A Network Containing a Periodically 



Operated Switch Solved by 



Successive Approximations 



By C. A. DESOER 



(Manuscript received June 15, 1956) 



This paper concerns itself with the analysis of a type of periodically 

 switched network that might be iised in time multiplex systems. The econom- 

 ics of the situation require that the ratio of the switch closure time r to the 

 switching period T he small. Using this assumption, the analysis is performed 

 by successive appro.vimations. More precisely the zeroth appro.vimation to 

 the transmission is obtained from a block diagram analogous to those used 

 in .sampled servomechanisms. From the convergence proof of the successive 

 appro.vimation scheme, it follows that when t/T is small, the zeroth approxi- 

 mation is very close to the exact transmission. A discussion of some examples 

 is included. 



CONTEXTS Page 



I. Introduction 1404 



II. Description of the system 1404 



III. Method of solution 1407 



IV. The zeroth approximation 1408 



4.1 Introduction 1408 



4.2 Description of the block diagram 1410 



4.3 Analysis of the bh>ck diagram 1411 



V. Transmission loss 1413 



VI. A simple example 1414 



VII. Numerical examples 1416 



VIII. The successive approximation scheme 1417 



8.1 Preliminary steps 1418 



8.2 Matrix description of the successive approximations 1420 



8.3 Convergence proof 1421 



IX. Modification of the block diagram to improve the zeroth approxima- 

 tion 1422 



X. Conclusions 1423 



Appendices 



I. Analysis of the resonant circuit 1423 



II. Study of the limiting case T -* 1425 



III. Zeroth approximation in the case where A'^i is not identical to N2 1425 



IV. Derivation of equation (24) 1426 



1403 



