Fractional-order Circuit Research Articles

Modeling and Analysis of Caputo–Fabrizio Definition-Based Fractional-Order Boost Converter with Inductive Loads

This paper proposes a modeling and analysis method for a Caputo–Fabrizio (C-F) definition-based fractional-order Boost converter with fractional-order inductive loads. The proposed method analyzes the system characteristics of a fractional-order circuit with three state variables. Firstly, this paper constructs a large signal model of a fractional-order Boost converter by taking advantage of the state space averaging method, providing accurate analytical solutions for the quiescent operating point and the ripple parameters of the circuit with three state variables. Secondly, this paper constructs a small signal model of the C-F definition-based fractional-order Boost converter by small signal linearization, providing the transfer function of the fractional-order system with three state variables. Finally, this paper conducts circuit-oriented simulation experiments where the steady-state parameters and the transfer function of the circuit are obtained, and then the effect of the order of capacitor, induced inductor, and load inductor on the quiescent operating point and ripple parameters is analyzed. The experimental results show that the simulation results are consistent with those obtained by the proposed mathematical model and that the three fractional orders in the fractional model with three state variables have a significant impact on the DC component and steady-state characteristics of the fractional-order Boost converter. In conclusion, the proposed mathematical model can more comprehensively analyze the system characteristics of the C-F definition-based fractional-order Boost converter with fractional-order inductive loads, benefiting the circuit design of Boost converters.

Analysis of Fractional Electrical Circuit Containing Two RC Ladder Elements of Different Fractional Orders

Abstract The study addresses the topic of different fractional orders in the context of simulation as well as experiments using real electrical elements of fractional-order circuit. In studying the two solutions of the resistance-capacitance (RC) ladder circuit of appropriate parameters, different fractional orders of the electrical circuit are considered. Two fractional-order (non-integer) elements were designed based on the Continued Fraction Expansion (CFE) approximation method. The CFE method itself was modified to allow free choice of centre pulsation. It was also proposed that when making individual ladder circuits, in the absence of elements with the parameters specified by the program, they should be obtained by connecting commercially available elements in series or parallel. Finally, the theoretical analysis of such a circuit is presented using state-space method and verified experimentally.

Insights into Non-uniform Aging Effects on Broadband Dielectric Behaviors for High-voltage XLPE Cable Insulation: A Novel Fractional-order Circuit Modeling Approach

Insights into Non-uniform Aging Effects on Broadband Dielectric Behaviors for High-voltage XLPE Cable Insulation: A Novel Fractional-order Circuit Modeling Approach

Passive Synthesis of Fractional Order Immittance of Three-Element-Kind

For the actual circuit with fractional order characteristics, the fractional order circuit model is more concise and precise than the classical circuit model. So studying the synthesis of fractional order immittance is of great significance. This paper investigates the passive synthesis method of the three-element-kind fractional order immittance function. It means designing a network with three kinds of passive fractional-order elements that exhibit a specific impedance function. This paper proposes a variable substitution method to solve this problem. The method consists of three steps: The first step of the process is to establish the correspondence between the fractional order and lossy reactance. This way, the fractional order immittance function is transformed into an integer order immittance function. Then the Brune procedure realizes the immittance function as a network with lossy reactance. Finally, a fractional order network is obtained by replacing the lossy reactance with the corresponding fractional reactance. A simulation example and an experiment are also given for verification. This method applies to all three-element-kind network impedance functions that meet the passive condition.

State of Charge Estimation of Supercapacitor Under Different Temperatures Using Particle Filter Algorithm Based on Fractional-Order Model

With the rise of new energy vehicles, supercapacitors (SCs) have been used as energy storage components for new energy vehicles due to their high-power density and good low-temperature performance. Accurate modeling and state of charge estimation of SC can ensure the safe operation of new energy vehicles. In order to explore the low-temperature performance of supercapacitors, this paper proposes a dual ZARC fractional-order circuit model to simulate the dynamic characteristics of SC. Using adaptive genetic algorithm for SC parameter identification, the model terminal voltage error is less than 6.5 mV. In addition, the SOC of SC at different temperatures and working conditions is estimated by using the fractional-order particle filter (FOPF) method and compared with the fractional-order extended Kalman filter (FOEKF). The experimental results show that the FOPF method has high estimation accuracy and robustness. Under the temperature of minus 40 °C, the maximum mean absolute error and maximum root-mean-square deviation of SOC estimation under different working conditions are less than 2%, showing good low-temperature performance.

