Neural and Nonlinear Information Processing
Neural and Nonlinear Information Processing EL ENG 129
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Date Created: 10/22/15
Chua s Circuit for High School Students Gandhi Gauruv Muthuswamy Bharathwaj2 and Roska Tamas 2Nonlinear Electronics Laboratory NOEL Department of Electrical Engineering and Computer Sciences University of California Berkeley Berkeley CA 94720 mbharateecsberkeleyedu Abstract A stepbystep procedure to build Chua s circuit using easily available and inexpensive components is presented We illustrate the classic breadboard method for building Chua s circuit For readers without access to an oscilloscope this paper proposes the use of a laptopPersonal Computer soundcard to capture the voltage waveforms generated from the circuit The reader can then plot the waveforms on a computer screen using the software provided by the authors 1 Introduction Have you ever noticed water dripping from a tap which is sometimes regular and sometimes irregular Or water owing through an obstacle in a way that can be either smooth laminar or turbulent At some point in time we all notice these phenomenons however they are usually very hard to explain It was only after Edward Lorenz Lorenz 1963 came to conclude that his computer simulated weather model is highly sensitive to initial conditions that the scientific community rigorously studied the phenomenon of chaos However it took more than 30 years to prove that the observations made by Lorenz were indeed chaotic Tucker 2000 Chaos along with Quantum Mechanics and Relativity has been hailed as one of the major discoveries of 20Lh century Chua s circuit was the first circuit implementation specially designed to exhibit chaos Chua 1984 and the first that was proved to be chaotic rigorously Chua et a1 1986 Chua s circuit is also the simplest Chua et a1 1986 physical system where chaos can be observed Here we show how easily high school students can build this simple inexpensive electronic circuit in less than an hour for approximately 10 The organization of this paper is first we give a simple nonmathematical introduction to Chua s circuit This is followed by the crux of the paper 7 building Chua s circuit This second section is further subdivided into a component checklist section a breadboard primer and building the actual circuit The third section shows how to use the PC s sound card as an oscilloscope The fourth section in this paper talks about a simple Chua s circuit kit that you can order for building Chua s circuit in 5 minutes This is followed by a FAQ section We conclude the paper with a references section The aim of this paper is that high school students can build Chua s circuit quickly and easily Therefore we have explicitly avoided the mathematical intricacies of Chua s emu We mcourage the reader to consult the references for the maLhemaucal detaxls of Chua s cxrcun and chaos 2 Intrnductiun Lu Chuz s circuit A schemaue of Chua s menu 15 shown m re 1 stor one shown m Fxgure 2 gu The cxrcu conslsts othree energy mg elemean two capa not c and c one 1nductor L one lmear resxstor R and 11011an a n M u n uby L L V AAA VVV V R R L gt lt v v gt CZ c2 R gt IL NR Figure 1 Chua39s 0mm 77sz Figme 2 Cmrenlvs vultage chamctmsucs nrune Chua Dmde 3 Building Chua s Circuit a Components required Table 1 A list of components for building Chua s circuit The inductor can be obtained from Coilcraft httpwwwcoilcraftcom part number PCH45186L The other components can be obtained from your local RadioShack store or ordered via DigiKey httpwwwdigikeycom1 The part number for the audio plug is SC1127ND You can download the software for turning your PC s soundcard into an oscilloscope at wwwchuacircuitcom Identify the different components and write their values on them You can consult Appendix I for identifying different component values With the help of Appendix 11 mark the pin con guration for the LMC6482 operational ampli er You can also use other opamps like the TL082 or the AD712 b A breadboard primer A breadboard is a reusable device for building an electronic circuit Breadboard Wiki 2007 Figure 3 shows Chua s circuit on a breadboard mponents are easy to change or replace on a breadboard Parts will not be damaged so they will be available for reuse a