RC AUTONOMOUS CIRCUITS WITH CHAOTIC BEHAVIOUR

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1 Radioengineering R Autonomous ircuits With haotic Behaiour Vol., No., June 00. BERNÁT, I. BALÁŽ eter BERNÁT, Igor BALÁŽ Department o Radio and Electronics Faculty o Electrical Engineering and Inormation Technology Sloak Uniersity o Technology in Bratislaa Ilkoičoa, 8 9 Bratislaa Sloak Republic Abstract The paper presents obtained results with applying the Semi-systematic procedure or producing the chaos rom quasi-sinusoidal oscillators. We applied this procedure to R autonomous circuit and obtained two new autonomous circuits with chaotic behaiour. The two obtained circuits are presented. The usage o arious non-linear deices is examined. Keywords R oscillators, chaotic behaiour, chaos. Introduction R AUTONOMOUS IRUITS WITH HAOTI BEHAVIOUR Today, our types o steady state behaiour in deterministic non-linear systems are known: D steady state, periodic state, quasi-periodic state and chaotic state also known as either deterministic chaos or simpler chaos. Since begin o 980s these types o behaiour hae been also known in non-linear electronic circuits. lenty o autonomous and non-autonomous electronic circuits with chaotic behaiour hae been presented and analysed. The many o these circuits were designed by trial and error approach. This joyless act is due to the reality that the suicient conditions or existence o chaotic behaiour in a system are still unknown. Only necessary conditions or existence o chaotic behaiour in a system are known. A little better approach is based on designing o circuit according to known set o dierential equations which solution is chaotic (e.g. Lorenz system, Rössler system, etc.). Limitation in this case is the type o the non-linear term used in equations. As the non-linear term is oten used the multiplication o ariables and arious complicated piecewise-linear unctions []. These non-linear terms are not typical in electronic circuits. Their circuit implementations could be problematic and limit usage o such designed circuits. Examples o autonomous electronic circuits with chaotic behaiour designed by aboe described approach are in [4]. In [] Elwakil deals with the possible simpliication o designing circuits with chaotic behaiour by suggested procedure. This procedure is shortly described in second part o this paper. In the third part o this paper are presented R oscillators with chaotic behaiour. We hae eriied the unctionality o suggested circuits by simulation in M6 simulation program.. Semi-systematic procedure or chaos generation in autonomous circuits Authors in [] introduced the semi-systematic procedure or design the autonomous circuits with chaotic behaiour by modiication o sinusoidal oscillators. Here is necessary to point out, that the authors in [] used the term sinusoidal oscillator or simple oscillator with the Wienbridge. It is well known that signals in this circuit are not sinusoidal but more quasi-sinusoidal. So since this point we will use the more exact term quasi-sinusoidal. The suggested procedure has three steps. The irst step is design the appropriate quasi-sinusoidal oscillator that ulils all circuit requirements like type o actie component (transistor, operational ampliier, current eedback operational ampliier, etc.), type o the energy storage components (inductors, capacitors), their connection to circuit (loating, grounded), etc. Then the relations or necessary condition o oscillations and requency o oscillations are deried by routine linear analysis. The second and third steps are not as systematic as the irst step. In the second step designers according to their preious experiences and deried relations or necessary condition o oscillations and requency o oscillations choose the appropriate position to insert a non-linear component with appropriate oltage-current characteristic. I the designed oscillator is the circuit o second order the designers hae to insert either inductor or capacitor into the designed circuit to obtain the circuit o third order. The alue and position o connected inductor or capacitor also depend on experiences o designers. The third step is adjusting the alues o components until the circuit behaes chaotically. The authors in [] also suggest to use the passie nonlinear component and recommend to use the either JFET or diode. JFET is appropriate to connect in serial combination with a capacitor and a diode in parallel combination with inductor. The actie component should work linear. Eidently, the procedure is not straightorward to desired circuit and greatly depends on preious experiences o designers. We tried to use the procedure to modiy R quasi-sinusoidal oscillator with current eedback op-amp (FOA) to autonomous circuits with chaotic behaiour.

