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Scaling Fractal-Chuan Fractance Approximation Circuits of Arbitrary Order

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Abstract

The scaling fractal-chuan fractance approximation circuit (SFCFAC), which can realize the rational approximation of arbitrary-order fractances and has an excellent approximation performance, is presented in this paper. For an original SFCFAC, the progression ratios of resistance and capacitance are limited to the range 0–1. However, it is possible for the values of both progression ratios to be greater than one. The impedance function of an SFCFAC can be represented by an irregular scaling equation. By solving the scaling equation approximately, the operational order of an SFCFAC can be obtained using both progression ratios as \(\mu = -\lg \alpha /\lg (\alpha \beta )\). Therefore, the SFCFAC has fractional operational characteristics, which is explained in theory. Oscillation phenomena are inherent to the SFCFAC. It is necessary to learn about these oscillation characteristics. The approximation performance can be improved by adding a series resistor and a series capacitor to the SFCFAC. The corresponding resistance and capacitance are determined by both progression ratios. The optimization has the advantages of simple operation, evident influence, and good practicality, which will make the SFCFAC competitive in future studies. Moreover, the values of the operational order of SFCFACs are extended from \(-1<\mu <0\) to \(0<|\mu |<2\) without using inductors, which is feasible for practical applications.

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Acknowledgements

The work was supported by the National Key Research and Development Program Foundation of China under Grants 2018YFC0830300 and the National Natural Science Foundation of China under Grants 61571312.

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Authors and Affiliations

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

    Qiu-Yan He & Yi-Fei Pu

  2. College of Physics and Engineering, Chengdu Normal University, Chengdu, 611130, China

    Bo Yu

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

    Xiao Yuan

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  1. Qiu-Yan He
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He, QY., Pu, YF., Yu, B. et al. Scaling Fractal-Chuan Fractance Approximation Circuits of Arbitrary Order. Circuits Syst Signal Process 38, 4933–4958 (2019). https://doi.org/10.1007/s00034-019-01117-x

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