Improved Implementation of Sprott s haotic Oscillators Based on urrent-feedback Op Amps Banlue Srisuchinwong and hun-hung Liou Sirindhorn International Institute of Technology, Thammasat University, Bangkadi ampus, Moo 5, Tiwanont Road, Muang, PaThum Tani, 000, Thailand, banlue@siit.tu.ac.th Abstract- Improved implementation of Sprott s chaotic oscillators based on current-feedback op amps (FOAs) is proposed. A Sprott s jerk function and four different types of nonlinear components are implemented using the attractively high-frequency features of the FOAs operating in both voltage and current modes. Four trajectories of the chaotic attractors are demonstrated. The chaotic spectrums are easily scaled and extended to higher frequencies by a factor of 77 to 80 In this paper, high frequency implementation of Sprott s chaotic oscillators is presented using current-feedback op amps (FOAs). The FOAs are currently recognized as versatile alternatives to the traditional op amps for their excellent performance in bandwidth and slew rates [9]. II. IRUIT IMPLEMENTATION I. INTRODUTION Over the past two decades there has been increasing interest in the study of chaotic oscillators [-]. haotic oscillators are useful tools not only for investigation of nonlinear phenomena, bifurcation and chaos, but also for a variety of applications such as synchronizations, control [4] and chaos-based communications systems [5]. hua s circuit [] is one of the best-known chaotic circuits but is difficult to scale to arbitrary frequencies because of the inductor with its frequencydependent resistive losses [6] although inductorless versions of hua s circuit have been possible [7]. Three reactive components (capacitors or inductors) and a nonlinear component are typically required for chaos systems with continuous flows so that the Kirchhoff representation of the circuit contains three first-order ordinary differential equations (ODEs). Recently, Sprott [6] has alternatively proposed chaotic oscillators based on a single third-order ODE in a simple form of d x/dt = F (d x/dt, dx/dt, x) called a jerk function (time derivative of acceleration). One of the Sprott s jerk functions in a general form is [6] x t x A t x t G ( x ) where G(x) is a nonlinear function and A = 0.6. The most straightforward implementation involves three successive active integrators to generate d x/dt, dx/dt and x from d x/dt coupled with a nonlinear element G(x) and feeds it back to d x/dt. Although Sprott s chaotic oscillators [6] based on () can be easily constructed using operational amplifiers (op amps) and be easily scaled to different frequencies, the operation has been somewhat delicate and the circuit has exhibited hysteresis because of the finite gain-bandwidth product and slew rates of the op amps [6, 8]. This problem has been circumvented by operating at a lower frequency of around.59 khz [6, 8]. () Figure. High frequency implementation of Sprott s chaotic oscillator using FOAs and each of G(x) shown in Figure.. Fig., shows the high frequency implementation of the Sprott s chaotic oscillator based on FOAs. The nonlinear components G(x) [6] can be implemented using FOAs as shown in each of the system in Figs. (a) to (d) where related parameters are also listed. In Fig., the FOA U 0 forms an integrator U 0 where the zero-db crossing (ZdB) frequency 0 = / 0 and 0 = R 0 0. The FOA U forms another integrator U where the ZdB frequency = / and = R. An expected integrator between U 0 and U is replaced by a simpler passive R filter. A routine analysis in Fig. reveals that the resulting jerk function at node N is K K K K G( x) x t t 0 X R R R X x 0 R K R R X 0 K R x t () ETI-ON 007 The 007 ETI International onference 8
Figure. Nonlinear components G(x) using FOAs. Figure. haotic attractors produced by Fig. using each of the nonlinear components G(x) shown in Fig.. ETI-ON 007 The 007 ETI International onference 9
