Chaos in a Switched-Capacitor Circuit

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-32, NO. 10, OCTOBER 1985 1083 whe e P=blockdiagonal(Z,,Z,;~~,~~,,Pi,Zi+,,~~~,ZP) (1.4 whe e Zk is k X k iden i y ma ix, and k is he dimension o componen 2,. Using he componen connec ion app oach, a model o he in e connec ed sys em Z is desc ibed wi h i=(F+GLH)z+GMu Wa) y=Hz W) whe e F=blockdiagonal(A,,A,;..,A,_,,~,A,+,,...,A,) (I3c) G=blockdiagonal(B,,B,,...,Bi_,,Gi,Bi+,,...,B,) (134 H=blockdiagonal(C,,C,;..,C,-,,H,,C,+,,...,C,). Equa ions (3c), (lOc), (12), and (13~) imply FP= PA. Equa ions (3d), (3e), (lOd), (lOe), (12), and (13d) imply ( GLH) P = PBLC. The e o e, (14a) and (14b) imply (F+GLH)P= P(A+BLC). Simila ly G=PB HP=C. (13e) (144 WI (154 (15b) (154 b) Necessi y: The condi ion s a ed in he heo em esul s om he de ini ion o he componen connec ion model. In his amewo k he s a e ec o o he composi e sys em is ob ained by s acking he s a e ec o s o he indi idual componen s. Rema k: I (lb) is modi ied o y, = C,x, + D, ; hen, (8b) needs o be modi ied o y = Hz + Ju and condi ions (9) will include J= D. (94 Consequen ly, he heo em s ill holds and he composi e sys em model is desc ibed by [6] i=(A+BQLC)x+BQMu y=(Z+DQL)Cx+DQMu whe e Q=(Z-LD)-‘. The agg ega e model o he composi e sys em is desc ibed by i=(F+GQLH)z+GQMu y=(Z+DQL)Hz+DQMu whe e he agg ega e model s a e a iables z a e as de ined abo e. V. CONCLUSION Whene e dealing wi h in e connec ed la ge-scale sys ems he size o he sys em becomes an issue. Model educ ion echniques a e used o alle ia e hese di icul ies. Mos o he echniques a e applied o he linea ized model o he composi e sys em model. ’ In his pape , i is shown ha in he componen connec ion model amewo k he composi e sys em is agg egable i and only i a leas one o i s componen s is agg egable. Because o he ela i ely small size o he indi idual componen s, i is easie o agg ega e he componen model a he han he composi e sys em model. The impo an consequence o his esul is ha when dealing wi h in e connec ed la ge scale sys ems in he componen connec ion model amewo k one does no need o agg ega e he la ge-scale composi e sys em bu a he can conside he agg ega- ion o each, ela i ely smalle scale componen , independen ly. 111 I21 [31 [41 PI I61 PI REFERENCES H. H. Happ, Diakop ics und Ne wo ks. New Yo k: Academic, 1971. E. J. Da ison, “Decen alized s abiliza ion and egula ion in la ge mul i- a iable sys ems,” in Di ec ions in La ge SC& Sys ems, H. and Mi e , Eds., pp. 303-323, 1975. A. Feliachi and A. P. Meliopoulos, “Modeling and Sim&a ion o La ge Scale In e connec ed Sys ems,” Ad ances in Modeling and Simulu ion, AMSE P ess, F ance, ol. 1, no. 1, pp. 33-49, 1984. N. R. Sandell, P. Va aiya, M. A hens, and M. G. Sa ono , “Su ey o decen alized con ol me hods o la ge scale sys ems,” IEEE T ans. Au oma . Con ., ol. AC-23, pp. 108-125, Ap . 1978. 