Simplified Analysis and Characterization of Dual Power Source Wireless Power Transfer System

Non-contact wireless power transfer systems require power regulation to ensure their proper operation at rated power due to the many degrees of freedom in non-contact coupling structures. Recently, a promising technology has been introduced for wireless power transfer systems, namely, the fractional-order technology with dual power sources to construct an equivalent impedance structure with both variable resistance and reactance. Unlike other impedance matching technologies, the fractional-order technology allows for the adjustment of the equivalent impedance of the resonant network without requiring precise selection of resonant parameters. However, this technology has not been widely adopted due to the abstract concepts involved in fractional calculus theory and control logic. In this study, we employed circuit theory to comprehensively analyze the electrical implications of fractional theory and to provide a detailed description of the inner features of Cα. This study reveals the underlying correlation between control variables and the order of Cα. Additionally, it is shown that the same fractional-order circuit can display both inductive and capacitive reactance, resulting in a significant extension of the impedance adjustment range of fractional-order components. A prototype was designed and constructed to validate the proposed approach, and the experimental results closely matched the theoretical calculations obtained from the proposed approach.

Theoretical analysis and experimental verification of fractional-order RC cobweb circuit network

Recent research has shown that ideal capacitors and inductors do not physically exist, and that the dynamics of real devices can be accurately described by fractional-order (FO) mathematical models. This paper investigates a class of 2×n order RC cobweb FO circuit network with central node. Based on the Kirchhoff’s laws, the impedance magnitude and phase between two points of the network are derived using difference equations and matrix transformations. Three impedance expressions are deduced, and their correctness is verified numerically and by simulations. The influence of various parameters of the electrical network, namely the resistance, capacitance, number of circuit units, frequency and fractional order, on the impedance is studied. Additionally, for the first time, physical experiments are presented to compare the effectiveness of FO and integer-order circuit networks for describing the impedance of actual physical circuits. These experiments confirm that FO circuit network models perform better than the integer-order ones for representing the characteristics of the impedance magnitude and phase.

On Cauchy Problems of Caputo Fractional Differential Inclusion with an Application to Fractional Non-Smooth Systems

In this innovative study, we investigate the properties of existence and uniqueness of solutions to initial value problem of Caputo fractional differential inclusion. In the study of existence problems, we considered the case of convex and non-convex multivalued maps. We obtained the existence results for both cases by means of the appropriate fixed point theorem. Furthermore, the uniqueness corresponding to both cases was also determined. Finally, we took a non-smooth system, the modified Murali–Lakshmanan–Chua (MLC) fractional-order circuit system, as an example to verify its existence and uniqueness conditions, and through several sets of simulation results, we discuss the implications.

Energy Consideration of a Capacitor Modelled Using Conformal Fractional-Order Derivative

Fractional order circuit elements have become important parts of electronic circuits to model systems including supercapacitors, filters, and many more. The conformal fractional derivative (CFD), which is a new basic fractional derivative, has been recently used to model supercapacitors successfully. It is essential to know how electronic components behave under excitation with different types of voltage and current sources. A CFD capacitor is not a well-known element and its usage in circuits is barely examined in the literature. In this research, it is examined how to calculate the stored energy of a CFD capacitor with a series resistor supplied from a DC voltage source. The solutions given in this study may be used in circuits where supercapacitors are used.

Electronic Circuit Implementations of a Fractional-Order Chaotic System and Observing the Escape from Chaos

Studies on fractional-order chaotic systems have increased significantly in the last decade. This paper presents Rucklidge chaotic system’s dynamical analyses and its fractional-order circuit implementations. Component values required for realizing the circuit of the fractional-order system are calculated for different fractional-orders. The feasibility of the attractor is examined by implementing its electronic circuit with a fractional-order module. The module is constructed based on the Diyi-Chen model since it is easier to implement and cost-effective. In electronic circuit implementations, it is observed that the system’s chaotic state disappears as the fractional degree decreases. Numerical and circuit simulation results are consistent well with the hardware experimental results.

