Stability and Bifurcation Analysis of Electronic Oscillators: Theory and some Experiments

Transcription

1 ISE'9 J. Kenne J.. hedjou K. Kyamakya University of Dschan (ameroon ( University of Klaenfurt (Austria ( Stability and Bifurcation Analysis of Electronic Oscillators: heory and some Eperiments Abstract. We consider the electronic structure of a shunt-fed olpitts oscillator. his oscillator is a feedback circuit built with a sinle bipolar junction transistor which is responsible of the nonlinear behavior of the complete circuit. he interest of investiatin the dynamics of this circuit is justified by its multiple potential applications in the field of enineerin and also by the strikin/tremendous and very comple behavior the oscillator can ehibit leadin to periodic multi-periodic and chaotic motions. We discuss some potential applications of the shunt-fed olpitts oscillator. he modelin process leads to the derivation of mathematical equations (ODEs to describe the behavior of the shunt-fed olpitts oscillator. Usin these equations the stability and bifurcation analyzes are carried out and basins of attraction of stable solutions are obtained as well as some bifurcation diarams showin the etreme sensitivity of the system to tiny chanes in the values of the electronic circuit components. Analytical numerical and eperimental methods/approaches are considered to et full insiht of the dynamical behavior of the proposed oscillator. Analytical and numerical results are validated by the results obtained from a real physical implementation (Hardware implementation of the oscillator. Keywords. Shunt-fed olpitts oscillator Loop ain Modelin Stability Bifurcation haos Introduction Durin the last decades a tremendous attention has been devoted to the analysis of the nonlinear behavior of sinusoidal oscillators [-8]. he focus on this analysis is eplained by the rich and comple behavior nonlinear oscillators can ehibit and also the various potential technoloical applications of such oscillators. Indeed in their nonlinear states these oscillators can be eploited in many applications. In their reular states the oscillators can be used for instrumentations and measurements while the chaotic behavior (irreular state ehibited by the oscillators can be used in securin communications and/or for random waves eneration as well just to name a few. Another interestin potential application of nonlinear oscillators can be related to findin solution to various problems in plasma physics such as desinin both microwave and transformer devices [7-8]. hese applications are classically realized by onestae bipolar junction transistor oscillators the nonlinearity in these oscillators bein a direct consequence of the nonlinear character of the transistor. Such applications require hihly stable oscillators with lare frequency bandwihs. Further these oscillators must be robust to noise at hih frequencies. he drive towards increasinly hiher operatin frequencies (by means of sinle-transistor L oscillators for communication products is advantaeous as it provides to circuit desiners the possibility of interatin as much as possible such oscillators on a sinle chip (i.e. interated circuit. oncernin the nonlinear dynamics of sinle-transistor L oscillators some interestin works have been carried out. Reference [] develops a methodoloical approach to the analysis and desin of a olpitts oscillator. A nonlinear approach for the two quasi-sinusoidal and chaotic operation modes is considered; in particular the eneration technique of reular and irreular (chaotic oscillations in terms of the circuit parameters is shown. Reference [] eploits a Lur e model to demonstrate the occurrence of chaos in the olpitts oscillator. omponents of the autonomous chaotic olpitts oscillator causin the instability of equilibrium points are identified. he study in [] is etended (by the same authors see [] to the case of a forced (non-autonomous olpitts oscillator. Both amplitude and frequency of the eternal ecitation are used for chaos control within the oscillator. he dynamic- maps of phase lockin transitions are depicted by measurements. Both simulation and eperimental results showin the ehibition (by the oscillator of the injection-locked dynamics are presented. Reference [4] considers the investiation of the nonlinear dynamical behavior of two different self-driven oscillators leadin to chaotic dynamics. A piecewise-linear model (PWL is derived to describe the dynamics of a transformer coupled oscillator and various bifurcation diarams are obtained to illustrate some strikin scenarios leadin to chaos. By eploitin a feedback model of such a system the loop ain of the oscillator is derived and its stron dependence with the nonlinear dynamics of the oscillator is demonstrated. Both Pspice based simulation and laboratory eperiment (real physical implementation are performed to confirm the numerical