Passive and Active Elements Using Fractional ${\rm L}_{\beta } {\rm C}_{\alpha }$ Circuit

This paper introduces a qualitative revision of the traditional LC tank circuit in the fractional domain. The paper can be divided into six major parts, aiming in turn to establish the various conditions under which L_βC_α impedance may act as a resistor, negative resistor, or a positive or negative pure imaginary inductor or capacitor, in accordance to new frequency definitions; illustrate the process by which the phase response chooses the shortest path from initial to final phase, and use this illustration to verify the cases discussed in part one; develop the generalized parameters for the bandpass filter response of the L_βC_α circuit, such as the resonance frequency and quality factor versus α- β plane; discuss sensitivity analyses with respect to the fractional orders, as well as the time domain analyses for the impulse and step responses with their analytical formulas; and lastly, to propose some possible applications for this generalized circuit. Mathematical and PSpice simulation results are included to validate the discussion.