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Research Article
 

OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators



J.M. Munoz-Pacheco, W. Campos-Lopez, E. Tlelo-Cuautle and C. Sanchez-Lopez
 
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ABSTRACT

Continuous time chaotic oscillators have been implemented by using several commercially available electronic devices. In this study, the generic circuit topologies based on Operational Amplifier (OpAmp), Current-Feedback Operational Amplifier (CFOA) and Operational Transconductance Amplifier (OTA), are summarized. These topologies allows an electronic designer to realize chaotic oscillators modeled with piecewise-linear functions, as it is shown herein by designing saturated function series.

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  How to cite this article:

J.M. Munoz-Pacheco, W. Campos-Lopez, E. Tlelo-Cuautle and C. Sanchez-Lopez, 2012. OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators. Trends in Applied Sciences Research, 7: 168-174.

DOI: 10.3923/tasr.2012.168.174

URL: https://scialert.net/abstract/?doi=tasr.2012.168.174
 
Received: October 19, 2011; Accepted: November 15, 2011; Published: February 16, 2012



INTRODUCTION

Nowadays, a wide number of publications dealing with chaotic oscillators and their electronic realizations have been introduced by Carbajal-Gomez et al. (2011) Elabbasy and El-Dessoky (2008), Fatehi Marj et al. (2009), Gonzales et al. (2000), Munoz-Pacheco and Cuautle (2009, 2010), Sanchez-Lopez et al. (2008, 2010, 2011), Tlelo-Cuautle (2011a, b) and Trejo-Guerra et al. (2009, 2010a-c, 2011). Among the active devices used in their implementation one can found operational amplifiers (OpAmps) (Munoz-Pacheco and Tlelo-Cuautle, 2009, 2010), unity-gain cells (Duarte-Villasenor et al., 2011; Sanchez-Lopez et al., 2008), current conveyors (Sanchez-Lopez et al., 2010; Ahmed and Soliman, 2011; Tlelo-Cuautle et al., 2010b; Trejo-Guerra et al., 2009), Current-Feedback Operational Amplifiers (CFOAs) (Carbajal-Gomez et al., 2011; Trejo-Guerra et al., 2010c) and Operational Transconductance Amplifiers (OTAs) (Garcia-Ortega et al., 2007; Gonzales et al., 2000). All these active devices can enhance the performances of the chaotic oscillators, when they are designed at the transistor level of abstraction, e.g., using metal-oxide-semiconductor field-effect-transistors (MOSFETs) (Duarte-Villasenor et al., 2011; Ibrahim et al., 2011; Rashtian et al., 2008; Riyadi et al., 2010; Tlelo-Cuautle et al., 2010a; Trejo-Guerra et al., 2011, 2010a). Unfortunately, very few information on the generic topologies being used in the realization of chaotic oscillators can be found in the literature. That way, this article summarizes the OpAmp-, CFOA- and OTA-based generic topologies used in the implementation of chaotic oscillators modeled by Piecewise-linear (PWL) functions. Some related works based on saturated function series can be found by Carbajal-Gomez et al. (2011), Munoz-Pacheco and Tlelo-Cuautle (2009, 2010); Sanchez-Lopez et al. (2010, 2011), Tlelo-Cuautle (2011a, b) and Trejo-Guerra et al. (2010b, c).

OPAMP-, CFOA- AND OTA-BASED GENERIC TOPOLOGIES

The OpAmp is a two-port device whose ideal behavior is described by:

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
(1)

where, Av is the voltage-gain and vin+ and vin- are the noninverting and inverting inputs. The CFOA has four terminals X, Y, Z and W (Duarte-Villasenor et al., 2011; Tlelo-Cuautle et al., 2010b). Y is an input port driving voltage, X is a bidirectional port sensing voltage from Y to X (vx = vy) and injecting current from X to Z (iz = ix). Z is a bidirectional port as X but sensing current from X to Z and injecting voltage from Z to W (vw = vz). W is an output port measuring voltage from Z. The OTA processes voltage to current. The transfer characteristic is denoted by the transconductance gm (Garcia-Ortega et al., 2007).

