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Simple Two-Transistor Single-Supply Resistor-Capacitor Chaotic Oscillator Lars Keuninckx, Guy Van Der Sande and Jan Danckaert†

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Simple Two-Transistor Single-Supply Resistor-Capacitor Chaotic Oscillator Lars Keuninckx, Guy Van Der Sande and Jan Danckaert† IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II, VOL. X, NO. Y, DECEMBER 2015 1 Simple Two-Transistor Single-Supply Resistor-Capacitor Chaotic Oscillator Lars Keuninckx, Guy Van der Sande and Jan Danckaerty Abstract—We have modified an otherwise standard one- adding a dimension to a predominantly two-dimensional limit transistor self-biasing resistor-capacitor phase-shift oscillator to cycle oscillator. In other cases, notably the Collpits oscilla- induce chaotic oscillations. The circuit uses only two transistors, tor [7], a chaotic regime is already present for certain values no inductors, and is powered by a single supply voltage. As such it is an attractive and low-cost source of chaotic oscillations for of the design parameters. Others directly translate a known many applications. We compare experimental results to Spice chaotic system of differential equations to electronics. The simulations, showing good agreement. We qualitatively explain necessary nonlinear terms are implemented by using dedicated the chaotic dynamics to stem from hysteretic jumps between analog multipliers as in reference [8] or operational amplifier unstable equilibria around which growing oscillations exist. based piece wise continious functions [9]. Being able to Index Terms—chaos, oscillator. nonlinear dynamics, RC- avoid operational amplifiers and dedicated multipliers is a ladder. plus in terms of cost and circuit complexity, as there is Copyright (c) 2014 IEEE. Personal use of this material is no connection between complexity -in terms of used parts- permitted. However, permission to use this material for any of the circuit and complexity of the chaotic oscillations -in other purposes must be obtained from the IEEE by sending terms of measurable quantitative properties such as entropy, an email to [email protected] attractor dimension, spectral content etc. Therefore chaotic oscillators using minimal and more basic components as in references [10] and [11] are desirable from an economic and I. INTRODUCTION logistic viewpoint. ECENTLY the interest in chaotic systems has been Elwakil and Kennedy conjecture that every autonomous R revived following the advent of novel and worthwhile chaotic oscillator contains a core sinusoidal or relaxation applications. Chaotic systems are now being used for so- oscillator [12]. Accordingly, one can derive a chaotic oscillator called chaos encryption, in which data signals can be secured from any sinusoidal or relaxation oscillator. Motivated by by hiding them within a chaotic signal. Relying on chaos this, we set out to modify the well known resistor-capacitor synchronization the original messages can then be safely (RC) phase shift oscillator, described as early as 1941 in recovered [1]. Also, the long-term unpredictability of the reference [13] to produce chaos. It has been used previously chaotic signal makes these systems well suited for true random as the basis for chaotic circuits. Hosokawa et al. report bit generation [2]. In robotics, chaotic signals are being used on chaos in a system of two such oscillators, coupled by in neural control networks or as chaotic path generators [3]. a diode [14]. They analyze the system as two nonlinearly Finally, from a more fundamental point of view, a shift in coupled linear oscillators. Ogorzalek describes a chaotic RC interest towards networks of interconnected chaotic unit cells phase shift oscillator based on a piecewise-linear amplifier can be observed [4]. In all these applications, a need exists built using operational amplifiers, however the RC-ladder is for flexible and preferably low cost chaotic circuits. Discrete unmodified [15]. In our approach we use a transistor based RC components are preferable over operational amplifiers and oscillator to which a small subcircuit is added, directly inter- specialized multiplier chips, the latter only being produced by acting with the RC-ladder itself. The purpose of the subcircuit a few suppliers. Single supply and low voltage operation widen is to add a nonlinearity in the RC-ladder. The attractiveness the applicability, as does a wide easily tunable frequency of the resulting circuit lies in its simplicity and low cost and range. Finally, using chaotic circuits as units in large networks parts count. No specialized parts such as dedicated multipliers adds the condition that the couplings between these units are or rare inductor values are needed. Neither the component dependable and stable. Requiring inductorless circuits will values nor the supply voltage is critical. The operation does not avoid such difficulties. depend on the dynamic properties of the active components, Since the