Fractional-order Capacitor Research Articles

A general emulator for fractional-order memristive elements with multiple pinched points and application

In this paper, X-controlled universal fractional-order memelements (FOMEs) emulator is proposed. The emulation circuit is realized using second-generation current conveyor (CCII) and analog voltage multiplier (AVM)/divider block with two switches to control the type of memelements and emulator mode. The effect of the fractional-order capacitor (FOC) on the pinched hysteresis loop (HL) area and the range of frequency is discussed at different fractional-order α. Additionally, the higher-order meminductor, memristor, memcapacitor, and inverse memristor are discussed presenting multiple pinched points (MPPs) in the φ-i,i-v,q-v,i-v plane, respectively. The circuit simulations for the proposed emulators have been verified using PSPICE simulations and validated experimentally at different orders. Finally, the proposed emulator is employed in Chua’s chaotic oscillator as an application investigating the effect of fractional-order on the chaotic response.

Reconnection–less reconfigurable low–pass filtering topology suitable for higher–order fractional–order design

The paper discusses a new design of a current–mode reconnection–less reconfigurable fractional–order (FO) low–pass filter of various orders. The filtering structure is based on a 4th–order leap–frog topology using operational transconductance amplifiers as basic building blocks. The resulting order of the filter is given by the setting of current gains (allowing the reconnection–less reconfiguration) alongside with the values of the fractional–order capacitors realized by the RC ladder networks. For this purpose, RC ladder networks of orders 0.3, 0.4, 0.5, 0.6 and 0.7 have been designed. The fractional–order form of the filter contains from one up to four FO capacitors (remaining capacitors (if there are any) are of integer–order) allowing to obtain low–pass functions of order of 3 + α, 2 + α, 1 + α, 2 + α + β, 1 + α + β, α + β, 1 + α + β + γ, α + β + γ and α + β + γ + δ. The proposed filter offers a wide variety of possible order combinations with an increasing degree of freedom as the number of fractional–order capacitors within the structure increases. The proposal is supported by the PSpice simulations of magnitude and phase characteristics, pole frequency adjustment and stability analysis. Moreover, the experimental measurements of the implemented filter were carried out and compared with the simulation results. The possibility of the electronic control of the fractional order is also discussed and presented.

Emulation circuits of fractional-order memelements with multiple pinched points and their applications

Abstract This paper proposes voltage- and current-controlled universal memelements emulators. They are employed to realize the floating and grounded fractional-order memelements. The proposed emulators are implemented using different active blocks such as the second-generation current conveyor (CCII), Differential input double output transconductance amplifier (DOTA + ), balanced output CCII, and Differential voltage current conveyor (DVCC) with analog voltage multiplier. One of the main characteristics of the memristive elements is hysteresis loop behaviour with one pinched point, and the higher-order memelements have multiple pinched points. The higher fractional-order memductance (FOM) and inverse memductance (FOIM) emulators are proposed, which achieve multiple pinched-off points. The coordinates of the multiple pinched-off points and the conditions to achieve them are discussed in the I-V plane. Additionally, the effect of different orders α of the fractional-order capacitor (FOC) on the memelements characteristic is discussed. The circuit simulations for the proposed emulators have been verified using PSPICE simulations and validated experimentally at different orders. Finally, the grounded proposed emulator is employed in Chua's chaotic oscillator as an application presenting the effect of fractional-order on the chaotic response. Also, the floating proposed emulator is applied to a relaxation oscillator, to show the reliability of the proposed emulator.

Exploring the impact of fractional-order capacitive behavior on the hysteresis effects of perovskite solar cells: A theoretical perspective

Perovskite-based solar cells are devices of increasing interest among new generation of photovoltaic technologies due to the outstanding power conversion efficiency (PCE) achieved in the last few years, and the favorable fabrication conditions. Among all these excellent photovoltaic properties, perovskite light-harvesting devices also show relevant issues such as the hysteresis mechanisms in the current density-voltage (J-V) characteristics, whose underlying origins have not yet been clarified from a single and universally accepted point of view. Here, we focus our attention on the ideal capacitive mechanisms which, on the other hand, are often modeled using fractional-order capacitors in the equivalent circuits used to fit impedance measurements. In particular, we provide the theoretical framework for the transient-photocurrent analysis described by fractional-order responses which involve the generalized Mittag-Leffler function. Importantly, the model introduces a connection between perovskite traps and defects, memory processes, fractional dynamics, and Cole-Cole behavior. Crucial dependences of non-ideal capacitive dynamics on fractional-order α and the selected scan rate were found in the J-V hysteresis behavior from the perspective of the non-integer order calculus. In this sense, it is necessary to point out that our numerical simulations of dark J-V curves by considering non-ideal capacitive effects reveal more prominent distortions than those of the ideal case, leading from free-hysteretic influences (α = 1) to significant hysteresis distortions (0<α<1) under certain voltage rates. Thus, this study can help to advance in the origin and the understanding of the experimental hysteresis mechanisms under pristine conditions or during degradation processes (α value decreases indicating a highly disordered morphology).

