TRANSMISSION LINE EQUATIONS 



133 



and 



dia 

 Zv 



dib 

 dx 



die 

 d^- 



= —YaaVa " YabVb — YacVc + ta{x) 



= —YbaVa — YbbVb — YbcVc + 4U') 



YcaVa — YcbVb — YccVc + tc(x) 



(1.2) 



where Zab = Zba , Yob = Yba , etc. If we are dealing with three parallel 

 wires la{x), lb{x), lc{x) are the longitudinal components of the electric force 

 of the impressed field at the w^ire surfaces; ta{x), tb{x), tdx) are specified 

 by the admittance of the direct leakage paths and the values of the im- 

 pressed potentials at the wires. If there are no direct leakage paths the 

 /'s are zero. 



In order to put these equations in matrix form^ we introduce the column 

 matrices 



V — 



and the symmetrical square matrices 



7 



Zba 



7 



^-'ca 



Zab 

 Zbb 

 Zcb 



7 



^ac 

 Zbc 



7 



(1.3) 



(1.4) 



The equations (1.1) and (1.2) may now be written as 



dv 



dx 



di 



dx 



= -Zi + l{x) 

 = -Yv-\-i(x) 



(1.5) 



and these are the equations to be solved. 



When there is no impressed field equations (1.5) give 



d'v 

 dx^ 



= ZYv 



dx- 



(1.6) 



2 Cf. L. A. Pipes, Phil. Mag., Vol. 24 (1937), p. 97. 



