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An Improved Realization Of The Chua’s Circuit Using RC-OP Amps
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Cherif Aissi1 and Demetrios Kazakos2
College of Engineering, University of Louisiana at Lafayette
Lafayette, LA 70504, USA
2
College of Engineering, University of Toledo
Toledo, OH 43606
Abstract: - In this paper, an improved implementation of the Chua’s circuit is proposed. The new realization
consists of only RC elements and op amps. Both computer simulations and laboratory measurements have
confirmed the chaotic behavior and show the existence of the double scroll attractor.
Key-Words: -Chua’s circuit, chaos, attractor, bifurcation, oscillator.
parameter, different shapes of the double scroll are
obtained.
1. Introduction
Chua’s circuit is a simple autonomous third-order
nonlinear electrical circuit that exhibits a variety of
dynamic behaviors including chaos, which has been
experimentally confirmed [1-6]. This oscillator
consists of a passive RLC1C2 block coupled to an
active nonlinear resistor (Chua’s diode). This
nonlinear resistor is implemented using six resistors
and two conventional op amps (VOAa) operating in
both their linear and nonlinear regions of operations
[1]. The improvement of the implementation of the
Chua’s diode, using current feedback op amp, has
been investigated [7]. Several studies to design
similar chaotic circuits have been reported [8-15].
2. Background
2. 1. The Chua’s circuit
The Chua's circuit (a third-order autonomous,
dissipative electrical circuit) has been investigated
thoroughly at the experimental, numerical and
analytical levels. This circuit, known for its rich
repertoire of nonlinear dynamical phenomena has
become a universal paradigm for chaos. Fig. 1(a)
shows the Chua's circuit that includes two
capacitors, a resistor, an inductor and a nonlinear
resistor NR (a pair of negative resistors).
In this work an improvement of the Chua’s circuit is
proposed. The new realization consists of only RC
elements and op amps. Since no inductor is used,
this realization can be easily implemented in a chip.
An active inductor, consisting of three resistors, one
capacitor and an op amp, replaces the passive
inductor.
In the following sections, some background
concepts are summarized; the circuit design and
implementation is reported. Finally, simulation
results using Electronics Workbench [16] are
shown. Experimental results confirm the existence
of the double scroll attractor. By varying the
Fig 1. (a) The unfolded Chua’s circuit
v1, v2 and i3 are the voltages across capacitor C1 and
C2, and the current through the inductor L,
respectively. Applying KCL and KVL, the Chua's
circuit is described by three differential equations:
dv1
dt
dv 2
C2
dt
di3
L
dt
C1



1
(v1  v 2 )  f (v1 )
R
1
(v1  v 2 )  i3
R
Where, Esat is the saturation voltage of the op amp.
(1)
 v2
Where the nonlinear Chua's function of the
nonlinear resitor NR is shown in Fig. 1(b). It is
described by
f (v R )  m o v R 

1
(m1  mo ) v R  B p  v R  B p
2
(2)

Fig.1(c). The realization of the Chua’s circuit [1]
3. Realization and simulation
The passive inductor L in Fig.1(b) is replaced by an
active inductor as shown in Fig.2(a).
f(vR)
mo
m1
-Bp1
-Bp2
Bp2
Bp1
vR
mo
Fig.2(a). Active inductor
The active inductance inductance L’ can be easily
derived as:
Fig.1(b) Chua’s nonlinear function
The realization of the Chua’s circuit is shown in Fig
1.(b). The constant mo , m1, and Bp can be easily
computed.
R
R
R
1 

 m1   2  5 , mo   2  , 
R1 R3 R4 R6
R1 R3 R4 



R3
R6
 B p1 

E sat , B p2 
R2  R3
R5  R6


L'  R8 R9 C3
The improved realization shown in Fig.2(b) consists
of only RC elements and op amps. Using Electronics
Workbench (EW) simulator, a double scroll attractor
is shown in Fig. 2(c) with C1=10nF, C2=100nF,
R1=R2=220,
R3=2.2k,
R4=R5=22k,
R6=3.3k, R7=10k, R8=2.6M, R9=0.68
C3=0.01F and using op amp TL082 for the Chua’s
diode and op amp TL074AC for the active diode.
Fig 2.(d) is obtained by varying R8 to 2.4M.
2
4. Experimental Results
Fig. 2(b). The improved Chua’s circuit realization
Fig. 2(c) EW simulation of the Vc1-Vc2 phase space
trajectory, R8=2.6M 
The circuit of Fig.2(b) was constructed with the
following parameters: C1=10nF, C2=100nF, R1=
R2=220, R3=2.2k, R4=R5=22k R6=3.3k,
R7=10k, R8=519k, R9=0.68 C3=0.01F and
using op amp TL082. Fig. 3(a) displays the V-I
characteristic of the Chua diode NR. Fig.3(b). shows
the existence of the double scroll obtained for R=
1.147k. By increasing the variable resistor R to
1.495k, a different shape of the double scroll
attractor was obtained as shown in Fig.3(c).
Fig 3(a). Measured V-I characteristic of the Chua
diode with R5 removed
Fig. 2(d) EW simulation of the Vc1-Vc2 phase space
trajectory, R8=2.4M
Fig. 3(b). Double Scroll attractor for R= 1.147k
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[6]
P. Bartissol and L.O. Chua, “The double
hook,” IEEE Transaction on Circuits and
Systems I, Vol.35, no.7, pp.1512-1522.,
December 1998.
[7]
A.S. Elwakil and M.P. Kennedy, “Improved
implementation of Chua’s chaotic oscillator
using current feedback op amp,” IEEE
Transaction on Circuits and Systems, Vol.47,
no.1, pp.76-75.,January 2000.
[8]
A. S. Elwakil, S. Ö. Guz, and M. P. Kennedy,
“Creation of a complex butterfly attractor
using a novel Lorenz-type system,” IEEE
Trans. Circuits Syst. I, vol. 49, pp. 527–530,
Apr. 2002.
[9]
J. Lü, G. Chen, and S. Zhang, “Dynamical
analysis of a new chaotic attractor,” Int. J.
Bifurcation Chaos, vol. 12, no. 5, pp. 1001–
1015, 2002.
Fig.3(c). Single scroll attractor for R=1.495k
6. Conclusions
An improved implementation of the Chua’s circuit
has been shown. Since this circuit contains only RC
elements and Op amps, it can easily be implemented
in a chip. The functionality of the circuit displaying
the double attractor was demonstrated. Other shapes
of the attractor were obtained by varying the resistor
R. Both computer simulations and laboratory
measurements have confirmed the chaotic behavior
and show the existence of the double scroll attractor.
References
[1]
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