The Analytics of Transmission Calculations The Theoretical Foundation of the Quaternary Linear Real Transformation Connecting Input and Output Quantities in the General Electric Circuit Together with Its Application by Algebraic and Geometrical Methods

This paper deals mainly with a more general and systematic method of handling certain classes of problems involving steady state transmission calculations in terms of real quantities than has hitherto been used. Part I is introductory, however, and shows that the binary linear transformation which the writer showed in 1919 to exist for all ladder circuits in terms of complex quantities, and with determinant unity, occurs also in operational form for the general input¿output network and likewise with determinant unity, thus including sine-wave as a particular case. Part II views the general case of input-output circuits under sine wave conditions as a quaternary linear transformation in real quantities, likewise with determinant unity, but subject to a certain condition (``absolute covariant´´). The subject matter involves on equal terms voltage-squared (z), current-squared (w), power (x), reactive voltamperes (y) for both input and output ends in the general circuit, and in the special case of a transmission line per se it also introduces mean values of these quantities throughout the length of the line, suggesting the engineering importance of some of them. For this latter purpose the sixteen real coefficients of the transformation are given as functions of s the length of the line leading to their mean values as tabulated. Twenty identities involving values of the ``a-set´´ of coefficients are given which are useful for checking numerical values and simplifying unnecessarily cumbrous results. The relations between the ``a-set´´ and the ``b-set´´ coefficients are given.