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An Improved Realization Of The Chua’s Circuit Using RC-OP Amps 1 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=0.01F 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=0.01F 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 3 [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] M.P. Kennedy, “Robust op amp realization of Chua’s circuit,” FREQUENZ, vol.46, pp.6680, Apr. 1992. [2] M.P. Kennedy, ``Three Steps to chaos-Part I: Evolution,''IEEE Transaction on Circuits and Systems I, Vol.40, no.10, pp.640-656, October 1993. [3] [4] [5] M.P. Kennedy, ``Three Steps to chaos-Part II: A Chua's Circuit Primer,'' IEEE Transaction on Circuits and Systems I, Vol.40, no.10, pp.657-674, October 1993. L.O. Chua and G.N. Lin, “Canonical realization of Chua's circuit family,” IEEE Transaction on Circuits and Systems I, Vol.37, no.7, pp.885-902., 1990. R. Madan, Chua’s Circuit: A Paradigm for Chaos. Singapore: World Scientific, 1993. [10] J. Lü, T. Zhou, G. Chen, and X. Yang, “Generating chaos with a switching piecewise-linear controller,” Chaos, vol. 12, no. 2, pp. 344–349, 2002. [11] M.E. Yalcin, J.A.K. Suykens, and J. Vandewalle, “Experimental confirmation of 3- and 5-scroll attractors from a generalized Chua's circuit”, IEEE Trans. Circuits Syst. I, vol. 47, pp. 425–429, March 2000. [12] C. W. Wu and L. O. Chua, “On the generality of the unfolded Chua's circuit,” in Int. J. Bifurcation Chaos, vol. 6, pp. 801–832, 1996. [13] A. S. Elwakil and M. P. Kennedy, “Construction of classes of circuitindependent chaotic oscillators using passiveonly nonlinear devices,” IEEE Trans. Circuits Syst. I, vol. 48, pp. 289–307, Mar. 2001 [14] J.E. Varrientos, and E. Sanchez-Sinencio, “A 4-D chaotic oscillator based on a differential hysteresis comparator,” IEEE Trans. Circuits Syst. I, vol. 45 Issue: 1 , pp. 3 –10, Jan. 1998. [15] U.F. Moreno, P.L.D. Peres, and I.S. Bonatti, “Analysis of piecewise-linear oscillators with hysteresis,” IEEE Trans. Circuits Syst. I, vol.50 Issue: 8 , pp. 1120 –1124, Aug. 2003. [16] “Electronics Workbench,” Interactive Image Technology Ltd., Ontario, Canada. 4