Chaotic electronic oscillator from single amplifier biquad

Int. J. Electron. Commun. (AEÜ) 66 (2012) 593–597

Contents lists available at SciVerse ScienceDirect

International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.de/aeue

LETTER

Chaotic electronic oscillator from single amplifier biquad Tanmoy Banerjee ∗ , Biswajit Karmakar, Bishnu Charan Sarkar Department of Physics, The University of Burdwan, Burdwan 713 104, West Bengal, India

a r t i c l e

i n f o

Article history: Received 22 February 2011 Accepted 29 November 2011 Keywords: Single amplifier biquad Nonlinear circuits Chaos Bifurcations

a b s t r a c t The present letter reports a simple chaotic electronic oscillator. A single amplifier biquad (SAB) based active high-Q Band Pass Filter (BPF) is converted into a chaotic oscillator by introducing a single passive nonlinear element in the form of a general purpose pn junction diode, and a storage element in the form of an inductor. The chaotic circuit is mathematically modeled, which is a set of four coupled first-order autonomous nonlinear differential equations. The behavior of the proposed circuit is investigated through numerical simulations and electronic hardware experiments. It is found that the circuit shows complex behaviors, like, bifurcations and chaos, for a certain range of circuit parameters. The chaotic behavior of the circuit is ensured qualitatively by bifurcation diagram, phase plane plot and experimentally obtained power spectrum, and quantitatively by Lyapunov exponents and Kaplan–York dimensions. © 2011 Elsevier GmbH. All rights reserved.

1. Introduction Design of chaotic electronic circuits and systems has been attracting the attention of researchers for at least three decades [1,2] mainly due to the following reasons: first, one can ‘observe’ chaos in laboratory by simply using off-the-shelf components (e.g., op-amps, passive components, diodes etc.); also, unlike other nonlinear systems, in a chaotic electronic circuit the control parameters are easily accessible. For example, a chaotic electronic circuit can be controlled by merely varying physically accessible parameters, like, resistors, voltage, etc. The second reason lies in its application potentiality in chaos based electronic communication systems, which is believed to be secure and interference free [3]. The third factor that motivates the researcher to design a chaotic electronic circuit is that we still do not know the sufficient condition(s) for designing a chaotic oscillator. After the advent of Chua’s circuit [4], different important chaotic circuits have been reported, e.g., chaotic Colpitts oscillator [5], hysteresis based chaotic circuit [6–8], one-diode chaotic circuit family [9,10], etc. Since then different chaotic oscillators based on sinusoidal oscillators have been reported [11]. Later, Elwakil and Kennedy [12] proposed a semi-sytematic methodology of designing a chaotic oscillator from a sinusoidal oscillator by introducing a suitable passive nonlinearity and a storage element. Using this

∗ Corresponding author. Tel.: +91 342 2657 800; fax: +91 342 2530 452. E-mail addresses: [email protected], [email protected] (T. Banerjee). 1434-8411/$ – see front matter © 2011 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2011.11.013

methodology many chaotic oscillators have been reported, e.g., chaotic Wien-bridge oscillator [12], current feedback op amp based chaotic oscillators [13], biquad based chaotic oscillator [14], etc. Investigations on sinusoidal oscillators are still on [15], and converting them into chaotic oscillator offers a great challenge to the circuit designers. In the present letter, we propose a new autonomous chaotic electronic circuit. A SAB [16,17] based active high-Q Band Pass Filter (BPF) is modified for generating chaotic oscillations using a general purpose pn junction diode as a passive nonlinear element, and a storage element in the form of an inductor. The proposed chaotic circuit is mathematically modeled, which is a set of four coupled first-order nonlinear autonomous differential equations. The behavior of the proposed circuit is investigated through numerical simulations and electronic hardware experiments. It is observed that the circuit shows complex behaviors like bifurcation and chaos for certain range of parameter values. Further, the chaotic behavior of the circuit is quantified by computing Lyapunov exponents and Kaplan–York dimension. 2. Proposed oscillator for chaos generation Fig. 1 shows the proposed chaotic oscillator. The circuit consists of two main parts: (i) a SAB based high-Q active Band Pass Filter (BPF) [16,17] and (ii) a parallel combination of a pn junction diode and inductor (DL), which is inserted between the V0 terminal (through a resistance R2 ) and the inverting terminal of the op amp. In the original SAB structure this DL arrangement is absent (i.e., V0 and the inverting terminal of the op amp are connected through R2 only). The SAB consists of an op amp, two capacitors having same