where = R, R X = (R 0 R )/( R 0 +R ), and X = R X. As () = (), therefore K =, K = A = 0.6 and K =. For simplicity, let R = R = R 0 = R T. onsequently, = A/(R T ), = /(AR T ) and 0 = R /R 0 =. Table I summarizes the calculated values of the capacitors for R T = and also, with slight modification or scaling, the practical values of the capacitors for R T = k. TABLE I alculated and practical values of resistors and capacitors. omponents (Fig. ) alculated Values Practical Values G(x) in Figs. (a), (b), (d) G(x) in Fig. (c) 0 0.0 F 0.5 F /A 0.50 F 5.0 F A/ 0.0 F.00 F R 0 k k R k k R k k III. SIMULATION RESULTS The performances of the circuits shown in Figs. and have been simulated through Pspice. Models AD844/AD and AD-845/AD are FOAs with and without current feedback terminal, respectively. By using the practical values shown in Table I, the simulated trajectories of the chaotic attractors in the x-(dx/dt) plane are shown in Figs. (a)-(d) for each of the nonlinear components G(x) shown in Figs. (a)-(d), respectively. In Table I, the practical values of capacitors 0,, may be scaled down until the simulated chaotic attractors shown in Fig. corrupt at which point Table II records the minimum values of such capacitors for each G(x) shown in Fig. whilst maintaining the value of R 0 = R = R = k. onsequently, the operating frequency f S where the chaotic spectrum is centered will be maximum as shown in Table II. Table II Minimum values of capacitors and the corresponding maximum operating frequencies f S. G(x) Types (F) (F) 0 (F) Fig. OAs 0.0 u 0.50 u 0.0 u.49 k (a) FOAs 0.05 n.5 n 5.0 p 544 k Fig. OAs.00 n 50.0 n.00 n 4.6 k (b) FOAs 5.00 p 5.0 p.50 p 4.05 M Fig. OAs.00 u 5.0 u 0.50 u 4.49 (c) FOAs.50 n 6.5 n.5 n 9. k Fig. OAs 0.0 u 0.5 u 5.00 n.8 k (d) FOA 5.0 p 65.0 p.5 p. M f S fs ( FOAs) (Hz) fs ( OAs) As an example in Table II where G(x) is shown in Fig. (d), Fig. 4 shows the corresponding output voltage waveform x and the chaotic frequency spectrum (dbm) centered around f S =. MHz. The latter is obtained through the fast fourier transform of x. For purposes of comparisons, Table II also includes the minimum values of the capacitors and the resulting frequency f S for the similar cases using op amps 65 77 80 7 (OAs). It can be seen from Table II that the proposed implementation using FOAs enables higher f S by a factor of 77 to 80. Figure 4. An example of the output voltage waveform and the chaotic spectrum (dbm) centered around. MHz indicated in Table II. IV. ONLUSIONS High frequency implementation of Sprott s chaotic oscillators has been presented using FOAs. The Sprott s jerk function and four different types of nonlinear components have been implemented using FOAs. Four trajectories of the chaotic attractors have been illustrated. The operating frequencies are easily scaled and extended by a factor of 77 to 80. AKNOWLEDGMENTS Authors are grateful to Mr Wimol San-Um who brings the topics of chaotic oscillators to the authors attentions. REFERENES [] Special issue on chaos in nonlinear electronic circuits, Part A : Tutorials and Reviews, IEEE Transactions on ircuits and Systems 40(0), 99. [] Delgado-Restituto, M. and Rodriguez-Vazquez, A. (00), Integrated haos Generators, Proceedings of the IEEE, vol. 90, No. 5, May, 00, pp. 747-767. [] hen, G. and Ueta T. ed., haos in ircuits and Systems, World Scientific, Singapore, 00. [4] Special issue on chaos synchronization, control, and applications, IEEE Transactions on ircuits and Systems 44(0), 997. [5] Mandal, S. and Banerjee, S., Analysis and MOS Implementation of a haos-based ommunication System, IEEE Trans. ircuits and Systems Part I 5(9), Sep. 004, pp. 708-7. [6] Sprott, J.. A New lass of haotic ircuits,, Physics Letters A, 66, 000, pp. 9-. [7] Morgul, O. Inductorless realization of chua oscillator, Electron. Lett. (7),995,pp.40-404. [8] Sprott, J.. Simple haotic Systems and ircuits, Am. J. Phys. 68(8), Auguest, 000, pp. 758-76. [9] Elwakel, A.S. and Kennedy, M.P. Improved Implementation of hua s haotic Oscillator Using urrent-feedback Op Amp, IEEE Trans. ircuits and Systems, Part I, 47(), 000, pp. 76-79. ETI-ON 007 The 007 ETI International onference 40
Whereas, previous II-based MSO scheme is only suitable for bipolar technology and not guarantee generate the sinusoidal at the high frequency. Therefore, the proposed MSO has more flexibility for I implementation and dealing with the applications than the previous II-based MSO. II. IRUIT DESRIPTIONS Fig. shows the symbol of the multiple-output OTA. An ideal OTA is a finite bandwidth voltage-controlled current source, with an infinite input and output impedance. The output currents of the ideal multiple-output OTA is given by Io gm( VV) () where is the transconductance gain, I o the output current, V + and V - is the non-inverting and inverting input voltages, respectively. For the case of OTA implemented with MOS transistors operating in saturation, the transconductance ( ) is proportional to (I abc ) / and it implemented with bipolar transistors, the is directly proportional to I abc. V + I abc I o I I () s Based on Fig., the generated circuit for realizing an MSO is shown in Fig.. Assuming that all OTA in Fig. are identical, by