SWITCHED XETWORK FOR TIME MULTIPLEX SYSTEMS 1411 



is the output voltage of Ni , when A''! is excited by the current source 

 /o and the switch S remains open at all times. 



4.3 Analysis of the Block Diagram 



For simplicity, vsuppose that the system starts from a relaxed condi- 

 tion (i.e., no energy stored) at i = 0. Let z{t) = £~\Z{p)]. Considering 

 the network A'': as driven by io and zVo , it follows that the voltage ^2(0 

 shown on Fig. 3 is given by 



620(0 = v(t) - [ iM)z{t - t') dt'. (15) 



Similarly 



Thus 



ez,{t) = f iM)z{t - t') dt'. (16) 



eM - eUt) = v(t) - 2 [ ir,{t')z{i - t') dt' . (17) 



''0 



These equations have been derived by considering Fig. 1. They could 

 have been also derived from the block diagram of Fig. 4 as follows: let 

 /ro(p) be the output of CSi{p). x\s a result, the output of the block 

 2Z(p) is 2Z(p)Iro{p). When this latter quantity is subtracted from V(p) 

 one gets V(p) — 2Z{p)Iro(p), which is the Jt^-transform of the right-hand 

 side of (17). Referring to the block diagram it is also seen that this 

 quantity is the input to the impulse modulator. 



Thus we see that if Iroip) is the output of CSi{p), then the input of 

 the impulse modulator is e^oit) — 630(0 by virtue of (17). If this is the 

 case the output of CSi(p) is given by Cho(O) - e^o{0)]si{t) , iorO ^t<T, 

 which, according to (9), is iro{t). 



Thus the block diagram of Fig. 4 is a convenient way of obtaining the 

 zeroth approximation to the periodic steady-state solution. 



In order to use the techniques developed for sampled data systems, • 

 we introduce the following notation.- If f{t) = ^~\F{p)], then we define 

 F*(p) by the relation 



F*ip) = I Z I'(P + Jru^s), (18) 



AN'here 



J- H = - 



CO. = ^ . (19) 



