Chaos from Two Modified Oscillator Configurations Using a ...1 w,, = ~ RC (3b) Figure 2 is a PSpice frequency spectrum simulation of this circuit’s output voltage for C = 1 nF1 R

Chaos from Two Modified Oscillator Configurations Using a Current Feedback Op Amp

AHMED S. ELWAKIL and AHMED M. SOLFMAN

Electronics and Communications Engineering Department. Cairo University, Cairo, Egypt

(Accepted 2 August 1996)

Abstract-Two oscillator configurations that employ the current feedback up amp (CFOA) as the active element are modified for chaos using a single discrete nonlinear element, namely a junction field effect transistor (JFET) operating in its triode region. Tote first oscillator is based on a passive Twin-T structure while the second oscillator is a Wien type oscillator. The generated chaotic signal from both circuits persists for a wide and continuous range of parameter values and in drfferent frequency bands that are easily adjusted by simple parameter scaling. The design equations of both configurations as ordinary sinusoidal oscillators are also shown to be useful when modified for chaos. Experimental laboratory results and PSpice simulations arc provided. 0 1997 Elsevier Science Ltd

1. INTRODUCTION

Since its appearance, Chua-Matsumoto’s circuit has been the main source for studying chaos in electronic systems. Extensive study of this circuit’s performance and its various applications have been introduced in the literature, notably in Refs [l-3].

Recently, there has been a growing interest in investigating chaos in conventional oscillator configurations [4]. Some of the well-known oscillators have been shown to exhibit chaotic performance as in Refs [5] and [6] where the Colpitts oscillator was studied, and as in Refs [7] and [X] where the Wien bridge oscillator was modified for chaos. On the other hand, growing interest has also been given to the current feedback op amp (CFOA) since it appears to be more flexible and versatile than traditional op amps [9, lo]. Among the important applications of the CFOA is the realization of sinusoidal signal generators [ll].

In this paper, two oscillator configurations that employ the CFOA as a noninverting voltage controlled voltage source (VCVS) are modified for chaos using a single discrete nonlinear element, namely a JFET operating in its triode region. The first oscillator is based on a passive Twin-T structure while the second oscillator is a Wien type oscillator, Although Wien type oscillators are basically of second order. interest in their modification for chaos is largely due to their popularity and ease of design. In Ref. [S], the proposed Wien bridge based chaos generator was formed by coupling a Chua diode with a Wien oscillator in parallel, thus requiring three op amps, eleven resistors and three capacitors, while in [7]. a simple and innovative modification was proposed requiring a single op amp, four resistors, three capacitors and a single discrete nonlinear element.

The two circuits proposed in this paper have the following attractive features:

(1) No coils are needed. (2) The generated chaotic signal has a large voltage swing and persists for a wide and

continuous range of parameter values. (3) A chaotic signal is characterized by a continuous broad band spectrum within a certain

frequency band. In the two circuits, however, this band is easily movable by changing

390 A. S. ELWAKIL and A. M. SOLIMAN

the values of the circuit capacitors and accordingly the circuit’s time constant. Chaotic spectrums in the Hz, kHz and up to the MHz bands can be obtained.

(4) The two circuits can still function as sinusoidal oscillators even when the nonlinear element exists. As a single grounded resistor is tuned, both circuits develop an amplitude controlled sinusoidal waveform before a period doubling route to chaos starts. In the chaotic region odd as well as even periodic orbits can be sustained.

(5) The linear case design equations are useful as a starting point for chaos modification,

For the PSpice simulations and the experimental work, the AD844 CFOA was used biased with Cl2 V in PSpice simulations and +9 V in experiments. Section 2 of this paper is devoted to the first Twin-T based oscillator, whiIe Section 3 discusses the second Wien type oscillator configuration.

