On the Possibility of Chaos Destruction via Parasitic Properties of the Used Active Devices ZDENEK HRUBOS Brno University of Technology Department of Radio Electronics Purkynova 8, 62 00 Brno CZECH REPUBLIC xhrubo00@stud.feec.vutbr.cz JIRI PETRZELA Brno University of Technology Department of Radio Electronics Purkynova 8, 62 00 Brno CZECH REPUBLIC petrzelj@feec.vutbr.cz Abstract: This paper reviews mathematical analysis, synthesis and simulation of algebraically simple chaotic flows and circuits implementations based on OTA elements and the possibility of chaos destruction via parasitic properties of the used active devices. The first part of article deals with mathematical analysis and calculations of eigenvalues. The second part of article deals with mathematical analysis and calculations of eigenvalues with thinking of influences of active elements parasitics. The last part deals with circuit realization and PSpice simulations. The main contribution of this article is in using of operational transconductance amplifiers working in mixed mode and study of influences of active elements parasitics. Performances of the proposed circuit are confirmed through PSpice simulations with consideration of influences of active elements parasitics. Key Words: Chaos, parasitic, eigenvalues, differential equations, OTAs elements Introduction Lately, several authors [], [2] have been successfully using the operational transconductance amplifier (OTA) as the main active element in continuous-time active filters. The OTA is an amplifier whose differential input voltage produces an output current. Thus, it is a voltage controlled current source (VCCS). There is usually an additional input for a current to control the amplifier s transconductance. OTA has only a single high-impedance node, in contrast to conventional op amps. This makes the OTA an excellent device candidate for high-frequency and voltage (or current) programmable analog basic building blocks. In this article is a simple authentication, how to simply realize a real physical systems electronically by using OTAs elements []. Simple system of three autonomous ordinary differential equations (ODEs) with any nonlinearity can exhibit chaos. When we talk about chaos motion we talk about a very specific solution of nonlinear dynamical systems which are widely exist in nature. Therefore, at the present time, research is focused onto relations between the real physical systems, its mathematical models and circuits realizations. From this perspective, electronic circuits can be used to modeling and observation of chaos. [5], [6], [7]. 2 Mathematical analysis Consider algebraically simple three-dimensional ODEs with six terms and one nonlinearity [8] as multiple of two state variables in the following general form ẋ = 0.36x + z ẏ = xz y () ż = x + y In the most publications [7], [9] the authors start with the mathematical model analysis together with the numerical solution of the system parameters. J. Sprott computed and described countless of simple chaotic flows [8]. For such systems, the positions of equilibria (critical) points are independent on the parameter and are located for first system at (0, 0, 0) and ( 2.5, 2.5, ). Investigation of vinicity around critical point is given by (4). det(λi J) = 0 (2) Jacobian matrix, characteristic polynomial and calculation of the eigenvalues of the first system can ISBN: 978--6804-3-9 204
be found in many literature [8], [5]. Jacobian matrix of this system is following J = 0.36 0 z x 0, (3) and is leading to the characteristic polynomial. For critical point (0, 0, 0) is following det(λi J) = λ + 0.36 4 0 0 λ 0 λ = = λ 3 + 0.64λ 2 + 0.64λ + = 0. and for critical point ( 2.5, 2.5, ) is following det(λi J) = λ + 0.36 4 0 λ 2.5 λ = = λ 3 + 0.64λ 2 + 3.4λ + 0.9 = 0. (4) (5) The local behavior of the system near the origin is determined by the eigenvalues: f : λ,2 = 0.8 ± 0.984i λ 3 =, (6) f 2 : λ 4,5 = 0.47 ± 0.985i λ 6 =.00, (7) The system have two real eigenvalue a and two complex-conjugate pair (a so-called saddle focus). 3 Influences of active elements parasitics V i Ii C in_ota R in_ota gmv i R out_ota I 0 C out_ota Figure : Non-ideal model of transconductance amplifier. Non-ideal active element is depicted in Fig.. Parasitic analysis deals mainly with input and output properties of used active element that cause significant problems in the state space. Important parasitic admittances of the circuit (signed as Y p ) caused by the real input and output properties of used active elements are shown in Fig.2. Common input and output small signal parameters OTA are R in OT A = 455kΩ, R out OT A = 54kΩ, C in OT A = 2.2pF, C out OT A = 2pF. g m x.z g m3 g m2 X Y Z C Yp C 2 G Yp2 C 3 Figure 2: Schematic of circuit realization with important parasitic influences. We suppose four locations (two nodes and two input diferences admittance) where parasitics cause the highest impact. These parasitic admittances (Fig.2) can be expressed as Y p = G p + sc p = (G pin + G pout + + G pout3 ) + s(c pin + C pout + C pout3 ) = = + + + R in OT A R out OT A R out OT A3 + s (C in OT A + C out OT A + C out OT A3 ) Yp4 Yp3 (8) Y p2 = G p2 + sc p2 = G pout2 + sc pout2 = (9) = + sc out OT A2 R out OT A2 ISBN: 978--6804-3-9 205
