The Calculation of the Long-Distance Transmission Line under Constant Alternating Voltage

By separating real quantities and imaginaries in the general solution of the differential equations which express the relations which exist between the current and voltage at any point of a transmission line, the components of current, I
₁
, and I
₂
, and the components of voltage, V
₁
and V
₂
, are expressed as rational functions of the four quantities coshα xcos ßx, sinhα xsin ßx, sinhα xcos ßx, coshα xsin ßx. Denoting these respectively by Q
₁
, Q
₂
, Q
₃
, Q
₄
, we have V
₁
= A
₁
Q
₁
-A
₂
Q
₂
+ B
₁
Q
₃
-B
₂
Q
₄
, V
₂
= A
₂
Q
₁
+ A
₂
Q
₂
+ B
₂
Q
₃
+ B
₁
Q
₄
, I
₁
= C
₁
Q
₁
-C
₂
Q
₂
+ D
₁
Q
₃
-D
₂
Q
₄
, I
₂
= C
₂
Q
_l
+ C
₁
Q
₂
+ D
₂
Q
₃
+ D
₁
Q
₄
, where x is the distance of any point in the line from some fixed point in the line chosen as origin, α and ß are certain functions of the line constants, and A
₁
, B
₁
, C
₁
, D
₁
, A
₂
, B
₂
, C
₂
, D
₂
are constants determined by the given terminal conditions of voltage, current, power and power factor. The constants C
₁
, C
₂
, D
₁
, D
₂
, are expressed in terms of the others by simple relations developed in the paper, thus reducing the problem to the determination of four constants only. There must then be at least four independent relations given involving the components of voltage and current at given points, the line constants being known.