The Bell System Technical Journal 



Vol. XX April, 1941 No. 2 



STEADY STATE SOLUTIONS OF TRANSMISSION 

 LINE EQUATIONS 



S. O.Rice 



Methods of obtaining the steady state voltages and currents in a 

 uniform transmission line consisting of several parallel wires are described 

 in Part I. This line maj^ or may not be acted upon by an externally 

 impressed field distributed along its length. A square matrix r, which is a 

 generalization of the propagation constant 7 for a single circuit, is intro- 

 duced. Matrix expressions obtained for the voltages and currents in- 

 volve r in much the same way as the corresponding single circuit expres- 

 sions involve 7. In Part II similar methods are described for obtaining 

 the voltages and currents in a transmission line composed of a number of 

 multi-terminal symmetrical sections connected in tandem. Expressions 

 for the voltages and currents in a line composed of unsymmetrical sections 

 are also given. These sections may or may not contain generators. 



THE transmission lines considered here are of two kinds, namely the 

 uniform transmission line, and the transmission line consisting of a 

 number of identical sections connected in tandem. The problem discussed 

 is that of determining the steady state electrical behavior of these lines 

 when the terminal conditions are given. Often there arises the problem 

 of determining the currents induced in a uniform transmission line by an 

 arbitrary impressed field of some fixed frequency or of determining the cur- 

 rents produced by generators placed in the branches of the sections if the 

 line is of the second kind. This is the type of problem with which we shall 

 be particularly concerned. 



In dealing with the uniform transmission line it is found convenient to 

 introduce a matrix T, which is a generalization of the propagation constant 7 

 for a single wire with ground return, or for a single circuit. This enables us 

 to obtain matrix expressions for the currents and voltages which are similar 

 in form to the single circuit expressions. 



A similar situation exists for the transmission line composed of a number 

 of symmetrical sections. However, when the sections are unsymmetrical 

 the corresponding procedure does not appear to yield a corresponding sim- 

 plification and the formulas are considerably more complicated than in the 

 symmetrical case. 



This paper is divided into two parts corresponding to the two kinds of 



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