ROBUST OP AMP REALIZATION OF CHUA'S CIRCUIT. Dublin 4 IRELAND. Abstract

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1 ROBUST OP AMP REALIZATION OF CHUA'S CIRCUIT Mchael Peter Kennedy Department of Electronc and Electrcal Engneerng Unersty College Dubln Dubln 4 IRELAND mpk@mdr.ucd.e Abstract Chua's crcut s a smple electronc network whch exhbts a arety of bfurcaton phenomena and attractors. The crcut conssts of two capactors, an nductor, a lnear resstor, and a nonlnear resstor. Ths paper descrbes the desgn methodology for a robust practcal op amp mplementaton of Chua's crcut. In addton, we present expermental results and SPICE smulatons for a workng crcut usng o-the-shelf components. Introducton Chua's crcut [], shown n Fg., s a smple oscllator crcut whch exhbts a arety of bfurcatons and chaos. The crcut contans three lnear energy-storage elements (an nductor and two capactors), a lnear resstor, and a sngle nonlnear resstor N R. The state equatons for the crcut are as follows: C d C dt C 2 d C2 dt L d L dt = G( C2 C ) g( C ) = G( C C2 ) + L = C2 () where G = R and g() s a pecewse-lnear functon dened by: g( R ) = m 0 R + 2 (m m 0 ) [j R + B p j j R B p j] (2) Ths relaton s shown graphcally n Fg. 2; the slopes n the nner and outer regons are m 0 and m respectely; B P denote the breakponts. The nonlnear resstor N R s termed oltage-controlled because the current n the element s a functon of the oltage across ts termnals. In the rst reported study of ths crcut, Matsumoto [] showed by computer smulaton that the system possesses a strange attractor called the Double Scroll. Expermental conrmaton of the presence of ths attractor was made shortly afterwards by Zhong and Ayrom [2]. Snce then, the system has been studed extensely; a arety of bfurcaton phenomena and chaotc attractors n the crcut hae been dscoered expermentally and conrmed mathematcally [3]{[5]. Most of the expermental studes of Chua's crcut hae appeared n the Crcut Theory lterature [6]{ [22]. Ths paper s drected at a broader audence of Electroncs Engneers who are excted by Nonlnear Frequenz, ol. 46, no. 3{4, March{Aprl 992, pp. 66{80.

2 R R L C2 C R C2 C L R Fgure : Chua's crcut conssts of a lnear nductor L, a lnear resstor R, two lnear capactors C and C 2, and a nonlnear resstor N R. R = g( R ) m0 m Bp -Bp m 0 R Fgure 2: Three-segment pecewse-lnear characterstc of the nonlnear resstor n Chua's crcut. The outer regons hae slopes m 0 ; the nner regon has slope m. There are two breakponts at B P. 2

3 Dynamcs but who hae wth lttle or no tranng n Nonlnear Crcuts. Our am s to prode the necessary crcut theoretcal background and practcal detals to assst the expermentalst n studyng Chua's crcut. Whle derental equatons and mechancal systems prode conenent frameworks n whch to examne bfurcatons and chaos, electronc crcuts are unque n beng easy to buld, easy to measure, and easy to model. Furthermore, they operate n real tme, and parameter alues are readly adjusted. The mportance of Chua's crcut and ts relates [23]{[28], s that they can exhbt eery type of bfurcaton and attractor whch has been reported to date n thrd-order contnuous-tme dynamcal systems [29]{[32]. Whle exhbtng a rch arety of complex dynamcal behaors, the crcuts are smple enough to be constructed and modeled usng standard electronc parts and smulators. In ths work, we show how to buld Chua's crcut usng o-the-shelf components. We descrbe n detal the desgn methodology whch has been followed for constructng the nonlnear resstor, and present expermental and smulaton results for an example crcut. 2 Practcal realzaton of Chua's Crcut Chua's crcut can be realzed n a arety of ways usng standard or custom-made electronc components. Snce all of the lnear elements (capactor, resstor, and nductor) are readly aalable as two-termnal deces, our prncpal concern here wll be wth crcutry to realze the nonlnear resstor. Seeral mplementatons of ths element already exst n the lterature; these use operatonal amplers [2], dodes [], transstors [33], and operatonal transconductance amplers [34]. The crcut whch we present n the followng sectons s for demonstraton, research, and educatonal purposes. Whle t may appear more complcated than earler mplementatons n that the nonlnear resstor comprses two operatonal amplers (op amps), t s possble to buy two op amps n a sngle package. Thus, our crcut uses a mnmum number of components: a par of op amps and sx resstors to mplement the negate resstor, two capactors, an nductor, and a arable resstor. 3 From computer smulaton to experment: scalng of current and tme Matsumoto et al. [6] hae shown by computer smulaton of equaton () that a Double Scroll attractor appears n Chua's crcut for the followng alues of the parameters: C = =9; C 2 = ; L = =7; G = 0:7; B P = ; m 0 = 0:5; m = 0:8: In these and hs earler smulatons, no unts were gen (or needed) for the state arables C, C2, and L snce Matsumoto was smply smulatng a set of derental equatons. If we rewrte the equatons n SI unts, then the oltages are measured n olts (), currents n Amperes (A), capactance n Farads (F), nductance n Henrys (H), and resstance n Ohms (); the recprocal of resstance, called conductance, s measured n Semens (S). Snce currents of mllamperes are easer to realze n electronc crcuts than Amperes, the rst step s to rescale all currents by a factor of 000; the eect s to reduce all capactances by a factor of 000 and to ncrease resstances and nductances by the same factor. Thus, wth C and C2 n unts of olts and L n mllamperes, Matsumoto's set of parameters becomes: C = =9 0 3 F C 2 = 0 3 F L = =7 0 3 H G = 0:7 0 3 S The slopes of the pecewse-lnear resstor are now -0.8 ms (ma/) and -0.5 ms; the breakponts reman unchanged at B P =. 3

