A Synthetic Inductor Implementation of Chua's Circuit

Bharathwaj MuthuswamyTamara BlainKyle Sundqvist

Electrical Engineering and Computer SciencesUniversity of California at Berkeley

Technical Report No. UCB/EECS-2009-20

http://www.eecs.berkeley.edu/Pubs/TechRpts/2009/EECS-2009-20.html

January 30, 2009

Copyright 2009, by the author(s).All rights reserved.

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A Synthetic Inductor Implementation of Chua’sCircuit

Bharathwaj Muthuswamy, Kyle Sundqvist and Tamara Blain

Abstract—We show how to build an inductorless versionof the classic Chua’s circuit. The goal is to build Chua’scircuit quickly and easily. To this end, only off-the-shelfcomponents are used. A suitable inductor for Chua’scircuit is often hard to procure. Here, the inductor isreplaced by a single op-amp synthetic inductor circuit.The synthetic inductor is novel in the sense that it is nota gyrator, it is a one port device and thus a simple black-box analogy to an inductor is possible. We illustrate therobustness of the synthetic inductor by synthesizing twodifferent inductor values.

I. INTRODUCTION

IN this paper, we propose an off-the-shelf imple-mentation of Chua’s circuit. This circuit is the

paradigm for generating chaotic attractors [1]. Aschematic of Chua’s circuit is shown in Fig. 1 [4].The iR-vR graph of the nonlinear resistor NR (alsocalled as the Chua diode) is shown in Fig. 2 [4].

Fig. 1. Chua’s Circuit Schematic. The circuit consists of a linearinductor L, a linear resistor R, two linear capacitors C1 and C2 anda nonlinear resistor NR.

Bharathwaj Muthuswamy, Kyle Sundqvist and Tamara Blain arewith the University of California, Berkeley, CA, 94720 USA. Contact:mbharat@cory.eecs.berkeley.edu,kylesun@cosmology.berkeley.edu,eb-9d9@eecs.Berkeley.edu

Fig. 2. The i-v characteristic of the nonlinear resistor NR. Everyphysically realizable nonlinear resistor is eventually passive - theoutermost segments must lie within the first and third quadrants ofthe v-i plane for sufficiently large |v| and |i|.

The state equations for Chua’s circuit are shownbelow.

C1dvC1

dt=

vC2 − vC1

R− iR (1)

C2dvC2

dt=

vC1 − vC2

R+ iL

LdiLdt

=−vC2

In (1), iR = g(vR) = g(vC1) is a piecewise-linearfunction defined by [4]:

g(vR) = m0vR +1

2(m1−m0)[|vR +Bp|−|vR−Bp|]

(2)From Fig. 2, it can be inferred that the slopes in theinner and outer regions are m1 and m0 respectively;±Bp denote the breakpoints. Good references forunderstanding Chua’s circuit are [5] and [4].

It is difficult to obtain precise values of the induc-tor needed to build the circuit in Fig. 1. Also, theinductor is quite bulky and it cannot be integratedon a chip. Moreover, many commercially availableinductors have a core that is added to increasethe inductance (and hence reduce the number of

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windings needed). But, this is known to have theeffects of adding distortion to the signal via hys-teresis [8]. Given the sensitive dependence on initialconditions for the chaotic circuit, this would belargely undesirable. One solution is to simulate theinductor using a gyrator, as demonstrated by [7],which uses a two op-amp implementation of thegyrator. In this paper, we use a simpler syntheticinductor implementation. Specifically:

1) We use only one op-amp for the syntheticinductor.

2) All components are off-the-shelf (a parts listis included at the end of this paper).

The organization of this paper is as follows: inSection II, we give a brief overview of the compo-nent values used in the 18mH inductor version ofChua’s circuit. In Section III, we give the expressionfor the impedance of the synthetic inductor (thederivation is given in the Appendix). In Section IV,we show simulation results using a 30-day fullyfunctional trial version of National Instruments’MultiSim suite [3]. This circuit simulator was cho-sen over PSPICE because MultiSim is easier touse. In Section V, we show experimental resultsusing oscilloscope waveforms from the physicalimplementation of the circuit. In Section VI, weimplement a different inductor value using the syn-thetic inductor (the 8.2 mHChua’s circuit from [5]).We conclude the paper with a parts list, suggestionsfor future work and acknowledgments.

II. CHUA’S CIRCUIT COMPONENT VALUES

Fig. 3 shows the realization of Chua’s circuitthat will be used in this paper. In comparison toFig. 4, the inductor in Fig. 3 has been replaced byour synthetic inductor and the Chua diode has beenimplemented using Kennedy’s two op-amp imple-mentation [4]. The component values for everythingbut the synthetic inductor were obtained from [4].The component values for the synthetic inductor aregiven in the next section.