development of Chua’s circuit [5], there has been as is often the case in even simpler chaotic circuits employing a considerable interest in the construction of autonomous only one transistor. The main oscillating frequency has a range chaotic circuits. Well known circuits, such as the Wien bridge of over five decades, from below 1 Hz up to several hundred oscillator, have been modified to produce chaos [6]. Often, kilohertz. which can be reached by scaling the capacitors these modifications involve the addition of an extra energy and/or resistors. storage element such as an inductor at the right place, thereby Additionally, the circuit has the attractive feature that the underlying core oscillator is clearly visible in the structure. yThe authors are with the Applied Physics Research Group (APHY), Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussel, Belgium, correspon- This makes it a great introductory pedagogical tool for students dence e-mail: [email protected] interested in chaos. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II, VOL. X, NO. Y, DECEMBER 2015 2 Vp consisting of Q2, R2, R3, R4 and C2 is responsible for the chaotic behaviour. This subcircuit adds a new equilibrium R 1 in which transistor Q1 is also biased as an active amplifier, enabling oscillations. It is clear that for low R4 or low Vp, v v i CE1 v1 2 vBE1 transistor Q2 does not conduct and the circuit reduces to B1 the unmodified phase shift oscillator. The base oscillation RRR1 frequency in the chaotic operating regime for R = 44 kΩ, CCC 4 Q1 Vp = 5 V is approximately 44 kHz. The circuit consumes 3:1 mW at 5 V. In Figure 2, we show the time trace of vCE1. The dynamical behavior consists of jumps between two states R 3 R 2 of high and low voltage of the collector of Q . In between v 2 i CE2 these jumps growing oscillations are observed on the collector vBE2 B2 of Q1. Note that the collector voltage of Q2 has an almost RCQ 4 2 2 binary distribution. In Figure 3(a), we show an oscilloscope picture of vCE1 vs. vCE2 for R4 = 44 kΩ and Vp = 5 V, from which the bistable oscillations around two unstable equilibria are clearly visible. Figures 3(b) and 3(c), showing vCE1 vs. Figure 1. A two-transistor chaotic RC phase shift oscillator. The components v2, and v1 vs. v2 respectively, also attest to this, where in the dashed line box cause chaotic dynamics in an otherwise standard self bi- the openings or ’eyes’ of the attractor indicate a possible asing RC oscillator. The components have the following values: R = 10 kΩ, unstable equilibrium at their center. To estimate the largest R = 5 kΩ, R = 15 kΩ, R = 30 kΩ, C = 1 nF, C = 360 pF, and 1 2 3 2 Lyapunov exponent, we use the Time Series Analysis ’Tisean’ VP = 5 V. package[16], from a timetrace consisting of 50000 points of vCE1, representing 10 ms, as λ ≈ (0:045 ± 0:005) =µs.A 2 positive Lyapunov exponent is a strong indication of chaotic vCE1 vCE2 dynamics. In Figure 4, we show oscilloscope pictures of v1 1.5 vs. vCE1 for increasing values of R4 at Vp = 5 V, indicates [V] the evolution of the dynamics to chaos. For R4 = 0 kΩ the 1 subcircuit is inactive such that a simple limit cycle exists (not CE2 v shown). Close to R4 = 30 kΩ [Figure 4(a)] we see a period 0.5 doubling bifurcation. At R4 = 31:6 kΩ [Figure 4(b)] a critical CE1 v slowing down occurs as shown by the point of high intensity 0 on the analog oscilloscope picture. This value marks the onset of a bistable operation. Subsequent period doublings lead to 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 chaos as seen in Figure 4(d) and Figure 4(e). The attractor t [ms] consists of oscillations around two unstable equilibria with lower and higher average vCE1. The asymmetry between the Figure 2. Timetrace of vCE1 (upper trace, red in color) and vCE2 (lower upper and lower opening of the attractors is caused by different trace, blue in color) for R4 = 44 kΩ, Vp = 5 V, other component values average collector current of transistor Q1, resulting in different as stated in the text. gain. This asymmetry is seen regardless of which projection is chosen. Further increase of R4 [Figure 4(f)] leads to a period II. CIRCUIT AND EXPERIMENTAL RESULTS halving sequence. Figure 1 introduces the circuit that we have designed. It III. MODEL AND SPICE SIMULATION consists of a standard single-supply self-biasing RC phaseshift The circuit of Figure 1 is described by: oscillator with an added subcircuit (inside the dashed line box) dv1 R RR3 interacting with the RC-ladder. Unless stated otherwise, the RC = − v1 1 + − (1) dt R1 R1(2R3 + R1) following component values are used: R = 10 kΩ, R1 = RR3 Vp vBE2 5 kΩ, R = 15 kΩ, R = 30 kΩ, C = 1 nF, C = 360 pF. + v2 + − iC1 + 2 3 2 2R + R R R The transistors Q and Q are of the type BC547C although 3 1 1 3 1 2 dv this is not critical. Both the first resistor of the RC-ladder and RC 2 = − 2v + v + v − i R; (2) dt 2 1 BE1 C2 the collector resistor of transistor Q1, have been chosen equal dvBE1 to 1=2R.
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