erwards Breadboards have many tiny sockets called 39holes39 ananged on a 01quot grid The leads of most components can be pushed straight into the holes Integrated circuits ICs are inserted across the central gap with their notch or dot to the left Figure 4 shows how the A n 39 39 all the way across The power supply is connected to these rows at the top and 0V zero volts at the bottom We suggest using the upper row of the bottom pair for W then you c u e the o e o lo 39 rr39 with 39 39 39 39 dual suppl e JV 0V 9V The other holes are linked vertically in blocks of5 with no link across the centre r39A c The hreadhnard Verdun nl39 China s circuit A Stzprhyrstzp Guide erreurt dlagram to thatyou follow the slmple rules below 1 When puttrng parts on a breadboard you rnust eoneentrate on therr ennneetlnns not therr posrtrons on the erreurt dlagram 2 Start the Wlnng wrth the ICs rst Place rt m the eentre of the breadboard and work round rt pm by on puttrng m all the eonneetaons and eornponents for eaeh pm m turn We wlll follow the two rules above to eonneet Chua39s erreurt Probably the rnost drmeult part m Wlnng up Chua39s erreurt ls the nonllnear elernent Chua dlode so we wlll start wth that rst 1 Flgure 5 shows how we ean reallze the eurrent vs voltage eharaetenstre from gure 2 We are uslng the two Opramp rrnplernentataon of the Chua dlode fr Kennedy Kennedy1992 orn N Equot gm Figures L quot 39 quot 39 MP Kenned 3 o p p I39 o39 The Chua diode is realized using operational ampli ers opamps Appendix 11 gives a brief introduction to opamps From Appendix 11 you can easily infer the connection diagram pinout of the opamp this is the most important piece of information you need for connecting the opamp The pinout is repeated below in gure 6 for convenience The opamp that we will be using is the LMC6482 V a OUTPUT A V 2 7 mvmms INPUT A ounu1 s NONTNVERTINGL LINVERTTNG INPUT A A INPUT 3 V 39 39 5NDNINVERTNG TNPUT a m 1 71314 Figure 6 LMC6482 opamp pinout Notice the triangle circuit symbol actually represents the opamp However using gures 5 and 6 you can easily wire up the opamp circuit for the Chua diode Before you proceed make sure that you have sorted out your resistors and capacitors refer to Appendix I For wiring up the Chua diode you will need two 22k resistors two 220 ohm resistors one 33k and one 22k resistor Lets go through the wiring stepbystep First insert the opamp across the middle of the breadboard refer to gure 7 22 Rgea 24 25 26 27 28 25 26 27 28 Figure 7 Opamp inserted on my breadboard As shown in gure 7 breadboards usually have column numbers on the sides so circuit wiring is easy I will use the column numbers shown in gure 7 number 21 throu Insert resistor R5 22k from gure 5 across pins 1 and 2 of the operational amplifier On the breadboard pin 1 is on line 24 and pin 2 on line 25 Refer to gure 8 5 Figure 8 39 39 39 1 5 Insert resistor R6 33k from gure 5 from pin 2 of the operational ampli er to the negative terminal of the battery On the breadboard pin 2 is on line 25 Notice that I have decided to use the vertical strips on the sides of my breadboard for positive power and negative ground terminals of my battery refer to the breadboard primer section for more information Refer to gure 9 the breadboard socket next to the blue strip Will be the negative terminal of the battery Figure 9 Inserting the 33k resistor from pin 2 to ground 0 Insert resistor R4 22k from gure 5 from pin 1 of the operational amplifier to pin 3 of the operational amplifier On the breadboard pin 1 is on line 24 and pin 3 on line 26 Refer to figure 10 ggmsa ae a O gilllllill39 Figure 10 Inserting the 22k resistor across pins 1 and 3 of the operational amplifier 1 This completes one opamp connection in figure 5 Now complete the second opamp connection using figure 5 as a reference Figure 11 shows the final result for the Chua diode 7WIIUII lii39i iliii 7 Intui 00 ABCQE FEHIJ Figure 11 Completed Chua s diode Figure 12 shows Chua s circuit with the Chua diode shown explicitly We will now put in the capacitors potentiometer and inductor Next we will finish the wiring and connect the batteries san lel 1 m lt C1 1 unF EQCD C2 100nF 1 arm Ra Ra l kn 22m 5 5 I39ll It a lt Figure 12 The version of Chua s circuit that we are building I will now start using the column labels