Simulation of Chua's Chaotic Oscillator Using Unity ... - IEEE Xplore

Proceedings of the 7th International Caribbean Conference on Devices, Circuits and Systems, Mexico, Apr. 28-30, 2008

Simulation of Chua’s Chaotic Oscillator Using Unity-Gain Cells C. S´anchez-L´opez,1 R. Trejo-Guerra,2 E. Tlelo-Cuautle2 1

Autonomous University of Tlaxcala (UAT) Calzada Apizaquito S/N, Apizaco, Tlaxcala, P.O.Box 90300, Mexico. 2 National Institute of Astrophysics, Optics and Electronics (INAOE) Av. Luis Enrique Erro No 1. Tonantzintla, Puebla, P.O.Box 51 & 216, 72000, Mexico Email: [email protected], [email protected], [email protected]

Abstract— A novel topology to design a chaotic oscillator which is based on Chua’s circuit, is proposed. The chaotic oscillator is realized with unity-gain cells, that is, the use of Voltage Followers (VF) and Current Followers (CF) to emulate the behavior of the two most important elements of the Chua’s circuit: the active three-segment voltage controlled nonlinear resistor and the grounded inductor. Furthermore, it is shown that a Negative Current Follower (CF-) can be implemented by using the IC AD844AN type Current Feedback Operational Amplif ier (CFOA). The approach shows that the proposed topology can generate chaotic oscillations, where the HSPICE simulations have been obtained by using the macromodel of the IC AD844AN in conf iguration of unity-gain cell. Therefore, chaotic behavior in the time domain and the state space are presented.

R

C2

L

Fig. 1.

C1

+

Chua’s chaotic oscillator circuit. IN

m1 m0

(m1-m2)E1

-E3

m 2 E1 -E2

E2

-E1

m0

(m2-m1)E1

˙ = VC2 − VC1 − Rf (VC1 ) RC1 VC1 ˙ = VC1 − VC2 − RIL RC2 VC2 LI˙L = VC2

(1)

1 f (VC1 ) = m1 VC1 + (m2 −m1 )[|VC1 +E1 |−|VC1 −E1 |] (2) 2 where m1 and m2 are the slopes of the outer and inner segments of the active three-segment voltage-controlled nonlinear resistor inside the range (−E2 , E2 ) and ±E1 , ±E2 are the breakpoints, as shown in Fig. 2. Moreover, from Fig. 1 one can see that the Chua’s oscillator also is composed on the coupling of a passive LC tank resonator circuit with a NR by using a RC passive f ilter [9]. One the other hand, although several topologies have been proposed in the literature to emulate the behavior of the NR and the grounded inductor or contrarily the passive LC tank resonator circuit [1]-[5], [7][9], they are based on the use of some active devices, such 978-1-4244-1957-9/08/$25.00 ©2008 IEEE.

VN E3

I. I NTRODUCTION It is well known that chaos phenomenon has been intensively studied in the last four decades ago. This is due principally to the possible commercial applications in areas such as medicine, biology and secure communication systems [10]. Among the chaotic circuits designed to date, Chua’s chaotic circuit is an extremely simple system that can easily be built and tractable mathematically [1], [3], [4], [7], [8], [9]. As shown in Fig. 1, Chua’s circuit contains three energy storage elements, a linear resistor and a nonlinear resistor (NR) called Chua’s diode. The set of equations that describe the behavior of the Chua’s circuit are def ined by:

IN

VN

m1

Fig. 2.

Nonlinear resistor characteristics.

as: CFOA [8], [9], [7], [11], Operational Amplif ier (OPAMP) [4], Operational Transconductance Amplif ier (OTA) [3] and Second Generation Current Conveyor (CCII±) [12]. Recently, the design of the NR by using VF and CF has been investigated [14]. However, the results presented in [14] only show the design of the NR and still the behavior of the grounded inductor is modeled ideally. Herein, an improved design of NR given in [14], is introduced. Because the inductor is a less desirable circuit element, several inductorless realizations of Chua’s chaotic oscillators can be found in the literature [8], [9], [12]. Instead of the passive inductor element, an active inductance emulator have been used. In general, OPAMP, CFOA, CCII and OTA building blocks are preferred as the active element [2], [5], [8], [9], [12]. Unlike of the classical topologies, a new topology to emulate the behavior of a passive grounded inductor which is designed with unitygain cells, also is introduced. That way, we demonstrate here that the NR and the passive grounded inductor can also be designed by using unity-gain cells. As a consequence, a novel topology of the Chua’s chaotic oscillator is proposed. The rest of the paper is organized as follows. In Section II the concept of unity-gain cells is described, where the VF and CF- are designed by using the IC AD844AN. In Section III,

VF +

V1

V2

I1

(a) V1

I1

_

V1

I2

V2

V1

I1

I2

V2

Fig. 4.