4 R = g( R ) m0 m Bp ^ Bp ^ -Bp -Bp m 0 R Fgure 3: Eery physcally realzable nonlnear resstor N R s eentually passe the outermost segments (whle not necessarly lnear as shown here) must le completely wthn the rst and thrd quadrants of the plane for sucently large jj and jj. It s easer to use capactances of nf and nductances of mh than Farads and Henrys, respectely. The eect of rescalng tme n equaton () by a factor k s to scale each nductance and capactance by the same factor k; resstances are unaected by a tme scalng. In partcular, slowng tme by decreases C, C 2, and L by the same factor. The resed parameters are now: C = =8 0 7 F = 5:56nF C 2 = =2 0 4 F = 50nF L = =4 0 H = 7:4mH G = 0:7 0 3 S = 0:7mS (whch corresponds to R = 428) The breakponts and slopes of the pecewse-lnear resstor N R are unchanged when tme s rescaled. These are the component alues whch Matsumoto et al. used to conrm ther computer smulatons expermentally [6]. Ths wll be our startng pont too, but wth a slght derence. Electronc components are aalable \o-the-shelf" n standard alues; 7.4 mh, 5.56 nf, 50 nf, and 428 are not standard alues. Therefore, we arbtrarly choose 8mH, 0nF, 00nF, and 800 as a nearby \standard alue" startng pont. Hang scaled current and tme, our next goal s to construct a nonlnear resstor wth the characterstc shown n Fg. 2. The mportant feature of ths s that t possesses two negate slopes m 0 and m. In order to understand the desgn methodology presented here, we rst rese some results from Nonlnear Crcut Theory. 4 Eentual Passty By denton, the Double Scroll attractor s bounded. Ths s mportant because all physcal resstors are eentually passe, meanng smply that for a large enough oltage across ts termnals, the power P (= ) consumed by a real resstor s poste. For large enough jj or jj, therefore, the characterstc must le only n the rst and thrd quadrants of the plane. Hence, any physcal realzaton of the threesegment characterstc speced n Chua's crcut must nclude at least two more segments whch return the characterstc to the rst and thrd quadrants (see Fg. 3). As long as the oltages and currents on the attractor are restrcted to the negate resstance regon of the characterstc, these outer segments wll not aect the crcut's behaor. In our dscusson of a practcal mplementaton of ths crcut, we wll show how to maxmze the extent of the negate resstance regons. 4