594

T. Banerjee et al. / Int. J. Electron. Commun. (AEÜ) 66 (2012) 593–597

value C and four resistors (R1 , R2 , Ra , Rb ). In the diode-inductor arrangement, the diode switches on and off according the voltage developed across the inductor. The inductor voltage appears across the parasitic transit capacitance (CD ) of the diode, and let that voltage be VCD . The circuit is mathematically modeled using conventional circuit analysis. Following is the mathematical model of the circuit, which is a set of four coupled first-order nonlinear autonomous differential equations in voltages (V0 , V1 and VCD ) and inductor current IL .

dV 0 2 (k + 1) 2(k + 1) = − V0 + V1 + VCD , R2 R1 R2 dt (2k + 1) k (2k + 1) dV 1 =− V0 + V1 + VCD , C R1 R2 dt (k + 1)R2 VCD dI L = , L dt dV CD 1 1 = V0 − IL − VCD − ID . CD R2 dt (k + 1)R2 C

Fig. 1. Circuit diagram of the proposed chaotic oscillator. A parallel diode (D)–inductor (L) arrangement has bean introduced between V0 (through resistor R2 ) and inverting end of the op amp.

(1)

Fig. 2. Phase plane plots in y–x plane for different values of r. (a) Period-1 at r = 1.4, (b) period-2 at r = 1.54, (c) period-4 at r = 1.57 and (d) chaotic attractor at r = 1.64 (k = 0.4, b = 0.36,  = 0.1, KD = 10).

T. Banerjee et al. / Int. J. Electron. Commun. (AEÜ) 66 (2012) 593–597

595

Here k = Rb /Ra . The nonlinear diode current ID has been modeled as a piece-wise linear function of voltage such that ID

1 (VCD − V) if VCD ≥ V , RD = 0 if VCD < V .

=

(2)

RD is the diode forward conductance resistance and V is the diode forward voltage drop. The sets of equations given in (1) and (2) can be written in the following dimensionless form: x˙ = −2x + r(k + 1)y + 2(k + 1)p, (2k + 1) y˙ = − x + rky + (2k + 1)p, (k + 1) ˙z = bp, 1 x − z − (1 + a)p + a. p˙ = (k + 1)

(3)

where we have defined the following dimensionless quantities:  = t/CR2 , u˙ = du/d, x = V0 /V , y = V1 /V , p = VCD /V , z = R2 IL /V , r = R2 /R1 , b = R22 C/L,  = CD /C, KD = R2 /RD . The quantity a is defined as, a

= KD if p ≥ 1, = 0 if p < 1.

Fig. 3. Chaotic attractor in x–z space for r = 1.64 (k = 0.4, b = 0.36,  = 0.1, KD = 10).

(4)

To find out the equilibrium values of (3) we have to consider two regions of operations of (4). These equilibrium points are given by (x0 , y0 , z0 , p0 ) = (0, 0, a, 0). Therefore, in the region p < 1 (i.e., for a = 0), there is a single trivial equilibrium point. In the region p ≥ 1, the equilibrium point is virtual, meaning that it lies outside this region [18]. 3. Numerical results To explore the dynamics of the system, numerical integration has been carried out on (3) using fourth order Runge–Kutta algorithm with step size h = 0.001. Fig. 2 shows the phase plane representation (in y–x plane) of the system for different values of the control parameter r, keeping the values of other parameters fixed at k = 0.4, b = 0.36,  = 0.1, KD = 10. For r < 0.36, it shows a single trivial equilibrium point at (0, 0, 0, 0); at r = 0.36 a stable limit cycle is formed through Hopf bifurcation. With the increase in r, amplitude of this limit cycle increases. Fig. 2(a) shows a limit cycle at r = 1.4. At r = 1.43, the period-1 cycle becomes unstable giving rise to a period2 oscillation. Fig. 2(b) shows period-2 behavior for r = 1.54. Further increase in value of r results in a period doubling route to chaos (Fig. 2(a)–(d)). Chaotic oscillation sets in at r = 1.59. Phase space representation of the chaotic behavior is shown in Fig. 2(d) (in the y–x plane), and Fig. 3 (in the x–z plane) at r = 1.64. To observe the detailed dynamical behavior of the system for a range of values of r, bifurcation diagram has been drawn using poincare section at y = 0.5 with dy/dt < 0, for different values of r excluding the transient values (with other parameters fixed at k = 0.4, b = 0.36,  = 0.1, KD = 10). Fig. 4 shows the bifurcation diagram of x; it shows the period doubling route to chaos. It can be seen that, Period-1 behavior persists up to r < 1.43; period-2 for 1.43 < r < 1.575, period-4 for r ≥ 1.575, and chaos occurs for r ≥ 1.59. Thus, bifurcation diagram agrees with phase plane plots. However, due to five dimensional parameter space, there exists a large number of choices of parameters for which the circuit shows chaotic behavior. Fig. 5 depicts the numerically computed power spectrum of x with normalized frequency. Fig. 5(a) shows period-1 behavior for r = 1.4, and Fig. 5(b) shows chaotic power spectrum at r = 1.64 (k = 0.4, b = 0.36,  = 0.1, KD = 10). Note that, the chaotic power spectrum is broad band in nature. Eigenvalues of (3) in the two regions have been calculated with the parameters used in Fig. 3. For p < 1, eigenvalues are (−1.767,