0. Wasynczuk and R. A. DeCa lo, “The componen connec ion model and s uc u e p ese ing model o de educ ion,” Au omu ica, ol. 17, no. 4, pp. 619-626,198l. R. A. DeCa lo and R. Saeks, In e connec ed Dynomicc l Sys em. New Yo k: Ma cel Dekke , 1981. M. A&i, “Con ol o la ge scale dynamic sys ems by agg ega ion,” IEEE T uns. Au oma . Con ., ol. AC-13, pp. 246-253, 1968. Chaos in a Swi ched-Capaci o Ci cui ANGEL B. RODRIGUEZ-VAZQUEZ, JOSE L. HUERTAS, AND LEON 0. CHUA Abs ac -We epo chao ic phenomena obse ed om a simple nonlin- ea swi ched-capaci o ci cui . The expe imen ally measu ed bi u ca ion ee diag am e eals a pe iod-doubling ou e o chaos. This ci cui is desc ibed by a i s -o de disc e e equa ion which can be ans o med in o he logis ic map whose chao ic dynamics is well known. Se e al nonlinea ci cui s which exhibi a ious ypes o chao ic phenomena ha e been epo ed ecen ly [l]--[5]. Ou objec i e in his le e is o epo an expe imen al esul showing he ubiqui ous chao ic phenomena can also occu in a swi ched- capaci o ci cui . Since swi ched-capaci o ci cui s a e impo an in VLSI echnology, any po en ial anomaly o ailu e mechanisms Manusc ip ecei ed May 28,1985. This pape was suppo ed in pa by he O ice o Na al Resea ch unde Con ac NOOO14-76-C-0572, by he Na ional Science Founda ion unde G an ECS-8542885, and by he Spain CAICYT unde Con ac 0245/81. A B. Rod igua-Vazquez and J. J. Hue as a e wi h he Depa men o Elec ical and Elec onic Enginee ing, Uni e si y o Secille, Se ille, Spain. L. 0. Chua is wi h he Depa men o Elec ical Enginee ing and Compu e Sciences, Uni e si y o Cali o nia, Be keley, CA 94720. 0098-4094/85/1000-1083$01.00 01985 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 01,2020 a 16:18:45 UTC om IEEE Xplo e. Res ic ions apply. 1084 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-32, NO. 10, OCTOBER 1985 +T,Y (b) Fig. 1. A nonlinea swi ched-capaci o ci cui and i s associa ed iming di- ag am. due o he onse o chaos should be ully analyzed. This chao ic ci cui is also o ci cui - heo e ic in e es because i s dynamic equa ion is equi alen o he well-known logis ic map [6] whose chao ic dynamics ha e been ex ensi ely s udied and is now well unde s ood. Since he logis ic map is he simples chao ic poly- nomial disc e e map, he chao ic ci cui o be desc ibed below is he simples chao ic ci cui desc ibed by a i s -o de disc e e map. Conside he swi ched-capaci o ci cui in Fig. l(a): i is made o a ba e y I’,, a linea capaci o C,, a nonlinea swi ched-capa- ci o componen [7], [8], and h ee analog swi ches. The s a e (on o o ) o he swi ches is con olled by a s anda d wo-phase clock de ined by he iming diag am shown in Fig. l(b). The swi ches labeled S’ ( esp., So) u n on in synch oniza ion wi h he ising edge o he clock signal @’ ( esp., +‘). The nonlinea swi ched-capaci o componen -hence o h called an FESC ( o wa d. Eule swi ched capaci o ) esis o -is de ined by Q, -Q,-, = kV,2_, A AQ, (1) whe e AQ, is he ne cha ge lowing in o he FESC esis o du ing he n h clock pe iod, V,-, is he ol age sampled ac oss he FESC esis o du ing he (n - 1) h pe iod, and K is an a bi a y posi i e cons an . We ha e buil he ci cui in Fig. l(a) wi h C, =l nF and k = 0.5 nF/V using o - he-shel componen s and obse ed he s eady-s a e ol age wa e o m samples V o di e en alues o he ba e y ol age ,. Con a y o ou in ui i e expec a ion o a single- alued ela ionship be ween V and V,, we ound he ela- ionship o be mul iple- alued o e some anges o he “pa ame- e ” y , and unde ined i.e., chao ic, o o he anges. This obse a ion is summa ized by he bi u ca ion ee measu ed expe imen ally om his ci cui . The amilia cascades p eceding he chao ic egion implies a pe iod-doubling ou e o chaos [6]. To de i e a ecu si e ela ionship o V,, we no e ha he ne cha ge AQ, + i lowing in o he FESC esis o du