From Fig. 3 and [4], the parameters m0, m1

and ±Bp for the Chua diode i-v in Fig. 2 can becomputed as:

m0 = −0.409mS, m1 = −0.756mS, Bp = 1.08V(3)

Fig. 4. Chua’s Circuit with component values for investigating chaos,the component values are from [4]. Compare with Fig. 3.

Fig. 5. The synthetic inductor circuit. The goal is to derive anexpression for Zin. We assume the op-amp is operating in the linearregion. The derivation is in the appendix

III. THE SYNTHETIC INDUCTOR IMPEDANCE

Fig. 5 shows our modified version of the syntheticinductor from [2].

If RL = 10−4Rg (so that RL << Rg) in Fig. 5,we have:

Zin ≈ RL + jωRLRgC (4)

The derivation of (4) is given in the Appendix. Inour case, RL = 10Ω, Rg = 100kΩ and C = 18nF ,so we have the equivalent circuit of Fig. 6 for an18mH inductor.

Fig. 6. 18mH synthetic inductor circuit.

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IV. MULTISIM SIMULATION RESULTS

MultiSim 10 has been used to simulate Chua’scircuit. A free 30-day evaluation version of Multi-Sim 10 can be downloaded from [3]. Please down-load the professional edition of MultiSim 10. Thisedition has the simulation models for the LMC6482op-amp. A MultiSim 10 simulation file for theChua’s circuit discussed in this paper can be down-loaded from [6].

Fig. 7 shows simulated attractors obtained fromMultiSim 10. We show a period-doubling route tochaos by varying C1 [4], the other parameters arefixed. Attractor periods shown are period-1, period-2, period-4 and a Double-Scroll Chua attractor.

V. EXPERIMENTAL RESULTS

Fig. 8 shows a series of measured attractors fromthe physical circuit. Note that the experimental C1

values used for illustrating the period doubling routeto chaos closely match the C1 values from thesimulated version (refer to Fig. 7).

VI. 8.2 MH INDUCTOR VERSION OF CHUA’SCIRCUIT

To examine the robustness of this circuit, let usimplement Chua’s circuit with component valuesin [5]: L = 8.2 mH, C2 = 55 nF, R =1.33 kΩ, C1 = 5.5 nF . However, the capacitor val-ues are not off-the-shelf, therefore we implementedL = 8.2 mH, C2 = 47 nF, C1 = 4.7 nF . The 2kpotentiometer has been set to 1.5 kΩ. The syntheticinductor capacitor has been changed to C = 8.2 nFto implement the 8.2 mH inductor. The schematicof this circuit is shown in Fig. 9. A simulated andexperimental Double-Scroll is also shown.

VII. CONCLUSIONS AND FUTURE WORK

A. Parts ListThe parts list for both the 18 mH and the 8.2 mH

circuit are given in Tables I and II. All resistors are5% tolerance. Capacitors are mylar and have 10%tolerance.

The op-amp that was used in this paper is theLMC6482 from National Semiconductor. However,we replaced the LMC6482 in the circuit with theTL082 and the AD822AN (pin-for-pin compatibleop-amps with the LMC6482). Both the TL082and the AD822AN circuits displayed bifurcation

TABLE IPARTS LIST FOR 18 mH AND 8.2 mH SYNTHETIC INDUCTOR

VERSIONS OF CHUA’S CIRCUIT

L RL C Rg C2 R C1

18 mH 10 Ω 18 nF 100 kΩ 100 nF 2 kΩ pot. 10 nF

8.2 mH 10 Ω 8.2 nF 100 kΩ 47 nF 2 kΩ pot. 4.7 nF

TABLE IICHUA DIODE PARTS LIST FOR 18 mH AND 8.2 mH SYNTHETIC

INDUCTOR

L R1 R2 R3 R4 R5 R6

18 mH 220 Ω 220 Ω 2.2 kΩ 22 kΩ 22 kΩ 3.3 kΩ

8.2 mH 220 Ω 220 Ω 2.2 kΩ 22 kΩ 22 kΩ 3.3 kΩ

and chaos phenomenon. Therefore, possible op-amps that can be used are the LMC6482, TL082and the AD822AN. Moreover, the TL082 and theAD822AN can be powered using ±9 V supplies.This makes them attractive for use with ±9 Vbatteries.

B. Future Work

In this paper, we discussed a single op-ampsynthetic inductor version of Chua’s circuit. Weimplemented Chua’s circuit for two synthetic valuesof inductance: 18 mH and 8.2 mH . An interestingproblem would be to explore the maximum possiblebandwidth of this circuit.

APPENDIXTHE SYNTHETIC INDUCTOR IMPEDANCE

DERIVATION

We will now derive the equivalent impedanceof the synthetic inductor, as seen from its inputterminals. Refer to Fig. 5.

