Implementation of a Chua’s Chaotic Oscillator Using “Roughly-Cubic-Like” Nonlinearity Banlue Srisuchinwong and Wimol San-Um Sirindhorn International Institute of Technology, Thammasat University, Bangkadi Campus, 131 Moo 5, Tiwanont Road, Muang, PaThum Tani, 12000, Thailand,
[email protected] Abstract— Implementation of a Chua’s chaotic oscillator is proposed using roughly-cubic-like nonlinear resistor. A simple current mirror and a single-ended differential pair shape a grounded nonlinear resistor and exhibit roughly cubic-like voltage-current characteristics. A simulated trajectory of the double-scroll chaotic attractor is demonstrated.
I.
INTRODUCTION
Over the past two decades chaotic oscillators [1-3] have received increasing interest as a useful tool not only for investigation of nonlinear phenomena, bifurcation and chaos, but also for a variety of applications such as synchronizations, control [4] and chaos-based communications systems [5]. Chua’s circuit [3] is one of the best-known chaotic oscillators using a nonlinearity known as a (piecewise-linear) “Chua Diode”. Several techniques including the “cublic” nonlinearity [6] or “cubic-like” nonlinearity [7, 8] have alternatively been proposed to replace the Chua diode. In this paper, an implementation of the Chua’s chaotic oscillator is presented using roughly-cubic-like nonlinearity. A single-ended differential pair and a simple current mirror are combined to form a compact four-transistor grounded nonlinear resistor that exhibit roughly cubic-like voltagecurrent characteristics. The circuit develops the double-scroll chaotic attractor. II.
For a large signal operation, RN can be distored in a nonlinear fashion because of the nonlinearity of the MOS transistors or eventually may enter into saturation.
CIRCUIT IMPLEMENTATION
Figure 1 shows the proposed Chua’s chaotic oscillator using a roughly-cubic-like nonlinear resistor RN. Inductor L1, capacitors C1 and C2 form parts of the Chua’s circuit. A single-ended differential pair (M1, M2), a simple current mirror (M3, M4) and a current source I1 form the nonlinear resistor RN. For simplicity, let the effects of the channel length modulation of transistors be neglected. Let vx and ix be the input voltage and input current to RN, i.e. RN = vx / ix at the gate of M1. A routine small-signal analysis suggests that ix = 2id where id ! gm1vx / 2 and gm1 is the transconductance of transistor M1 or M2. Consequently, for a small-signal operation, RN exhibits a negative resistance i.e.
RN !
1 g m1
(1)
Figure 1 Chua’s circuit using roughly-cubic-like nonlinearity.
III.
SIMULATION RESULTS
The performances of the circuits shown in Fig. 1 have been simulated through Pspice using models IRF150 and M2N6849 for NMOS and PMOS transistors, respectively. The aspect ratio of all transistors are W/L = (100 "m)/(2 "m) and the threshold voltage of the NMOS VTH = 2.831 V. The values of I1 = 800 "A, L1 = 18 mH, C1 = 10 nF, C2 = 100 nF, R1 = 1.38 k#. The gate-source DC voltage of M1 and M2 is VGS1 = 3.71 V. Consequently, gm1 = I1/(VGS1-VTH) = 910.12 "#-1 and, for the small-signal analysis, the calculated value of RN = 1.09 k#. Figure 2 shows the simulated voltage-current characteristics of the nonlinear resistance RN where vx in the x axis is between -5 V to 5 V and ix in the y axis is between -800 "A to 800 "A. For the small-signal analysis, the measured value of RN = vx / ix = ( 40 mV)/(35 "A) = 1.14 k# and is consistent with the calculated value. It can be seen from Fig. 2 that, for the small signal operation, the negative resistance RN is evident through the negative slope whilst, for the large signal operation, RN exhibits nonlinearity. The curve of RN shown in Fig. 2 looks like a roughly coarse version of a “cubic-like” voltage-current characteristics and hence the name “roughly-
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cubic-like” nonlinearity. For purposes of comparison, a simulated example of a “cubic-like” voltage-current characteristics is shown in Fig. 3. Fig. 4 depicts the simulated trajectory of the double-scroll chaotic attractor developed in Fig. 1 in the VC1 VC2 plane where VC1 and VC2 are the voltages across the capacitors C1 and C2, repectively.
IV.
CONCLUSIONS
Implementation of a Chua’s chaotic oscillator has been presented using the roughly “cubic-like” voltage-current nonlinearity. The combination of a single-ended differential pair and a simple current mirror with a feedback has enabled a roughly cubic-like nonlinear resistor. The simulated doublescroll chaotic attractor has been demonstrated. ACKNOWLEDGMENTS Authors are grateful to Mr. Chun-Hung Liou for his useful assistant. REFERENCES [1] [2]
[3] Figure 2 Voltage-current characteristics of RN looks like a roughly coarse version of the “cubic-like” characteristics.
[4] [5]
[6]
[7]
[8]
Special issue on chaos in nonlinear electronic circuits, Part A : Tutorials and Reviews, IEEE Transactions on Circuits and Systems 40(10), 1993. Delgado-Restituto, M. and Rodriguez-Vazquez, A. (2002), “Integrated Chaos Generators”, Proceedings of the IEEE, vol. 90, No. 5, May, 2002, pp. 747-767. Chen, G. and Ueta T. ed., Chaos in Circuits and Systems, World Scientific, Singapore, 2002. Special issue on chaos synchronization, control, and applications, IEEE Transactions on Circuits and Systems 44(10), 1997. Mandal, S. and Banerjee, S., “Analysis and CMOS Implementation of a Chaos-Based Communication System,” IEEE Trans. Circuits and Systems – Part I 51(9), Sep. 2004, pp. 1708-1722. Zhong, G.Q., “Implementation of Chua’s Circuit with a Cubic Nonlinearity”, IEEE Trans. Circuits and Systems-1, 41(12), 1994, pp. 934-941. Eltawil, A.M. and Elwakil, A.S., “Low-Voltage Chaotic Oscillator With an Approximate Cubic Nonlinearity,” Int. J. Electron. Commun. (AEUE), 53(3), 1999, pp. 11-17. Donoghue, K.O., Kennedy, M.P. and Forbes, P., “A Fast and Simple Implementation of Chua’s Oscillator using a “Cubic-Like” Chua diode,” Proceedings of the 2005 European Conference on Circuit Theory and Design, 2005.
Figure 3 An example of a ”cubic-like” voltage-current characteristics
Figure 4 A simulated trajectory of the double-scroll chaotic attractor developed in Fig. 1
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