bmf"^ (35) 



After some algebraic work, the desired formula comes out of (34) in the 

 following form: 



h_\l - 2m{t) sin Ut + |) + ^'(0 J ^ > 



where 



(a2 + A'^y 



m{i) = 2e~"' sin I (36) 



The discussion so far has dealt with a single impedance (or network) and has 

 been concerned with obtaining formulas for the voltage across the imped- 

 ance, and the envelope thereof, when a frequency-modulated current is sent 

 through it. It is necessary now to refer to the construction of the balanced 

 frequency detector, which is the particular object of our study. Figure 1 

 shows two impedances having the variation with frequency sketched in 

 Fig. 2. The carrier current is driven through the two impedances in series 

 and linear rectifiers are connected across each in such polarity that their 

 low-frequency output voltages are opposed. We assume that the output 