results. his paper considers a shunt-fed structure of the olpitts oscillator. he advantae of such a structure (amonst many others can be found in the practical realization. Indeed this structure is simple to realise. Further the ood stability of the fundamental characteristics (i.e. amplitude and frequency of the waveforms enerated by such a structure is due to the fact that the biasin current is not flowin throuh the oscillatory network as observed in the series-fed structure. Another advantae of the shunt-fed olpitts oscillator is the possibility of derivin correspondin equations from which can be shown the capability of the oscillator to perform in a lare frequency bandwih ranin from low frequencies (e.. the order of Hertz to hih frequencies (e.. the order of iahertz. his could be achieved by performin an appropriate time scalin of the equations which are derived to describe the nonlinear dynamical behavior of the oscillator. Another interestin feature of this oscillator is the intrinsic nonlinear character of its analo device (e.. the current-voltae characteristics of the bipolar junction transistor is eponential; this character can easily ive rise to reular (periodic oscillations or chaotic dynamics dependin upon a suitable choice of the system parameters. We consider the shunt structure of the olpitts oscillator in Fi.. his particular structure was recently introduced in reference [5]. In this reference we used a fourth-order nonlinear equation to investiate the bifurcation structures ehibited by the shunt olpitts oscillator. Usin one of the feedback divider capacitors as control parameter various bifurcation diarams were obtained showin the richness of modes ehibited by the oscillator and some scenarios leadin to chaos. Both Pspice based simulations and laboratory eperimental measurements were performed to validate the results of the modelin process. Nevertheless an improvement of these results was found necessary to facilitate both the desin and implementation processes and increase the performance of the oscillator as well. his work addresses these issues. Some key problems investiated 7

2 could be reducin both the number of parameters involved and the state space dimension as well. It is worth noticin that such an approach is of reat importance both for analysis and desin purposes. Our aim in this paper is to contribute to the eneral understandin of the nonlinear dynamical behavior of the shunt-fed olpitts oscillator and complete the results obtained so far by (a carryin out a systematic and methodoloical analysis of the dynamical behavior of the oscillator; (b providin both theoretical and eperimental tools which will be of precious use for the desin and control purposes as they could provide full insiht of/on the nonlinear dynamical behavior of the oscillator; (c pointin out some of the unknown and strikin behavior of the shunt-fed olpitts oscillator. he rest of the paper is oranized as follows. In section we use a simplified equivalent model of the bipolar junction transistor for the modelin process. he state equations of the circuit desined (i.e. the shunt olpitts oscillator are obtained. Section investiates the stability of fied points. he analysis of local bifurcations of fied points is considered. Different types of bifurcations likely to occur are found. onditions for the occurrence of Hopf bifurcations are derived. Section 4 deals with numerical simulation. Various bifurcation diarams associated to their correspondin raphs of larest one-dimensional (D numerical Lyapunov eponents are obtained showin both comple and strikin scenarios to chaos. Section 5 is concerned with the eperimental study. he desin and real physical implementation of the oscillator are considered usin electronic components. Eperimental results are compared with both analytical and numerical results and a very ood areement is obtained. Section 6 deals with conclusions and proposals for further works. Mathematical modelin of the shunt-fed olpitts Oscillator. ircuit description Fis. ( show the electronic structure proposed for the shunt-fed olpitts oscillator. Fi. (a is the complete circuit of the oscillator while Fi. (b is a simplified model of the bipolar junction transistor used in the modelin process. he bipolar junction transistor used in the commonbase confiuration plays the role of nonlinear ain element. he resonant network uses split capacitance and is an identification feature of the olpitts oscillator. he couplin capacitor is supposed to have very low impedance at the operatin/runnin frequencies. he bias is provided by the D voltae source V and the current source I. he resistor R is used to set the equilibrium point of the circuit. he main difference between the series-fed and the shuntfed olpitts oscillators is the bias current which doesn t flow trouh the tank coil when dealin with the fed-shunt olpitts. his situation justifies