5 o d A d o d (a) (b) Fgure 4: (a) oltage-controlled oltage Source (CS): d = 0 and o = f( d ); (b) oltage transfer characterstc of lnear CS wth gan A. 5 Negate Resstance Conertor There are many ways to synthesze a negate resstance, one of whch s by connectng three poste lnear resstors to a oltage-controlled oltage source to form a negate-resstance conertor. Ths arrangement s attracte from an expermentalst's pont of ew because t s readly mplemented by means of an operatonal ampler (op amp). 5. oltage-controlled oltage Source (CS) A oltage-controlled oltage source (CS) s an deal crcut element whch has two nput termnals and two output termnals (see Fg. 4(a)). It s characterzed by two propertes: no current ows nto or out of the nput termnals, and the oltage o whch appears across the output termnals s a functon of the potental derence d between the nput termnals. The smplest non-tral functonal relaton between the output and nput oltages of a CS occurs when o depends lnearly on d,. e. o = A d. Ths s llustrated n Fg. 4(b). 5.2 Negate Resstance Conertor A two-termnal negate resstance conertor can now be produced by connectng three poste resstances around a oltage-controlled oltage source as shown n Fg. 5(a). Let us assume that the CS n Fg. 5(a) s lnear wth oltage transfer functon o = A d. When A s sucently large, ths negate resstance conertor has the followng relaton (whch s dered n Appendx A): By choosng R 2 = R, ths reduces to: = R2 R R 3 = R 3 Thus, lookng between the nput termnals of ths element N R, one sees a resstance of R 3. 5

6 R d 2 o 3 R2 m R3 0 R (a) (b) Fgure 5: (a) Negate resstance conertor usng a oltage-controlled oltage source; (b) characterstc of negate resstance conertor usng a lnear CS wth lnear oltage transfer functon o = A d. The h ( A)R slope of the characterstc s gen by m = 2+R 3 R [R 2+(+A)R 3] ; for sucently large A, R2 R R 3. 6

7 + Esat o A d + d - o OS d - -Esat (a) (b) Fgure 6: (a) Operatonal ampler wth assocated power supples; (b) oltage transfer characterstc of op amp. 5.3 Operatonal Amplers An operatonal ampler (op amp) prodes us wth a real-world approxmaton to a oltage-controlled oltage source. Consder the crcut shown n Fg. 6(a), whch conssts of an operatonal ampler and ts assocated power supples + and (depcted as batteres). A oltage appled between the non-nertng and nertng nput termnals (labeled \+" and \{") produces a potental derence between the output termnal and the reference termnal (usually the common pont of the power supples). Ths real op-amp-and-batteres crcut module draws a small current d at ts nput termnals; let us assume that d = 0. When the derental nput oltage d of a real op amp s sucently large n magntude and negate, the output s approxmately constant at E sat ; ths s called the negate saturaton regon. When the nput s small n magntude, the output ares almost lnearly wth the nput; ths s called the lnear regon. The gan n the lnear regon s usually greater than 0 5 /. In addton, the characterstc s oset from the orgn by an nput oset oltage OS, whch s typcally a few m. When the nput oltage s large and poste, the output assumes a maxmum alue of E + sat ; ths s called the poste saturaton regon. The dc oltage transfer functon of a real op amp s thus closely approxmated by a three-segment pecewse-lnear characterstc, as shown n Fg. 6(b). Because a real op amp contans compensaton and parastc capactances, a complete model of the dece should nclude dynamc elements. We assume here that the op amp behaes purely resstely at the frequences of nterest n Chua's crcut. Ths can always be ensured by approprately scalng tme as ndcated n secton 3. Thus, we neglect all frequency-dependent eects n the op amp and treat t as purely resste. We assume too that the output mpedance of the op amp s sucently small that t can be neglected. Thus, for our purposes, the output of the op amp looks lke an deal oltage source and ts nput looks lke an open crcut. We can therefore model the op amp by a CS: d = 0; o = f( d ), where f() s as shown n Fg. 6(b). The adantage of ths pecewse-lnear model s that we can now determne the behaor of a crcut contanng op amps and other components by analyzng each lnear regon of operaton (negate saturaton, lnear, and poste saturaton) separately. For further dscusson and worked examples of op amp crcuts We consder only the case when the oltages at the non-nertng and nertng termnals relate to the reference termnal are wthn the common-mode range of the op amp. 7

