132 BELL SYSTEM TECHNICAL JOURNAL 



transmission lines. The first part discusses the uniform line. After a 

 statement of the transmission equations in matrix form, expressions for the 

 voltages and currents are given. Two methods of evaluating these ex- 

 pressions are described. The first is based upon a property possessed by 

 many transmission systems, namely that the various modes of propagation 

 have nearly the same speed. The second method is based upon equations 

 which may be obtained by the formal application of a theorem due to 

 Sylvester. The first part concludes with the proof that these two methods 

 lead to the correct results. 



After a short introduction the second part discusses the difference equa- 

 tions which govern the transmission in a line composed of multi-terminal 

 sections. The sections may contain generators. Expressions for the volt- 

 ages and currents in a symmetrical section line, i.e. a line whose sections are 

 symmetrical, are stated and proved in much the same order as the corre- 

 sponding expressions for the uniform line. A discussion of the unsymmetri- 

 cal section line concludes the second part. 



A sketch of the solution of the uniform transmission line equations by 

 the classical method is given in Appendix I. In Appendices II and III 

 methods are described for solving the symmetrical section line difference 

 equations. These methods are similar to the one of Appendix I. The 

 method of Appendix III uses section constants which may be obtained from 

 measurements made at one end of a typical section. 



Part I 



Uniform Transmission Lines 



1.1 Diferential Equations 



For the sake of convenience in writing down equations w^e shall assume 

 that the particular line under consideration consists of three parallel wires 

 with ground return, or of three parallel circuits, denoted by the subscripts 

 a, b, and c respectively. The differential equations for this line in an arbi- 

 trary impressed field are 



— J ^aa'^a ^ab^b ^ac^-c 1 l^aK-^) 



ax 



-r- = —Zbaia — ^bbib — Zbclc + lb{x) (l.l) 



ax 



~% ^^ ^ca^a ^cb^b ^cc^c I ''c\X) 



ax 



1 These equations are given in substance by J. R. Carson and R. S. Hoyt, B.S.T.J., 

 Vol. 6, pp. 495-545 (1927). Equations (1.2) are equivalent to their equation (90) and 

 equations (1.1) may be obtained by combining their equations (83), (84), and (94). We 

 shall use the term "impressed field" to mean a field distributed along the line. According 

 to our convention there is no impressed field when the line is energized only at the terminals. 



