B.S.T.J. Briefs: Realizability Conditions for the Impedance Function of the Lossless Tapered Transmission Line

Realizability Conditions for the Impedance Function of the Lossless Tapered Transmission Line By P. L. ZADOR (Manuscript received August 2,1966) In the study of tapered transmission lines or accoustical horns, an unsolved problem of great practical interest is the determination of the taper function (inductance or capacitance per unit length as a function of distance; it is assumed that the product of these quantities is unity) for the structure which will possess a prescribed driving point impedance function. For the case where the structure may be modeled by a cascade of sections of uniform transmission line segments, physical realizability conditions and a synthesis procedure have been given by B. K. Kinariwala. 1 For the case of continuous taper no results of a general nature are known. In this note, we shall give an almost complete characterization of driving point impedances for structures possessing once continuously differentiable taper functions. Although the proof of the realizability theorem will not be given here, the author wants to point out that the sufficiency is, in fact, proved by a construction of the taper function. However, this construction is too unwieldly to be of practical use. The mathematical formulation of the problem is as follows. Suppose that for all complex s y(x,s), 0 ^ x ^ I is the solution of the Horn equation (c(x)i/(x,s)Y = s?c(x)y(x,s) (1)