First-Order Filters Generalized to the Fractional Domain

Traditional continuous-time filters are of integer order. However, using fractional calculus, filters may also be represented by the more general fractional-order differential equations in which case integer-order filters are only a tight subset of fractional-order filters. In this work, we show that low-pass, high-pass, band-pass, and all-pass filters can be realized with circuits incorporating a single fractance device. We derive expressions for the pole frequencies, the quality factor, the right-phase frequencies, and the half-power frequencies. Examples of fractional passive filters supported by numerical and PSpice simulations are given.