Fig. 4. Bifurcation diagram of x with r as a control parameter. It can be seen that the diagram shows a period doubling route to chaos (k = 0.4, b = 0.36,  = 0.1, KD = 10).

Fig. 5. Numerical power spectrum (Sf , dB) of x variable of the system with normalized frequency (f, Hz). (a) Period-1 oscillation at r = 1.4, (b) chaotic oscillation at r = 1.64 (k = 0.4, b = 0.36,  = 0.1, KD = 10).

596

T. Banerjee et al. / Int. J. Electron. Commun. (AEÜ) 66 (2012) 593–597

Fig. 6. The oscilloscope trace of experimentally obtained phase-plane plots in the V1 –V0 space, and time series plots of V0 . Phase plane plots: (a) period-1 at R2 = 850 , (b) period-2 at R2 = 900 , (c) period-4 at R2 = 940 , (d) chaos at R2 = 1.1 k, (e) time series plots of period-1 of V0 corresponding to (a), (f) time series plots of chaotic V0 corresponding to (d). (R1 = 300 , Ra = 2 k, Rb = 500 , C = 1 nF and L = 7.8 mH.) For (a)–(d): V1 (x-axis): 0.5 v/div, V0 (y-axis): 1 v/div. For (e) and (f) V0 (y-axis): 1 v/div, time (x-axis): 10 ␮s/div.

− 0.473, 0.244 ± j0.435) indicating an unstable focus. Eigenvalues corresponding to the virtual equilibrium point (i.e. in the region p ≥ 1) are (−11.146, − 0.02, − 0.284 ± j0.923), which indicates that the virtual equilibrium point is stable. Further, to quantify the chaotic behavior, Lyapunov exponents (LE) of the system has been computed using algorithm proposed in [19]. For the parameters used in Fig. 3, we have derived the following values of LEs: 1 = 0.0192, 2 = − 0.00008 (∼ =0), 3 = − 0.2568, 4 = − 5.7137. Consequently, Kaplan–York dimension (DKY ) of the

system is DKY = 2.0729, which is a fractional number. Presence of a positive LE, and fractional value of DKY confirms the occurrence of chaotic behavior in the system. 4. Experimental results Electronic hardware experiment is carried out using discrete components. Op amp used in the experiment is TL082 with ±12 V power supply; diodes are general purpose 1N1183 diodes.

T. Banerjee et al. / Int. J. Electron. Commun. (AEÜ) 66 (2012) 593–597

Fig. 7. Experimental power spectrum of chaotic oscillation of V0 (circuit parameters are same as described in Fig. 6(d)). (Frequency span: 5–500 kHz.)

597

Although, the circuit is a 4-D autonomous circuit but, due to the small value of  (i.e. small value of the parasitic capacitance of the pn junction diode) the circuit effectively lives in a 3-D subspace (determined by 3 storage elements). Also, we have considered more practical model of the pn-junction diode (unlike [10], where diode has been considered as an ideal switch). Numerical integration and real world hardware experiments agree with each other, and confirm that the circuit shows bifurcation and chaos for a certain range of parameter values. Occurrence of chaotic behavior has been ensured by computing Lyapunov exponents, Kaplan–York dimension and experimental power spectrum. This circuit may be useful as a simple and well characterized low-dimensional chaos generator in chaos based electronic communication systems. Also, this type of piece-wise linear (PWL) circuits show slow-fast dynamics (as is evident from the small value of ), which is also shown by many biological systems (e.g., neural dynamical systems); that is why, studies on this PWL circuit may be important for better understanding of the dynamics of biological oscillations. Acknowledgement

