Apparent impedances of very long multiconductor transmission lines

Classical transmission-line theory predicts that the locus of the input impedance of a short-circuited line is a symmetrical spiral in the complex impedance plane, with antiresonance frequencies at odd multiples of a quarter wavelength. In the present work, modern multiconductor transmission-line theory is used to study the problem of faulted transmission lines in greater detail than has hitherto been possible. By this means, it is shown that the short-circuit loci for unbalanced faults are unsymmetrical, and furthermore that resonances are liable to occur for line lengths considerably lower than those previously predicted. It is indicated that this is most probably due to the interaction between inductance of the faulty phase and the capacitance of the sound phases. The topic is developed by considering the practical problem of the protection of long lines when taking into account the complex nature of the short-circuit-impedance locus. In conclusion, it is indicated that digital-computer studies are, in all probability, necessary to establish the parameters necessary to apply distance relays to the protection of very long lines.