Y p3 = G p3 = G pin3 = Y p4 = G p4 = G pin2 = R in OT A3 (0) R in OT A2 () ẋ = (A + A p + A 2p + A 3p ) x + Bxx T C (2) A = 0.36 (g m g m3 ) 0 g m3 0 G 0 g m2 g m2 0 B = A p = 0 0 A 2p = A 3p =, C = G p 0 0 0 0 G p2 G p3 0 G p3 G p3 0 G p3 G p4 G p4 0 G p4 G p4 0 0 0, (3) (4), (5), (6), (7) The local behavior of the system 2 near the origin with consideration of influence of parasitic elements is following J p = 0.3637 0.0022.0022 z.0022 x 0.9978 0.0207, (8) Characteristic polynomial for critical point (0, 0, 0) is following det(λi J p ) = = λ 3 + 0.707λ 2 + 0.697λ + 0.993 = 0, and for critical point ( 2.5, 2.5, ) is following det(λi J p ) = = λ 3 + 0.707λ 2 + 3.95λ + 0.805 = 0. Now, we get a new values af eigenvalues (9) (20) f 2 : λ 4,5 = 0.453 ±.782i λ 6 = 0.266. (2) f 4 : λ 0, = 0.47 ±.787i λ 2 = 0.236. (22) (C + C p ) du dt = (g m g m3 G p G p3 G p4 )u + G p4 u 2 + (g m3 + G p3 )u 3 C 2 du 2 dt = G p4u (G + G p4 )u 2 + u u 3 (C 3 + C p2 ) du 3 dt = ( g m2 + G p3 )u + + g m2 u 2 (G p2 + G p3 )u 3 (23) Figure 3: Influence of parasitic elements on given system () - projection X-Y (red-with parasitic, bluewithout parasitic). 4 Circuitry realization and simulation The circuit design procedure is based on classical circuit synthesis [4]. Parasitic properties of the active devices are. The advantage is also evident: a small number of passive and active circuit elements. Operational transconductance amplifiers OPA860 are used for circuitry implementation of mathematical models. Nonlinearities are formed by connection of fourquadrant analog multipliers AD633. The schematics of the oscillators are shown in Fig. 2. Values of used passive elements were chosen C = C 2 = C 3 = 47nF, R = k (variable). Simulation results are shown in Fig. 5 resp. Fig. 6. ISBN: 978--6804-3-9 206
2.0V.0V 0V -.0V -2.0V -3.0V -3.0V -2.0V -.0V 0V.0V 2.0V 3.0V V(x) -V(z) Figure 4: Numerical analysis of given system (2) for g m = 0.4 - projection X-Y. Figure 5: Circuit simulation - influence of parasitic elements on given system (). 2.0V 5 Conclusion We showed calculations of eigenvalues and circuitry implementations of nonlinear circuits using OTAs elements in this article. In this article are shown calculations of eigenvalues with thinking of influences of active elements parasitics. Simple method as to electronically implement nonlinear circuits using OTAs elements is also shown. The suitability of OTAs as the main active element to obtain basic building blocks for the design of nonlinear circuits was verified, but very important is thinking of influences of active elements parasitics. The solution is in change of any parameter to compensate influences of active elements parasitics. In our case we change value of g m. Acknowledgements: The support of the project CZ..07/2.3.00/20.0007 WICOMT, financed from the operational program Education for competitiveness, is gratefully acknowledged. The described research was performed in laboratories supported by the SIX project; the registration number CZ..05/2..00/03.0072, the operational program Research and Development for Innovation. The research leading to these results has also received funding from the European Community Seventh Framework Programme (FP7/2007-203) under grant agreement no. 23026. The research is a part of the COST action IC 0803, which is financially supported by the Czech Ministry of Education under grant no. OC0906. The research is also part of the specific research grant de-.0v 0V -.0V -2.0V -3.0V -3.0V -2.0V -.0V 0V.0V 2.0V 3.0V V(x) -V(z) Figure 6: Circuit simulation - with compensate of influence of parasitic elements on given system (). noted FEKT-S--3. References: [] E. Sanchez-Sinencio, J. Ramirez-Angulo, B. Linares-Barrancom and A.Rodriguez- Vazquez, Operational Transconductance Amplifier-Based Nonlinear Function Syntheses, IEEE Journal of Solid-State Circuits, Dec. 989, vol. 24, no. 6, p. 576 586. [2] R. Sotner, J. Jerabek, T. Dostal and K. Vrba, Multifunctional Adjustable Current Mode Biquads Based on Distributed Feedback Voltage Mode Prototype with OTAs, International Jour- ISBN: 978--6804-3-9 207
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