New class of fractal elements with log-periodic corrections: Confirmation on experimental data

Abstract

In this paper, the authors propose a theory to describe the behavior of the new type of fractal element which exihibit complex-conjugated power-law exponents. To explain the impedance behaviour of these elements, log-periodic corrections are employed. So far, the impedance of fractal elements are studied which shows power law behaviour in a limited frequency band; and they are modeled as band limited constant phase element. However, the proposed theory is able to describe the impedance behaviour of the fractal element in the entire frequency zone of measurement. To validate the theory, NanoTubes (MWCNTs) based 10 fractal elements are fabricated which exihibit sinusoidal type phase response in the frequency zone from 1 Hz to 4 MHz (it is to be noted that the measurement range of the instrument is 10 mHz to 4 MHz). The derived fitting formula is expressed in the squared log-normal term and contain log-periodic corrections. The proposed fitting formula, incorporated in the paper, will facilitate to explore the deterministic nature of the log-periodic oscillations in complex systems. Earlier these oscillations were considered random or counterintuitive and is misinterpreted as the results of measurement artifacts. Besides, it will find application in modelling the spatially extended systems and also to the analysis of nonlinear dynamics in a complex system. Moreover, one can combine the conventional resistance, capacitance and inductance in one “hybrid” combination using the theory.