SWITCHED NETWORK FOR TIME MULTIPLEX SYSTEMS 1425 



right-hand capacitor is ^(t) = C'[(?2(0) — f:i(0)] and as a result at time 

 I = T the right-hand capacitor has a voltage CoCO) and the left-hand 

 capacitor has a voltage fsCO). Considering now the network of Fig. 9, 

 the ecjuation is 



dr C C 



Assuming all initial conditions* to be zero we get, 



/Xp)=^^^=M+A:W. (42) 



p2 -f coo- 2 



Appendix II 



STUDY OF THE LIMITING CASE T -^ 



We expect that if the sampling period T — > 0, which is equivalent to 

 stating that the sampling freciuency w^ ^ ^ , then the inductance ^ -^ 

 and as a result the voltage e-i{t) will be infinitely close, at all times, to 

 the voltage e-2{t). Thus, in the limit, everything happens as if the termi- 

 nal pairs (2) of A^i and A^o were directly connected. In that case the gain 

 of the system is 



2Z(^) ^"^P'- 



as is easily seen by referring to the Thevenin equivalent circuit of A^i . 

 Let us show that as T — > 0, (21) leads to the same result. First note 

 that both Z12/0 and Z»Si go to zero at least as fast as l/p" for p -^ x . 

 Hence the summations in (21) reduce to the term corresponding to 

 71 = 0. Therefore, 



„ / V C IZ12/0J *J1 rj C Ziolo^n^^l ^12 T 



1 + 2C[ZSi]* T + 2CZS1 2Z "■ 



Appendix III 



ZEROTH approximation IN THE CASE WHERE A^i IS NOT IDENTICAL TO A^2 



Let, for k = 1,2; Ck be the shunt capacitor at the terminal pair 2 of 

 Nk , Zk{p) be the driving point impedance of Nk , and Zu''\p) be the 

 transfer impedance of N'k • In the present case the capacitors Ci and C-i 

 are in series in the resonant circuit of Fig. 2. It can be shown that the 



* Their contribution has been found in (40). 



