Measurement Units and Physical Dimensions of Fractance-Part I: Position of Purely Ideal Fractor in Chua’s Axiomatic Circuit Element System and Fractional-Order Reactance of Fractor in Its Natural Implementation

Here and in the companion paper (Part II), 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. Until now, however, the measurement units and the physical dimensions of fractance have not been proposed, which is a challenging theoretical problem. Motivated by this need, in this paper, we studied the position of purely ideal fractor in Chua's circuit axiomatic element system and the fractional-order reactance of fractor in its natural implementation. A state-of-the-art mathematical method, fractional calculus, is used to analyze the proposed conceptual framework. First, the position of purely ideal fractor in Chua's axiomatic circuit element system is introduced. Second, through mathematical and circuit analysis, the 1/2-order reactance of the 1/2-order fractor in its natural implementation is studied, following which the fractional-order reactance of the arbitrary-order fractor in its natural implementation is proposed. Third, the branch-current of the arbitrary-order fractor in its natural implementation is analyzed. Fourth, the order-sensitivity characteristics of fractor are proposed. Finally, the temperature effect of fractor and the influence of the fractional-order of an arbitrary-order purely ideal fractor on its intrinsic electrical characteristics are analyzed in detail experimentally. This paper discusses fundamental issues, whereas Part II is devoted to the measurement units and physical dimensions of fractance and the rules for fractors in series and parallel.