Using Thevenin’s theorem, we can write an ex-pression for Zin:

Zin =Vin(jω)

Iin(jω)

=Vin(jω)

I1(jω) + I2(jω)

=Vin(jω)

Vin(jω)−Vn(jω)RL

+ Vin(jω)

Rg+ 1jωC

(5)

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Assuming the op-amp is operating in the linearregion:

Vn(jω) = Vp(jω)

Vn(jω) =Rg

Rg + 1jωC

· Vin(jω) (6)

Substituting for Vn(jω) in (5) from (6) and sim-plifying:

Zin =Vin(jω)

Vin(jω)− Rg

Rg+ 1jωC

·Vin(jω)

RL+ Vin(jω)

Rg+ 1jωC

=1

1− Rg

Rg+ 1jωC

RL+ 1

Rg+ 1jωC

=RL

1− Rg

Rg+ 1jωC

+ RL

Rg+ 1jωC

(7)

Let RL = 10−4Rg (so that RL << Rg). Thensubstituting for RL in the denominator of (7):

Zin =RL

1− Rg

Rg+ 1jωC

+ 10−4Rg

Rg+ 1jωC

≈ RL

1− Rg

Rg+ 1jωC

=RL(Rg + 1

jωC)

Rg + 1jωC

−Rg

=RL(Rg + 1

jωC)

1jωC

= jωRLC(1

jωC+ Rg)

= RL + jωRLRgC (8)

Thus, if RL = 10−4Rg (so that RL << Rg), wehave:

Zin ≈ RL + jωRLRgC (9)

Fig. 10 shows the circuit equivalent of (9).If RL = 10Ω, Rg = 100kΩ and C = 18nF , we

have the equivalent circuit of Fig. 6 for an 18mHinductor.

Fig. 10. The synthetic inductor circuit from Fig. 5 can be modelledas a parasitic resistance RL in series with an inductance RLRgC.

ACKNOWLEDGMENT

Many thanks to Prof. Pravin Varaiya for insightfuldiscussions on chaos. Prof. Joos Vandewalle wasvery helpful in providing valuable comments. Manythanks to the students in EE100 Summer 2007 at theUniversity of California, Berkeley. They providedvaluable feedback on how to build a simple off-the-shelf version of Chua’s circuit.

REFERENCES

[1] L. O. Chua, “The genesis of chua’s circuit,” Archiv for Elektronikand Uebertragungstechniko, vol. 46, no. 4, pp. 250–257, 1992.

[2] P. Horowitz and W. Hill, The Art of Electronics. Cambridge,Massachussetts: Cambridge University Press, 1989.

[3] N. Instruments. (2008, February) Multisim professional edition30-day trial version. [Online]. Available: http://www.ni.com/multisim

[4] M. P. Kennedy, “Robust op-amp realization of chua’s circuit,”Frequenz, vol. 46, pp. 66–80, 1992.

[5] T. Matsumoto, L. O. Chua, and M. Komuro, “The double scroll,”IEEE Transactions on Circuits and Systems, vol. CAS-32, no. 8,pp. 798–818, August 1985.

[6] U. of California Berkeley. (2008, February) Nonlinear elecotrnicslaboratory chaos in chua’s circuit homepage. [Online]. Available:http://nonlinear.eecs.berkeley.edu/chaos/chaos.html

[7] L. Torres and L.A.Aguirre, “Inductorless chua’s circuit,” Elec-tronics Letters, vol. 36, no. 23, pp. 1915–1916, 2000.

[8] O. S. University. (2008, May) Physics 517/617: Introduction toelectronics. [Online]. Available: http://www.physics.ohio-state.edu/∼durkin/phys617/

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Fig. 7. Simulated attractors from MultiSim 10 with component values RL = 10 Ω , Rg = 100 kΩ , C = 18 nF , C2 = 100 nF , R =1.83 kΩ. C1 is varied to show the period-doubling route to chaos. Horizontal axis is vC1 ; Vertical axis is vC2 .

(a) C1 = 13.2 nF (b) C1 = 12.7 nF

(c) C1 = 12.2 nF (d) C1 = 10 nF

Fig. 8. Measured attractors (using an HP54645D oscilloscope) with component values RL = 10 Ω , Rg = 100 kΩ , C = 18 nF , C2 =100 nF , R = 1.83 kΩ. C1 is varied to show the period-doubling route to chaos. Horizontal axis is vC1 ; Vertical axis is vC2 . Scales areVertical axis: 1.00 V/div for (a), 0.5 V/div for (b) and (c), 1.00 V/div for (d); Horizontal axis: 0.2 V/div for (a), (b) and (c), 1.00 V/div for(d).

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