8 R d A 2 o 3 m 0 m R2 OS + Bp Esat + R3 - -Esat - -Bp 0 m 0 R NEGATIE SATURATION LINEAR POSITIE SATURATION (a) (b) Fgure 7: (a) Operatonal ampler-based negate resstance conertor; (b) characterstc of op amp negate resstance conertor, assumng the three-segment pecewse-lnear oltage transfer characterstc n Fg. 6(b). m 0 = R h h, B + P = R2+(+A)R3 A(R 2+R 3) E + ( A)R sat + OS, m = 2+R 3 R [R 2+(+A)R 3], B P = h h R2+(+A)R 3 E A(R 2+R 3) sat + OS, and OS = A(R2+R 3) ( A)R 2+R 3 OS. When the oset OS s reduced to zero, h h the gan A s large, and R 2 = R, then m 0 = + R R, B P 3 +, m, B P h R3 R 2+R 3, and OS = 0. R 2+R 3 R 2 R R 3 usng pecewse-lnear technques, see [35] and [36]. 5.4 Negate Resstance Conertor usng an op amp We can now buld a negate resstance conertor usng an op amp as shown n Fg. 7(a). The drng pont ( ) characterstc s shown n Fg. 7(b) (See Appendx B for the detals of the calculaton). The relaton s pecewse-lnear and conssts of three segments. As before, we assume that A s large. The central porton then has slope m R2 R R 3 and the outer regons (correspondng to saturaton of the op amp a consequence of eentual passty) hae slopes m 0 = R. If we set R 2 = R then m = R 3. In the followng, we assume that the saturaton leels of the op amp are equal n magntude and that the oset OS s zero. Later, we wll dscuss technques for accomplshng ths. Thus, + = and E sat =, by assumpton, and the breakponts occur at R3 R 2+R 3. The op amp negate resstance conertor (NRC) wll be the core buldng block for the nonlnear resstor n Chua's crcut. What happens f we now connect two such NRCs n parallel [35]? 8

9 2 2 R 2 R Fgure 8: Parallel connecton of two nonlnear resstors. If = f ( ) and 2 = f 2 ( 2 ), then = f ()+f 2 (). 6 Puttng the blocks together Two nonlnear resstors are connected n parallel, as shown n Fg. 8. Let us assume that both are oltagecontrolled. The current whch ows n the two-termnal resstor N R, when a oltage s appled across ts termnals, s dened by = f ( ). Smlarly, a current 2 = f 2 ( 2 ) ows n N R2. The total current owng nto the par s gen by = g() where g() = f () + f 2 (). Thus, the parallel combnaton of two (or more) oltage-controlled nonlnear resstors s also a oltage-controlled nonlnear resstor. We can determne the shape of g() graphcally by addng and 2 for each, as ndcated n Fg. 9. The process s smpled consderably f the consttuent functons f () and f 2 () are pecewse-lnear. For more extense dscusson of seres and parallel combnatons of nonlnear resstors, see [35]. It should now be clear how a e-segment physcally realzable pecewse-lnear resstor of the type needed n Chua's crcut can be constructed smply by connectng n parallel two negate resstance conertors wth approprately shaped characterstcs. 7 Realzaton of nonlnear resstor for Chua's crcut usng two op amp oltage-controlled negate resstance conertors Fg. 0 shows an op amp mplementaton of Chua's crcut. The desred characterstc s produced by connectng two oltage-controlled negate resstance conertors N R and N R2 n parallel. Nonlnear resstor N R has a three-segment pecewse-lnear characterstc wth slopes m 0 and m and breakponts B P (as n Fg. 9(b)). Smlarly, N R2 has slopes m 02 and m 2 and breakponts B P 2 (Fg. 9(a)). The compound e-segment characterstc has slopes m, m 0, and m and two pars of breakponts at B P and B P 2 (as n Fg. 9(c)). We hae seen from our dscusson of the op amp negate resstance conertor that specfyng R 2 = R h n Fg. 7(a) yelds slopes R and =R 3, wth breakponts at R 3 R 2+R 3. Thus, wth R 2 = R, m 0 = R m = R 3 Smlarly, R 5 = R 4 ges B P = R 3 m 02 = R 4 9

10 2 m 0 m m 02 m 2 Bp 2 Esat 2 Bp Esat -Esat -Bp 2 m 02 -Esat -Bp m 0 (a) (b) m - m 0 m Bp 2 Bp Esat -Esat -Bp -Bp 2 m - m 0 (c) Fgure 9: Graphcal combnaton of pecewse-lnear oltage-controlled resstors: (a) characterstc of N R2 ; (b) characterstc of N R ; (c) characterstc of N R2 n parallel wth N R. 0

11 2 R R R4 R + + L A2 5 A 3 C2 C R - - C2 C 6 4 R5 R2 R6 R 2 R3 R R 0 Fgure 0: Realzaton of Chua's crcut usng two op amps and sx lnear resstors to mplement N R. The two op amps are aalable n a sngle eght-pn DIP.

