Amr M. AbdelAty

On the Analysis and Design of Fractional-Order Chebyshev Complex Filter

This paper introduces the concept of fractional-order complex Chebyshev filter. A fractional variation of Chebyshev differential equations is introduced based on Caputo fractional operator. The proposed equation is solved using fractional Taylor power series method. The condition for fractional polynomial solutions is obtained and the first four polynomials scaled using an appropriate scaling factor. The fractional-order complex Chebyshev low-pass filter based on the obtained fractional polynomials is developed. Two methods for obtaining the transfer functions of the complex filter are discussed. Circuit implementations are simulated using Advanced Design System (ADS) and compared with MATLAB numerical simulation of the obtained transfer functions to prove the validity of the two approaches.

A fractional-order dynamic PV model

A dynamic model of Photo-Voltaic (PV) solar module is important when it is utilized in conjunction with switching circuits and in grid connected applications. In this paper, a fractional-order dynamical model of a PV source is introduced. The model includes both a fractional series inductor and a parallel capacitor which are in general of two different orders allowing for extra degrees of modeling freedom. An expression for the load current is derived and the step response is investigated for different orders. It is found that the nature of the connections has a dominant effect on the response in comparison with the nature of the PV itself. The abstract goes here. The length of the abstract should not exceed 150 words.

Low pass filter design based on fractional power chebyshev polynomial

This paper introduces the design procedure for the low pass filter based on Chebyschev polynomials of fractional power of any order. The filter order is considered in intervals of width two. Only the first two intervals are considered along with their pole locus produced by varying the filter order and the magnitude response. A general formula for constructing the filter from its s-plane poles is suggested. Numerical analysis and circuit simulations using MATLAB and Advanced Design System (ADS) based on the proposed design procedure are presented. Good matching between the circuit simulation and the numerical analysis is obtained which proves the reliability of the proposed design procedure.

Transient and Steady-State Response of a Fractional-Order Dynamic PV Model Under Different Loads

In this paper, a fractional-order dynamic model of the photovoltaic (PV) solar module is introduced. Dynamic modeling of PV solar modules is useful when used in switching circuits and grid-connected situations. The dynamic elements of the proposed model are a fractional-order inductor and capacitor of two independent orders which allow for two extra degrees of freedom over the conventional dynamic model. The step response and transfer function of the load current are investigated for different orders under resistive and supercapacitor loading conditions. Closed-form expressions for the time response of the load current at equal orders of capacitor and inductor are derived. Stability analysis of the load current transfer function is carried out for different orders and loading conditions. The regions for pure real and pure imaginary input admittance scenarios are calculated numerically for both resistive and supercapacitor load cases. It is found that the order of the inductor has a dominant effect on the responses. As a proof of concept, the model is fitted to experimental data to show its flexibility in regenerating the actual response. The fitted fractional-order model response is compared to optimized integer-order ones from literature showing noticeable improvement.

Modeling and analysis of fractional order DC-DC converter

Due to the non-idealities of commercial inductors, the demand for a better model that accurately describe their dynamic response is elevated. So, the fractional order models of Buck, Boost and Buck-Boost DC-DC converters are presented in this paper. The detailed analysis is made for the two most common modes of converter operation: Continuous Conduction Mode (CCM) and Discontinuous Conduction Mode (DCM). Closed form time domain expressions are derived for inductor currents, voltage gain, average current, conduction time and power efficiency where the effect of the fractional order inductor is found to be strongly present. For example, the peak inductor current at steady state increases with decreasing the inductor order. Advanced Design Systems (ADS) circuit simulations are used to verify the derived formulas, where the fractional order inductor is simulated using Valsa Constant Phase Element (CPE) approximation and Generalized Impedance Converter (GIC). Different simulation results are introduced with good matching to the theoretical formulas for the three DC-DC converter topologies under different fractional orders. A comprehensive comparison with the recently published literature is presented to show the advantages and disadvantages of each approach.

