Design and application examples of CMOS fractional-order differentiators and integrators
Introduction
Fractional-order calculus is utilized in a variety of interdisciplinary applications [[1], [2], [3]], and towards this goal differentiation and integration stages are essential building blocks for performing the required signal processing. Various applications of fractional-order circuits in filter design [[4], [5], [6], [7], [8], [9], [10], [11], [12]], oscillator design [13,14], biological tissue modeling [[15], [16], [17], [18], [19], [20], [21], [22], [23]] as well as in automatic control [24,25] have been introduced in the literature. Fractional-order differentiators and integrators offer attractive characteristics compared to their integer-order counterparts, including scaling of the time-constants as well as phase difference between input and output which is ±απ∕2, with (0 < α < 1). The straightforward way for implementing such blocks is the substitution of capacitors in the conventional (i.e. integer-order) structures by fractional-order capacitors, known also as Constant Phase Elements (CPEs). A fractional-order capacitor is characterized by two parameters , where Cα is the pseudo-capacitance expressed in units of Farad∕ sec1−α and 0 < α < 1 is the (fractional) order [26]. The impedance of a CPE is described in the s-domain by (1), where the relation between pseudo-capacitance and the conventional capacitance (in Farad), can be expressed as in (2)This direct substitution technique is not easily realizable, due to the absence of commercially available fractional-order elements despite the growing effort and clear progress towards this goal [[27], [28], [29], [30]]. Thus, appropriately configured RC networks [31] have been employed to approximate the behavior of fractional-order capacitors. However, this technique's major drawback is that the complete RC network must be re-designed in order to change the approximated fractional capacitor's pseudo-capacitance and/or its order. An alternative technique is the utilization of integer-order transfer functions, derived through an appropriate method, for approximating the integro-differential Laplacian operator (sα) [19]. Discrete IC component implementations of fractional-order differentiators/integrators, where Operational Amplifiers (op-amps) or Current Feedback Operational Amplifiers (CFOAs) were used, have been presented in the literature, but they suffer from the absence of electronic tuning of their characteristics [32]. Only integrated CMOS implementations [33,34] offer such feature, as a result of employing, for example, the current-controlled small-signal transconductance parameter (gm) of Operational Transconductance Amplifiers (OTAs) or Current-Mirrors (CMs). However, the CMOS implementations already reported in the literature suffer from increased complexity which limits the possibility of increasing the order of the underlying integer-order approximation.
In order to overcome this obstacle, CMOS topologies which perform 2nd- and 5th-order approximations of fractional-order differentiators/integrators while offering significantly reduced circuit complexity, are presented in this paper. While some preliminary results have been reported in Ref. [35], here we develop a new systematic method for performing higher-order approximations of fractional-order differentiators/integrators and, in addition, validate the proposed design in application examples using both 2nd-order and 5th-order approximations.
The paper is organized as follows: in Section 2, the topology of fractional-order differentiator/integrator, derived by implementing a 2nd-order transfer function that approximates the Laplacian operator, is demonstrated. Designs of fractional-order lowpass/highpass filters and the Leaky-Integrate-and-Fire Mihalas-Niebur neuron model are presented in Section 3. In Section 4, the implementation of a 5th-order transfer function that approximates the fractional Laplacian operator is demonstrated. As an application example, the realization of emulators of fractional-order capacitors and inductors is presented in Section 4.1. The behavior of all the proposed designs is verified in Cadence using the Design Kit provided by the Austria Mikro Systeme (AMS) 0.35 μm CMOS process.
Section snippets
Realization of a fractional-order differentiator/integrator using a 2nd-order approximation
The transfer function of a fractional-order differentiator/integrator is given by:where r = α (0 < α < 1) is the order of the differentiator, while for the integrator r = −α (0 < α < 1). Also, the unity-gain frequency (ωo) of the stage is defined as ωo = 1∕τ, where (τ) is a time-constant. Using (3), the magnitude and phase frequency responses are given by the expressions and ∠H(ω) = πr∕2, respectively [36].
