1404 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



I. INTRODUCTION 



One main contributor to the cost of transmission circuits is the trans- 

 mission medium itself. Thus it is important to share the transmission 

 medium among as many messages as possible. One possible method is 

 the frequency multiplex where each message vitilizes a different frequency 

 band of the whole band available in the medium. An alternate method 

 is the time multiplex where each message is assigned a time slot of dura- 

 tion T and has access to that time slot once every T seconds. It is obvious 

 that the economics of the situation requires that r be as small as possible 

 and T as large as possible so that the largest possible number of messages 

 are transmitted over the medium. For this very reason the analysis of 

 periodically switched networks is of special interest in the case where 

 t/T is small. 



W. R. Bennett'' has published an exact analysis of this problem without 

 any restrictions either on the network or on the ratio r/T. It is believed, 

 however, that the analysis presented in this paper will, in most practical 

 cases, give the desired answer with a considerable reduction in the 

 amount of calculations. The simplification of the analysis is mainly a 

 result of the assumption that t/T is small. 



First the successive approximation method of solution will be discussed 

 in general terms. Next it will be shown that the zeroth approximation 

 to the transmission through the network can be obtained from the gain 

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

 servomechanisms. The nature of the zeroth approximation is further 

 clarified by some general discussion and some examples. Next it is shown 

 that the successive approximations converge. The convergence proof then 

 suggests some slight modifications of the block diagram to obtain a more 

 accurate solution. 



II. DESCRIPTION OF THE SYSTEM 



The system under consideration is shown on Fig. 1. It consists of two 

 reactive networks Ni and A''2 connected through a switch S which is it- 

 self in series with an inductance /. A^i is driven at its terminal pair (1) 



Fig. 1 — System under consideration. 



I 