Two implementations of fractional-order relaxation oscillators

This work proposes general formulas for designing two different topologies of fractional-order relaxation oscillators. One topology contains an Operational Amplifier and the other one relies on an Operational Trans-Resistance Amplifier. The design procedure hinges on the general fractional-order natural and step responses of RC, which is proved in this work depending on Mittag Leffler function. The proposed topologies can be controlled to generate symmetrical and non-symmetrical square wave signals. They also benefit from the employment of fractional-order capacitors (FOCs), which makes it possible to obtain higher frequencies using simple components. Furthermore, these topologies are verified through numerical solutions, circuit simulations, and experimental implementations. This encourages the authors to build circuit emulators for the FOC, which is achieved by applying the Foster-I synthesizing technique to the Matsuda’s approximation of $$s^\alpha$$.

Fractional-order time-sharing-control-based wireless power supply for multiple appliances in intelligent building

The wired power transmission is usually adopted to supply power for the devices in the traditional buildings. With the development of intelligent buildings, the way of wired power supply would greatly increase the complexity and consumption of laying the lines. To improve the flexibility of power supply and reduce the cost of wiring, wireless power transfer technology has been used in smart buildings. However, it remains a fundamental challenge to create a simple wireless power transfer system in which power can be wirelessly transferred to multiple appliances. Therefore, this paper proposes a wireless power transfer scheme based on fractional-order time-sharing control for a variety of household appliances in intelligent building. In the proposed scheme, only one fractional-order capacitor in the transmitter is needed to realize the time-sharing resonant charging. In contrast, the traditional multiple-receiver systems require complicated control scheme, for example, controlling a plurality of sets of series-parallel capacitors through a series of relay switches. To demonstrate the method, a 150 W LED TV with 300 kHz and a 5 W mobile phone charger with 127 kHz serve as the actual loads. The experimental results show that the proposed system can supply power to the TV and the mobile phone by a time-sharing way wirelessly.

Integer- and Fractional-Order VCO Using Non-Inertial Amplitude Stabilization and Modern Active Elements

This paper introduces design of the linearly tunable quadrature voltage-controlled oscillator (VCO) using modern off-the-shelf active elements suitable for the design of electronically adjustable applications. The simplified topology was achieved using non-inertial stabilization of amplitude. It allows targeting adjustability of the oscillation frequency only into lossless integrator part of the topology. This arrangement simplifies the design and the tenability also. Derived symbolical expressions indicate that gain adjustment of used amplifiers (that are still in single path) does not influence amplitude and phase shift ratio of generated waveforms, which is a beneficial feature. The performances of the circuit are tested experimentally in band of units of MHz and results confirmed expected behavior. Initial study of the oscillator with fractional-order capacitors is presented and discussed. Results are supported by laboratory measurements.

Compact Fractional-Order Model of On-Chip Inductors With BCB on High Resistivity Silicon

Fractional-order calculus exhibits clear advantages over modeling the nonlinear and complex physics phenomena. In this article, fractional-order calculus is introduced to the field of on-chip inductor modeling. A novel fractional-order 1-π inductor equivalent circuit model is proposed to characterize on-chip inductors based on silicon-benzocyclobutene (Si-BCB) technology. The proposed model, which contains six integer-order circuit elements and three fractional-order elements, is sufficient to describe frequency-dependent effects, such as the skin effect, proximity effect, and distributed effect. Three fractional-order elements (one fractional-order inductor and two fractional-order capacitors) introduced in the above model are major contributors to improving the model's accuracy and bandwidth. The parameter extraction method is discussed in this article, and the direct simulation method is employed to solve fractional-order equations. The accuracy of the proposed fractional-order model is validated up to 40 GHz. An excellent agreement is observed between the model and the measurements.

Nonlinear charge-voltage relationship in constant phase element

The constant phase element (CPE) or fractional-order capacitor is an electrical device that has an impedance of the form Z(s)=1/Cαsα, where Cα is the CPE parameter and α is a fractional dispersion coefficient of values between 0 and 1. Here we show that in the time-domain the classical linear charge-voltage relationship of ideal capacitors, q=C·v, is not valid for CPEs. In fact the relationship is nonlinear and can be expressed as q=C(v;Cα,α)·v. We verify our findings using (i) circuit simulations of an integer-order emulator of a CPE, and (ii) experimental results from a commercial supercapacitor as well as an in-house fabricated device based on carbon film materials.

