930 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



oiiit response; i.e., a fluctuation about the steady state value derived 

 from the first component. 



The third component consists of a train of dipulses. Eacli dipulse 

 consists of a pair of pulses of amplitude 1 and —1, displaced by an 

 interval ±5. The response of the resonant circuit to this component 

 gives the effect of random displacements ±5 in the oiiginal "on-off" 

 pulse train. 



2 Im'pedance of ResonaJit Circuit 



The impedance of a resonant circuit consisting of R, L and C in 

 parallel is 



Ziiw) = Z{.i<S)e-'\ (1) 



— 7? 



[1 + Q\w/oi(i — oif^/wYY-'^ 



where 



and 



tan ip = Qioi/wo — coo/co), (3) 



Q = cooi?C = Loss constant, (4) 



ojci = {\/LC) ' = Resonant frequency. (5) 



The above expressions also apply for the admittance of a resonant 

 circuit consisting oi R, L and C in series, except that in this case Q = 

 ojoL/R. 



3 Impulse Response of Resonant Circuit 



When a rectangular pulse of unit amplitude and sufficiently short 

 duration 5o is applied to a resonant circuit, the impulse response for 

 Q >>> 1 is of the form 



P(t) = P(0) cos wo^e"""'''*', (6) 



where 



P(0) = c^o8oR/Q. (7) 



P{t) designates voltage in response to an impulse current in the case 

 of a parallel resonant circuit, or the current in response to an impulse 

 voltage in the case of a series resonant circuit. 



