This paper presents a generalization of six well-known quadrature third-order oscillators into the fractional-order domain. The generalization process involves replacement of three integer-order capacitors with fractional-order ones. The employment of fractional-order capacitors allows a complete tunability of oscillator frequency and phase. The presented oscillators are implemented with three active building blocks which are op-Amp, current feedback operational amplifier (CFOA) and second generation current conveyor (CCII). The general state matrix, oscillation frequency and condition are deduced in terms of the fractional-order parameters. The extra degree of freedom provided by the fractional-order elements increases the design flexibility. Eight special cases including the integer case are illustrated with their numerical discussions. Three different phases are produced with fixed sum of 2p which can be completely controlled by fractional-order elements. A general design procedure is introduced to design an oscillator with a specific phase and frequency. Two general design cases are discussed based on exploiting the degrees of freedom introduced by the fractional order to obtain the required design. Spice circuit simulations with experimental results for some special cases are presented to validate the theoretical findings. © 2017 World Scientific Publishing Company.
Simple floating voltage-controlled memductor emulator for analog applications
The topic of memristive circuits is a novel topic in circuit theory that has become of great importance due to its unique behavior which is useful in different applications. But since there is a lack of memristor samples, a memristor emulator is used instead of a solid state memristor. In this paper, a new simple floating voltage-controlled memductor emulator is introduced which is implemented using commercial off the shelf (COTS) realization. The mathematical modeling of the proposed circuit is derived to match the theoretical model. The proposed circuit is tested experimentally using different excitation signals such as sinusoidal, square, and triangular waves showing an excellent matching with previously reported simulations.
Single and dual solutions of fractional order differential equations based on controlled Picard’s method with Simpson rule
This paper presents a semi-analytical method for solving fractional differential equations with strong terms like (exp, sin, cos,Â…). An auxiliary parameter is introduced into the well-known Picard’s method and so called controlled Picard’s method. The proposed approach is based on a combination of controlled Picard’s method with Simpson rule. This approach can cover a wider range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for higher order differential equations. The proposed approach can be effectively applied to Bratu’s problem in fractional order domain to predict and calculate all branches of problem solutions simultaneously. Also, it is tested on other fractional differential equations like nonlinear fractional order Sine-Gordon equation. The results demonstrate reliability, simplicity and efficiency of the approach developed. © 2017 University of Bahrain
Boundary Dynamics of Memcapacitor in Voltage-Excited Circuits and Relaxation Oscillators
This paper discusses the boundary dynamics of the charge-controlled memcapacitor for Joglekar’s window function that describes the nonlinearities of the memcapacitor’s boundaries. A closed form solution for the memcapacitance is introduced for general doping factor (Formula presented.)p. The derived formulas are used to predict the behavior of the memcapacitor under different voltage excitation sources showing a great matching with the circuit simulations. The effect of the doping factor (Formula presented.) on the time domain response of the memcapacitor has been studied as compared to the linear model using the proposed formulas. Moreover, the generalized fundamentals such as the saturation time of the memcapacitor are introduced, which play an important role in many control applications. Then the boundary dynamics under sinusoidal excitation are used as a basis to analyze any periodic signal by Fourier series, and the results have been verified using PSPICE simulations showing a great matching. As an application, two configuration of resistive-less memcapacitor-based relaxation oscillators are proposed and closed form expressions for oscillation frequency and conditions for oscillation are derived in presence of nonlinear model. The proposed oscillator is verified using PSPICE simulation showing a perfect matching. © 2015, Springer Science+Business Media New York.
Modified methods for solving two classes of distributed order linear fractional differential equations
This paper introduces two methods for the numerical solution of distributed order linear fractional differential equations. The first method focuses on initial value problems (IVPs) and based on the ?th Caputo fractional definition with the shifted Chebyshev operational matrix of fractional integration. By applying this method, the IVPs are converted into simple linear differential equations which can be easily handled. The other method focuses on boundary value problems (BVPs) based on Picard’s method frame. This method is based on iterative formula contains an auxiliary parameter which provides a simple way to control the convergence region of solution series. Several numerical examples are used to illustrate the accuracy of the proposed methods compared to the existing methods. Also, the response of mechanical system described by such equations is studied. © 2017 Elsevier Inc.
Resistive-less memcapacitor-based relaxation oscillator
Recently, the realization of the conventional relaxation oscillators was introduced based on memristors. This paper validates the concept using two series memcapacitors in general which is applicable for a capacitor and memcapacitor as well. Furthermore, the necessary conditions for oscillation are introduced, and a generalized closed-form expression for the oscillation frequency is derived. Two special cases are introduced and verified using PSPICE simulations showing a perfect matching. Copyright © 2014 John Wiley & Sons, Ltd.
Optimal Charging and Discharging of Supercapacitors
In this paper, we discuss the optimal charging and discharging of supercapacitors to maximize the delivered energy by deploying the fractional and multivariate calculus of variations. We prove mathematically that the constant current is the optimal charging and discharging method under R s -CPE model of supercapacitors. The charging and round-trip efficiencies have been mathematically analyzed for constant current charging and discharging. © 2020 The Electrochemical Society (“ECS”). Published on behalf of ECS by IOP Publishing Limited.
Memcapacitor response under step and sinusoidal voltage excitations
Recently, mem-elements have become fundamental in the circuit theory through promising potential applications based on the built-in memory-properties of these elements. In this paper, the mathematical analysis of the memcapacitor model is derived and the effect of different voltage excitation signals is studied for the linear dopant model. General closed form expressions and analyses are presented to describe the memcapacitor behavior under DC step and sinusoidal voltage excitations. Furthermore, the step and sinusoidal responses are used to analyze the memcapacitor response under any periodic signal using Fourier series expansion where the effect of the DC component on the output response is investigated. In addition, the stored energy in the memcapacitor under step, sinusoidal and square wave excitations is discussed. Moreover, the analysis of series and parallel connection of N non-matched memcapacitors in general is introduced and special cases of matched memcapacitors are discussed. The derived equations are verified using SPICE simulations showing great matching. © 2014 Elsevier Ltd. All rights reserved.
First-order filters generalized to the fractional domain
Traditional continuous-time filters are of integer order. However, using fractional calculus, filters may also be represented by the more general fractional-order differential equations in which case integer-order filters are only a tight subset of fractional-order filters. In this work, we show that low-pass, high-pass, band-pass, and all-pass filters can be realized with circuits incorporating a single fractance device. We derive expressions for the pole frequencies, the quality factor, the right-phase frequencies, and the half-power frequencies. Examples of fractional passive filters supported by numerical and PSpice simulations are given. © 2008 World Scientific Publishing Company.
On some generalized discrete logistic maps
Recently, conventional logistic maps have been used in different vital applications like modeling and security. However, unfortunately the conventional logistic maps can tolerate only one changeable parameter. In this paper, three different generalized logistic maps are introduced with arbitrary powers which can be reduced to the conventional logistic map. The added parameter (arbitrary power) increases the degree of freedom of each map and gives us a versatile response that can fit many applications. Therefore, the conventional logistic map is considered only a special case from each proposed map. This new parameter increases the flexibility of the system, and illustrates the performance of the conventional system within any required neighborhood. Many cases will be illustrated showing the effect of the arbitrary power and the equation parameter on the number of equilibrium points, their locations, stability conditions, and bifurcation diagrams up to the chaotic behavior. © 2012.