12 m 2 = R 6 B P 2 = R 6 R 5 + R 6 From graphcal consderatons of the compound characterstc, we hae: m + m 02 = m 0 m + m 2 = m Wth these obseratons, we can dere a desgn strategy for determnng the approprate alues of the components R {R 6 from m 0 ; m, and B P 2. The method s deeloped n Appendx C. 7. Desgn procedure s determned by the power supples and nternal structure of the op amps. We do not necessarly know ts alue a pror but t can be measured. The shape of the desred characterstc determnes B P 2 ; m 0, and m. We are free to choose B P or m. Choose R large enough that t wll not sgncantly load the op amp (say 330 ). Calculate B P = m E R sat. If B P s not large enough that the dynamcs of the attractor wll reman wthn the negate-resstance regon of the characterstc, reduce R and try agan. One must trade o the length of the negate resstance regon and the sze of R. Choose R 2 = R. Ealuate Calculate R 3 = (B P 2 )m 0 B P 2 m R 4 = B P 2 (m 0 m ) Set R 5 = R 4. Ealuate R 6 = ( B P 2 )(m 0 m ) 7.2 Practcal mplementaton of Chua's crcut { worked example Fg. shows a practcal mplementaton of Chua's crcut usng an Analog Deces AD72 dual BFET op amp, two 9 batteres, and sx resstors to mplement the negate resstor. Usng two 9 batteres to power the op amps ges + = 9 and = 9. From measurements of the saturaton leels of the AD72 outputs, 8:3. The desred nonlnear characterstc s dened by m 0 = 0:409mS, m = 0:756mS, and B P 2 = :08. Our slopes and breakponts are chosen (wth hndsght) to be slghtly derent from those used by Matsumoto et al. [6] because we wsh to use only o-the-shelf components n ths example. Followng the desgn strategy aboe, we dere a complete component lst for ths crcut. 2

13 R R R4 - L R C2 C2 C C R R6 R5 + R2 R3 9 C 9 C R Fgure : Practcal realzaton of Chua's crcut usng an eght-pn dual op amp ntegrated crcut. Component Lst Element Descrpton alue Tolerance A Op amp ( AD72, TL082, or equalent) 2 R W Resstor 220 5% 4 R 2 W Resstor 220 5% 4 R 3 W Resstor 2.2 k 5% 4 A 2 Op amp ( AD72, TL082, or equalent) 2 R 4 W Resstor 22 k 5% 4 R 5 W Resstor 22 k 5% 4 R 6 W Resstor 3.3 k 5% 4 C Capactor 0 nf 5% R Potentometer 2 k C 2 Capactor 00 nf 5% L Inductor (TOKO type 0RB or equalent) 8 mh 0% In addton to the components lsted, we recommend that a bypass capactor C of at least 0:F be connected across each power supply, as shown n Fg., as close to the op amp as possble. The purpose of these capactors s to mantan the power supples at a steady dc oltage. 7.3 Expermental ercaton of characterstc The characterstc of the nonlnear resstor N R can be measured n solaton by means of the crcut shown n Fg 2. Resstor R S, known as a current-sensng resstor, s used to measure the current R whch ows nto the negate resstor N R when a oltage R s appled across ts termnals. An approprate choce of R S n ths example s 00. Current R owng n R S then causes a oltage R = 00 R to appear across the 3

14 R R4 - R S R R6 R5 + R2 R3 9 C 9 C R R R S Fgure 2: The characterstc of negate resstor N R can be measured by applyng a trangular oltage waeform S to the seres combnaton of N R and a small current-sensng resstor R S. Plot R (/ R ) ersus R. The eght-pn dual op amp package s shown from aboe n schematc form. The reference end of the package s ndcated by a dot or a semcrcle (shown here). 4

