134 BELL SYSTEM TECHNICAL JOURNAL 



and the analogy with the one circuit case leads us to put 



p ^ ZY, T = VZY (1.7) 



where F is a square matrix representing a generalization of the propagation 

 constant. Putting aside for the moment the question of interpreting the 

 square root, we note that interchanging the rows and columns in T^ = ZY 

 gives 



r'2 = Y'z' ^ Yz, v = Vyz (1.8) 



where the primes denote transposition. Y' and Z' are equal to Y and Z 

 respectively because of their symmetry. We thus expect T' to be associated 

 with the propagation of i in the same way that F is associated with the 

 propagation of v. 

 1.2 Statement of Results for an Infinite Line — No Impressed Field 



It is shown that when there is no impressed field the voltages and currents 

 at any point a; in a transmission line extending from x = to x = co are 

 given by 



v{x) = e^"^ v{o) = e~^ Zoi{o) 



i{x) = e-^'^'iio) (1.9) 



v{x) = Zoi{x) 



where e ""^ is the square matrix defined by the convergent series of matrices 



c^ F^ x^ F^ 

 1 ! ' "IT " "3 ! 



^-i' = / _ -^ + lil _ -^r + . . . (1.10) 



and e '^^' is the transposed oi e ""^ . I denotes the unit matrix. Zo is a 

 square matrix and is called the characteristic impedance matrix: 



Z„ = T~'Z = FI'^' (1.11) 



Additional expressions of the same type for Zo are given by equations 



(1.45). The matrix e~''^Zo, being of the nature of a transfer impedance, 



is symmetrical. 



The matrices e""^ and Zo may be computed in several ways, the choice 



depending upon the circumstances. The first method to be described is 



useful when .v is not too large and when the propagation constants of the 



various modes of propagation are nearly equal to each other. In the case 



of open-wire lines these propagation constants are grouped around the 



value /co/r where v is of the order of 180,000 miles per second. The second 



method may be used for all cases, including those for which the series in 



3 Frazer, Duncan and Collar, "Elementary Matrices," Cambridge University Press, 

 §2.5. In the work which follows, this text will be referred to as "F.D.C." 



