International Journal of Electronics and Computer Science Engineering 223 Available Online at www.ijecse.org ISSN: 2277-1956 A new high frequency realization of Chua s chaotic circuit using current feedback operational amplifiers Prem Bhushan Mital 1, Umesh Kumar 2, Rai Sachindra Prasad 3 1 United College of Engineering& Research, Greater Noida U.P. India 2 Department of Electrical Engineering IIT New Delhi India 3 Inderprastha Engineering College Ghaziabad U.P. India Abstract: A new implementation of Chua s chaotic circuit using current feedback operational amplifier (CFOAs) and giving high frequency oscillations is presented. Both the circuit topologies based on CFOA S i.e. CFOA based nonlinear resistor and CFOA based synthetic inductor are used together The new circuit is operated at high frequencies in which the CFOAs have excellent performance. The workability of the proposed realization is verified using P-spice simulation. Key Words: Chaos; Chua s circuit; current feedback operational amplifier; high frequency chaotic oscillations. I-INTRODUCTION Since Chua s circuit is structurally the most simple circuit ( 3 energy storage elements, one linear resistor and one nonlinear resistor) yet dynamically the most complicated, exhibiting a rich variety of chaos and bifurcations, the same has been very active topic of research and studied extensively and accepted as a paradigm for generating and studying chaos. Several realizations of Chua s circuit have been proposed in the literature. Chaotic signals have been used successfully in modern communication systems including secure communication, spread spectrum communication, optical communication, signal encryption, chaos based modulation Schemes viz chaotic on-off keying (COOK), chaos shift keying (CSK), Differential CSK (DCSK), Quadrature chaos shift keying (QCSK) etc., Figure.1 the Chua s circuit and v R -i R characteristics of the Chua s diode The nonlinearity of the Chua s diode N R is expressed as G b v R + (G b G a )E 1, if v R E 1 g(v R ) G a v R if v R < E 1 (1 ) G b v R + (G a G b )E 1 if v R E 1
IJECSE,Volume1,Number 2 Prem Bhushan Mital et al. Where G a and G b are the slopes of inner segment and two outer segments and ±E 1 are the break points. If V1, V 2 and i L are the voltage across capacitor C 1, the voltage across the capacitor C 2 and the current in the inductor L respectively, the state equations of the Chua s circuit are written as: dv 1 /dt 1/C 1 [G(v 2 v 1 ) g(v 1 )] dv 2 /dt 1/C 2 [G(v 1 v 2 ) + i L ] (2) di L /dt 1/L v 2 Rescaling these equations by making substitutions as below, x v 1 /E 1 τ tg/c 2 α C 2 /C 1 y v 2 /E 1 a Ga/G β C 2 /(LG 2 ) (3) z i l / (E 1 G) b Gb/G The above equations are written as where, bx + b a, x α[y x f(x)] y x y + z (4) z βy if x 1 f(x) ax, if x < 1 (5) Simplifying we have bx + a b, if x 1 x α[y h(x)] y x y + z (6) z βy These are known as Chua s equations which are used in a more convenient form for the analytical treatment 1.1 Frequency of operation In case of a sinusoidal oscillator, the output power is concentrated in a single frequency component around the operating frequency of the sinusoidal oscillator (ω ). However, as the system is moved around the chaotic region, the power is spread to more frequency components aroundω. The linearized form of Chua's Circuit is given by + + s+ 0 (7) where + ; + ; (8) ;
A new high frequency realization of Chua s chaotic circuit using current feedback operational amplifiers 225 Here ; for slope of inner segments ( -0:8 normally) ; for slope of outer segments ( -0:5 normally) The condition for system to oscillate is given by ( ) G - and the frequency of oscillation is given by (ω a ); (9) ω 1 (10) The frequency of oscillation of the sinusoidal oscillator is given by Estimated frequency range is given by ω (11) ω ω 1 (12) Thus, whereas L and C2 specify the central frequency of operation, L, G, mi and C1 are responsible for the range covered by the spectrum. II. PROPOSED CHUA S CHAOTIC CIRCUIT CFOA s are recognized as versatile alternatives to the traditional voltage mode OP amps ( VOA s) due to their high performance with almost constant bandwidth ( up to 100 MHz) available independent of the closed loop gain and very high slew rate around 2000V/µs. Commercial types available are AD844/AD811 manufactured by M/s Analog Devices USA. These have been used instead of VOA s in new implementation of Chua s circuit giving high frequency performance. In the proposed CFOA-based Chua s circuit (inductor less), both the circuit topologies based on CFOA s i.e. CFOA based non linear resistor and CFOA based synthetic inductors are used together in the same design so that the novel circuit exhibits original