Ahmed S. Elwakil

Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices

Two generic classes of chaotic oscillators comprising four different configurations are constructed. The proposed structures are based on the simplest possible abstract models of generic second-order RC sinusoidal oscillators that satisfy the basic condition for oscillation and the frequency of oscillation formulas. By linking these sinusoidal oscillator engines to simple passive first-order or second-order nonlinear composites, chaos is generated and the evolution of the two-dimensional sinusoidal oscillator dynamics into a higher dimensional state space is clearly recognized. We further discuss three architectures into which autonomous chaotic oscillators can be decomposed. Based on one of these architectures we classify a large number of the available chaotic oscillators and propose a novel reconstruction of the classical Chuas circuit. The well-known Lorenz system of equations is also studied and a simplified model with equivalent dynamics, but containing no multipliers, is introduced.

Fractional-Order Sinusoidal Oscillators: Design Procedure and Practical Examples

Sinusoidal oscillators are known to be realized using dynamical systems of second-order or higher. Here we derive the Barhkausen condition for a linear noninteger-order (fractional-order) dynamical system to oscillate. We show that the oscillation condition and oscillation frequency of some famous integer-order sinusoidal oscillators can be obtained as special cases from general equations governing their fractional-order counterparts. Examples including fractional-order Wien oscillators, Colpitts oscillator, phase-shift oscillator and LC tank resonator are given supported by numerical and PSpice simulations.

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.

ON THE GENERALIZATION OF SECOND-ORDER FILTERS TO THE FRACTIONAL-ORDER DOMAIN

This work is aimed at generalizing the design of continuous-time second-order filters to the non-integer-order (fractional-order) domain. In particular, we consider here the case where a filter is constructed using two fractional-order capacitors both of the same order α. A fractional-order capacitor is one whose impedance is Zc = 1/C(jω)α, C is the capacitance and α (0 < α ≤ 1) is its order. We generalize the design equations for low-pass, high-pass, band-pass, all-pass and notch filters with stability constraints considered. Several practical active filter design examples are then illustrated supported with numerical and PSpice simulations. Further, we show for the first time experimental results using the fractional capacitive probe described in Ref. 1.

Improved implementation of Chua's chaotic oscillator using current feedback op amp

An improved implementation of Chuas chaotic oscillator is proposed. The new realization combines attractive features of the current feedback op amp (CFOA) operating in both voltage and current modes to construct the active three-segment voltage-controlled nonlinear resistor. Several enhancements are achieved: the component count is reduced and the chaotic spectrum is extended to higher frequencies. In addition, a buffered and isolated voltage output directly representing a state variable is made available. Based on a linearized model of Chuas circuit, the useful tuning range of the major bifurcation parameter (G) and the expected frequency of oscillation, are estimated.

Creation of a complex butterfly attractor using a novel Lorenz-Type system

A novel Lorenz-type system of nonlinear differential equations is proposed. Unlike the original Lorenz system, where the chaotic dynamics remain confined to the positive half-space with respect to the Z state variable due to a limiting threshold effect, the proposed system enables bipolar swing of this state variable. In addition, the classical set of parameters (a, b, c) controlling the behavior of the Lorenz system are reduced to a single parameter, namely a. Two possible modes of operation are admitted by the system; switching between these two modes results in the creation of a complex butterfly chaotic attractor. Numerical simulations and results from an experimental setup are presented.

Field programmable analogue array implementation of fractional step filters

In this study, the authors propose the use of field programmable analogue array hardware to implement an approximated fractional step transfer function of order (n+α) where n is an integer and 0 < α < 1. The authors show how these filters can be designed using an integer order transfer function approximation of the fractional order Laplacian operator sα. First and fourth-order low- and high-pass filters with fractional steps from 0.1 to 0.9, that is of order 1.1–1.9 and 4.1–4.9, respectively, are given as examples. MATLAB simulations and experimental results of the filters verify the implementation and operation of the fractional step filters.

Measurement of Supercapacitor Fractional-Order Model Parameters From Voltage-Excited Step Response

In this paper, we propose using a numerically solved least squares fitting process to estimate the impedance parameters of a fractional order model of supercapacitors from their voltage excited step response, without requiring direct measurement of the impedance or frequency response. Experimentally estimated parameters from low capacity supercapacitors of 0.33, 1, and 1.5 F in the time range 0.2-30 s and high capacity supercapacitors of 1500 and 3000 F in the time range 0.2-90 s verify the proposed time domain method showing less than 3% relative error between the simulated response (using the extracted fractional parameters) and the experimental step response in these time ranges. An application of employing supercapacitors in a multivibrator circuit is presented to highlight their fractional time-domain behavior.

On the practical realization of higher-order filters with fractional stepping

We propose the use of a compact integer-order transfer function approximation of the fractional-order Laplacian operator s^@a to realize fractional-step filters. Lowpass and bandpass filters of orders (n+@a) and 2(n+@a), where n is an integer and 0 5.9) is given as an example with its characteristics compared to 5th- and 6th-order Butterworth filters. Spice simulations and experimental results are shown.

A family of Wien‐type oscillators modified for chaos

SUMMARY A family consisting of four Wien-type oscillator circuits are modified for chaos by direct replacement of one of the linear resistors with an asymmetrical-type non-linearity introduced by a junction field e⁄ect transistor (JFET) operating in its triode region and the addition of a single capacitor. The internal op amp dominant pole is found to play a major role in understanding the chaotic behaviour of the proposed circuits. Mathematical models that describe the observed behaviours are derived. The well known Wien bridge oscillator design equations are shown to be useful as a starting point for chaos modification. Experimental laboratory results, PSpice simulations and numerical simulations of the mathematical models are provided for this family of autonomous RC chaos generators. ( 1997 John Wiley & Sons, Ltd.