1422 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



and since r « 1 we get from (38) and (39) N(V) = aN{X) with a < 1. 

 Hence the convergence is estabhshed. 



IX. A MODIFICATION OF THE BLOCK DIAGRAM TO IMPROVE THE ZEROTH 

 APPROXIMATION 



In principle it is possible to obtain a block diagram whose transmission 

 characteristic is equal to the first approximation. In many cases it is 

 not necessary to go that far. The first approximation takes into account 

 the effect of the currents ino{t) and z„o'(0 on the resonant circuit of Fig. 2. 

 Since during the switch closure time the currents t„o and t,,n' cannot vary 

 much, let us assume that they remain constant for the duration of the 

 switch closure. 



Referring to the analysis of Appendix I and to (42) in particular, we 

 see that the current ir is increased by 



8in{p) = 



C^O^ 4(0 — ) + in\0 — ) 



p2 + 0)0^ 2p 



or 



3i(0 = C ^^(Q-) - ^3(0-) (1 _ ,^, ^^^) o^t<r. 



Defining S^ip) = ^ {^{1 - cos cooO[;'(0 - u{t - t)]}, or 



1 



S2{p) = 



Wo 



{2p' + Wo") -pT 



e , 



2 p{p' + W) 2p{p' + coo^) 



I 



( 



E4(P) 



c[z„(p)Io(p)]*S,(p)+c[pZ,g(p)Io(p)]*S3(p) 

 l + 2c|[z(p)S,(p)]*+ [pZ(p)S2(p)]' 



Fig. 8 — Modified lilock diagram. 



