On power supplies used for laboratory demonstration of chaoticelectronic oscillatorsProdyot Kumar Roya)Department of Physics, Presidency College, Calcutta 700 073, IndiaArijit Basurayb)Research Division, Neo Tele-Tronix Pvt. Ltd., 6/7 Bijoygarh, Calcutta 700 032, IndiaEmanuel Gluskinc)The Academic Institute of Technology, Holon 58102, Israel, and Department of Electrical Engineering,Ben-Gurion University, Beer-Sheva, 8415, Israel共Received 9 November 2004; accepted 4 March 2005兲For a simple oscillator capable of producing chaotic as well as periodic signals we discuss the roleof the voltage source used to power the oscillator. This choice may sometimes be crucial to theoperation of the oscillator. We have shown experimentally that a poor choice may suppress chaosaltogether. ©2005 American Association of Physics Teachers.关DOI: 10.1119/1.1900100兴I. INTRODUCTIONAfter publication of a paper1describing a chaotic genera-tor, one of us 共EG兲 submitted to the AJP editor a short com-munication as a comment on Ref. 1. He pointed out thatwhen dealing with chaotic circuits one should mention2thenature and role of the impedance of the voltage sources usedin the circuit. Although it is the normal practice of numericalexperimentalists to assume the voltage sources to be ideal,we were surprised to learn that papers describing actual ex-periments often provide little detail of the voltage sourcesused. A simple survey of literature shows that this is indeedthe case.3–9The editor then suggested that it would be useful to haveexperimental evidence in support of his comments. Thepresent paper is the result of our joint study of the role of theimpedance of the voltage sources used to power chaotic elec-tronic oscillators. A few simple cases are considered to illus-trate the various situations that may arise in the course of anexperiment. We present here the results of our laboratoryobservations along with PSPICE simulations to stress ourpoints.II. THEORYFigure 1 shows a block diagram of the complete oscillator,which consists of 共1兲 a power supply 共shown as a separateblock in the diagram兲 that serves as power source and 共2兲 anoscillator circuit that is the load for the power source. Asalready mentioned, it is normal practice to place a powersource symbol 共indicating an ideal power source兲 in the cir-cuit instead of a separate block for a nonideal power supply.DC voltage sources are generally used in the laboratory forsuch oscillator systems, but chaos may also be obtained inperiodically driven systems; in both cases, it is essential togive a proper definition of the system, including its powersource, if chaos is to be observed and understood.2In practice, the power source impedance contains someinductive and capacitive contributions in addition to resistivecontributions. In a good source the inductive and capacitivecontributions are very small throughout the range of operat-ing frequencies.Fig. 1. Block diagram of a complete electronic oscillator. The power sourceis shown as a separate block at the left and the remaining part of the oscil-lator is shown in the block at the right.APPARATUS AND DEMONSTRATION NOTESJeffrey S. Dunham, EditorDepartment of Physics, Middlebury College, Middlebury, Vermont 05753This department welcomes brief communications reporting new demonstrations, laboratory equip-ment, techniques, or materials of interest to teachers of physics. Notes on new applications of olderapparatus, measurements supplementing data supplied by manufacturers, information which, while notnew, is not generally known, procurement information, and news about apparatus under developmentmay be suitable for publication in this section. Neither the American Journal of Physics nor the Editorsassume responsibility for the correctness of the information presented. Submit materials to Jeffrey S.Dunham, Editor.1082 1082Am. J. Phys. 73 共11兲, November 2005 http://aapt.org/ajp © 2005 American Association of Physics TeachersBut the range specified by manufacturer of the power sup-ply may be narrower than the range that arises in the circuitwith the source when the chaotic phenomena develop. It canbe argued2that some additional terms should appear in thesystem equations to account for these properties of the powersource; otherwise, the system equations will provide only apoor representation of the actual system. This can be verifiedby following consideration.Let Z(␻) be the impedance of the power source whoseoutput voltage; V0(t), is a known function of time that maybe constant or periodic. If the output current of the powersource, i(t), is written in terms of the spectral density, i(␻),through the Fourier transformi共t兲⫽冕⫺ ⬁⫹ ⬁i共␻兲ej␻td␻, 共1兲then the output voltage of the source, Vout(t), can be writtenasVout共t兲⫽ V0共t兲⫺冕⫺ ⬁⫹ ⬁Z共␻兲i共␻兲ej␻td␻. 共2兲Obviously, Vout(t) is different from the ideal value V0(t).The last term expresses the change in the input voltage to theoscillator circuit due to the nonideal properties of the powersource.Our observations show that well-regulated power suppliesavailable commercially operate as almost ideal voltagesources over the range of operating frequencies typical ofmany chaotic oscillators.1This is illustrated in case 1 of Sec.IV. We have also found that chaos may be suppressed ordisappear altogether in the case of a poor choice of the volt-age source. Cases 2, 3, and 4 of Sec. IV give examples of theuse of such sources.III. EXPERIMENTAL APPARATUSIt is common practice to use a regulated dc power supplyfor experiments with simple oscillators. Alternatively, onemay opt for simple dry cells. The selection of a battery or anelectronic power supply obviously depends on the nature ofthe load. If the load has low impedance, then the sourceought to have low internal impedance in order that there notbe considerable voltage drop within the power source itself.The case becomes more complex when the source is oscilla-tory and the driven circuit has a nonlinear frequency re-sponse with an extended frequency range.We keep the present investigation relatively simple bystudying a chaotic oscillator circuit that is driven by a simpledc source 共dual supply兲. In order to make the power sourceimperfect, we deliberately introduce additional resistances inseries 共R2 and R3 in Fig. 2兲 with the regulated dc powersupply, thereby modifying the nature of the power source, asshown in Fig. 2. In