TRANSMISSION LINE EQUATIONS 



171 



Sometimes it is desirable to terminate a line consisting of a finite number 

 of sections by a network which simulates an infinite line. As is known, the 

 elements in one such network may be obtained from I'^ • Every terminal 

 is joined to every other terminal, including the return terminal (denoted 

 by the subscript o), by the branches of this network. The admittance of 

 the branch connecting terminal / to terminal /, i 9^ 0,j 9^ 0, is — y.-y where 

 y,y is the element in the zth row and 7th column of Yo ■ The admittance 

 of the branch connecting terminal i to terminal is yn + yi2 + • • • + yu 

 + • • • + Vim , i.e., it is the sum of all the elements whose first subscript is i. 



-II 



APPENDIX III 



Classical Solution of Symmetrical Section Line Equations- 



When the electrical properties of a typical symmetrical section are to be 

 determined by measurement, equations (A2.1) and (A2.2) show that Zn 

 and Fu may be obtained by mxasurements at one end. In order to obtain 

 1^2 and Z12 measurements have to be made at both ends. Expressions for 

 v(n) and /(«) will now be given which depend only upon Zn and I'n and 

 hence are useful in case the measurements are restricted to one end. 



The method is based upon the equations 



v(n + 2) + v(n) = Zn[i{n) - i{n + 2)] 



iin + 2) + i(n) = Yn[v(n) - v(n + 2)] 



which may be derived from (A2.1) and (A2.2). Combining these equations 

 leads to 



[/ - ZnVnMn + 2) + v{n - 2)] + 2 [/ + ZnViMn) = 

 [/ - YnZu][i(n + 2) + i{n - 2)] + 2 [/ + YnZn]i{n) = 

 The first step in the solution is to solve the equation 



I m/ - ZiiFii 1 = (A3.2) 



for its roots ^i , i"2 , • • • Mm which we shall suppose are distinct. The diag- 

 onal matrices M and M' are defined by 



(A3.1) 



M = 



Ml 

 







M2 







M^ = 



Ml 

 







M2 







(A3.3) 



and A is defined as in (A2.3) where Xr is given by 



X - .AT^i M -P + 'H' 



(A3.4) 