Chaotic Oscillations in a Fractional-Order Circuit with a Josephson Junction Resonator and Its Synchronization Using Fuzzy Sliding Mode Control

A linear resistive capacitive inductive shunted model of a Josephson junction with a topologically nontrivial behaviour is considered in this paper. We have considered a fractional-order flux-controlled memristor to effectively model the feedback flux effects across the Josephson junction (JJ). The mathematical model of the proposed JJ oscillator is derived, and the dimensionless model is used to study the various dynamical properties of the oscillator. The stability plot shows that the proposed oscillator has both stable and unstable regions of oscillations for different choices of equilibrium points and fractional order. The bifurcation plots are presented to understand the route to crisis, and we have also shown that the oscillator has regions of coexisting attractors. We have also achieved the synchronization of the proposed oscillator using fuzzy sliding mode control with the master and slave systems considered with different parameter sets. The chattering amplitude is estimated by using the fuzzy logic, and it is used in the synchronization algorithm to minimize the error.

Analysis of an Oscillation Circuit with a Linear Time-invariant Inductor and a Capacitor Modelled with Conformal Fractional Order Derivative

Fractional order circuit elements are being examined by researchers unremittingly. They are ever becoming more popular in the literature. The Conformable Fractional Derivative has been proposed and gained importance in the last decade. Examination of an LC tank circuit or an LC oscillator can be found in almost all undergrad physics books. There’s a considerable number of studies on fractional-order capacitor circuits but, to the best of our knowledge, examination of an oscillator made of a linear time-invariant inductor and a supercapacitor modeled with Conformable Fractional Derivative has not been found in literature. In this paper, a lossless oscillator circuit containing a linear time-invariant inductor and a supercapacitor modeled with Conformable Fractional Derivative is examined for the first time in the literature. Natural response of the circuit has been found analytically. Its behavior has been illustrated with simulations for different initial conditions.

Fractional-order circuit design with hybrid controlled memristors and FPGA implementation

A hybrid memristor circuit consisting of a magnetic-controlled and a charge-controlled is designed. In order to construct a relatively complex dynamical system, an absolute value and a square root algorithm are embedded in the magnetic-controlled model specially. This novel five-dimensional system is first analyzed with integer order for the existence of chaotic phenomena, and then implemented the fractional-order memristor system by the Grunwald–Letnikov algorithm. Five fractional order values of qi(i=1,2,3,4,5) are taken as identical and different in the numerical simulation, respectively. The bifurcation diagram, 0-1 test, SALI algorithm, and complexity analysis reveal the existence of nonlinear dynamic characteristics to the hybrid fractional-order memristor system. Finally, the commensurate and non-commensurate fractional-order systems based on the frequency domain method are implemented by FPGA technology, their accuracy can reach to 14 decimal places. The results of numerical calculation and circuit simulation have the same phase trajectory, which can verify that the hybrid memristors have practical values in the future.

Oscillators Based on Fractional-Order Memory Elements

This paper deals with the new oscillator structures that contain new elements, so-called memory elements, known as memristor, meminductor, and memcapacitor. Such circuits can exhibit oscillations as well as chaotic behavior. New mathematical models of fractional-order elements and whole oscillator circuits are proposed as well. An illustrative example to demonstrate the oscillations and the chaotic behavior through the numerical solution of the fractional-order circuit model is provided.

A New Adaptive Robust Sliding Mode Control Approach for Nonlinear Singular Fractional-Order Systems

This article focuses on designing an adaptive sliding mode controller via state and output feedback for nonlinear singular fractional-order systems (SFOSs) with mismatched uncertainties. Firstly, on the basis of extending the dimension of the SFOS, a new integral sliding mode surface is constructed. Through this special sliding surface, the sliding mode of the descriptor system does not contain a singular matrix E. Then, the sufficient conditions that ensure the stability of sliding mode motion are given by using linear matrix inequality. Finally, the control law based on an adaptive mechanism that is used to update the nonlinear terms is designed to ensure the SFOS satisfies the reaching condition. The applicability of the proposed method is illustrated by a practical example of a fractional-order circuit system and two numerical examples.

Plant stem tissue modeling and parameter identification using metaheuristic optimization algorithms

Bio-impedance non-invasive measurement techniques usage is rapidly increasing in the agriculture industry. These measured impedance variations reflect tacit biochemical and biophysical changes of living and non-living tissues. Bio-impedance circuit modeling is an effective solution used in biology and medicine to fit the measured impedance. This paper proposes two new fractional-order bio-impedance plant stem models. These new models are compared with three commonly used bio-impedance fractional-order circuit models in plant modeling (Cole, Double Cole, and Fractional-order Double-shell). The two proposed models represent the characterization of the biological cellular morphology of the plant stem. Experiments are conducted on two samples of three different medical plant species from the family Lamiaceae, and each sample is measured at two inter-electrode spacing distances. Bio-impedance measurements are done using an electrochemical station (SP150) in the range of 100 Hz to 100 kHz. All employed models are compared by fitting the measured data to verify the efficiency of the proposed models in modeling the plant stem tissue. The proposed models give the best results in all inter-electrode spacing distances. Four different metaheuristic optimization algorithms are used in the fitting process to extract all models parameter and find the best optimization algorithm in the bio-impedance problems.