2 R Autonomous ircuits With haotic Behaiour Radioengineering. BERNÁT, I. BALÁŽ Vol., No., June 00. Inestigated circuits As a base R quasi-sinusoidal oscillator we chose the circuit presented in Fig.. The requency o oscillation is determined by eqn.. The condition o oscillation is determined by eqn.. I = R J V V V V V < V The shape o characteristic with one breakpoint is similar to oltage-current characteristic o diode. The dierence is in the alues o resistance or the irst harmonic or small oltages and large oltages. In case o diode the resistance or small oltages is greater than resistance or large oltages. In case o JFET the situation is reerse. () R R Fig. Quasi-sinusoidal oscillator with FOA = () π R R R R = () This circuit had been already used in [] to design circuit with chaotic behaiour. In [] the resistor R is replaced by JFET which works as oltage controlled resistor. As the third energy storage deice the authors used inductor in serial connection with resistor R. In our modiications we use capacitor instead o inductor. O course its connection to circuits hae to be dierent. RN R Fig. First inestigated circuit As a irst possible modiication we inestigated the circuit shown in Fig.. We considered using as a non-linear component either JFET, which operates as oltage controlled resistor or ordinary diode. The oltage-current characteristics o these deices are presented in Fig.. In [] authors in the mathematical models o their circuits approximated the oltage-current characteristic o JFET by piecewiselinear unction with one breakpoint (). V is breakpoint o characteristic (approximately 0.7 V), R J is small signal resistance close to coordinate system origin (approximately 750 Ω). These mathematical models reliable represented behaiour o their circuits. R Fig. Voltage-current characteristics o considered non-linear deices The appropriate shape o non-linearity should enable birth o oscillation in circuit and should limit their unbounded rise. The proper type o characteristic is possible to ind ater analysis o magnitude-requency characteristic and phase-requency characteristic o open loop circuit with connected non-linear resistor and third capacitor. This analysis is possible to perorm by computer simulation (A Analysis in M6) or exact analytical technique. Using the analytical technique leads to enormous mathematical expressions. The obtained results are complicated and blind. Thereore we used the simpler computer simulations. We disconnected the circuit in non-inerting input o FOA. We chose the alues o R, R,, according to linear condition or oscillation and required requency area. The alue o capacitor we chose close to alues o capacitors and. Instead o non-linear resistor we used a linear resistors R with dierent alues. We obsered inluence o resistor R alues to open loop gain. In inestigated circuit in case R <.5 kω open loop gain o circuit is greater than (birth o oscillation), in case R >.5 kω open loop gain is lesser than (limitation o oscillation). According to the obtained results we chose JFET as the non-linear component in oscillator. O course, this analysis and chosen non-linear deice do not guarantee the existence o chaotic behaiour in circuit. In the ollowing step by computer simulation (M6 program) o suggested circuit we were able to quickly ind a chaotic behaiour or arious alues o components presented in Tab.. Fig. 4 shows projection o inestigated circuit state trajectory onto plane V,V or the case = 0 nf, = 0 nf, = 8 nf. Open loop gain equals

