Comparative Study of Discrete Component Realizations of Fractional-Order Capacitor and Inductor Active Emulators

Abstract

Due to the absence of commercially available fractional-order capacitors and inductors, their implementation can be performed using fractional-order di®erentiators and integrators, respectively, combined with a voltage-to-current conversion stage. The transfer function of fractional-order di®erentiators and integrators can be approximated through the utilization of appropriate integer-order transfer functions. In order to achieve that, the Continued Fraction Expansion as well as the Oustaloup's approximations can be utilized. The accuracy, in terms of magnitude and phase response, of transfer functions of di®erentiators/integrators derived through the employment of the aforementioned approximations, is very important factor for achieving high performance approximation of the fractional-order elements. A comparative study of the accuracy o®ered by the Continued Fraction Expansion and the Oustaloup's approximation is performed in this paper. As a next step, the corresponding implementations of the emulators of the fractional-order elements, derived using fundamental active cells such as operational ampli¯ers, operational transconductance ampli¯ers, current conveyors, and current feedback operational ampli¯ers realized in commercially available discrete-component IC form,

Reduced Active Components Count Electronically Adjustable Fractional-Order Controllers: Two Design Examples

“…The corresponding error plots are provided in Figure 7c, where the error of the magnitude is less than 2% for the whole range and the maximum error in phase is about 10%. As the observed deviation in phase is observed at the lower limit, it is mainly caused by the limitations imposed by the order of the employed approximation [21]. The values of the gain crossover frequency and phase margin are 129.5 mHz and 42.68 • , close to the theoretically predicted values of 158 mHz and 45 • , respectively.…”

“…Using the 5th-order Oustaloup's approximation [20,21], the rational transfer function which approximates (7) in the range [10 −3 rad/s, 10 +3 rad/s], is…”

Minimum MOS Transistor Count Fractional-Order Voltage-Mode and Current-Mode Filters

“…Comparing (1) with (14) and (6) with (15), it is readily observed that the topology in Figure 2a implements low-pass and high-pass filter functions, with the time-constant given by (16) Adding a fractional-order capacitor C β (0 < β < 1) in the topology in Figure 2a, as it is depicted in Figure 2b, the topology implements the following band-pass filter function…”

“…Due to the absence of commercially available fractional-order capacitors [6][7][8], fractional-order filters can be derived through: (a) the substitution of the conventional (i.e., integer-order) capacitors in the well-known integer-order filters with RC networks (e.g., Foster or Cauer) [9,10], and (b) the implementation of the rational integer-order transfer functions, which are derived through the substitution of the Laplacian fractional-order operator with a suitable expression offered by approximation formulas such as Oustaloup, Continued Fraction Expansion, Matsuda, El-Khazali etc. [11][12][13][14][15][16]. Comparing the aforementioned methods, the main derivation is that the first one offers a quick design procedure but the resulting filter structures suffer from the absence of electronic tuning of the employed RC networks; the second one offers fully electronic tunability of the characteristics of the fractional-order filters.…”

An ultra-broadband single-component fractional-order capacitor using MoS2-ferroelectric polymer composite

“…It is important to mention here that, previously developed FOCs do not only suffer from narrow CPZs but also have other shortcomings that limit their use in modern electronic devices and systems. For instance, liquid electrode based (LEB) FOCs cannot be integrated with microelectronics, 6 the CPA of fractal tree (FT) FOCs cannot be tuned, 11 the FOCs making use of operational trans-conductance amplifiers (OTAs) are power hungry, [47][48][49] and the CPA of the FOCs fabricated using carbon-ferroelectric polymer composites is very sensitive to the filler ratio of the carbon 1,3 and it has a low dynamic range. 46 On the other hand, the FOCs proposed in this work are fully compatible with PCBs, passive, and have a stable CPA over a broader frequency range.…”