Dynamical Analysis of a Fractional-Order Boost Converter with Fractional-Order Memristive Load

In this paper, a boost converter emulator with a memristive load instead of a resistive load is proposed, and a fractional-order model of the memristive boost converter is created. Firstly, the fractional-order models of the memristor and the memristive boost converter are established respectively, and then based on different switching states, the circuit equations of the fractional-order memristive boost converter operating in continuous conduction mode are derived. Secondly, the dynamic characteristics of the fractional-order system are analyzed by numerical simulations and are compared with the integer-order memristive boost converter. Furthermore, the effects of the memristive load on the system is studied by comparing it to the traditional boost converter with resistive load. The results indicate that the fractional-order memristive boost converter can exhibit rich dynamic behavior by adjusting bifurcation parameters, and the fractional-order circuit system expands the stable working regions of the integer-order circuit system. Meanwhile, the addition of the memristive load significantly widens the normal working regions. Finally, simulations of the fractional-order system circuit based on PSIM further verify the correctness of the theoretical analysis.

Adomian Decomposition, Dynamic Analysis and Circuit Implementation of a 5D Fractional-Order Hyperchaotic System

In this paper, a class of fractional-order symmetric hyperchaotic systems is studied based on the Adomian decomposition method. Starting from the definition of Adomian, the nonlinear term of a fractional-order five-dimensional chaotic system is decomposed. At the same time, the dynamic behavior of a fractional-order hyperchaotic system is analyzed by using bifurcation diagrams, Lyapunov exponent spectrum, complexity and attractor phase diagrams. The simulation results show that with the decrease of fractional order q, the complexity of the hyperchaotic system increases. Finally, based on the fractional-order circuit design principle, a circuit diagram of the system is designed, and the circuit is simulated by Multisim. The results are consistent with the numerical simulation results, which show that the system can be realized, which provides a foundation for the engineering applications of fractional-order hyperchaotic systems.

An experimental analog circuit realization of Matsuda’s approximate fractional-order integral operators for industrial electronics

Analog circuit realization of fractional order (FO) elements is a significant step for the industrialization of FO control systems because of enabling a low-cost, electric circuit realization by means of standard industrial electronics components. This study demonstrates an effective operational amplifier-based analog circuit realization of approximate FO integral elements for industrial electronics. To this end, approximate transfer function models of FO integral elements, which are calculated by using Matsuda’s approximation method, are decomposed into the sum of low-pass filter forms according to the partial fraction expansion. Each partial fraction term is implemented by using low-pass filters and amplifier circuits, and these circuits are combined with a summing amplifier to compose the approximate FO integral circuits. Widely used low-cost industrial electronics components, which are LF347N opamps, resistor and capacitor components, are used to achieve a discrete, easy-to-build analog realization of the approximate FO integral elements. The performance of designed circuit is compared with performance of Krishna’s FO circuit design and performance improvements are shown. The study presents design, performance validation and experimental verification of this straightforward approximate FO integral realization method.

Structure Identification of Fractional-Order Dynamical Network with Different Orders

Topology structure and system parameters have a great influence on the dynamical behavior of dynamical networks. However, they are sometimes unknown or uncertain in advance. How to effectively identify them has been investigated in various network models, from integer-order networks to fractional-order networks with the same order. In the real world, many systems consist of subsystems with different fractional orders. Therefore, the structure identification of a dynamical network with different fractional orders is investigated in this paper. Through designing proper adaptive controllers and parameter updating laws, two network estimators are well constructed. One is for identifying only the unknown topology structure. The other is for identifying both the unknown topology structure and system parameters. Based on the Lyapunov function method and the stability theory of fractional-order dynamical systems, the theoretical results are analytically proved. The effectiveness is verified by three numerical examples as well. In addition, the designed estimators have a good performance in monitoring switching topology. From the practical viewpoint, the designed estimators can be used to monitor the change of current and voltage in the fractional-order circuit systems.