Outline of a theory of non-uniform transmission lines

Complex differential equations relating to non-uniform transmission lines are studied, and the corresponding two-parameter families of solution curves are investigated with the aim of establishing some important properties, e.g. the relationship which exists between the singular points of the differential equation for the normalized impedance and the non-reflective impedances of the corresponding non-uniform line. Because of this relationship it is possible to determine the non-reflective impedances of an arbitrary tapered line by purely algebraical method and to construct the tangents to the impedance loci at a given point of the impedance plane. Moreover, reflection and transmission coefficients are uniquely determined by the singularities, and differential equations can be readily deduced for these coefficients. Methods of synthesis of non-uniform lines are also indicated.