Fractional Order Capacitor in First-Order and Second-Order Filter

Abstract

Background: Fractional order Butterworth and Chebyshev (low-pass filter circuits, highpass filter circuits and band-pass filters circuits) types of first and second order filter circuits have been simulated and their transfer function are derived. The effect of change of the fractional order α on the behavior of the circuits is investigated. Objective: This paper presents the use of fractional order capacitor in active filters. The expressions for the magnitude, phase, the quality factor, the right-phase frequencies, and the half power frequencies are derived and compared with their previous counterpart. Methods: The circuits have been simulated using Orcad as well as MATLAB for the different value of α. We have developed the fractional gain and phase equations for low pass filter circuits, high pass filter circuits and band pass filter circuits in Sallen-Key topology. Results: It is observed that the bandwidth increases significantly with fractional order other than unity for the low pass as well as high pass and band pass filters. Conclusion: We have also seen that in the frequency domain, the magnitude and phase plots in the stop band change nearly linearly with the fractional order. If we compare the fractional Butterworth filters for low-pass and high-pass type with conventional filters then we find that the roll-off rate is equal to the next higher order filter.