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

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.

Fractional-order electric double-layer capacitors with tunable low-frequency impedance phase angle and energy storage capabilities

In this study, we report the electrical response of two sets of solid-state fractional-order electrochemical capacitors for which the low-frequency impedance phase angle can be tuned from −69° to −7°. The configuration makes use of a gel electrolyte in which carbonaceous additives (graphite or reduced graphene oxide) are dispersed at different proportions. Such a disordered electrolyte structure results in subdiffusive charge transport and thus a frequency dispersive capacitive-resistive behavior typical of a constant phase element, which can be useful for both frequency applications and energy storage purposes.

A Fractional-Order Resonant Wireless Power Transfer System With Inherently Constant Current Output

Wireless power transfer (WPT) using magnetic-coupling resonant has been proved to be a promising transmission method. At present, most of the existing resonant WPT systems adopt integer-order inductors and capacitors. In fact, in addition to integer-order components, there also exists fractional-order inductors and capacitors, which possess richer characteristics than integral-order inductors and capacitors. However, there are few studies on the application of fractional-order components in resonant WPT system. Therefore, this paper proposes a fractional-order resonant WPT system, which is based on a fractional-order resonant circuit only incorporating a fractional-order capacitor. The coupled-mode theory model of the proposed WPT system is established, and the impact of different loads and orders on transfer characteristics is explored. The theoretical analysis demonstrates that the proposed system can achieve constant current output that is independent of load, just by choosing appropriate parameters of the fractional-order capacitor. Finally, an experimental prototype of the fractional-order resonant WPT system is built, and the experiment results verify the correctness of the theoretical analysis.

Development of Fractional Order Modeling of Voltage Source Converters

This paper proposes a mathematical fractional order modelling of Voltage Source Converters (VSC). Fractional behaviour helps to model and describe a real object more accurately than the integer order model. State-space equations of VSC is obtained using fractional calculus in d-q domain. The Caputo derivative method is employed to eliminate complex definitions of fractional calculus and to get approximate solution with optimal design of parameters. A detailed analysis is performed on developed mathematical fractional order model of VSC including small-signal analysis and DC analysis. The stability and time-domain transient analysis is carried out to validate the fractional order VSC for its application as Distribution STATtic COMpensator (DSTATCOM) in power distribution system, using real-time simulator. Fractional-order capacitor and inductor are designed and modelled for verification of fractional order VSC through simulations. The real-time simulation results verify the effectiveness of proposed model. It shows that fractional order model can describe the operating characteristics of VSC more accurately. The comparative performance analysis demonstrates superior transient response in case of fractional order devices as compared to integer order.

Fractional-Order Modeling and Analysis of a Three-Phase Voltage Source PWM Rectifier

In this paper, the fractional-order modeling and analysis of a three-phase voltage source PWM rectifier (VSR) are researched considering the fractional-order characteristics of actual inductors and capacitors. While the Caputo fractional calculus operator is used to describe the fractional-order characteristics of an inductor and a capacitor, the fractional-order model of a three-phase VSR is established in the three-phase static coordinate system. With the coordinate transformation, the fractional-order model is expressed in the two-phase static coordinate system and in the synchronous rotation coordinate system, respectively. Simulation results verify the effectiveness of the established models, and indicate that the fractional-order models can describe the operating characteristics of a three-phase VSR more accurately. Besides, compared with the traditional integer-order three-phase VSRs, the three-phase VSRs which contain the fractional-order inductors and the fractional-order capacitor can present better static and dynamic performance indexes (such as smaller overshoot and shorter regulation time) by appropriately selecting the orders of the inductors and the capacitor.

Assessment of Fractional-Order Arterial Windkessel as a Model of Aortic Input Impedance.

Goal: Fractional-order Windkessel model is proposed to describe the aortic input impedance. Compared with the conventional arterial Windkessel, the main advantage of the proposed model is the consideration of the viscoelastic nature of the arterial wall using the fractional-order capacitor (FOC). Methods: The proposed model, along with the standard two-element Windkessel, three-element Windkessel, and the viscoelastic Windkessel models, are assessed and compared using in-silico data. Results: The results show that the fractional-order model fits better the moduli of the aortic input impedance and fairly approximates the phase angle. In addition, by its very nature, the pseudo-capacitance of FOC makes the proposed model's dynamic compliance complex and frequency-dependent. Conclusions: The analysis of the proposed fractional-order model indicates that fractional-order impedance yields a powerful tool for a flexible characterization of the arterial hemodynamics.