on the breadboard for clarity On my breadboard the columns are indicated by the letters A through J So I will refer to a connection point on the breadboard as D24 Notice D24 refers to the line where pin 1 of the opamp is connected 9 Insert the 10 nF capacitor across E23 and F23 Also insert the potentiometer on the breadboard Refer to gure 13 Notice the potentiometer has three legs make sure all three are inserted properly On my breadboard the legs are at A19 A21 and G20 Figure 13 Capacitor Cl 10 nF has been inserter across E23 and F23 Also shown is the potentiometer 10 Next insert the 100 nF capacitor across E17 and F17 Refer to gure 14 filich FGHIJ Figure 14 Capacitor c2 100 n1 inserted across E17 anan 11 Insert the inductor across 117 and 116 We will also start connecting the two capacitors to the two ends ofthe potentiometer Refer to gure 15 Alaan FGHlJ Figure 15 n and 115 ofthe potennometer the blue wrre and the yellow ere 12 In gure 15 we have wlred one end of the inductor to the common negative terminal ofthe battery t AMEquot EH Q Figure 15 Wrre one end othe rnohetor to the negatave battery tennrna1 Iuse F15 to D15 then B15 to the negatave battery tennrna1 13 Connect the ends of the capacitor to the common battery terminal refer to gure 17 113an FGHIJ Figure 17 Conneet 1317 andB23 to the negatave battery terrninal 14 Now unscrew the audio plug housing This should reveal three wires refer to gure 18 for the correct connection The middle wire goes to the negative terminal ofthe battery the other two go to the capacitor ABCUE FGHIJ Figure 13 39 39 39 the other two t i i gotoG17and123 15 Connect the positive power supply terminal ofthe opamp Refer to gure 19 ABCUE FGHIJ Figure 19 Positive tenninai ofthe opamp is connected from 624 viaDZO and A20 to the positive end of the battery 15 Connect the negative power supply ofthe opamp refer to gure 20 if BCUE FGHIJ Figure 20 V mm de29 17 The most important connection is to hook up the Chua diode to the rest of the circuit 39m 39 39 39 I 39 avoid confusion The connection is shown by the two yellow wires from 23 to 26 and 27 Figurell Connecting the Chua dwde 18 Wire up the batteries as shown in gure 22 r 39 i a of the CAUTION We assume that the s the positive termin battery and the black wire is the negative terminal Be extremely carelul in this step You are done FGHIJ Figure 22 Positive anoinegatiye terminals of the battery green dots 4 Measurement on PC as an oscilloscope 1 2 3 Plug the microphone in the Line1n orMicrophone jack ofyour computer Select appropriate input LineIn Microphone from the volume control of your system observe the voltage waveform time series andor double scroll strange attractor onthe choose sound card capture option in the so ware You can also use several otherpc oscilloscope soltvwares available on intemet Vary the potentiometer between the two capacitors and observe the evolution I er route to chaos and listen to the sou d of chaos through the computer s speak An osqoop screenshot is shown in gure 23 Elle Elsplav Qlannd DMESKElE Wager Elugms ng mm 5m XAXlSFreszaleFannr nzs WWWWW m l vl w i Jasquap usqaap asqaup clad slot inherin fmm QWvdgcl Description Figure 23 Screenshot of osqoop 5 Chua s Circuit Kit If you want to build Chua s circuit in just 5 minutes you can write to us for a pre designed PnP plug and play kit where you just have to plug the components and observed the wave forms on laptopdesktop PC computer The kit contains the plug and play Printed Circuit Board for Chua s circuit as shown below along with required components that you need to insert into the PCB guy f p39 39 quot 39EV i39g w Figure 24 Chua s circuit printed circuit board The kit will look as follows after you will insert the components to it Figure 25 Completed Chua s circuit kit You need to buy a pair of ninevolt batteries to power the kit and insert the microphone jack to the laptop 6 FAQs 393quot Unable to arrange components You can request a complimentary complete snapkit based PnP Chua s circuit kit by writing to us at WRITE PROF ROSKA S LAB ADDRESS HERE I Will Chua s circuit work for values mentioned in the paper A complete set of Monte Carlo simulations was performed to analyze the robustness of the circuit The complete results will be published in another paper We have used