D1

D2

IN

VF

+ VN

II. U NITY- GAIN C ELLS D ESIGN Both circuits, unity-gain VF and unity-gain CF have become versatile analog building blocks in the analog signals processing, since they are recognized for its excellent performance in wider bandwidth, low bias voltage, low power consumption, simpler architecture compared with others more complex analog building blocks and high accuracy while operating in open-loop conf igurations, since their bandwidth is not inversely related to the closed loop gain [6], [13]. The unitygain VF shown symbolically in Fig. 3a, is characterized by the following set of equations: 


0 0 V1 I1 = (3) 1 0 V2 I2 In the same manner, the symbolical representation of unitygain CF± is shown in Fig. 3b and its behavior is described by Eq. (4). 


0 0 I1 V1 = (4) 1 0 I2 V2 These analog building blocks can be designed in CMOS technology [6], [13], [14]. Instead of designing the unity-gain followers with CMOS technology, the HSPICE macromodel of the commercial IC AD844AN is adopted to implement the VF and CF±. Thus, by connecting the inverting input with the output, a VF can easily be obtained. On the other hand, the IC AD844AN is internally composed by two VFs and one CF+. Hence, to obtain a CF+ only the noninverting input should be grounded while the current output is delivered by the TZ terminal. In order to obtain a CF-, two CF+ connected in cascade should be used, as shown in Fig. 4.

I2

Tz

Symbolic representation to: (a) unity-gain VF (b) unity-gain CF.

the NR is designed by using one VF and two CF-, which have been introduced in Section II. HSPICE simulations are given to conf irm the suitability to implement the NR. Section IV includes the design of the active grounded inductor, which make use of two VF, one CF- and one CF+. Section V shows the design of the novel Chua’s chaotic oscillator circuit by using the NR and the active grounded inductor proposed herein, where simulation results by using HSPICE in the time domain and phase plane conf irm that the proposed circuit can generate chaotic oscillations. Finally, the conclusions are given in Section VI.

V2

_

Realization of a CF- using the IC AD844AN.

(b)

Fig. 3.

AD844AN

Tz

CF-

CF-

CF+

+

AD844AN

+ R2

+

V0 R1

R3

CF-

R4

CF-

Z1

Z2 r0

Fig. 5.

+ Vx

r0

+ Vy

Design of the NR with unity-gain cells.

III. N ONLINEAR R ESISTOR D ESIGN Several topologies to design the NR have been reported in the literature. All these topologies have been designed with several active devices, such as: OPAMP [4], CFOA [8], [9], [7], [11], OTA [3] and CCII± [12] and recently by using VF and CF [14]. An improved version of the NR presented in [14] is introduced herein. Hence, this NR can be simply realized using one VF, two CF- and four resistors, as shown in Fig. 5. A VF can be modeled as voltage-controlled voltage source and a CF as current-controlled current source both with unity-gain, thus, the node voltages Vx and Vy are given by:     2 4 Vy ≈ VN 1 + R (5) Vx ≈ VN 1 + R R1 , R3 The currents f lowing into R2 and R4 are given as: iR2 =

1 R2 (VN

− Vx ),

iR4 =

1 R4 (VN

− Vy )

(6)

where iN = iR2 + iR4

(7)

Further, the input current associated to the CF’s are provided by V0 /R1 and V0 /R3 , where V0 is the output voltage of the VF. On the other hand, the maximum output current of a CF is limited by the bias current and by the maximum output voltage swing. Thus, the maximum positive output voltage is constant at E+ sat (positive saturation region), the maximum negative output voltage is -E− sat (negative saturation region) and when the input current is small in magnitude, the output varies almost linearly with the input (linear region). In HSPICE simulation this output voltage is between the range ± = 5.4656V . Assuming this concept, it can easily be of Esat shown that the breakpoints and slopes, which are shown in Fig. 6, of the corresponding resistors D1 and D2 from Fig. 5 are given as:

IIN

IN m0D1 m1D1

m0D2

+

-E3

E1 -E2

E2

m1D2

-E1

VF1

CF+

VN

+

E3 m0D2

VIN

m0D1

R1

R2 +

CF-

Fig. 6.

VF2

Negative resistance behavior of D1 and D2 from Fig. 5. C

+ Vx

m1

iR1

Fig. 8. m2

m0

Grounded inductance simulator with unity-gain cells.

m0

iR2 iN

breakpoints are approximated by Eqs. 9 and 11. Hence, by using the numeric values listed above lines, the slopes and the breakpoints are given as:

m1

-E3

-E1

-E2

Fig. 7.

E1

E2

E3

V-I characteristic of the nonlinear resistor.