15 (a) (b) Fgure 3: Measured characterstc of negate resstor. (a) S s a trangular waeform wth zero dc oset, ampltude 7 peak-to-peak, and frequency 30 Hz. Horzontal axs: R (/d); ertcal axs: R (00m/d); (b) S s a trangular waeform wth zero dc oset, ampltude 5 peak-to-peak, and frequency 30 Hz. Horzontal axs: R (2/d); ertcal axs: R (00m/d). sensng resstor. Thus, we can measure the characterstc of N R by applyng a oltage S as shown and plottng R (/ R ) ersus R. Ths s acheed by connectng R to the Y-nput and R to the X-nput of an osclloscope n X-Y mode. The resultng characterstc for the components lsted n the table s shown n Fg 3. Note that we hae plotted R ersus R ; ths s possble f your osclloscope permts nerson of the Y-nput n X-Y mode. 8 Bfurcatons and Chaos 8. R bfurcaton sequence By reducng the arable resstor R n Fg. from 2000 towards zero, Chua's crcut exhbts a sequence of bfurcatons from dc equlbrum through a Hopf bfurcaton and perod-doublng sequence to a Rossler-type attractor and the Double Scroll strange attractor, as llustrated n Fg. 4. A two-dmensonal projecton of the attractor s obtaned by connectng C and C2 to the X and Y channels, respectely, of an X-Y osclloscope. Notce that aryng R n ths way causes the sze of the attractors to change: the perod-one orbt s large, perod-two s smaller, the Rossler-type attractor s smaller agan, and the Double Scroll shrnks consderably before t des. 8.2 C bfurcaton sequence An alternate way to ew the bfurcaton sequence s by adjustng C. In ths case, x the alue of R at 800 and ary C. Montor C and C2 as before. The full range of bfurcatons from equlbrum through Hopf, perod-doublng, Rossler, and Double Scroll can be obsered as C s reduced from 2.0 nf to 6.0 nf. 9 Smulaton of Chua's crcut These expermental obseratons may be conrmed by smulaton usng a specalzed Nonlnear Dynamcs smulaton package such as INSITE [37]. Alternately, one can smulate Chua's crcut on a general-purpose 5

16 (a) (b) (c) (d) (e) (f) (g) (h) () Fgure 4: Typcal R bfurcaton sequence n Chua's crcut (component alues as n the table aboe). Horzontal axs C (a){(h) /d, () 2/d; ertcal axs C2 (a){(h) 500m/d, () 2/d. (a) R = 2:00k, dc equlbrum; (b) R = :88k, perod-; (c) R = :85k, perod-2; (d) R = :84k, perod-4; (e) R = :825k, perod-3 wndow; (f) R = :79k, Rossler-type attractor; (g) R = :74k, Double Scroll attractor; (h) R = :49k, Double Scroll attractor; () R = :40k, large lmt cycle correspondng to outer segments of the characterstc. 6

17 crcuts smulator such as SPICE. 9. SPICE smulatons The op amps n our realzaton of the crcut may be modelled usng macro-model subcrcuts whch are aalable from a number of ntegrated crcuts manufacturers. Analog Deces' macro-models [38] are compatble wth SPICE release 2G6 and later whle Texas Instruments [39] use PSpce [40] polynomal controlled sources whch are ncompatble wth Berkeley SPICE [4]. Three attractors n the C bfurcaton sequence were smulated usng SPICE 3d2 wth the nput deck shown n Fg. 5; the results are shown n Fg. 6. The AD72 op amp [42] s modelled wth Analog Deces' AD72 SPICE macro-model [38]. A real nductor has a non-zero seres resstance whch we hae ncluded n the SPICE model; we measured RL = 3.5. Node numbers are as n Fg. 0. The power rals are and 222; 0 s the \nternal" node of our physcal nductor where ts seres nductance s connected to ts seres resstance. We note n passng that the prncpal derence between the R and C bfurcaton sequences s that whle the sze of the Double Scroll attractor ares consderably wth R, adjustng C has lttle eect on ts magntude. 0 The eects of non-dealtes on Chua's crcut In our dscusson of the SPICE model, we ntroduced a small parastc seres resstor to account for the dc resstance of a physcal nductor. The eect of ths parastc s small. In contrast, non-dealtes n the op amp hae a major eect so t s mportant to be aware of these nuences. We noted earler that the outer regons of the characterstc hae no eect on the shape of the attractor f the oltages and currents on the attractor reman sucently small. If the attractor grows too large n magntude, t wll be at best be clpped. Ths eect s llustrated n Fg. 7. By reducng the alue of C from 0 nf to 8 nf, the ampltude of the attractor n the R bfurcaton sequence grows too large and s clpped. The breakponts n the nonlnear resstor's characterstc are proportonal to the saturaton leels of the op amps. The saturaton leels n turn are determned by the power supply oltages and by the nternal archtecture of the op amps. If the leels are derent, as they typcally are, the resultng characterstc wll be asymmetrc. Ths results n a Double Scroll attractor whch has one lobe bgger than the other; the eect s llustrated n Fg. 8. Here, we hae reduced the negate power supply oltage from 9 to 7 n magntude. The result s to moe the left breakpont of the characterstc, whch n turn produces an asymmetry n the attractor. We also saw that the nput oset oltage OS of an op amp causes a shft n the characterstc when t s used as a negate resstance conertor (refer back to Fg. 7(b)). Whle asymmetry may be aesthetcally unpleasng, t has lttle eect on the bfurcaton sequence or on the nature of the attractor. If you wsh, the asymmetry due to saturaton leel msmatch may be corrected by adjustng the poste and negate power supply oltages untl symmetry s acheed. For example, the negate saturaton leel mght be 0.7 less n magntude than the poste leel. Ths could be corrected by usng power supples of 9 and -9.7 nstead of 9. Normally, t s not possble to zero the oset n an eght-pn dual op amp such as the AD72 or the TL082. In fact, we delberately chose the AD72 because t s draws neglgble nput current, by rtue to ts FET nput stage, and has a guaranteed maxmum nput oset oltage of.0m (AD72K). If the oset s dsturbng, one may substtute for the dual op amp two sngle op amp equalents such as the AD7 and TL08; these hae oset balancng pns to enable the user to set OS precsely equal to zero. A superor realzaton of Chua's crcut (n the sense that the breakponts are ndependent of the saturaton leels of the op amp and the slopes and breakponts can be set ndependently) s descrbed n [43]. Fnally, we note also that the regon of the negate resstance conertor n Fg. 7(a) exhbts a resstance of R 3 only when R 2 = R. Therefore, one should try to match the resstor pars (R ; R 2 ) and (R 4 ; R 5 ) n Fg. 0 as closely as possble. 7

