Rigorous methods of solving long transmission-line problems

Two methods of solving transmission-line problems which avoid the use of charts or tables of hyperbolic functions of complex angles are outlined. In the first method the voltage and current at any point are shown to be due to two waves, one travelling forward, and one backward, attenuating as they travel. The forward and backward wave vectors are calculated from receiving-end conditions. From these the wave vectors at any other point are derived. These, when compounded, lead to the voltage and current at that point. In the second method the (complex) angle of the load is determined from the forward and backward wave vectors. To this is added the (complex) angle of the line. The voltage and current are proportional to the hyperbolic cosine and sine respectively of the total angle. The magnitudes of these can be calculated, using circular and hyperbolic functions of real angles. In continuation of this, a semi-graphical method of finding the current and voltage at several points on the line is developed.