using () the loop again can be expressed as L s s According to the Barkhausen criterion, the condition for the proposed circuit to provide sinusoidal oscillation of frequency is or s N N gm S jo N N N g ( ) 0 o m () (4) j (5) By expanding (5), it is show that (5) would have a solution only if the value of N is odd (N). By equating the imaginary and real parts to zero, respectively, the frequency of oscillation (FO) and the condition of oscillation (O) can be expressed as V - -I o FO o tan n (6) Fig.. ircuit symbol of the multiple-output OTA. I Fig.. Basic building block of the proposed multiphase oscillator. -I +I O tan n From (6) and (7), the FO and O of a three-phase sinusoidal oscillator (N=) can be given as and f o (7) (8) gm (9) (OTA) (OTA) (OTA) N -I o gm +I o -I o gm +I o Fig.. Generalized circuit for realizinultiphase oscillator. -I on +I on Fig. shows the basic block of the proposed multiphase oscillator circuit. It consists of a multiple-output OTA and two grounded capacitors. The transfer functions between the output and input terminal of the circuit in Fig. can be given by The frequency of oscillation and the condition of oscillation for realizing N-phase sinusoidal oscillator, equal in amplitude and equally spaced in phase, are summarized in Table. From Fig., the use of multiple output-ota provides an inverted of the output current. Thus, there are n=6, 0, 4 even-phase available output currents. Therefore, the MSO circuit of Fig. can generate both oddnumber and even-number of phase by a single circuit. From (8) and (9), the frequency of oscillation and the condition of oscillation can be orthogonally controllable. The oscillation condition can be adjusted the capacitor and the frequency condition can be tuned by electronically the transconductance through the bias current. The high frequency oscillation can be obtained without the effect of OTA bandwidth in term of oscillation condition. Since the output impedance of the OTA is very high, the MSO current ETI-ON 007 The 007 ETI International onference 4
outputs can be directly connected to the next stage without the using additional current followers. The active and passive sensitivities of proposed MSO circuit have low, approximately - to. ONDITION AND FREQUENY OF OSILLATION OF MULTIPHASE SINUSOIDAL OSILLATOR. Number of phase (N) ondition of oscillation Frequency of oscillation ( o ) =.7 / 5 =.7 0.78 / 7 =. 0.48 / 9 =.06 0.64 / III. SIMULATION RESULT To the theoretical analysis of the proposed multiphase sinusoidal oscillator, a MOS design example has been simulated through PSPIE simulation program. The PSPIE model parameters for NMOS and PMOS transistor are standard 0.5m MOS process of MOSIS. The multiple-output plus/minus OTA schematic is modified from well know single-ended OTA structure [9]-[0], as shown in Fig. 4. It consists of a source-coupled pair with identical MOS devices (M-M) operating in the saturation region [9] where the output current is replicated using the current mirrors. The MOS transistors aspect ratios are: 0m/m for M, M; 40m/m for M-M6, M9-M0, M-M4, M7-M8; and 46m/m for M7-M8, M- M, M5-M6, M9-M0. The power supply is V DD =-V SS =.5V. Fig. 5 presents the simulation results of proposed MSO circuit with =0pF, =0.pF, I abc =50A ( =0.7mS) for N= where was designed to be larger than times of to ensure the oscillation will start. Fig. 6 shows sinusoidal waveform for six phases. Fig. 5. The simulated output waveform of three phase oscillator of Fig.. Fig. 6. Simulated output waveform of six phase oscillator of Fig.. 8 7 -I o M7 M M9 M5 M M4 M6 M0 M4 M8 V - V + -I o M M +I o +I o I abc Oscillation Frequency, MHz 6 5 4 Simulated Theoretical M M7 M8 M 0 00 00 00 400 500 600 Bias urrent, A M5 M6 M9 M0 Fig. 4. MOS multiple-output OTA implementation used in simulation. Fig. 7. Variation of the oscillation frequency with the bias current. Fig. 7 presents the simulation results of oscillation frequency of Fig. by varying the value of the bias current I abc (i.e. 50A to 500A or equal from 0.98mS to 0.48mS) with =0pF and =0.pF. The relationship between and oscillation frequency is shown in Fig. 8, which is varied by on (8). It shows that the proposed ETI-ON 007 The 007 ETI International onference 4