Following values of components are used throughout the experiment: C = 1 nF (±5%) and L = 7.8 mH (±10%). All the resistors have ±5% tolerance. We have fixed the following resistance values: Ra = 2 k, Rb = 500  and R1 = 300  (using a 1 k potentiometer (POT)). R2 has been varied (using a 2 k POT) for exploring different behavior of the circuit. Fig. 6 shows the circuit behavior for the variation of R2 . For R2 < 870 , we observed sinusoidal oscillation (period-1). With the increase of R2 this sinusoidal oscillation becomes distorted. Fig. 6(a) shows the real time oscilloscope trace of phase plane plot of period-1 oscillation in V1 –V0 space, and the corresponding time series of V0 is shown in Fig. 6(e) at R2 = 850 ; the frequency of oscillation is 54.13 kHz. At R2 = 870 (approx.) the period-1 oscillation becomes unstable and period-2 oscillation emerges. Fig. 6(b) shows the period-2 behavior for R2 = 900 . With the increase of R2 , further period doubling occurs (Fig. 6(c)), and the circuit finally goes into chaotic mode of oscillations at R2 = 960 (approx.). Fig. 6(d) shows the oscilloscope trace of this chaotic attractor in the V1 –V0 space at R2 = 1.1 k. It can be noted that numerical simulation results (Fig. 2) and experimentally obtained results qualitatively agrees well with each other. Time series plot of V0 of the chaotic attractor of Fig. 6(d) is shown in Fig. 6(f). Note that, the real time plot seems to be random in nature. Fig. 7 shows the power spectrum (measured with Agilent E4411B spectrum analyzer) of the chaotic waveform V0 of Fig. 6(d); the broad frequency spectrum indicates the occurrence of chaotic oscillations. 5. Conclusion A simple but dynamically complex chaotic electronic oscillator has been reported in this letter. The circuit is simple, because to design the circuit one needs only one operational amplifier as an active element, a passive nonlinear element in the form of a general-purpose pn junction diode and three storage elements (two capacitors and one inductor). Mathematical modeling reveals that the circuit can be represented by four coupled first-order nonlinear autonomous differential equations in voltages and current.

One of the authors (T. Banerjee) gratefully acknowledges the financial support provided by the University Grants Commission (UGC), India (Project No. F No. 34-506/2008 (SR)). References [1] Ogorzalek MJ. Chaos and complexity in nonlinear electronic circuits. World Sci Ser Nonlinear Sci Ser A 1997;22. [2] Ramos J. Introduction to nonlinear dynamics of electronic systems: tutorial. Nonlinear Dyn 2006;44:3–14. [3] Banerjee T, Sarkar BC. Chaos and bifurcation in a third order phase-locked loop. AEU Int J Electron Commun 2006;62:86–91. [4] Chua LO. The genesis of Chua’s circuit. AEU Int J Electron Commun 1992;46:250–7. [5] Kennedy MP. Chaos in the Colpitts oscillator. IEEE Trans Circuits Syst I Fundam Theory Appl 1994;41:771–4. [6] Saito T. An approach toward higher dimensional hysteresis chaos generators. IEEE Trans Circuits Syst I Fundam Theory Appl 1990;37:399– 409. [7] Saito T. Reality of chaos in four-dimensional hysteretic circuits. IEEE Trans Circuits Syst I Fundam Theory Appl 1991;38:1517–24. [8] Mitsubori K, Saito T. Reality of chaos in four-dimensional hysteretic circuits. IEEE Trans Circuits Syst I Fundam Theory Appl 1991;38:1517–24. [9] Saito T. A chaotic circuit family including one diode. Electron Commun Jpn 1989:52–9. [10] Saito T. A simple hyperchaos generator including one ideal diode. IEICE Trans Fundam Electron Commun Comput Sci 1992;E75-A:294–8. [11] Namajunas A, Tamasevicius A. Modified Wien-bridge oscillator for chaos. Electron Lett 1995;31:335–6. [12] Elwakil AS, Kennedy MP. A semi-systematic procedure for producing chaos from sinusoidal oscillators using diode-inductor and fet-capacitor composites. IEEE Trans Circuit Syst I 2000;47:582–90. [13] Elwakil AS, Kennedy MP. Chaotic oscillators derived from sinusoidal oscillators based on the current feedback opamp. Analog Int Circuits Signal Process 2000;24:239–51. [14] Banerjee T, Karmakar B, Sarkar BC. Single amplifier biquad based autonomous electronic oscillators for chaos generation. Nonlinear Dyn 2010;62: 859–66. [15] Psychalinos C, Souliotis G. A log-domain multiphase sinusoidal oscillator. AEU Int J Electron Commun 2008;62:622–6. [16] Deliyannis T. High-Q factor circuit with reduced sensitivity. Electron Lett 1968;4:577–678. [17] Friend JJ. A single operational-amplifier biquadratic filter section. IEEE Int Symp Circuit Theory 1970:189–90. [18] Chua LO, Desoer CA, Kuh ES. Linear and nonlinear circuits. McGraw-Hill; 1987. [19] Wolf A, Swift JB, Swinney HL, Vastano JA. Determining Lyapunov exponents from a time series. Physica D 1985;16:285–317.