In Table 1-4, we summarize the generic topologies for realizing linear operations.

The saturated function series can be modeled by PWL functions (Munoz-Pacheco and Tlelo-Cuautle, 2009, 2010). For instance, Eq. 1 is a PWL approximation of a saturated function serie, as already shown by Carbajal-Gomez et al. (2011), Munoz-Pacheco and Tlelo-Cuautle (2009, 2010) and Trejo-Guerra et al. (2010b, c):

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
(2)

Using the finite gain model of the OpAmp, as shown in Fig. 1 (Munoz-Pacheco and Tlelo-Cuautle, 2010), Eq. 2 can be implemented using electronic devices. The description of Fig. 1 is given by Eq. 3. Equation 4 describes a negative shift operation, requiered to generate the saturated functions.

Table 1: OPAMP-, CFOA- and OTA-based inverter
Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators

Table 2: OPAMP-, CFOA- and OTA-based integrator
Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators

Table 3: OPAMP-, CFOA- and OTA-based adder
Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators

Table 4: OPAMP-, CFOA- and OTA-based subtractor
Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
Basic topologies to implement saturated function series

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
Fig. 1: Finite gain behavior of the OpAmp

Some circuit realizations of PWL functions are already given by Carbajal-Gomez et al. (2011), Munoz-Pacheco and Tlelo-Cuautle (2010), Sanchez-Lopez et al. (2010), Tlelo-Cuautle (2011a, b) and Trejo-Guerra et al. (2010c):

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
(3)

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
(4)

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
Fig. 2(a-b): (a) OpAmp basic cell and (b) Transforming voltage to current through Rc

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
Fig. 3(a-b): (a) CFOA basic cell and (b) Transforming voltage to current through Rc

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
Fig. 4(a-b): (a) OTA basic cell and (b) Transforming voltage to voltage through g2

The basic cell to implement the saturated function series using OpAmps is shown in Fig. 2, where, Ei indicates positive or negative shift described by E in Eq. 3 (Munoz-Pacheco and Tlelo-Cuautle, 2010). The basic cell to implement the saturated function series using CFOA and OTA are shown in Fig. 3 and 4, respectively. For the last case, the basic cell in Fig. 4(a), i(x) denotes the output current Io which is described by:

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
(5)

SCROLLS CHAOTIC OSCILLATOR

Here, we just review the realization of multi-scroll chaotic oscillators from (Tlelo-Cuautle, 2011a, b). Lets us consider the dynamical system described by the state equations (Chattopadhyay et al., 2011; Munoz-Pacheco and Tlelo-Cuautle, 2009, 2010), given by Eq. 5, where, f (x: k, h, p, q) is defined by Eq. 1 and x, y and z are the state variables, with a = b = c = d = real positive constants.

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
Fig. 5: Multi-scroll chaotic oscillator implemented with CFOAs

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
Fig. 6: 6-scrolls attractor from Fig. 5

Image for - OpAmp-, CFOA- and OTA-Based Configurations to Design Multi-Scroll Chaotic Oscillators
(6)

The CFOA-based realization is shown in Fig. 5, where the PWL function named SNLF is implemented using the basic cell shown in Fig. 3. The experimental result is shown in Fig. 6. This chaotic oscillator can be used to implement secure communication systems as the ones designed and shown by Carbajal-Gomez et al. (2011), Gonzales et al. (2000), Munoz-Pacheco and Tlelo-Cuautle (2010), Tlelo-Cuautle (2011a, b) and Trejo-Guerra et al. (2009).

CONCLUSION

This study was devoted to show the generic circuit topologies based on OpAmps, CFOAs and OTAs and used in the design of multi-scroll chaos generators. The realization of saturated functions series was also described by using the three active devices, from a PWL function approach. As a result, an electronic designer has at his disposal three kinds of circuit topology realizations of multi-scroll chaotic oscillators.

ACKNOWLEDGMENTS

This study is partially supported by UATLXPTC-088 and UPPUE-PTC-033 funded by PROMEP-México, 131839Y funded by CONACyT and by project TIC-2532 funded by Consejería de Innovación, Ciencia y Empresa, Junta de Andalucía, Spain.

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