A Note on the Transmission Line Equation in Terms of Impedance

Especially, work by Schelkunoff in extending the impedance concept shows that impedance can be quite as general and exact a means for expressing electromagnetic relations as are current, voltage, electric and magnetic fields, and vector and scalar potentials. In reformulating certain problems in terms of impedance the content and ultimate solution must of course be equivalent. There may, however, be a considerable change of procedure and sometimes a simplification. For instance, in many cases a single impedance condition can replace the usual two boundary conditions for voltage and current. One very simple case in which it is perhaps easiest to deal directly with impedance is in the derivation of the transmission line equation on a distributed- constant basis. In the usual derivation, two linear second order differential equations are obtained, one for voltage and one for current. The impedance, in terms of which the engineer expresses many of his results, is obtained as a ratio from solutions for voltage and current. In treating the transmission line from the impedance point of view, without dealing with currents and voltages, a first order non-linear differential equation in terms of impedance and distance is obtained. This impedance equation is a Ricatti equation and could be obtained from the usual line equations. It is simpler, however, to derive it directly. As the principal interest of such a treatment lies in the method and in the fact that the line may be tapered, rather than in losses, the derivations will be carried out for lossless lines.