An Improved Realization Of The Chua’s Circuit Using RC-OP Amps

Cherif Aissi1 and Demetrios Kazakos2

1College of Engineering, University of Louisiana at Lafayette

Lafayette, LA 70504, USA

2College 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.

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].

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 parameter, different shapes of the double scroll are obtained.

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).

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:

(1)

Where the nonlinear Chua's function of the nonlinear resitor NR is shown in Fig. 1(b). It is described by

(2)

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.

Where, Esat is the saturation voltage of the op amp.

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).

Fig.2(a). Active inductor

The active inductance inductance L’ can be easily derived as:

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=220W, R3=2.2kW, R4=R5=22kW, R6=3.3kW, R7=10kW, R8=2.6MW, R9=0.68W C3=0.01mF 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 2.4MW.

Fig. 2(b). The improved Chua’s circuit realization

Fig. 2(c) EW simulation of the Vc1-Vc2 phase space trajectory, R8=2.6M W

Fig. 2(d) EW simulation of the Vc1-Vc2 phase space trajectory, R8=2.4MW

4. Experimental Results

The circuit of Fig.2(b) was constructed with the following parameters: C1=10nF, C2=100nF, R1= R2=220W, R3=2.2kW, R4=R5=22kW R6=3.3kW, R7=10kW, R8=519kW, R9=0.68W C3=0.01mF 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= 1.147kW. By increasing the variable resistor R to 1.495kW, 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. 3(b). Double Scroll attractor for R= 1.147kW

Fig.3(c). Single scroll attractor for R=1.495kW

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

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[16] “Electronics Workbench,” Interactive Image Technology Ltd., Ontario, Canada.

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