chaotic behaviour Chua s circuit and also high frequency chaotic oscillations as shown in Fig. 2. All CFOA s (AD844 type) are biased with ±9V.. The new circuit is operated at high frequencies in which the CFOA has excellent performance. With the proposed realization, several enhancements are achieved. Firstly, in addition to a current output signal, a buffered output voltage (V O ) is also available which represents a state variable for direct access which is isolated from all other components.v O equals V C which is by connection forced to V t. However V t is just the voltage across C. Hence V O directly represents a state variable. Secondly the operating frequency is extended i.e. chaotic spectrum is extended to higher frequencies i.e. upto 50MHz making use of excellent performance of CFOA s at higher frequencies. The circuit parameters used for very high- frequency operation mode are: R1 R2 1K, RN1 RN2 2K RN3 2K, RN4 2.2 K C 1 1pF C 2 10 pf C 3 1nF L 1 µh CFOA s are biased with ±9V. ISSN: 2277-1956/V1N2-223-229
IJECSE,Volume1,Number 2 Prem Bhushan Mital et al. III. SIMULATION RESULTS The plots of voltage across C1 vs time and voltage across C2 vs time are shown in Fig.3 and Fig.4 respectively. The plots of VC1 vs VC2 are shown in Fig.5. Fast Fourier transform of the chaotic signal has been obtained through P-spice simulation and is shown in Fig.6. It is seen that although the circuit is operating at wide range of frequencies, it is mostly concentrated around 50 MHz which is the fastest for any double scroll reported so far. Figure.2 Proposed CFOA based Chua s circuit Figure3: Voltage across C1 Vs time
A new high frequency realization of Chua s chaotic circuit using current feedback operational amplifiers 227 Figure 4 Voltage across C2 Vs time ISSN: 2277-1956/V1N2-223-229
IJECSE,Volume1,Number 2 Prem Bhushan Mital et al. Figure 5: Plots of VC1 Vs VC2 Figure 6. Fast Fourier Transform (FFT) of the chaotic signal obtained from proposed circuit IV- CONCLUSION A new implementation of Chua s chaotic circuit using current feedback operational amplifier (CFOAs) and giving high frequency oscillations of the order of 50 MHz is proposed. Both the circuit topologies based on CFOA s i.e.
A new high frequency realization of Chua s chaotic circuit using current feedback operational amplifiers 229 CFOA based nonlinear resistor and CFOA based synthetic inductor are used together. All the state variables Vc1, Vc2 and Il are made available in a direct manner. The new circuit is operated at high frequencies in which the CFOAs have excellent performance. The workability of the proposed realization is verified using P-spice simulation which confirms that the proposed circuit exhibits the original chaotic circuit. The proposed implementation offers very attractive and useful circuit structure for high frequency chaos applications. The design provide advantages as high frequency of operation, simple design and availability of current across inductor. REFERENCES [1] R. Senani and S.S.Gupta, Implementation of Chua s chaotic circuit using CFOA s Electronic letters, vol 34,pp 829-839 April 1998. [2] U.Cam, A new high performance realization of mixed-mode chaotic circuit using current feedback operational amplifiers. Computer and Electrical Engineering, 30 pp 281-290, 2004.. [3] A.S.Elwakil and M.P.Kennedy, High frequency Wein type chaotic oscillator, Electronic letters,vol 32 pp 1161-1162 June 1998. [4] A.S.Elwakil and M.P.Kennedy, Chua s circuit decomposition: A systematic design approach for chaotic oscillators, Journal of the Franklin Institute,337 pp251-265, 2000. [5] A.S.Elwakil and M.P.Kennedy, Improved Implementation of Chua s chaotic oscillator using current feedback op amp. IEEE Trans. On C&S part I vol 47,no.1 Jan.2000. [6] Kilic R, Study of CFOA-based inductorless Chua's circuit," International Journal of Bifurcation and Chaos, vol. 14, pp 1369-1374, 2004. [7] Kilic R, On current feedback operational amplifer-based realization of Chua's circuit," Circuits, systems, and signal processing, vol. 22, No. 5, pp 475-491, 2003. [8] Soliman A M, Applications of the current feedback operational amplifiers, Analog integrated circuit and signal processing, 14, pp 265-302, 1996. [9] Bales J, A low power,high speed current feedback operational amplifier with a novel class AB high current output stage, IEEE J.Solid state circuits, Vol.32, pp 1470-1474, 1997. [10] Toumazou C and Lidgey J, Current feedback op-amp : a blessing in disguise?. IEEE Circuits devices magazine, 10 (1), pp 34-37, 1994. [11] Kilic R and Yildiram F, CFOA based inductorless mixed-mode chaotic circuit Int. J. Bif. and chaos, Vol.16, No.3, pp 709-714, 2006. ISSN: 2277-1956/V1N2-223-229