2. THE FIRST OSCILLATOR CONFIGURATION

Figure l(a) represents the generalized oscillator configuration with the passive network N shown in Fig. l(b) [lo]. The characteristic equation of this oscillator is given by

where (la)

(lb)

1 T

RA I-LI R3

=!= r

Fig. l(a)

c3 R2 c3 R2

2

Fig. l(b)

I---- 1. Rl

‘I Cl

Fig. l(c)

Fig. I. (a) Generalized oscillator configuration. (h) Twin-T network N. (c) Twin-T network N modified for chaos.

Current feedback op amp

and S is the complex frequency variable

Tzds) = 4RCS

R2C2S2 + 4RCS t 1 (2)

i.e. it is assumed that a symmetrical Twin-T is used with R, = R2 = R, R, = R/2, C, = C, = C and C3 = 2C. Thus the condition of oscillation is given by

K=l (3a)

and the radian frequency of oscillation is given by:

1 w,, = ~ RC

(3b)

Figure 2 is a PSpice frequency spectrum simulation of this circuit’s output voltage for C = 1 nF1 R = 1.5 kQ and K = 1.155, which indicates a single harmonic frequency.

The network N modified for chaos is shown in Fig. l(c) where the linear resistor R3 has been repIaced with a JFET having its gate and drain terminals shorted. The nonlinearity introduced by this element during circuit operation is plotted in Fig. 3 which represents the gate-to-source voltage of the JFET versus its drain-to-source current. A JFET of the type J2N4338 was chosen and used for all simulations and experiments. As seen from Fig, 3, this JFET acts as a resistor of about 750 Q resistance in its linear portion of operation.

Considering the symmetrical Twin-T case, and since R3 is now around 750 SZ. then RI and Rz should be taken as around 1 .S kQZ. The values of the circuit capacitors are arbitrary chosen to select the frequency band of interest which can be estimated from eqn (3b). In order to change the circuit dynamics, the gain K must be slightly increased from the

[ 8.0” I

I i (

Fig. 2. PSpice frequency spectrum simulation for C = I nF, R = 1.5 kQ and A’ = 1.155

39: A. S. ELWAKIL and A. M. SOLIMAN

1. o*

0.5mA i’

Fig. 3. JFET’s gate-to-source voltage versus drain-to-source current.

nominal value indicated by (3a). This is achieved by tuning one of the grounded resistors RA or RB upon which the circuit’s output develops from a decaying sinusoid into an amplitude controlled sinusoid before a period doubling route to chaos starts.

Figure 4(a) and (b) represents PSpice simulations of the CFOA output voltage for C = 100 nF, R = 1.592 kQ and K = 1.18 with Fig. 4(a) demonstrating the time domain behavior and Fig. 4(b) the corresponding spectrum. For the set of values C = 0.1 nF, R = 1.592 kQ and K = 1.19, Fig. 5(a), (b), (c) and (d) h s ows PSpice simulations demonstrating a chaotic behavior with Fig. 5(a) illustrating the time waveform and Fig. S(b). (c) and (d) illustrating the phase space trajectories.

Denoting the voltage across the grounded capacitor Cr as X and the CFOA output voltage as Y, the circuit is described by the following pair of differential equations:

R!R,C&ii - KR,R&&f - K(R, + R&j

+ [(RI + R2)C2 - R1C3(K - l)]I’ + Y = 0 (4a)

and

C,I’ - K(C, + C&t + KZ(X, Y) = 0 (4b)

where Z(X, Y) is the drain-to-source current flowing in the JFET and is plotted in Fig. 3 versus the JFET’s gate-to-source voltage (Y-X).

For the symmetrical Twin-T configuration, the above equations become

2R2C2y - 2KR2C2x - 2KRCk - 2RC(K - 2)%’ + Y = 0 (54

and

0’ - 2KCk + KI(X, Y) = 0 (sbl

Curreur feedback op amp

Fig. 4. (a) PSpice simulation of CFOA output, and (b) PSpice frequency spectrum simulation, for C = 100 nF. R = 1.592 kQ and K = 1.18.

394

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A. S. ELWAKIL and A. M. SOLIMAN

G 1

Fig. 5.(a) and (b). Continued opposirr


I.0”

-0.0”

-..O’,

-2.0”.

