TRAXS MISSION LINE EQUATIONS 169 



multiplying factor.) The matrix S is then formed by taking Si to be the 

 tirst column, ^2 the second and so on. 



S = [si, So, • • • s„,] (A2.6) 



Similarly let the row matrix tr be proportional to any nonvanishing row of 

 F(s>) '^nd form the matrix T where 



T ^ [t,,h,...Q (A2.7) 



in which ir is the column matrix obtained by transposing tr . 



Solving our difference equations for the passive case by the customary 

 method gives the expressions 



v{n) = PA"a + PA""a 



(A2.8) 

 i(n) = QA"a - QA "a 



for the voltages and the currents. P and Q are square matrices and a and a 

 are column matrices whose elements are determined by the boundary con- 

 ditions, a and a are of the same nature as constants of integration. The 

 minus sign over a indicates that it is associated with propagation in the 

 negative direction, i.e., in the direction of n decreasing. 



P and Q may be chosen in a number of ways, each choice requiring 

 different values of a and a to represent the same system. In all cases, 

 however, the rth column of P may be expressed as arSr where ar is a scalar 

 multiplier which may depend upon r. Similarly the rth column of Q may be 

 expressed as /?,/, . Wlien either P or Q has been chosen the other one is 

 fixed since equations (A2.2) and (A2.1) require 



Q = YnP + I'i2PA = -YnP - YnPA-' 



(A2.9) 

 P = ZnQ - ZuQA = -ZuQ + ZnQA 



Some useful choices are, 



\. P = S, Q = -YuSZ = B'^SX 



2. P = SZ Q = ZV2S = CS 



(A2.10) 



3. Q = T, P = ZnTZ = C 'TH 



4. Q = TX P = -YnT = BT 



The particular choice to be made depends upon the system of difference 

 equations which is being used. In choices 1 and 2, T is not required and in 

 3 and 4, S is not required. However, if both ^S and T are known some of 



" Methods of determining Sr and l'^ are available. A description will be found in 

 F.D.C. §4.12. 