18 MODEL + 0 DC DC 9 L RL R C N C N XA AD72 R R R XA AD72 R R R * * The AD72 SPICE Macro-model s aalable from Analog Deces, Inc. *.WIDTH OUT=80.IC (2)=2.0 ()=0.TRAN 0.0MS 00.0MS 80.0MS.PRINT TRAN (2) ().PLOT TRAN (2) ().END Fgure 5: SPICE deck to smulate the transent response of our dual op amp mplementaton of Chua's crcut. Node numbers are as n Fg. 0. The op amps are modelled by the AD72 macro-model from Analog Deces. RL models the seres resstance of the real nductor L. 8

19 (a) (b) (c) Fgure 6: Attractors n Chua's crcut from SPICE 3d2 smulatons of the C bfurcaton sequence. Horzontal axs C 2/d; ertcal axs C2 200m/d. (a) C = 0:nF, Rossler-type attractor; (b) C = 9:5nF, Double Scroll attractor; (c) C = 8:5nF, Double Scroll attractor. (a) (b) Fgure 7: When the attractor s too large, we cannot neglect the eects of the outer regons of the esegment characterstc. (a) C = 0nF: Double Scroll attractor s restrcted to the regons of negate slope; (b) C = 8nF: attractor s dstorted by the outer poste-gong segments. 9

20 (a) (b) Fgure 8: Asymmetry nduced by unequal poste and negate op amp saturaton leels. (a) left breakpont of the characterstc s smaller (n magntude); (b) attractor s asymmetrc. Closng Remarks Because Chua's crcut can exhbt a wde arety of nonlnear behaors, t presents an attracte paradgm for expermental nestgaton of dynamcal system. In ths paper, we hae descrbed a robust mplementaton of Chua's crcut. We hae resed the crcut theoretc concepts underlyng the desgn of negate resstors and hae descrbed a desgn strategy for syntheszng a e-segment pecewse-lnear resstor by connectng two op amp negate resstance conertors n parallel. Our example yelds standard alues allowng a workng crcut to be assembled from o-the-shelf electronc components. We strongly encourage the reader to buld ths crcut and to explore the exctng world of Nonlnear Dynamcs. 2 Acknowledgements Ths work s supported n part by the Natonal Scence Foundaton under Grant MIP , by the Oce of Naal Research under Grant N J-402, and by the Semconductor Research Corporaton under Contract 90-DC-008. The bblography of papers on Chua's crcut was prepared by Ms. G. Horn. I am grateful to C.-W. Wu, K. Eckert, and N. Hamlton for erfyng the robustness of the crcut presented here by test-buldng t for me. Thanks to Bert Sh for makng t possble to typeset the photographs. 20

21 Appendx A CS-based negate resstance conertor Krchho's Current Law (KCL) at node n Fg. 5(a) ges: Krchho's oltage Law (KL) around loop yelds: = R ( o ) (3) = d + The transfer functon of the CS s gen by Hence, from equatons (4) and (5), Equalently, = o = Substtutng for 0 n equaton (3) ges = R 3 o (4) o = A d (5) R2 + ( + A)R 3 o A( ) A(R2 + R 3 ) R 2 + ( + A)R 3 ( A) R [R 2 + ( + A)R 3 ] For large A, Further, choosng R = R 2 ges R2 R R 3 R 3 Ths result s summarzed graphcally n Fg. 5(b). 2