Fig. 5. (a) PSpice simulation of CFOA output for C= 0.1 nF, R = 1.592 kR and K = 1.19; (b) Vcl YWSUS V, trajectory; (c) V,, versus Vc3 trajectory: (d) V, versus VC~ trajectory.

3% A. S. ELWAKIL and A. M. SOLIMAN

The circuit was experimentally tested with the following precision components: RI = Rz = 2 kL2 pots and RA = 1 kP; RB is a 1 kQ resistor in series with a 500 Q pot and using different sets of capacitor values. Figure 6(a), (b), (c) and (d) shows oscilloscope printouts of the CFOA output voltage with C = 0.01 PF, where period 2, period 3, period 5 and chaotic signals can be observed, respectively.

3. THE SECOND OSCILLATOR CONFIGURATION

Figure 7 represents the Wien oscillator using the CFOA as the active element [II]. The characteristic equation of this oscillator is given by eqn (la) with K given by (lbj. The passive network N of this oscillator is shown in Fig. 7(a), from which it is found that

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(b)

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Fig. 6 Osciiloscope printouts of (a) period 2, (b) period 3 and (c) period 5 and (d) chaotic signals with C = 0.01 IIF.

T,,(S) = R,C2S

R,R&CzS2 + (R,C, + RIC2 + R&S -I- 1 (6)

A well-known design equation for the Wien type oscillator is to take equal R and equal C, i.e.

RI = R2 = R and c, = cz = c Oa!

In this case the condition of oscillation is given by

K=3 0)

and the radian frequency of oscillation is given by

398 A. S. ELWAKIL and A. ti. SOLIMAN

Fig. 7. (a) Wien bridge network N; (b) Wien bridge nehvork N modified for chaos

1 wJ=--

RC

Figure 7(b) illustrates the proposed network N modified for chaos, where the linear resistor R, has been replaced with a JFET having its gate and drain terminals shorted and with an additional capacitor C, added. Figure 8 plots the gate-to-source voltage of the J2N4338 JFET versus its drain-to-source current during circuit operation, from which it can be seen that the JFET acts as a linear resistance of about 500 52 in its linear portion of operation.

Adopting the design equation given by (7a), and since R2 is now around 500 Q, then R, should be taken as around 500 Q. The sum of the two capacitors Cz and C3 should be taken as equal to C,; an easy choice is thus to take C? = Ci = C and C, = 2C where C is

Current feedback op amp 3%

arbitrarily chosen to define the signal’s frequency band of interest, which can be estimated from equation (7~). In order to change the circuit dynamics, the gain K should be slightly increased from the nominal value given by (7b). This is achieved by tuning one of the grounded resistors R, or RB. It is worth noting, however, that the circuit’s behavior is sensitive to small changes in K, As can be seen, the overall modified circuit requires one CFOA. three resistors, three capacitors and a single discrete nonlinear element. Figure 9(a) and (b) shows two time waveform PSpice simulations, given, respectively, for the following values:

C = 0.5 nF, R = 500 R and K = 3.15

C = 500 nF1 R = 500 Q and K = 3.15

Figure IO(a), (b), (c) and (d) re p resents PSpice simulations for the set of values C = 5 nF. R = 500 R and K = 3.135: where Fig. 10(a) is a time waveform, Fig. 10(b) is the corresponding spectrum and Fig. 10(c), (d) are the phase space trajectories.

Denoting the voltages of the two nonlinear device terminals as X and Y, where Y is

also the voltage across the grounded capacitor Cl, the system is described by the following pair of differential equations:

and

-[R&(K - 1) + R,(KC; - C,)]i’ + R&i’ + Y =0

KC,li - c,2 - Z(X. Y) =o

(84

tab)

where, Z(X, Y) is the drain-to-source current flowing in the JFET and is plotted in Fig. 8 versus the JFET’s gate-to-source voltage (X-Y). For the adopted design equation, the above equations become

and

RC(3-2K)tj+RCk+ Y=O Pa 1

KC?-C&Z(X:Y)=O (9b)