these values several times and every time it turned out that the system is working without any problem In Can Ioperate Chua s circuit at di remfrequenaes Yes All you need to do is to replace the components by appropriate components However it maybe possible that your soundcard doesn t support those frequency values so you may need to observe the results on an oscilloscope I What else can I do with Chua s circuit voltage waveforms There are lot more things that you can achieve with these waveforms Chaos and chaotic systems have applications in science engineering economics nance music etc You can sample the waveforms into a commercial mathematical package like MATLAB and do signal analysis Check out the wikipedia dedicated to Chua s circuit chaos theory and nonlinear dynamics at wwwchuacircuit com and share your experiences queries and suggestions 3941quot More questions If you have more queries you can referdiscuss them in FAQ section of ChaosWiki at wwwchuacircuitcom 7 Conclusion A stepbystep guide for highschool students to build Chua s circuit has been proposed The goal is not only expose students to the exciting field of Chaos but also help them in basic circuit construction techniques understanding different passive components understanding DualInlinePackage DIP numbering and basic concepts behind a bread board DISCLAIMER Authors are not responsible for any damage that may be caused due to the use of material presented in the paper kits or the software provided 9 N E 4 V39 O gt1 Electronics Club 2007 httpwww knsec freenk J References Chua 1984 The Genesis of Chua s Circuit Chua Leon 0 Archiv fur Elektronik und Uebertragungstechnik Vol 46 No 4 pp 250257 1992 Lorenz 1963 Deterministic Nonperiodic ow Lorenz Edward N J Atmos Sciences Vol 20 pp 130 7 141 1963 Tucker 2000 The Lorenz Attractor Exists Tucker Warwick Nature Vol 406 No 6799 pp 948949 2000 Chua et al 1986 The double scroll family Parts I and II Chua Leon 0 Matsumoto T and Komuro M IEEE Trans On Circuits and Systems Vol CAS33 No 11 pp 107271200 1986 Breadboard Wiki 2007 httpenwikinedia nrg wikiI 139 J August 15Lh 2007 Last accessed hrendhrd ndf Last accessed August 15Lh 2007 Kennedy 1992 Robust Opamp Realization of Chua s Circuit Kennedy Michael P Freqz Vol 46 no 37 4 pp 66 7 80 March 7 April 1992 Appendix I Decnding the Cnmpnnent Values Decnding Resistnr Values M a m Red Red Orange orang OrangeRed Red Red Brown RedRed Red 22 m 3 3 m 220 ohms 2 m F39gm 26 Dxffexent 125mm valuesusedmchua39s uncut Decnding Czpzcitnr Values Capamorvalues are marked on the capamtor Referto gure 27 100 nF Figure 27 Capsular valuesta be usedm Chua39s mm Appendix 11 Operatinnal Ampli ers OprAmp Numbering anonal ampllfler Opramp ls an eleetronle devlce that can p rforrn a y ety of ms o mphy slgnals You don39t 7 wor s all you need to do ls ere rt up Note that ln elreult dlagrams opramps are represented by mangle syrnbols The operauonsl ampllfler and other lntegrated chlps eorne ln dlfferent packaglng The numbenng of these plns ul quotrm 9 l m r n VInJ me packages ls shown ln gure 28 lt JI F39gm m Opurnp my package numbenng Always refer to the datasheet of operauon a1 ampllfler to know the loeauon of posluye and negatlve power supply plns Most o the datasheets are avallable at http www datasheeteatslog eorn 1n the paper the LMC6482 operauonsl ampllfler has been used However you can also use the TL082 or AD712 opramps In case you are not uslng LMC6482 but some otherlc ensure the followlng s P ye power supply pm shouldreplace pm 8 ofLMC6482 B Ne a ye power supply pm shouldreplace pm 4 c 2 Pl 2 fL C6482 sh ldbe re laced by lnyerung represented P by mmus slgn ln datasheet nondnverh g represented by posl the datasheet and output the thrrd end of the mangle of one operauonal ampllfler othe chlp D Pm 67 and 8 of TL082 should be replaced by negatwe posluye and output of another operauonsl ampllfler othe chlp uve slgn m The connection diagrams are shown below LMcmsz connection diagram ourPur A mvmwc mm A owuv a unnmvmmc 6 mvmms mm A V vatRTms um a Mmcm TLOSZ connection diagram U v 2 1 uva Imur n quotumquot I A a unnmvzvmuc 3 quot 5 INFquot n INVERYIMG II PIIYB v 5 NnImvEnmc mun AD712 Connection diagram AMPLIFIER N01 AMPUHEH No2 Figure 29 Camm an OprAmp pmauts
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