the name shunt-fed. his shunt when performed keeps a hih circuit s quality factor and results in better frequency stability. he resonant frequency of the unloaded tank circuit which is an estimator of the fundamental frequency of the oscillator is iven by: ( + f = π L In this oscillator the only nonlinear component (necessary for settin up chaotic oscillations is the bipolar junction transistor (BJ. he rest of circuit components are assumed linear. Fis.: (a ircuit diaram of the shunt-fed olpitts oscillator and (b the BJ model consistin of one current-controlled source and a sinle diode.. State equations Accordin to the qualitative theory in nonlinear dynamics we select a minimal model [8] for the circuit. he idea is to consider as simple a circuit model as possible which maintains the essential feature ehibited by the real oscillator. o carry out our modellin process the followin simplifyin hypotheses are considered: Ideal bias source i.e. the bias current in the emitter of the bipolar junction transistor is provided by an ideal current source I. Passive and reactive elements are assumed ideal. he bipolar junction transistor is modeled simply by a (voltae-control nonlinear resistor R E and a linear current source (see Fi.a. he current-voltae (V-I characteristic of R E is modeled by an eponential function namely V BE V BE I E= f ( VBE = IS ep ISep V V for V BE > V ; where I S is the inverse saturation current ( and V 6mV at room temperature. he common-base forward short-circuits current. his α is approimately equal to unity ( α ain F F corresponds to nelectin the base current. Parasitic effects are nelected (e.. in ce and be. In case these effects are considered they will be added in parallel with and respectively. We emphasize that the assumptions above do not alter the qualitative dynamics of the system (i.e. they do not add etra dynamics to the system and we suppose that only the quantitative dynamics (e.. position of the attractors in the parameter space can be affected as discussed in [ ]. We use the electrical circuit theory and some assumptions to derive the followin model of the shunt oscillator: (a dv R = V cc V V V Rα F I E RI L (b dv R = V cc V V (( V + R α I F L E I I (c di L L = V + V Eqs. ( describe the dynamics of the shunt-fed oscillator. I L denotes the current flowin trouh the inductor; V i i = stand for the voltaes across capacitors ( 74

3 i. For the sake of simplicity It can be obtained from Eqs. ( the followin normalized sets of ordinary differential equations (Eqs. (4 to describe the dynamical behavior of the shunt-fed olpitts oscillator: d (4a = k( + ( [ ( ] + n k d k (4b ( (4c d k = + k = ( k ( + n = ep (4d ( ( (4e k = + Lω = and R LI = V R( + and are normalized voltaes across capacitors and and represents the normalized current flowin in the inductor L. Notice the remarkable simplicity of the nonlinear term in the proposed model. Indeed only the first equation (Eq. (4a contains a nonlinear term. It can be shown by means of some mathematical alebraic manipulations that Eqs. (4 are equivalent to the followin Jerk model: d ( + d + n( d n( + + = (5 ( Further the followin equilibrium point can be derived from the precedin equations describin the nonlinear dynamics of the shunt olpitts oscillator: ( ln ln (6 ( V V i = V ( I I V ( I I L S S As it appears in Eq. (5 the proposed structure of the shunt olpitts oscillator is modeled by a third order nonlinear ordinary differential equation which is very sensitive to the parameters and. Another interestin remark is related to the parameter k which does not sinificantly affect the behavior of the Shunt olpitts oscillator. In fact the parameter k acts like a scalin factor/coefficient for the attractors ehibited by the shunt olpitts oscillator. It should be worth noticin that the parameters and can lead to a precise physical interpretation in case the issue is related to the desin of an ideal/optimal structure of the olpitts oscillator. Specifically the quality factor measures the selectivity of the resonant circuit while admits a precise physical meanin in the contet of Barkhausen criterion related to the series-fed olpitts oscillator [] which defines the oscillator condition accordin to classical linear analysis.. Stability analysis he stability analysis is carried out by transformin Eqs. (4 in the form of Eq. (7. dv = (7 F( V μ Where ( and = F ( k V = is an autonomous vector field μ is an element of the parameter space. Eq. (7 enerates a flow { φ } φ = on the phase space R S and there eist a lobal map: P : Vp P Vp = φ ( ( { } ( where ( represents the coordinates of the projection of the attractors onto the Poincaré cross section defined by = {( R S }. he perturbation analysis [7-8] is used to investiate the stability of oscillatory solutions in the Poincaré map. hus Eq. (4 is perturbed by addin a small perturbation δ V P = ( δ δ δ to the steady state PO ( V. he variational differential equation (Eq. (8a is derived with the correspondin variational matri defined in Eq. (8b. dδv (8a P = DF ( V (8b PO δv P ( ep k k + ( k ( k DF ( V = ( ( PO k k k k( k k( k he stability of periodic motion is obtained accordin to the real parts of the roots of the followin characteristic equation ( det DF ( V PO λid = : (9 λ + λ + ( + λ + = obtained by considerin very small amplitudes of the steady states. herefore the states of the oscillator (e.. stable or unstable states Hopf bifurcation and subscritical bifurcation etc are obtained by the roots (i.e. eienvalues solutions of Eq. (9. 