The transfer function in (3) can be approximated, around the
Filter design example
The Functional Block Diagram (FBD) of a fractional-order low/highpass filter, using a differentiator/integrator with unity gain frequency ωo = 1∕τ as active core, is demonstrated in Fig. 2. In the case of r = α, then the highpass filter function given by (11) is implemented, while when r = −α the transfer function of the realized lowpass filter is given by (12). The desirable inverted output is also available, using an extra current-mirror at the output, as depicted in the full circuitry shown
Fractional-order differentiator/integrator using a 5th-order approximation
Although the 2nd-order CFE approximation offers circuit simplicity, its accuracy for an error in phase less than 10%, is restricted to the range [fo∕10, 10fo], where fo is the center frequency of the CFE approximation. In order to extend this frequency range, a higher-order approximation must be utilized. The integer-order transfer function in the case of a 5th-order CFE approximation is given byThe values of coefficients Ai (i = 0, 1,
Conclusions
Application design examples including the realization of fractional-order filters, a fractional-order neuron model and emulators of fractional-order capacitor/inductor have been demonstrated and verified in this work. These applications rely on optimized and reduced complexity fractional-order differentiator/integrator stages that are also electronically tunable. Compared to already published CMOS structures, there is a significant reduction in the number of transistor employed which
Acknowledgment
This work is supported by the General Secretariat for Research and Technology (GSRT) and the Hellenic Foundation for Research and Innovation (HFRI).
This article is based upon work from COST Action CA15225, a network supported by COST (European Cooperation in Science and Technology).
References (48)
- et al.
A new collection of real world applications of fractional calculus in science and engineering
Commun. Nonlinear Sci. Numer. Simulat.
(2018) - et al.
On the practical realization of higher-order filters with fractional stepping
Signal Process.
(2011) - et al.
Fractional order filter with two fractional elements of dependant orders
Microelectron. J.
(2012) - et al.
Fractional-order filters based on low-voltage DDCCs
Microelectron. J.
(2016) - et al.
Analog realization of fractional filters: Laguerre approximation approach
AEU-Int. J. Electron. Commun.
(2017) - et al.
Two-port two impedances fractional order oscillators
Microelectron. J.
(2016) - et al.
Fractional order oscillators based on operational transresistance amplifiers
AEU-Int. J. Electron. Commun.
(2015) - et al.
Emulation of an electrical-analogue of a fractional-order human respiratory mechanical impedance model using OTA topologies
AEU-Int. J. Electron. Commun.
(2017) - et al.
Emulation of current excited fractional-order capacitors and inductors using OTA topologies
Microelectron. J.
(2016) - et al.
New analog implementation technique for fractional-order controller: a DC motor control
AEU-Int. J. Electron. Commun.
(2017)
Studies on fractional order differentiators and integrators: a survey
Signal Process.
A low power current controllable single-input three-output current-mode filter using MOS transistors only
AEU-Int. J. Electron. Commun.
First-order adjustable transfer sections for synthesis suitable for special purposes in constant phase block approximation
AEU-Int. J. Electron. Commun.
Synthesis and design of constant phase elements based on the multiplication of electronically controllable bilinear immittances in practice
AEU-Int. J. Electron. Commun.
Fractional Signal Processing and Applications
Fractional order calculus: basic concepts and engineering applications
Math. Probl Eng.
Optimization of fractional-order RLC filters
Circ. Syst. Signal Process.
Electronically tunable fractional-order low-pass filter with current followers
Current conveyors in current-mode circuits approximating fractional-order low-pass filter
Design of CMOS Analog Integrated Fractional-order Circuits: Applications in Medicine and Biology
Reconfigurable fractional-order filter with electronically controllable slope of attenuation, pole frequency and type of approximation
J. Circ. Syst. Comput.
Fractional electrical impedances in botanical elements
J. Vib. Contr.
Bioimpedance and Bioelectricity Basics
Modeling and simulation of equivalent circuits in description of biological systems-a fractional calculus approach
J. Electr. Bioimpedance
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