3 Radioengineering R Autonomous ircuits With haotic Behaiour Vol., No., June 00. BERNÁT, I. BALÁŽ.9. In simulation we used model o JFET type N48 and model o FOA type AD844. = 5.6 nf. Open loop gain equals.7. We used model o JFET type N48 and model o FOA type AD844. Two presented R autonomous circuits with chaotic behaiour suggest that it is not complicated to design R autonomous circuits with chaotic behaiour. In the ollowing part o study we inestigated the possibility to use the diode and resistor connected in parallel combination instead o JFET by the similar manner as we used JFET in oscillators described aboe. According to our best knowledge a modiication o quasi-sinusoidal oscillator to R oscillator with chaotic behaiour by usage one diode has not been presented yet. As a base R quasi-sinusoidal oscillator we chose the circuit in Fig.. R R Fig. 4 rojection o state trajectory o the oscillator in Fig. 0 As the ollowing modiication we inestigated the circuit depicted in Fig. 5. We perormed analysis o open loop circuit by manner similar to analysis o preiously analysed circuit. R R 00 nf 00 nf 80.0 nf 0 Ω 500 Ω 0 nf 0 nf 8.0 nf 0 Ω 500 Ω nf nf.8 nf 0 Ω 500 Ω 00 pf 00 pf 80.0 pf 0 Ω 500 Ω Tab. According to the obtained results we chose the JFET as a proper non-linear deice. In the subsequent computer simulation by M6 program we were able to ind the chaotic behaiour in circuit. The alues o components or chaotic behaiour are shown in Table. haotic behaiour is presented in Fig. 6 by projection o state trajectory o analysed circuit onto plane V,V or case = 0 nf, = nf, 00 nf 0.0 nf 56.0 nf 0 Ω 700 Ω 0 nf.0 nf 5.6 nf 0 Ω 700 Ω nf. pf pf 0 Ω 700 Ω 00 pf 0.0 pf 68.0 pf 0 Ω 700 Ω Tab. We tried arious possible positions or connection o nonlinear two-terminal network built up by serial connection o capacitor with parallel connection o diode and resistor. The possibility o origin and limitation o generated signal amplitude we inestigated by analysis o magnitude-requency characteristic and phase-requency characteristic o open loop circuit with connected non-linear resistor and third capacitor. Instead o non-linear resistor we used linear resistors R with dierent alues. The process was same as a process that we described aboe. The result o this analysis was design o circuits presented in Fig. 7, Fig. 8. RN R R Fig. 6 rojection o state trajectory o oscillator in Fig. 5 Fig. 5 Second inestigated circuit = ( R R ) ( R R ) FOA FOA ( ) = ( ) ( ) (4) RFOA R RFOA R R ( ) =

4 4 R Autonomous ircuits With haotic Behaiour Radioengineering. BERNÁT, I. BALÁŽ Vol., No., June 00 As a irst candidate or possible chaos generation we inestigated the circuit shown in Fig. 7. We tried to ind the chaotic behaiour in designed circuit by trial and error approach by M6 simulation program. We did not ind chaotic behaiour. Thereore we decided to ind suitable alues o components by systematic method. We described this circuit by set o state equations (4). We considered linear model o FOA with input resistance o inerting input equals R FOA = 0 Ω. We approximated the oltagecurrent characteristic o diode by relation (5). I S = pa, α = 0.875, V T = 5.8 mv, V is oltage on diode, I is current through diode I ( V ) = I [ exp( α V V ) ] (5) = s T Then we inestigated behaiour o circuit or all possible combinations o the components alues that are shown in Tab.. The alue o capacitor = 50 nf was constant during all simulations. We ound plenty o combinations o components alues that cause the origin o quasi-periodic behaiour in circuit. We ound combinations components alues that cause the unbounded rise o oltages on capacitors and combinations o components alues that cause the attenuation and expiration o initial oscillations (open loop gain lesser than ). But we did not ind any chaotic behaiour. R RD D R Fig. 8 Fourth inestigated circuit R RD D As a ollowing step we inestigated the circuit shown in Fig. 8. The obtained results are like to results obtained by analysis o preious circuit. We did not ind any chaotic behaiour by M6 simulation program. R Fig. 7 Third inestigated circuit The initial conditions or all 5400 simulations were the same: V = mv, V = 0 V, V = 0 V. The type o behaiour in circuit was automatically identiied according to the distribution o intersections o state trajectory with oincaré plane during a long time interal. We used onesided oincaré plane dedicated by equation: = 0. For mentioned alues o components the open loop gain was in interal <0., 