On the Generalization of Second-Order Filters to the Fractional-Order Domain

Abstract

This work is aimed at generalizing the design of continuous-time second-order filters to the non-integer-order (fractional-order) domain. In particular, we consider here the case where a filter is constructed using two fractional-order capacitors both of the same order α. A fractional-order capacitor is one whose impedance is Zc = 1/C(jω)α, C is the capacitance and α (0 < α ≤ 1) is its order. We generalize the design equations for low-pass, high-pass, band-pass, all-pass and notch filters with stability constraints considered. Several practical active filter design examples are then illustrated supported with numerical and PSpice simulations. Further, we show for the first time experimental results using the fractional capacitive probe described in Ref. 1.

On Digital Realizations of Non-integer Order Filters

“…In their celebrated paper [34], Radwan et al have presented a non-integer generalization of classical second-order filter section. In this paper, we focus on the low-pass variant (known also as bi-fractional filter).…”

“…In [34], Radwan et al generalize standard second-order filters. In particular, they analyze filters of various characteristics (i.e., low pass, high pass, band pass, notch) constructed using two fractional-order capacitors of order α.…”

Fractional-step Tow-Thomas biquad filters

“…This approach of replacing traditional capacitors with fractional capacitors has previously been investigated for both the Sallen-Key filter and the Kerwin-Huelsman-Newcomb biquad [12]. However, the work in [12] examined the special case when fractional capacitors of the same order were used. Here, we replace both C 1 and C 2 in the Tow-Thomas Biquad with fractional capacitors of impedance…”

“…where K m is the desired magnitude scaling factor, K f = ω α is the frequency scaling factor [12], and ω is the desired frequency to be shifted to. The element values required to realize the transfer function (15) magnitude scaled by a factor of 1000 and frequency shifted to 1 kHz when α 2 = 0.1, 0.5, and 0.9, a = 1 and b and c are selected for minimum passband error are given in Table III.…”

“…While there are no physical analogies to these derivatives, like slope or area under the curve, they are still applicable to physical systems with varying applications including materials theory [3,4], control theory [5,6], electromagnetics [7], robotics [8,9] and many more. The import of these concepts into circuit theory is relatively new [10] with much recent progress regarding filter theory [11,12], analysis [13] and implementation [14][15][16]. This emerging field has introduced fractional calculus into analog filter design to achieve continuous time filtering circuits with fractional step stopband attenuations.…”