Realization of fractional order circuits by a Constant Phase Element

Fractional-order calculus has been used for generalizing many modern and classical control theories including the well establish PID paradigm. The obtained controllers, of non-integer order, must be approximated with high order integer ones, in order to be realized. Successively, analog or digital implementations are used for the real world applications. This approach offers the hip to a classical criticism to fractional calculus. Why design a fractional-order system, which is usually of low order, if you need a high order system to implement it? In order to face this problem, in this paper, a fractional-order capacitor, more specifically a Constant Phase Device, is applied for implementing a first order fractional transfer function. Due to the intrinsic nature of the realized device, just one capacitor is needed for the implementation, avoiding therefore the need of high order RC approximation. Furthermore a fractional-order Wien oscillator and a chaotic Duffing circuit are presented confirming the potentiality of the proposed device in realizing fractional order circuits.

Electronically Adjustable Emulator of the Fractional-Order Capacitor

This paper presents a design of the controllable emulator of the FOC (Fractional-Order Capacitor) and its application. The circuit is based on 5th-order RC topology (type Foster I), where the passive elements in the topology are replaced by electronically adjustable components. The proposed emulator is based on OTA (Operational Transconductance Amplifier) and VDCC (Voltage Differencing Current Conveyor). The electronically controllable resistors are implemented by OTAs. The electronically tunable capacitors are implemented using capacitance multipliers, which employ VDCCs. The proposed structure provides the electronic control of the order and electronic shifting of the frequency band of the approximation validity. The proposed FOC emulator is also used for fractional-order filter design. The proposed circuits are verified using PSpice simulations.

Comparative Study of Riemann–Liouville and Caputo Derivative Definitions in Time-Domain Analysis of Fractional-Order Capacitor

The fractional-order capacitor applied in power electronic converters, filters or other circuits has become a hot spot due to benefits, such as high-performance and flexibility. Two typical kinds of fractional-order derivative definitions, Riemann-Liouville (RL) or Caputo derivatives are popularly used to analyze the characteristics of fractional-order capacitor. However, it remains a fundamental challenge to find that when and which fractional-order derivative definition is more suitable in the time-domain analysis of fractional-order capacitor. In this brief, comparative investigations about the transient and steady analysis of a fractional-order capacitor adopting these two definitions are performed. It is determined that difference comes from description of dynamic process. In particular, the difference is especially obvious for the cases of step-up input. Based on the analysis, this brief proposed that RL derivative is a more accurate choice for modeling and analyzing fractional-order capacitor in time domain. Moreover, RL derivative is also more suitable for solving fractional-order circuits with fractional-order capacitor. The results from simulations and experiments verify the conclusion.

Quantification of memory in fractional-order capacitors

In this study we quantify and interpret the inherent memory in fractional-order capacitors when subjected to constant current charging/discharging waveforms. This is done via a finite difference approximation of the fractional order rate equation relating current to voltage in these devices. It is found that as the fractional exponent decreases, the weight of the voltage memory trace that results from the contribution of past voltage activity increases, and thus the measured response of the device at any time is increasingly correlated to its past. Ideal capacitors, on the other hand, which are described by the first order differential equation , do not exhibit such behavior and are memoryless devices.

A Fractional-Order Wireless Power Transfer System Insensitive to Resonant Frequency

Since electric power was first transferred wirelessly by Tesla over a hundred years ago, wireless power transfer (WPT) systems have always been composed of integer-order components, such as inductors and capacitors. Recent studies have shown that efficient WPT can be achieved in the range of tens of centimeters or even meters by using magnetic-coupling of two or more integer-order resonant circuits. However, their transfer efficiency and output power are highly sensitive to the resonant frequency. Unfortunately, the resonant frequency is easily disturbed by metal or nonresistive load and so on in real-world application, which will unavoidably lead to transfer efficiency reduction and unstable output power in these integer-order circuits. Therefore, this article proposes and experimentally demonstrates a fractional-order WPT system containing a fractional-order capacitor, in which the transfer efficiency and output power are inherently stable and insensitive to the resonant frequency. The results show that the efficiency and output power only change within 1% when the receiver resonant frequency is detuned by ±5%. In contrast, the WPT systems that are sensitive to the resonant frequency suffer from fall in the efficiency of around 30%, and a largely unstable output power. The available experiments verify the effectiveness of the proposed system.