SELF-TIMING REGENERATIVE REPEATERS 931 



4 Response to Steady State Impulse Train 



Let a long sequence of impulses of amplitude ^ and the same polarity- 

 impinge on a resonant circuit at uniform intervals T. The response after 

 N impulses is then 



AM) =li:P(t- nT) (8) 



2 r!=0 



= — ^ 2^ COS coo(« - nT)e . (9) 



The subscript s indicates a systematic component. 



The above series is conveniently summed by taking the real part of 

 the series 



2 ri=0 



With t = NT -V t,,Q <t,<T: 



2 7i=o 



When .V -^ x , the stead}' state responses becomes 



A '(A - ^^ ia-o(o -"o«o/2Q 1 (12) 



The interval between pulses can be written 



T = 27r/co, (13) 



where w is the fundamental frequency of the impulse train, or the pulse 

 repetition frequency. 

 Let 



o)(\ = 0) — Aco, 

 so that 



CO = 1^ (1 - Aco/c). (14) 



The following approximations then apply : 



2irj —'IriAaloi 



= e e 



^ 1 - 2iriA(jo/co; 



—tIQ +(7r/Q)Au/a) 



^ 1 - t/Q when ir/Q « 1. 



3 ° = e e = e 



-uor/2Q _ -vlQ +{TrlQ)Aulu 



(15) 

 (16) 



