SWITCHED NETWORK FOR TIME MULTIPLEX SYSTEMS 1423 



and recalling that the input of the impulse modulator of Fig. 4 is 

 eo{0) — ^3(0), it becomes ob\'ious that the modified block diagram should 

 be that given by Fig. 8. The output of the modified block diagram is 

 given by^ 



E (n) = <"[^i2(?>)/o(p)]*-^i(?>) + C[pZr,(p)Io(p)]*SM 



'^^' 1 + 2C{[Z{v)S,{p)]* + [pZ{j>)SM]*\ ' 



X. CONCLUSION 



Let us compare the method of solution presented above with the more 

 formal approach proposed by Bennett. The latter method leads to the 

 exact steady-state transmission through a network containing periodi- 

 cally operated switches. This method is perfectly general in that it does 

 not require any assumption relative to the properties of the network 

 nor to the ratio of t/T. As expected this generality implies a lot of de- 

 tailed computations. In particular it rec^uires, for each reactance of the 

 network, the computation of the voltage across it due to any initial con- 

 dition. The method presented in this paper is not so general because it 

 assumes first that the ratio t/T is small; second the value of the induct- 

 ance ( is very much smaller than that of L„ (see Fig. 3). The result 

 of these assumptions is that the system of time varying equations 

 may be solved by successive approximations with the further advantage 

 that the convergence proof guarantees that, for very small r T, the 

 zeroth approximation will be a close estimate of the exact solution. 



The zeroth approximation may conveniently be obtained by consider- 

 ing a block-diagram analogous to those used in the analysis of sampled 

 servomechanisms. Further the proposed method leads directly to some 

 interesting results, for example, as far as the zeroth approximation is 

 concerned, the dc transmission may be achieved with as small a loss as 

 desired provided the lossless networks Ni and N-i are suitably designed. 

 Another advantage of the proposed method is that the simplicity of the 

 analj'sis permits the designer to investigate at a small cost a large num- 

 ber of possible designs. 



Finally it should be pointed out that this approach to the solution of 

 a system of time-varying linear differential equations may find applica- 

 tions in many other physical problems. 



Appendix I 



ANALYSIS OF THE RESONANT CIRCUIT 



Consider the re.sonant circuit of Fig. 2. Suppose that at t = 0, the left- 

 hand capacitor has a potential ro(0) and the right-hand capacitor has 