Experimental tests were done with the following values: R = 500 Spot: R, = 1 k8 and R, = 5 kQpot, with two capacitor sets: C, = C3 = 0.1 @. C, = 0.2 PF and Cz = C, = 1 nF, C, = 2 nF. OscilIoscope printouts for the first capacitor set are given in Figs, 11(a), (b). (c) and (d) with the first two figures demonstrating two different forms of periodicity and the latter two demonstrating chaos. For the second capacitor set, Fig. 12(a)-(g) shows oscilloscope printouts illustrating signals of periods 2, 4, 3, 5, 7, 9, 6, respectively. while Fig. 12(h), (i) and (j) shows chaotic signals,

4. COKCLUSIONS

Two new RC chaos generators have been reported. The two circuits are based on

ordinary sinusoidal oscillators that have been modified for chaos, which is an addition to recent research concerned with chaos investigation in conventional oscillators. It has been shown that the design equations of both circuits as ordinary oscillators are useful in the chaotic case for determining the start values for all components and estimating the chaotic frequency band. Laboratory experimental results and PSpice simulations have been in- cluded.


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‘1 I u

h

Fig. 9. PSpice simulation of CFOA output for (a) C = 0.5 nF, R = 500 Q and K = 3.15: (b) C = 500 nF, R=500QandK=3.15.


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Fig. IO.(a) and (b). C’onfhwd overleaf.

-SO&---- --1comv “Y l”“rn”

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Ailurn” .^^_. .

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“VU”

0 “(c3,1,-“(c3:2~

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Current fedhack op amp

DATE: 28-86-1996 TIME: 89:51:41

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4(14 A. S. ELWAKIL and A, M. SOLIMAN

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Fig. 11. (a), (h) Oscilloscope printout of two periodic signals with C = 0.1 $ and R = 500 Q. (c). (d) Oscilloscope printout of two chaotic signals with C = 0. I [AF and K = 500 R.


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A. S. ELWAKIL and A. .M. SOLIMAN

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Acknowledgements-The authors wish to thank Dr A. Namajunas and Dr A. Tamasevicius of the Lithuanian Semiconductor Institute for providing the JFET that has been used in experiments.

REFERENCES

1. Special issue on chaos in nonlinear electronic circuits, Part A: Tutorials and Reviews. IEEE Tran.r. CAS-40 (1993).

?. Special issue on chaos in nonlinear electronic circuits, Part B: Bifurcation and chaos. IEEE rwt~. CAS-40 (1993‘).

3. Special issue on chaos and nonlmear dvnamics. J Franklin Inst. 331B, 6 (1994). 1. Y. Ued a survey of regular and chaotic phcnomcna in the forced Duffinn oscillator. C. S. 8i F vol. I, No. 3

(1991) pp. 199-i31. - 5. M. P. Kennedy, Chaos in the Colpitts oscillator. IEEE Trans. CAS-41, 771 (1994). h. G. Sarafian and B. Z. Kaplan, Is the Colpitts oscillalor a relative of Chua’s circuit? EEE Trrans. CAS-42. 373

(1995). 7. A Namajunas and A. Tamasevicius, Modified Wien-bridge oscillator for chaos. Elccrronicr Lerr. 31, 335

(1995). X. 0. Morgul, Wien bridge based RC chaos generator. Electronics Len. 31: 2058 (1995). 9. C. Toumazou. J. Lidgy and A. Payne, Emerging Techniques for High Frequency BJT AmpIificr LIesip; .A

C’urrwr Mode Perspecrive. Parchment press Ltd, Oxford (1994). 19. A. M. Soliman, Applications of current feedhack operattonal amplifiers. Analog Integrated circuits and signal

processing. (To be published). 11. A. M. Soliman. Wien oscillators using current feedback op amps. (Submitted for publication).

Documents

Chaos from Two Modified Oscillator Configurations Using a ...1 w,, = ~ RC (3b) Figure 2 is a PSpice frequency spectrum simulation of this circuit’s output voltage for C = 1 nF1 R