Passive realization of the fractional-order capacitor based on fractional-order inductor and its application

This paper investigates a new implementation of a passive fractional-order capacitor (FOC) based on a single-component fractional-order inductor (FOI). The significance of this approach lies in providing the fractional-order passive circuit to synthesize an FOC, which avoids the approximation that only depends on the integer-order transfer function and reflects truly the infinite-dimensional nature of fractional-order impedance characteristic. For this purpose, the realizable conditions of this passive FOC are obtained and the main ideas of the proposed method are described by examples. The sensitivity of the impedance magnitude and phase angle on required components parameters is also analyzed in order to facilitate the design of the FOC emulator. Furthermore, the bandwidth is extended by modular structures connected in series, and the proposed approach shows an improvement over some previous methods in terms of accuracy. Also, based on the fractional calculus Grunwald-Letnikov definition, a single-component FOI circuit element is packaged to build simulation circuit in Matlab/Simulink, and the effectiveness of the proposed method are validated by simulated and experimental results. In the application of an FOC, the results from in the series-parallel fractional-order RLC beta bandreject filter circuit indicate that the fractional-order counterpart can reduce the ripple and improve the suppression level at a certain frequency, and the peak amplitude and the overshoot decrease initially with decreasing order. Moreover, under the fixed inductance, capacitance, and resistance, the rise time varies almost inversely with order, whereas the setting time is on the decrease as the order is decreasing.