Dissymmetrical electrical conducting networks

IT is well known that a symmetrical T-connection of three simple impedances a o, o b, and o q, Fig. 1, in which the two arms a o, and b o are electrically equal, and with their common terminal o connected to the ground or return conductor g h through the impedance o q, can be completely replaced,^∗ at any single assigned alternating-current frequency, by a certain symmetrical π, a′ b′ g′ h′, Fig. 2, in which the two pillars a′ g′ and b′ h′, have equal impedances; also that either of these two symmetrical systems can be completely replaced, at the same frequency,† by a single smooth line conductor a″ b″, Fig. 3, having uniformly distributed series impedance in its conductor and also uniformly distributed lateral leak admittance between the conductor and the return conductor g″ h″. Such a line conductor may be regarded as subtending or possessing, at the frequency in question, a certain complex hyperbolic angle θ, and also a certain single surge impedance or “characteristic impedance,” z0 ohms. Consequently, both the symmetrical T conducting system of Fig. 1, and the symmetrical π of Fig. 2, may be regarded as possessing the same complex angle θ and surge impedance z0, by virtue of equivalence. In any network of conductors, carrying alternating currents of the assigned frequency in the steady state, any of the three equivalent systems of Figs. 1, 2 and 3 may be interchanged, without disturbing the distributions of potential, current and power, at and outside the terminals of the system.