Positive Saturation Region m0D1 = E2 =

+ Esat R 1+ R2

,

1 R2 ,

E1 =

1

m0D2 = + Esat R 1+ R4

,

3

1 R4 + E3 = Esat

(8) (9)

Negative Saturation Region m0D1 = −E2 = −

− Esat R 1+ R2 1

1 R2 ,

, −E1 = −

m0 = 10mS, m1 = −0.3882mS, m2 = −1mS

(14)

±E1 = ±1.77V, ±E2 = ±5.16V, ±E3 = ±5.4656V

(15)

IV. ACTIVE G ROUNDED I NDUCTOR D ESIGN In this section, we introduce a novel and simple architecture for the implementation of a grounded active inductance simulator based on unity-gain cells. The basic scheme of the proposed topology is depicted in Fig. 8. Considering ideal properties, the equivalent inductance value can be computed as follows: VIN = R1 IR1 = sCR1 Vx = sCR1 R2 IR2 (16) = sCR1 R2 IIN from which:

m0D2 = − Esat R 1+ R4 3

1 R4

− , −E3 = −Esat

(10) (11)

Linear Region 1 R1 − r0 IN ≈− = = m1D1 VN R1 (R2 + r0 ) R1

(12)

1 R3 − r0 IN ≈− = = m1D2 VN R3 (R4 + r0 ) R3

(13)

where r0 is the output impedance of the CF-, which will generally be about 3MΩ. Hence, the NR can be obtained by the parallel connection of D1 and D2 as shown in Fig. 5, under the condition: r0 >> R1 , R3 . HSPICE simulation is shown in Fig. 7 with R1 = 1.7kΩ, R2 = 100Ω, R3 = 2.4kΩ, R4 = 5kΩ, using ±9V to bias the IC AD844AN. As one can see, the proposed topology also show nonlinear behavior characteristics compared with those topologies implemented with others active devices [3], [4], [7], [8], [9], [11], [12]. From Fig. 7, the slopes of the NR are given as: m0 = m0D1 +m0D2 , m1 = m0D2 + m1D1 , m2 = (m1D1 + m1D2 ) and the

VIN = sCR1 R2 = sLeq (17) IIN This equation is related to ideal unity-gain cells. Considering the parasitic input-output impedances and capacitors associated to each VF and CF±, the equivalent input impedance is modif ied to: 1 1 IIN = + Z4 ≈ (18) VIN Z1 Z2 Z3 Z1 Z2 Z3 where 1 1 + rin,V Z1 = sCA + rout,CF − F2 Z2 = R1 + rin,CF − + rout,V F1 Z3 = R2 + rin,CF + + rout,V F2 (19) 1 1 Z4 = sCB + rin,V + rout,CF F1 + CA = C + Cout,CF − + Cin,V F2 CB = Cout,CF + + Cin,V F1 Since the VF and CF± are implemented with CFOA’s, the typical values of parasitic components of the IC AD844 are aproximated to: rin,CF ± = 50Ω, rout,CF ± = 3M Ω, Cout,CF ± = 4.5pF , rin,V F1,2 = 10M Ω, rout,V F1,2 = 50Ω, Cin,V F1,2 = 2pF . As one can see, the consideration of the parasitic impedances does not modify drastically the behavior of the inductor.

IL

VF

CF+

R5

R6

CF-

IIN

R VC2

VC1

+

+

C2

VF + R2

C1

R1

R3

CF-

CF-

R4

VF r0

C

r0

Fig. 11.

Behavior dynamic of VC1 (t) and VC2 (t) with R = 1180Ω.

The improved Chua’s chaotic oscillator using unity-gain cells.

the NR has been presented. The equations that control the behavior of the slopes and breakpoints associated to the NR have been deduced. Second, a novel and simple topology to model a grounded inductor, which is modeled with unitygain cells, also has been presented. Therefore, by substituting these topologies in Chua’s circuit, a new chaotic oscillator is obtained. HSPICE simulations have been realized by using the macromodel IC AD844AN in conf iguration of unity-gain cell, where the behavior in the time domain and the double scroll attractor in the state space, have been observed.

V(1)

Fig. 9.

R EFERENCES

V(2)

Fig. 10. The double scroll attractor, projection in the VC2 −VC1 state space.

V. C HAOTIC O SCILLATOR D ESIGN Our novel topology of chaotic oscillator circuit is designed only with unity-gain cells as active elements to implement both: the NR and the active grounded inductor, as shown in Fig. 9. A HSPICE simulation of a double scroll attractor is shown in Fig. 10, with R1 = 1.7kΩ, R2 = 100Ω, R3 = 2.4kΩ, R4 = 5kΩ, R5 = 1kΩ, R6 = 1kΩ, R = 1.18kΩ, C1 = 20nF , C2 = 1uF , C = 15nF and using the macromodel IC AD844AN biased with ±9. Initial conditions to the capacitances are VC1 = 0, VC2 = 0.01V . The inductance value is approximated to L = 15mH, where the parasitic impedances of the VF and CF have been considered. Also, the chaotic waveforms in the time domain are depicted in Fig. 11. ACKNOWLEDGMENT The HSPICE macromodel IC AD844AN used in this work were supplied by Analog Devices. This work has been partially supported by CONACyT, MEXICO, under the projects J56673 and 48396-Y. VI. C ONCLUSION A novel chaotic oscillator based on unity-gain cells has been introduced. First, an improved version on the design of

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