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Measurement Units and Physical Dimensions of Fractance-Part II: Fractional-Order Measurement Units and Physical Dimensions of Fractance and Rules for Fractors in Series and Parallel

Here and in the companion paper (Part I), a novel conceptual framework on the measurement units and physical dimensions of fractance and rules for fractors in series and parallel is mainly discussed. The term fractor arose following the successful synthesis of a fractional-order capacitor or a fractional-order inductor in an analog circuit. Fractor is actually a promising fractional-order circuit element that is a core component for the hardware implementation of the fractional-order circuits and systems. The term fractance, as a portmanteau of the fractional-order impedance, refers to the fractional-order impedance of a fractor. Up to now, however, no effective measurement units and physical dimensions of fractance have been proposed for fractor, which is a challenging theoretical problem. Motivated by this need, in this paper, we studied the fractional-order measurement units and physical dimensions of fractance and rules for fractors in series and parallel. We use the state-of-the-art mathematical method, fractional calculus, to analyze the proposed conceptual framework. In particular, the fundamental issues introduced in the companion paper (Part I) are combined with an analysis for the realistic requirement of the fractional-order measurement units of fractance, and a proposal for the fractional-order measurement units for capacitive fractance and inductive fractance, respectively, as well as the fractional-order physical dimensions of fractance, together with the rules for fractors in series and parallel, respectively. Finally, an arbitrary-order fractor in the form of an analog circuit is achieved by the binomial theorem-based approximate implementation, the implementations of capacitance and resistance of arbitrary value are discussed, and approximately implemented fractors in series and parallel are analyzed in detail experimentally.