RESONANT FREQUENCY DISTRIBUTION OF NON-UNIFORM LINES

Abstract

The article defines the conditions that must be satisfied by the resonant frequencies (spectrum) of non-uniform transmission lines loaded with a reactive lumped load consisting of a finite number of lumped inductances and capacitances. To determine the conditions for the physical realizability of non-uniform lines, the theory of continued fractions is applied. In this approach, the input impedance of the loaded non-uniform line is represented as an infinite ladder network, obtained through a limit transition from a finite continued fraction to an infinite one.

To determine the line delay time and fulfill the physical realizability conditions, a Richards' rod circuit is utilized (a multi-stage line consisting of a cascade connection of uniform transmission line segments with different characteristic impedances and equal delay times for all stages). A methodology has been developed to determine the values of the lumped load elements, allowing the elements of the continued fraction to be expressed through the frequency spectrum of the loaded line. The core of this methodology lies in the fact that the reactive load can be represented as the last element of a ladder network comprising both the line and the load. Consequently, to determine the load elements, a ladder network is first constructed, and the impedances of the final elements are expressed via the line spectrum. To transition to a loaded long line, a limit transition to an infinite ladder network is performed. This approach has yielded analytical expressions in the form of series for determining the elements of the lumped load circuit. Based on the obtained relations, the construction of convergents for lossless distributed circuits is considered. The results enable the synthesis of resonant systems with a required distribution of resonant frequencies.

Significant attention is paid to the deformation of resonant frequency spectra: conditions are established under which new resonant frequencies can be added or existing ones excluded within a limited frequency range. Furthermore, the variation of load elements is studied, demonstrating that spectrum deformation leads not only to changes in the values of the continued fraction elements but also to a change in the total number of load elements.