722 . BELL S YSTEM TECH NIC A L JOURNA L 



For the present purpose, only a pair of terms of (24) need be considered. 

 This will provide a specific solution of the balanced frequency detector for the 

 case, previously used for illustration, where the impedances are two simple 

 resonant circuits of parallel R, L and C. At the same time, this solution 

 can be extended to more complicated circuits by superposing a number of 

 such elementary solutions, as is clearly possible with the type of impedance 

 development indicated by (24). 



The impedance of i?, L and C in parallel, written in the form of (23) and 

 (24), is 



Z(faj) = — 7-T — , ... -T = + 



C (ioj — p\){i(ii — pi) io) — p\ i(j3 — p2 



(26) 



+ 



where 



— p\ = y = a + i^ 



— p2 = y = a —i^ 



2C\ ^ (3/ 



_ 1 R-./i 1 _ /-^ 2 _ -J— 



"'IRC ^'VlC 4i?2C2 ~ ^"«-«' """VZC- 



The voltage response of the circuit to a unit impulse of current is then 



H{t) = r ("^-4— + ^-A-^) e'^' df = ke-'' + ke-'\ t > 0. (27) 

 J-oo \io3 + 7 103 -\- y/ 



To find the envelope function of the voltage drop when the frequency 

 modulated current 



is applied to the circuit, we make use of the general formula, (17), which 

 states that the spectrum, or Fourier transform, of this envelope function is 



Ml) = r Ht) + i6(/)le-'"' dt = *(/)Z(/ + /„) (17) 



J— 00 



