158 BELL SYSTEM TECHNICAL JOURNAL 



lie is a matrix satisfying the conditions of §2.3, namely, (a) e~^ satisfies 

 the equation 



2 cosh T = e^ -{- e~^ = 2A (2.14) 



and (b), every element in e~" approaches zero as n^ cc , and if Z,, is 

 defined by v{o) and i(o) for an infinite line as in (2.13), then 



1. In an infinite line 



v(n) = e~" v(o), 



-nv (2.32) 



i(n) = e " i{o), 



vi'fi) = Zoi(n) (2.33) 



2. The characteristic impedance matrix Zg is given by 



Zo = (sinh T)~'B = 5(sinh T')~' = C~' sinh V = sinh T C~' 



(2 34) 

 Z-' = B~' sinh T = sinh T' B~' = (sinh T'y'C = C(sinh T)~' 



3. The matrices Zg , B and C obey the commutation rules 



*(/)Z„ = ZMe^') 



$(/)5 = 5$(/') (2.35) 



C$(/) = $(/')C 



where <J>(g ) is a square matrix representable as a sum of powers of e^^ . 

 The matrices $(d ) Zo, $(e ) B, and C$(e ) are symmetrical. 



To prove these statements we proceed as follows: By direct substitution 

 into (2.31) it is seen that v{n) = e~" v{o) is a solution by virtue of condi- 

 tion (a) satisfied by e . Since, by condition (b), v{n) ^ o as n — > oo it fol- 

 lows that v{n) is the voltage in an infinite line. Similarly, i{n) — e~''^ i{o) 

 is the current in such a line. Substituting the expressions (2.32) for v(n) 

 and i{n) into the difference equations (2.10), setting n = 0, using the 

 definition of Zo , and regarding v(o) and i{o) as arbitrary columns gives 



e~^ = A - BZo' 

 r, (2.36) 



Applying condition (a) in the form of (2.14) to these equations gives 



BZ~^ = sinh T, CZo = sinh T' (2.37) 



Since the sections are symmetrical, B and C are symmetrical matrices, and 

 from the reciprocal theorem for networks it follows that Zg is also sym- 

 metrical. These remarks and (2.37) lead to (2.34). Setting the expres- 

 sions (2.32) for v(n) and i{n) in the second of the difference equations (2.10) 



