Fractal Impedance Element Synthesis Based on Multilayer Resistance-Capacitance Environment with C-R-Nc Layer Structure

Abstract

The idea of fractural calculus appeared almost at the same time as the development of conventional differential calculus began. However, modern scientists became familiar with it only from pioneering works of B. Mandelbrot on fractal geometry of nature in 1980s. Since then, numerous scientific works were published on various aspects of fractural calculus and its application in science and engineering, and the interest to this area remains high. However, for physical implementation of fractural integration operators two-terminal elements with fractional power and frequency law (the fractural impedance elements - FIE) are required. Commercial FIE that could be produced industrially have not been developed so far. The greatest potential, in this case, have FIE built on the basis of multilayer resistance-capacitance element with distributed parameters (RC-distributed elements). One of the baseline designs of such FIEs with R-C-NR layer structure was manufactured as a prototype model. However, when manufacturing and testing of this element, flaws that may prevent its application were detected. The article suggests making FIE based on RC-distributed elements with dual C-R-NC layer structure where, as it is assumed, it is possible to avoid flaws found in FIE with R-C-NR layer structure. However, to solve this problem it is necessary to develop an algorithm and a program of structure synthesis of new type FIE enabling production elements with parameters not worse than those of the existing FIE with R-C-NR layer structure. For this purpose, an algorithm of frequency characteristic analysis of equivalent circuit FIE based on RC-distributed parameters having C-R-NC layer structure and being the component of synthesis algorithm, was developed. Principal stages of synthesis algorithm based on search optimization genetic algorithm were formulated. Ways of information coding of internal structure and parameter C-R-NC FIE were defined. Method of function calculation of individual fitness within population of object representation and ways of genetic operators (selection, crossing-over and mutation) realization in the process of genetic algorithm performance was described. Thereupon the synthesis program was developed and the results of its operation are given in the article. Synthesis result validity was verified by means of circuit simulation software OrCAD by application of Pspice-based C-R-NC-line model. The obtained FIE equivalent circuits integral C-R-NC structures, taking into account chosen materials and layer parameters, were designed. Comparative analysis results of synthesis program operation of C-R-NC FIE and R-C-NR FIE for synthesis FIE with the same requirements to frequency impedance FIE characteristics are given, showing that synthesizing C-R-NC FIE; the FIE with phase uniformity phase-frequency impedance characteristics from - 45° to about - 80° without galvanic coupling between poles and synthesizing R-C-NR FIE, the FIE with phase uniformity of phase-frequency impedance from -10° to about -45° with galvanic coupling between the pole are more likely to be realized.