
Research Article


OpAmp, CFOA and OTABased Configurations to Design MultiScroll Chaotic Oscillators 

J.M. MunozPacheco,
W. CamposLopez,
E. TleloCuautle
and
C. SanchezLopez



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), CurrentFeedback Operational Amplifier (CFOA) and Operational Transconductance Amplifier (OTA), are summarized. These topologies allows an electronic designer to realize chaotic oscillators modeled with piecewiselinear functions, as it is shown herein by designing saturated function series.





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 CarbajalGomez
et al. (2011) Elabbasy and ElDessoky (2008),
Fatehi Marj et al. (2009), Gonzales
et al. (2000), MunozPacheco and Cuautle (2009,
2010), SanchezLopez et al.
(2008, 2010, 2011), TleloCuautle
(2011a, b) and TrejoGuerra
et al. (2009, 2010ac,
2011). Among the active devices used in their implementation
one can found operational amplifiers (OpAmps) (MunozPacheco
and TleloCuautle, 2009, 2010), unitygain cells (DuarteVillasenor
et al., 2011; SanchezLopez et al.,
2008), current conveyors (SanchezLopez et al.,
2010; Ahmed and Soliman, 2011; TleloCuautle
et al., 2010b; TrejoGuerra et al.,
2009), CurrentFeedback Operational Amplifiers (CFOAs) (CarbajalGomez
et al., 2011; TrejoGuerra et al., 2010c)
and Operational Transconductance Amplifiers (OTAs) (GarciaOrtega
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 metaloxidesemiconductor
fieldeffecttransistors (MOSFETs) (DuarteVillasenor et
al., 2011; Ibrahim et al., 2011; Rashtian
et al., 2008; Riyadi et al., 2010;
TleloCuautle et al., 2010a; TrejoGuerra
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 OTAbased generic topologies used in the implementation of chaotic oscillators
modeled by Piecewiselinear (PWL) functions. Some related works based on saturated
function series can be found by CarbajalGomez et al.
(2011), MunozPacheco and TleloCuautle (2009, 2010);
SanchezLopez et al. (2010, 2011),
TleloCuautle (2011a, b) and TrejoGuerra
et al. (2010b, c).
OPAMP, CFOA AND OTABASED GENERIC TOPOLOGIES The OpAmp is a twoport device whose ideal behavior is described by:
where, A_{v} is the voltagegain and v_{in+} and v_{in
}are the noninverting and inverting inputs. The CFOA has four terminals
X, Y, Z and W (DuarteVillasenor et al., 2011;
TleloCuautle et al., 2010b). Y is an input port
driving voltage, X is a bidirectional port sensing voltage from Y to X (v_{x}
= v_{y}) and injecting current from X to Z (i_{z} = i_{x}).
Z is a bidirectional port as X but sensing current from X to Z and injecting
voltage from Z to W (v_{w} = v_{z}). W is an output port measuring
voltage from Z. The OTA processes voltage to current. The transfer characteristic
is denoted by the transconductance g_{m} (GarciaOrtega
et al., 2007).
In Table 14, we summarize the generic
topologies for realizing linear operations.
The saturated function series can be modeled by PWL functions (MunozPacheco
and TleloCuautle, 2009, 2010). For instance, Eq.
1 is a PWL approximation of a saturated function serie, as already shown
by CarbajalGomez et al. (2011), MunozPacheco
and TleloCuautle (2009, 2010) and TrejoGuerra
et al. (2010b, c):
Using the finite gain model of the OpAmp, as shown in Fig. 1
(MunozPacheco and TleloCuautle, 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 OTAbased inverter 

Table 2: 
OPAMP, CFOA and OTAbased integrator 

Table 3: 
OPAMP, CFOA and OTAbased adder 

Table 4: 
OPAMP, CFOA and OTAbased subtractor 

Basic topologies to implement saturated function series 

Fig. 1: 
Finite gain behavior of the OpAmp 
Some circuit realizations of PWL functions are already given by CarbajalGomez
et al. (2011), MunozPacheco and TleloCuautle
(2010), SanchezLopez et al. (2010), TleloCuautle
(2011a, b) and TrejoGuerra et
al. (2010c):

Fig. 2(ab): 
(a) OpAmp basic cell and (b) Transforming voltage to current
through R_{c} 

Fig. 3(ab): 
(a) CFOA basic cell and (b) Transforming voltage to current
through R_{c} 

Fig. 4(ab): 
(a) OTA basic cell and (b) Transforming voltage to voltage
through g_{2} 
The basic cell to implement the saturated function series using OpAmps is shown
in Fig. 2, where, E_{i} indicates positive or negative
shift described by E in Eq. 3 (MunozPacheco
and TleloCuautle, 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 I_{o} which is described by:
SCROLLS CHAOTIC OSCILLATOR
Here, we just review the realization of multiscroll chaotic oscillators from
(TleloCuautle, 2011a, b). Lets
us consider the dynamical system described by the state equations (Chattopadhyay
et al., 2011; MunozPacheco and TleloCuautle,
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.

Fig. 5: 
Multiscroll chaotic oscillator implemented with CFOAs 

Fig. 6: 
6scrolls attractor from Fig. 5 
The CFOAbased 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 CarbajalGomez et al. (2011),
Gonzales et al. (2000), MunozPacheco
and TleloCuautle (2010), TleloCuautle (2011a, b)
and TrejoGuerra 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 multiscroll 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 multiscroll
chaotic oscillators.
ACKNOWLEDGMENTS This study is partially supported by UATLXPTC088 and UPPUEPTC033 funded by PROMEPMéxico, 131839Y funded by CONACyT and by project TIC2532 funded by Consejería de Innovación, Ciencia y Empresa, Junta de Andalucía, Spain.

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