4 Results 4. Analytical results We investiate analytically the stability of fied points and the stability of oscillatory states in the weakly nonlinear response. Fi. shows in the comple plane ( Re ( Im ( λ λ the representation of the eienvalues (roots of the characteristic equation. hese roots are obtained (usin the Newton-Raphson alorithm for 75

4 [..8] and [.5 5]. From Fi. one can have an idea on both the stability of periodic solutions and the different types of bifurcation likely to appear in the system. Since DF ( V PO is a real matri comple eienvalues in comple conjuate pairs are responsible of the observed symmetry alon the real ais. hus if the real parts of the eienvalues (λ are all neative the rate is of the contraction type or otherwise of the epansion. If the eienvalues are all real the contraction or the epansion is observed near the steady state while the comple values of the eienvalues show the contraction or epansion of the spiral. If there eist eienvalues havin real parts with different sin the equilibrium state is called saddle; an equilibrium point whose eienvalues have nonzero real parts is called hyperbolic. On the other hand period doublin bifurcation is observed if there eists an eienvalue ( λ = while bifurcations of the Hopf type are observed if the followin conditions are satisfied: (a there eists a pair of pure imainary comple eienvalues ( λ = ± jim( ; dλ d (b ( (non zero crossin speed α bein α α = α the bifurcation control parameter. α is the critical value for the occurrence of Hopf bifurcation obtained from the equation Re ( λ =. onsiderin the previous results it clearly appears that our system for the iven parameters and can undero various types of bifurcations namely: saddle bifurcation period-doublin bifurcation and Hopf bifurcation. We have derived the followin relationships between the parameters of the model (describin the dynamics the oscillator for the occurrence of Hopf bifurcation: (a (b ω ω = osc ( = By constructin the Routh-array of the oscillator it can be shown that the equilibrium point of the system is stable if the followin conditions are satisfied: (a Or (b ( < ] [ > (he equilibrium point is always stable plane. 4. Numerical results Eqs. (4 are solved numerically to define routes to chaos. We use the four-order Rune-Kutta alorithm for the sets of parameters used in this work the time step is always Δt =. 5 and the calculations are performed usin real variables and constants in etended mode. he 6 interation time is always. Here the types of motion are identified usin two indicators. he first indicator is the bifurcation diaram the second bein the larest D numerical Lyapunov eponent denoted by where (a (b λ ma = lim ( / t ln ( d( t t ( = ( δ + ( δ + ( δ d t and computed from the variational equations obtained by perturbin the solutions of Eqs. (5 as follows: + δ + δ and + δ. d (t is the distance between neihbourin trajectories [5]. Asymptotically d( t = ep( λmat. hus if λ ma > neihborin trajectories divere and the state of the oscillator is chaotic. For λ ma < these trajectories convere and the state of the oscillator is non-chaotic. λ ma = for torus states of the oscillator. In the precedin section we have shown that the dynamics of the oscillator is overned only by the couple of parameters and the third parameter k bein a scalin factor with no sinificant effect on the system s attractor. Settin the value of =. 78 and k =.67 we analyze the effects of the control parameter on the behavior of the system. he scannin process is performed to investiate the sensitivity of the oscillator to tiny chanes in. he use of as the control parameter revealed the richness and the variability of the bifurcation modes in the olpitts oscillator within its nonlinear reime. he etreme sensitivity of the oscillator to tiny chanes in is observed. Reions of non oscillations reions of diverence solutions and reions of oscillations are obtained by monitorin. In the latter case various types of oscillations are obtained: periodic quasi-periodic and chaotic oscillations. Such oscillations are due to the presence of the nonlinearity in the olpitts oscillator. Fi. presents a bifurcation diaram showin a transition to chaos throuh period- doublin scenarios. It is also shown the etreme sensitivity of the oscillator to tiny chanes in. Widows of chaotic dynamics alternate with tiny windows of reular dynamics. Fi.: Representation of the eienvalues solutions in the comple 76