MSO can be generated the frequency high up to 0MHz according with theoretical. The circuit be generated the frequency higher than 0MHz but the simulation results not confirm with theory, this error cause from the parasitic capacitor of OTA. Finally, Fig. 9 shows the output currents against varied bias current I abc. Oscillation Frequency, MHz Output urrent, A p-p 0 5 0 5 0 5 0 0 0 40 60 80 00 0, pf Simulated Theoretical Fig. 8. Variation of the oscillation frequency with capacitor. 500 450 400 50 00 50 00 50 00 50 0 00 00 00 400 500 600 Bias urrent, A Fig. 9. Output currents of proposed multiphase sinusoidal oscillator again varying bias current with =0pF and =0.pF. IV. ONLUSIONS In this paper, a new electronically tunable MSO circuit has been presented. The proposed MSO circuit has a simple configuration which uses a multiple-output OTA and two grounded capacitors per section. The MSO circuit can be configured to provide an odd-number of equal-amplitude equally special in-phase output current. The frequency and condition of oscillation are independent controlled. The proposed MSO enjoys simple structure, an electronically tunable and suitable for I implementation as both MOS and bipolar technologies. Simulation results, which confirm the theoretical analysis, are obtained. REFERENES [] E. Sanchez-Sinecio, J. Ramirez-Angulo, B. Linares-Barranco and A. Rodriguez-vazquez, Operational transconductance amplifier-based non-linear function syntheses, IEEE Journal of Solid-State ircuits, vol. 4, pp. 576-586, 989. [] A. Rodriguez-Vazquez, B. Linares-Barranco, J. L. Huertas and E. Sanchez-Sinencio, On the design of voltage-controlled sinusoidal oscillators using OTA s, IEEE Transactions on ircuits and Systems, vol. 7, pp. 98-, 990. [] I. A. Khan, M. T. Ahmed and N. Minhaj, Tunable OTA-based multiphase sinusoidal oscillators, International Journal of Electronics, vol. 7, pp. 44-450, 99. [4] J. Wu, urrent-mode high-order OTA- filters, International Journal of Electronics, vol. 78, pp. 9-6, 994. [5] T. Tsukutani, M. Ishida, S. Tsuiki and Y. Kukui, Versatile currentmode biquad filter usinultiple current output OTAs, International Journal of Electronics, vol. 80, pp. 5-54, 996. [6] Y. Sun and J. K. Fidler, Structure generation and design of multiple loop feedback OTA-grounded capacitor filter, IEEE Transaction on ircuits and Systems, vol. 44, pp. -, 997. [7] P. Prommee and K. Dejhan, An integrable electronic-controlled quadrature sinusoidal oscillator using MOS operational transconductance amplifier, International Journal of Electronics, vol. 89, pp. 65-79, 00. [8] B. Z. Kaplan and S. T. Bachar, A versatile voltage controlled three phase oscillator, IEEE Transaction on Industrial Electronics and ontrol Instrumentation, vol. 6, pp. 9-95, 979. [9] A. Rahman and S. E. Haque, A simple three-phase variablefrequency oscillation, International Journal of Electronics, vol. 5, pp. 8-89, 98. [0] V. P. Ramamurti and B. Ramaswami, A novel three-phase reference sinewave generator for PWM inverter, IEEE Transactions on Industrial Electronics, vol. 9, pp. 5-40, 98. [] W. B. Mikhael and S. Tu, ontinuous and switched-capacitor multiphase oscillators, IEEE Transactions on ircuits and Systems, vol., pp. 80-9, 984. []. Hou and B. Shen, Second-generation current conveyor-based multiphase sinusoidal oscillators, International Journal of Electronics, vol. 78, pp. 7-5, 995. [] D. S. Wu, S. I. Liu, Y. S. Hwang and Y. P. Wu, Multiphase sinusoidal oscillator using second-generation current conveyors, International Journal of Electronics, vol. 78, pp. 645-65, 995. [4] D. S. Wu, S. I. Liu, Y. S. Hwang and Y. P. Wu, Multiphase sinusoidal oscillator using the FOA pole, IEE Proceeding on ircuits, Devices and Systems, vol. 4, pp. 7-40, 995. [5] S. J. G. Gift, Multiphase sinusoidal oscillator system using operational amplifiers, International Journal of Electronics, vol. 8, pp. 6-67, 997. [6] S. J. G. Gift, Multiphase sinusoidal oscillator using inverting-mode operational amplifiers, IEEE Transactions on Instrumentation and Measurement, vol. 47, pp. 986-99, 998. [7] M. T. Abuelma atti and M. A. Al-Qahani, A new currentcontrolled multiphase sinusoidal oscillator using translinear current conveyor, IEEE Transactions on ircuits and Systems-II: Analog and Digital Signal Processing, vol. 45, pp. 88-885, 998. [8] M. Bhusan and R.W. Newcomb, Grounding of capacitors in integrated circuits, Electronics Letters, vol., pp. 48-49, 967. [9] M. Tan and R. Schaumann, Simulation general L-ladder filters for monolithic realizations with only transconductance elements and grounded capacitors, IEEE Transaction on ircuits and Systems, vol. 6, pp. 99-07, 989. [0] S. Szczepanski, A. Wyazynski, and R. Schaumann, High linear voltage-controlled MOS transconductances, IEEE Transaction on ircuits and Systems, vol. 40, pp. 58-6, 99. ETI-ON 007 The 007 ETI International onference 44