4.5>. We did not inestigated the cases with open loop gain lesser than. R R R D 50 nf 50 nf 00 Ω 470 Ω.0 kω 80 nf 80 nf 60 Ω 680 Ω. kω 00 nf 00 nf 0 Ω 80 Ω 5.0 kω 50 nf 50 nf 0 Ω 00 Ω 5.0 kω 00 nf 00 nf 90 Ω 00 Ω 50.0 kω 00 nf 00 nf kω Tab. = ( RFOA R ) ( RFOA R ) = ( RFOA R ) ( RFOA R ) R R R R D 50 nf 50 nf 0 Ω 90 Ω.0 kω 80 nf 80 nf 80 Ω 470 Ω. kω 00 nf 00 nf 0 Ω 560 Ω 5.0 kω 50 nf 50 nf 70 Ω 80 Ω 5.0 kω 00 nf 00 nf 0 Ω 800 Ω 50.0 kω 00 nf 00 nf kω Tab. 4 We described the circuit by the set o state eqns. (6) and inestigated the circuit or all possible combinations o the components alues that are shown in Table 4. The alue o capacitor = 00 nf was constant during simulations. For mentioned alues o components the open loop gain was in interal <0.4, 4.>. We did not inestigated the cases with open loop gain lesser than. We ound only quasi-periodic behaiours, unbounded rises o oltages on capacitors and expirations o initial oscillations (open loop gain lesser than ). We did not ind any chaotic behaiour. The same manner, we tried to examine circuit in Fig. with parallel combination o diode and resistor as nonlinear component. We tried 6480 arious combinations o components alues without inding any chaotic behaiour. ( ) ( )

5 Radioengineering R Autonomous ircuits With haotic Behaiour 5 Vol., No., June 00. BERNÁT, I. BALÁŽ ( ) = (6) R 4. onclusion We examined design the simple R autonomous circuits with chaotic behaiour by Semi-systematic procedure or producing the chaos rom sinusoidal oscillators. Firstly we tried to use the aised non-linearity introduced by JFET. Although mentioned procedure is not straightorward to the acquisition o circuit with chaotic behaiour, it is not complicated to design such circuit by usage o this procedure. The main problem is to choose a right position or connecting o JFET. rocedure, howeer, does not concern with the attributes o designed system like Lyapuno exponents, basis o attraction, statistical attributes o generated signals, power spectrums o generated signals etc. I designed autonomous circuit has to ulil some o that attributes procedure cannot be used. It does not also concern with design o hyper-chaotic systems. We inestigated the possibility to replace the aised JFET by diode. The modiication o quasi-sinusoidal oscillator to circuit with chaotic behaiour by means o one diode has not been presented yet. But none o inestigated circuits behaed chaotically. It could be caused by seeral reasons like usage improper topology o inestigated circuits, incorrect alues o components (we inestigated only inite number o combinations o components alues), incorrect initial condition (all simulation was done or one initial condition) etc. This possible modiication is urther inestigated. Reerences [] BERNÁT,., BALÁŽ, I. A Twin-T Oscillator Modiied or haotic Behaiour. In roc. 00 URSI Int. Symposium on Signals, Systems, Electronics. Tokyo (Japan), 00, p. -6. [] ELWAKIL, A., KENNEDY, M.,. haotic Oscillator Deried rom Sinusoidal Oscillator Based on the urrent Feedback Op Amp. Analog Integ. ir. Signal roc. 000, ol. 4, p. 9-5 [] ELWAKIL, A., S., KENNEDY, M.,. A Semi-systematic rocedure or roducing haos rom Sinusoidal Oscillator Using Diode-Inductor omposites and FET-apacitor omposites, IEEE Transactions on AS-I. 000, ol. 47, no. 4., p [4] SROTT, J.,. A new class o chaotic circuit. hysics Letters A. 000, p. 9- About authors... eter BERNÁT was born in 975 in iešťany. He receied Ing. degree in electrical engineering rom Faculty o Electrical Engineering and Inormation Technology, Sloak Technical Uniersity in Bratislaa in 000. urrently he is h.d. student at the Dept. o Radio and Electronics. He is interested in non-linear electronic circuits. Igor BALÁŽ was born in 99. He receied the h.d. degree and DrSc. degree both in electrical engineering in 97 respectiely 988. urrently he is Full roessor at the Department o Radio and Electronics at Faculty o Electrical Engineering and Inormation Technology, Sloak Technical Uniersity in Bratislaa.