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Classic Scaling Fractal Fractance Approximation Circuits: Optimization Principle Analysis and Method

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Abstract

This paper presents the optimization principle and law of classic scaling fractal fractance approximation circuits (FACs). The scaling extension of FACs with negative half-order operational performance can facilitate the design of scaling fractal FACs with arbitrary-order fractional operators. This report summarizes the operational performance and mathematics describing of typical scaling fractal FACs. The scaling iteration algorithm was used to numerically calculate the impedance–admittance function of arbitrary real-order scaling fractal FACs, the features and defects of frequency-domain curves of the scaling fractal FACs were analyzed. Moreover, the methods of optimizing arbitrary-order scaling fractal FACs were analyzed theoretically, and symmetrical resistor–capacitor T-sections with certain universality were developed for FAC optimization. By comparing the approximation performances of FACs before and after optimization, the functions and indices for quantitatively analyzing the effects of circuit optimization were obtained and verified using examples. Fractance devices and active devices such as operational amplifiers can be combined to develop active fractional-order circuits and systems. Moreover, \(-\,0.2\)-order FACs before and after optimization were selected to construct fractional-order operational circuits and to obtain the results of the fractional-order differentiation and integration of a periodic square wave. The experimental simulation results agreed with the theoretical analysis. The test results prove that the FAC optimization proposed herein is theoretically correct, and that the circuit optimization methods are universal; these methods provide valuable references for solving the problem of FAC optimization.

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Authors and Affiliations

  1. College of Electronics and Information Engineering, Sichuan University, Chengdu, 610065, China

    Yue-Rong Zhang & Xiao Yuan

  2. College of Computer Science, Sichuan University, Chengdu, 610065, China

    Qiu-Yan He

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  1. Yue-Rong Zhang
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Zhang, YR., He, QY. & Yuan, X. Classic Scaling Fractal Fractance Approximation Circuits: Optimization Principle Analysis and Method. Circuits Syst Signal Process 40, 2659–2681 (2021). https://doi.org/10.1007/s00034-020-01606-4

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