22 Appendx B Op amp-based negate resstance conertor The op amp s modeled as a CS wth a three-segment pecewse-lnear oltage transfer characterrstc, as shown n Fg. 6(b). Ths model accounts for the nonzero dc oset OS, nte gan A n the lnear regon, and (possbly derent) saturaton leels and Esat+. NEGATIE SATURATION o = E sat d Esat A + OS LINEAR o = A( d OS ) Esat A + OS d Esat A + POSITIE SATURATION o = d Esat+ A + OS KCL at the non-nertng termnal of the op amp (node ) n Fg. 7(a): + + OS = R ( o ) (6) KL around loop -3-0-: R 3 = d + o (7) We consder the three lnear regons of the transfer characterstc separately. Op amp n poste saturaton Then, substtutng for o n equaton (6) ges The op amp s n poste saturaton for o = + = R R + d + A + OS Ths s called the aldatng equaton for the poste saturaton regon. Now Thus, the aldatng equaton becomes: = d + + Esat A + OS + R2 + ( + A)R 3 A( ) R 3 o R OS Ths corresponds to the rghtmost segment of the characterstc n Fg. 7(b). The breakpont s dened by B + R2 + ( + A)R 3 P = + + OS A( ) and the slope by m 0 = R For large A, B P + R OS 22

23 Op amp n negate saturaton Substtutng o = E sat for o = + n the aboe analyss yelds the leftmost segment of the characterstc n Fg. 7(b). m 0 = R as before, and the aldatng equaton ges B P = R2 + ( + A)R 3 A( ) as the upper bound of the negate saturaton regon. + OS Op amp n lnear regon In the lnear regon, Substtutng for o n equaton (6) ges o = A( d OS ) = R R A( d OS ) (8) Now, from equaton (7), Rewrtng d n terms of ges: and d = = d + R 3 o R 3 = d + A( d OS ) R2 + ( + A)R 3 = AR 3 d OS R 2 + ( + A)R 3 d OS = + R 2 + ( + A)R 3 Substtutng for ( d OS ) n terms of n equaton (8) ges: For large A, = ( A) R [R 2 + ( + A)R 3 ] The op amp s n ts lnear regon when A R2 R R 3 Equalently, substtutng for d from equaton (9), E sat A + OS d = R 2 + ( + A)R 3 Hence, the op amp s n ts lnear regon for R2 + ( + A)R 3 A( ) + AR 3 R 2 + ( + A)R 3 ( OS ) A( ) R [R 2 + ( + A)R 3 ] R2 + R 3 + R R 3 OS + OS d E + sat A + OS + + OS 23 AR 3 R 2 + ( + A)R 3 R2 + ( + A)R 3 A( ) OS (9) OS OS + A + + OS + OS

24 For large A, ths reduces to R 3 Consder once agan Fg. 7(b). We hae that + OS m = R 3 ( A) R [R 2 + ( + A)R 3 ] + + OS and and For large A, OS = A(R2 + R 3 ) ( A) m OS R2 R R 3 R2 + R 3 R 2 OS OS 24

25 Appendx C Parallel connecton of two negate resstance conertors We connect n parallel two negate resstance conertors N R2 and N R as shown n Fg. 8. Nonlnear resstor N R has a three-segment pecewse-lnear characterstc wth slopes m 0 and m and breakponts B P (as n Fg. 9(b)). Smlarly, N R2 has slopes m 02 and m 2 and breakponts B P 2 (Fg. 9(a)). The compound e-segment characterstc has slopes m, m 0, and m and two pars of breakponts at B P and B P 2 (as n Fg. 9(c)). From the contnuty of the characterstc of N R2, we hae B P 2 m 2 = ( B P 2 )m 02 (0) Thus But, from equaton (3), gng From equaton (0), m 02 = B P 2 (m 2 m 02 ) m 02 m 2 = m 0 m m 02 = B P 2 (m 0 m ) m 2 = B P 2 B P 2 m 02 = B P 2 (m 0 m ) From equaton (3), m = m m 2 = B P 2 m + ( B P 2 )m 0 From the contnuty of the characterstc of N R, Thus B P m = ( B P )m 0 m 0 = B P B P m 25

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