ing he (n + 1) h clock pe iod mus be equal o he ne cha ge lowing ou o he linea capaci o C, (cha ge conse a ion p inciple), and hence: V n+l = , - g ,‘. (4 s We can ans o m (2) in o se e al mo e amilia equi alen o ms by de ining X n+l = uV,+~ + b. I we choose a = l/y and b = 0, we would ob ain (3) X n+I =l- xx, Fig. 2. Bi u ca ion ee. + V 4 Fig. 3. O - he-shel ealiza ion o he FESC esis o in Fig. l(a). whe e X=k /;/C,. I we choose and b = i, we would ob ain he well-known logis ic map X n+1=4Gl(1- 41) whe e (5) k XP- 4aC, . Bo h equa ions, (3) and (5) ha e been in ensi ely s udied [6] and hei global quali a i e beha io s a e now well classi ied and unde s ood. Consequen ly, Fig. l(a) ep esen s he i s eal physical ci cui whose chao ic dynamics can be compIe ely analyzed. Fo eade s in e es ed in epea ing ou expe imen s, he FESC esis o in Fig. l(a) can be ealized by he ci cui shown in Fig. 3. ACKNOWLEDGMENT The au ho s would like o hank G eg Be ns ein o s imula ing discussions which led o his pape . REFERENCES [l] Y. S. Tang, A. I. Mees, and L. 0. Chua, “Synch oniza ion and chaos,” IEEE T ans. Ci cui s Sys ., ol. CAS-30, pp. 620-626, Sep . 1983. [2] A. Azzouz, R. Duh , and M. Hasle , “T ansi ion o chaos in a simple nonlinea ci cui d i en by a sinusoidal ol age sou ce,” IEEE T ans. Ci cui s Sys ., ol. CAS-30, pp. 913-914, Dec. 1983. [3] T. Ma sumo o, L. 0. Chua, and S. Tanaka, “Simples chao ic nonau ono- mous ci cui ,” Phys. Reu. A, ol. 30, pp. 1155-1157, Aug. 1984. [4] T. Ma sumo o, “A chao ic a ac o om Chua’s ci cui ,” IEEE T uns. Ci cui s Sys ., ol. CAS-31, pp. 1055-1058, Dec. 1984. [5] G-Q. Zhong and F. Ay om, “Expe imen al con i ma ion o chaos om Chua’s ci cui ,” In . J. Ci cui Theo y App ., ol. 13, pp. 93-98, 1985. [6] P. C i ano ich (Ed.), Uni e si y in Chaos: A Repo Selec ion. B is ol, England: Adam Hilge , 1984. [7] J. L. Hue as, L. 0. Chua, A. B. Rod iguez-Vazquez, and A. Rueda, “Nonlinea swi ched-capaci o ne wo ks: Basic p inciples and piecewise- linea design,” IEEE T ans. Ci cui s Sys ., ol. CAS-32, pp. 305-319, Ap . (4) 1985 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 01,2020 a 16:18:45 UTC om IEEE Xplo e. Res ic ions apply. [8] A. Rod iguez-Vazquez, J. L. Hue as and L. 0. Chua, “On a class o SC esis o s and i s applica ion o he syn hesis o nonlinea d i ing-poin and ans e -cha ac e is ic plo s,” In . J. Ci cui Theo y Appl., o be published. [9] A. Rod iguez-Vazquez, J. L. Hue as, and L. 0. Chua: “Ra ional SC ne wo ks,” in p epa a ion. Nodal Vol age Simula ion o Ac i e RC Ne wo ks P. V. ANANDA MOHAN, V. RAMACHANDRAN, AND M. N. S. SWAMY Abs ac --I is shown ha he loa ing nodes p esen in ac i e RC ne wo ks can he elimina ed by nodal ol age simula ion, leading o new ac i e RC ne wo ks. The new ac i e RC ne wo ks hus gene a ed can be easily used o ealize s ay-insensi i e SC ne wo ks. Fqi alence o ce ain ac i e RC opologies, esul ing om applica ion o his me hod, is also demons a ed. I. INTRODUCTION The heo y and design echniques o ac i e RC il e s a e conside ably ma u e a p esen . In eg a ed ci cui implemen a ion necessi a ed ci cui s using g ounded capaci o s, o elimina e he e ec o bo om-pla e pa asi ic capaci ances [l]. Wi h he ad en o swi ched-capaci o echnique, i has become necessa y o elimina e he e ec o pa asi ic capaci ances al oge he , since p edis o ion as well as imming