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, $$-\,0.2$$ - 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.

中文翻译：

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经典标度分形分形逼近电路：优化原理分析与方法

本文介绍了经典标度分形分形近似电路 (FAC) 的优化原理和规律。具有负半阶操作性能的 FAC 的缩放扩展可以促进具有任意阶分数运算符的缩放分形 FAC 的设计。本报告总结了典型缩放分形 FAC 的操作性能和数学描述。采用标度迭代算法数值计算任意实阶标度分形FAC的阻抗-导纳函数，分析标度分形FAC频域曲线的特点和缺陷。此外，从理论上分析了优化任意阶缩放分形FAC的方法，并开发了具有一定普遍性的对称电阻-电容T型截面用于FAC优化。通过比较优化前后FAC的逼近性能，得到了定量分析电路优化效果的函数和指标，并通过实例进行验证。Fractance器件和运算放大器等有源器件可以结合起来开发有源分数阶电路和系统。此外，选取优化前后的$$-\,0.2$$-0.2-order FACs 构建分数阶运算电路，得到周期方波分数阶微分和积分的结果。实验模拟结果与理论分析一致。测试结果证明本文提出的FAC优化在理论上是正确的，电路优化方法具有通用性；