Applications of Continuous-time Fractional Order Chaotic Systems

Abstract The study of nonlinear systems and chaos is of great importance to science and engineering mainly because real systems are inherently nonlinear and linearization is only valid near the operating point. The interest in chaos was increased when Lorenz accidentally discovered the sensitivity to initial condition during his simulation work on weather prediction. When a nonlinear system is exhibiting deterministic chaos, it is very difficult to predict its response under external disturbances. This behavior is a double-edged weapon. From a control and synchronization point of view, this proposes a challenge. On the other hand, from a communications and encryption perspective, this provides a higher level of security. This chapter is a survey of the recent contributions in engineering applications of fractional order chaotic continuous-time systems. The applications include but not limited to: communication and encryption, FPGA implementations, synchronization and control, modeling of electric motors, and biomedical applications.

On the Approximation of Fractional-Order Circuit Design

Abstract Despite the complex nature of fractional calculus, it is still fairly possible to reduce this complexity by using integer-order approximation. Each integer-order approximation has its own trade-offs from the complexity, sensitivity, and accuracy points of view. In this chapter, two different fractional-order electronic circuits are studied: the Wien oscillator and the CCII-based KHN filter with two different fractional elements of orders α and β. The investigation is concerned with changes in the response of these two circuits under two approximations: Oustaloup and Matsuda. A detailed review of each approximation technique is provided as well as its design procedure. Oscillator and filter responses are simulated using MATLAB. Foster-I realization is used to implement the approximated Wien oscillator and filter transfer functions as circuits in order to simulate them in PSpice. The responses are compared to the exact solution to investigate which achieves the lowest error. For oscillators, the comparison is based on oscillation condition and oscillation frequency while for filters, the focus is on filter fundamental frequencies. This is a big issue in filter design: maximum or minimum frequency, right phase frequency, and half-power frequency.

Effect of Different Approximation Techniques on Fractional-Order KHN Filter Design

Having an approximate realization of the fractance device is an essential part of fractional-order filter design and implementation. This encouraged researchers to introduce many approximation techniques of fractional-order elements. In this paper, the fractional-order KHN low-pass and high-pass filters are investigated based on four different approximation techniques: Continued Fraction Expansion, Matsuda, Oustaloup, and Valsa. Fractional-order filter fundamentals are reviewed then a comparison is made between the ideal and actual characteristic of the filter realized with each approximation. Moreover, stability analysis and pole movement of the filter with respect to the transfer function parameters using the exact and approximated realizations are also investigated. Different MATLAB numerical simulations, as well as SPICE circuit results, have been introduced to validate the theoretical discussions. Also, to discuss the sensitivity of the responses to component tolerances, Monte Carlo simulations are carried out and the worst cases are summarized which show good immunity to component deviations. Finally, the KHN filter is tested experimentally.

FPGA realization of Caputo and Grünwald-Letnikov operators

This paper proposes a hardware platform implementation on FPGA for two fractional-order derivative operators. The Grünwald-Letnikov and Caputo definitions are realized for different fractional orders. The realization is based on non-uniform segmentation algorithm with a variable lookup table. A generic implementation for Grünwald-Letnikov is proposed and a 32 bit Fixed Point Booth multiplier radix-4 is used for Caputo implementation. Carry look-ahead adder, multi-operand adder and booth multiplier are used to improve the performance and other techniques for area and delay minimization have been employed. A comparison between the two presented architectures is introduced. The proposed designs have been simulated using Xilinx ISE and realized on FPGA Xilinx virtex-5 XC5VLX50T. The total area of 2515 look up tables is achieved for Caputo implementation, and maximum frequency of 54.11 MHz and 1498 slices are achieved for Grünwald-Letnikov architecture.

On the analysis of current-controlled fractional-order memristor emulator

In this paper, a current-controlled fractional-order memristor model and its emulator are proposed. The emulator is built using two second generation current conveyor (CCII) and fractional-order capacitor. It is shown that the effect of the fractional order is clearly noticeable in the circuit response. PSPICE simulations are introduced for different values of the fractional order showing noticeable variations of the pinched-loop hysteresis curves. The fractional order model shows wider frequency of operation and larger pinched loop hysteresis area than the integer one.