Abstract
In this paper, general analytical formulas are introduced for the determination of equivalent impedance, magnitude, and phase, i.e., order, for ${n}$ arbitrary fractional-order capacitors (FoCs) connected in series, parallel, and their interconnection. The approach presented helps to evaluate these relevant quantities in the fractional domain, since the order of each element has a significant effect on the impedance of each FoC and their equivalent capacitance cannot be considered. Three types of solid-state fractional-order passive capacitors of different orders, using ferroelectric polymer and reduced graphene oxide-percolated P(VDF-TrFE-CFE) composite structures, are fabricated and characterized. Using an impedance analyzer, the behavior of the devices was found to be stable in the frequency range 0.2 – 20 MHz, with a phase angle deviation of ±4°. Multiple numerical and experimental case studies are given, in particular for two and three connected FoCs. The fundamental issues of the measurement units of the FoCs connected in series and parallel are derived. A MATLAB open-access source code is given in the Appendix for easy calculation of the equivalent FoC magnitude and phase. The experimental results are in good agreement with the theoretical assumptions.
Highlights
F RACTIONAL-(non-integer)-order circuits and systems offer unique benefits by enabling broader impedance matching and allowing the tunability of frequency response of electronic circuits
Note that this paper reports the first experimental verification of series- and parallel-connected fractional-order capacitors (FoCs) by fabricated solidstate passive FoCs, in contrast to using RC ladder structures [31] or active IC emulators [32]
The magnitude and phase responses of equivalent FoCs obtained using the Agilent 4294A precision Impedance Analyzer were proved by theoretical calculations via the MATLAB open access source code given in Appendix A
Summary
F RACTIONAL-(non-integer)-order circuits and systems offer unique benefits by enabling broader impedance matching and allowing the tunability of frequency response of electronic circuits. The series and parallel connections of FoCs are the two elementary and most important structures that allow us to calculate the impedance of equivalent capacitance. The main contributions of the present work are: 1) The general formulas for impedance, magnitude, and phase response of the series and parallel arbitrary-order n FoCs according to the main definition of the FoC are given as a complete study. The rest of the paper is organized as follows: In Section II, the general formulas for impedance, magnitude, and phase responses of series- and parallel-connected n FoCs are derived. Fabrication process and experimental characterization of three types (orders 0.69 (TP2), 0.92 (P2), 0.62 (G2)) of solid-state compact and stable-in-phase (in the measured frequency range 0.2 MHz − 20 MHz) electric passive FoCs, in particular those based on an actual dielectric material, are explained in Section III A. Based on the proposed fractional-order measurement units and physical dimensions of FoC, the rules for in-series and parallel connected FoCs are presented
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