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BELL SYSTEM TECHNICAL JOURNAL 



sets of 2w linear equations which may l)e conveniently written in matrix 

 form. One such set is 



v{n) = Zni(.n) 

 v{n + 1) = Z2ii{n) 



Zni(ii + 1) + v°in) 

 Z22i{n + 1) + u°(n) 



(2.1) 



Zn , Zn , Z21 , Z22 are square matrices of order m whose elements are im- 

 pedances. v{n) and i{ii) are the column matrices 



v{n) = 



V2(n) 



iin) = 



ii(n) 

 i2{n) 



The column matrices v°{ii) and u°{n) arise from generators which may 

 be acting within the «th section. If both ends of the section are open 

 circuited so that i(n) = i(n + 1) = the equations show that v(n) = 

 v°{n), v(n -)-!) = n°{n). Consequently, v°{n) and u°{n) give the open 

 circuit voltages produced on the left and right ends of the ;?th section by 

 the internal generators. If the section is a passive network then v°{n) = 

 ii°{ii) = and they do not appear in the equations. The subscripts on 

 the square matrices, the Z's, are chosen so as to preserve the analogy for 

 the simple case m = 1, where the left and right ends of the section are 

 denoted by the subscripts 1 and 2, respectively. 



Solving the equation (1.1) for i{n) and i{n + 1) gives 



i{n) = Ynv{:n) + Yi2v{n + 1) + i°{n) 

 -i(n + 1) = Y2iv{n) + Y22vin + 1) - f(n) 



(2.2) 



where the elements of the F's are admittances and i°(n), j°{n) are the 

 currents produced by the internal sources when the terminals on the right 

 and left are short-circuited so that v(n) = v(it + 1) = 0. 

 A third set of equations is 



vin) = Av(n + 1) + Bi{n + 1) - Bf(n) 

 i{n) = Cv{n + 1) + Di{n + 1) - Cu°{n) 

 Solving these equations for v{;n -\- 1) and ii^n +1) gives 

 v{n + 1) = D'v{n) - B'i{n) + 5'i°(«) 

 i{n + 1) - -C'v{n) + A'i{n) -f C'v°in) 



(2.3) 



(2.4) 



There are a great many relations between the square matrices appearing 

 in the equations (2.1) to (2.4). These are discussed in Appendix IV. 