5 Fi. : Bifurcation diaram of the shunt-fed olpitts oscillator and the raph of D larest numerical Lyapunov eponent. he positive value of λ ma is a sinature of chaos. 4. Eperimental results his section provides sample results to validate the analytical and numerical results. Usin the same values of the system parameters we determine the correspondin electronic coefficients to desin and implement the electronic circuit of Fi. 4. his fiure is the desin of the eperimental setup for the implementation of the shunt-fed olpitts oscillator built on a breadboard. Fi.4: Eperimental setup for the measurements on the shunt type olpitts he followin values of circuit components are used which correspond to the same set of the parameters values used in the analytical and numerical methods: = 66nF = 4nF = μf L = 47μH and V = V. he electronic network consistin of the operational amplifier U is an implementation of the ideal current enerator. U is used in a voltae circuit inverter. A Ω resistor is added in series to the inductor L to sensor it s current i L. he eperimental results are obtained by observin the time evolution of the voltae across the capacitor and by plottin phase portraits ( V i L. Eperimental results were enerally close to those obtained analytically and numerically. Basically some bifurcations obtained analytically and numerically were observed eperimentally. In Fis. 5 is shown some sample results of the comparison between numerical waveforms (a and eperimental waveforms (b and the comparison between numerical phase portraits (c and eperimental phase portraits (d. Both qualitative and quantitative very ood areements were obtained between different methods. 5 onclusion his paper has dealt with the analysis of the shunt fed olpitts oscillator. he analytical study of the oscillator has revealed various strikin bifurcations. hese bifurcations were confirmed analytically and eperimentally. Fi.5: ualitative comparison of eperimental and numerical results: (a and (b: numerical and eperimental wave forms; (c and (d numerical and eperimental phase portraits of the oscillator REFERENES [] G. M. Maio O. De Feo and M. P. Kennedy Nonlinear analysis of the olpitts oscillator and applications to desin IEEE ransaction on ircuits and Systems AS-46 pp [] J.. Liu H.. hou and J. H. hou larifyin the chaotic phenomenon in an oscillator by Lur e system form Microwave and Opt. ech. Letters Vol. pp [] J.. Liu H.. hou and J. H. hou Non-autonomous chaotic analysis of the olpitts oscillator with Lur e system Microwave and Opt. ech. Letters Vol. 6 pp [4] J. Zhan Investiation of chaos and nonlinear dynamical behaviour in two different self-driven oscillators Ph.D. thesis University of London. [5] J.. hedjou K. Kyamakya Van Duc Nuyen I. Moussa and J. Kenne Performance evaluation of analo systems simulation methods for the analysis of nonlinear and chaotic modules in communications ISAS ransactions on Electronics and Sinal processin N vol. 8 pp7-8. [6] G. A. Kriesmann Bifurcation in classical bipolar transistor circuits SIAM (Studies Applied Maths Vol. 49 pp [7] M. Kennedy haos in the olpitts oscillator. IEEE ransaction on ircuits and Systems- vol.4 pp [8] M. P. Kennedy On the relationship between chaotic olpitts oscillator and hua s circuit IEEE ransaction on ircuits and Systems AS-4 pp [9] G. M. Maio. Kennedy and M. P. Kennedy Eperimental manifestations of chaos in the olpitts oscillator in Proc. ISS 97 Derry Ireland June 997 pp [] J. Zhan X. hen and A. Davis. Hih frequency chaotic oscillations in a transformer- coupled oscillator. Proc. of NDES 99 Ronne Denmark pp []. Weene and M. Kennedy. RF chaotic olpitts oscillator. Proc. Of NDES 95 Dublin Ireland pp [] Y. Hosokawa Y. Nishio and Akio Ushida R-ransistor chaotic circuit usin phase-shift oscillators Proc. Int. Symp. On Nonlinear heory and its application (Nolta 98 Vol.( [] Nikolai F. Rulkov and Aleander R. Volkovskii Generation of broad-band chaos usin bockin oscillator. IEEE ransaction on ircuits and Systems- vol.48 No.6. [4] A. Najumas and A. amasevicius. Modified Wien-bride oscillator for chaos. Electronics letter pp [5] A. A. Andronov A. A. Vitts and S. E. Khaikin heory of oscillations. New York Perammon 996. [6] L. O. hua. W. Wu A. Huan and G.. Zhon A universal circuit for studyin and eneratin chaos-part I: Routes to chaos IEEE ransaction on ircuits and Systems- vol.4 No. pp [7] hance M. Glenn Synthesis of a Fully-interated Diital Sinal Source for communication from haotic Dynamicsbased Oscillations Ph.D thesis Johns Hopkins University January. [8] G. M. Maio O. De Feo and M. P. Kennedy A eneral method to predict the amplitude of oscillations in nearly sinusoidal oscillators IEEE ransactions on ircuits and Systems Vol.5 Auust 4 pp