a e una ac i e [2]. I is na u- al, he e o e, o a emp o de i e s ay-insensi i e SC opologies om ac i e RC il e s using g ounded capaci o s. Th ee ech- niques iz., pa asi ic compensa ion [3], [4], nodal ol age simula- ion [5], [6], and s ay-capaci ance elimina ing ans o ma ions [7] ha e been epo ed in he li e a u e. In his le e , we examine he second app oach in de ail. I may be no ed ha his app oach has been used in de i ing bilinea SC ladde il e s om doubly e mina ed LC il e s [8]. O he a ia ions such as nodal ans e unc ion simula ion [9], [lo] a e also a ailable in he li e a u e. II. NODALVOLTAGESIMULATION METHOD We s udy he nodal ol age simula ion echnique wi h e e - ence o speci ic ac i e RC ne wo ks, in wha ollows, in o de o demons a e i s u ili y. A. Applica ion o Mul iple Feedback Type Single -Ampli ie Low - Pass Fil e s Conside he mul iple- eedback ac i e RC low-pass il e o Fig. l(a). The in e nal node x in his ne wo k is desc ibed by KCL as Y , [ 1 1 1 F+F= ” x+R+R+SC,. I (1) 1 3 1 2 3 I is easy o ealize V, using OA’s wi h g ounded nonin e ing inpu as shown in Fig. l(b), by he ci cui wi hin do ed lines. Augmen ing his ne wo k by he emaining ci cui consis ing o Manusc ip ecei ed Ap il 10, 1984. P. V. Ananda Mohan was wi h he Depa men o Elec ical Enginee ing, Conco dia Uni e si y, Mon eal, Canada. He is now wi h T ansmission R&D, Indian Telephone Indus ies, Bangalo e, India. V. Ramachand an and M. N. S. Swamy a e wi h he Depa men o Elec i- cal Enginee ing, Conco dia Uni e si y, Mon eal, Canada. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-32, NO. 10, OCTOBER 1985 1085 L-------------l (b) Fig. 1. (a) Mul iple eedback low-pass il e . (b) Ac i e RC ne wo k equi - alen o Fig. l(a). R,, C, and OA A, o ealize V, om V,, we ob ain he ne wo k o Fig. l(b), which in e es ingly, is he well-known Tow-Thomas biquad [ll]. The a ailabili y o band-pass ou pu a node x and low-pass ou pu a V0 in he ci cui o Fig. l(a) is ans e ed o he ne wo k in Fig. l(b). No e ha , bo h hese ci cui s a e based on induc ance simula ion [12], [13]. Fu he , we obse e ha he ci cui o Fig. l(a) is ha ing damping due o esis o s R,, R,, and R,. This damping can be educed by mul iplying he e ec o R, and R, h ough a nega i e esis ance o g ound a node x [14]. The esul ing ac i e RC il e is shown in Fig. 2(a). P oceed- ing in he same manne as in he case o Fig. l(a), we ob ain he equi alen RC ac i e il e o Fig. 2(b). The ans e unc ion ealized is gi en by Kl - l/R, -= K (4 s2ClC2 R, - SC, R, Thus bo h low-pass and band-pass ans e unc ions a e s ill a ailable, while la ge Q’s can be ob ained h ough he use o nea ly equal R, and R, alues. The Q-sensi i i ies a e hus la ge: and $--=1-F. 5 No e ha Fische and Moschy z [15] used such a modi ica ion o Fleische -Lake ’s biquad [16] o educing he capaci ance sp ead. In applica ions whe e equal capaci o alues a e p e e ed in he ci cui o Fig. l(a), posi i e eedback can be applied [17]. This leads o he modi ica ion o he Tow-Thomas biquad, as shown in Fig. 3(a). The ans e unc ion ealized is , l/R, -= Y s2ClC2 R, s s!p$)+(&AL) (3) No e, howe e , ha he band-pass ans e unc ion is no ealiz- able a he ou pu s o he OA’s A, and A,. The design equa ions 0098-4094/85/1000-1085$01.00 01985 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 01,2020 a 16:18:45 UTC om IEEE Xplo e. Res ic ions apply.