On the Chaotic Behaviour of Buck Converters

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1 On the Chaotc Behaour of Buck Conerters A. Mehrz-San, W. Knsner, and S. Flzadeh Department of Electrcal and Computer Engneerng Unersty of Mantoba Wnnpeg, Canada {mehrz knsner Abstract Power electronc crcuts exhbt nonlnear dynamcal behaour due to ther nherent nhomogenety and swtchng. Among power electronc conerters, the DC/DC buck conerter s studed wth constant-frequency pulse-wdth modulaton feedback control n contnuous conducton mode. Phase-space and tme-doman plots for seeral perodc and chaotc orbts are presented. The bfurcaton dagram s studed together wth perodc orbts and chaotc behaour of the crcut. Seeral smulaton methods ncludng exact soluton and smulaton n an EMTP-type program are used and the mportance of accurate modelng s justfed. Fnally, a method for computaton of yapuno exponents n dscontnuous systems s reewed and mplemented. Index Terms Bfurcaton dagram, buck conerter, chaos, yapuno exponents, symbolc analyss, transent smulaton. P I. INTRODUCTION OWER EECTRONICS has changed the way electrc energy s used and processed. DC/DC conerters that are used to regulate and step down (buck conerters), step up (boost conerters), or both step down and step up (buck-boost or Ćuk conerters) are among the most-wdely used power electronc crcuts. One of the most mportant characterstcs of power electronc crcuts s ther hghly nonlnear behaour. Ths nonlnearty s due to both nonlnear elements used n these deces (e.g., dodes, BJTs, transformers, and control crcutry employed such as comparators and pulsewdth modulators) and the swtchng operaton, whch changes the topology of the crcut [1], [2]. The tradtonal method for dealng wth systems wth slght nonlnearty s to lnearalze the system equatons around the operatng pont. Ths technque, howeer, s good only for a small neghbourhood of the operatng pont, whch n turn causes dffcultes n smulaton of a nonlnear crcut n ts entre operatng range, and s not sutable for modelng of swtched-mode DC/DC conerters that are both nonlnear and tme-aryng dynamcal systems. Other efforts for desng conentonal lnear models for power electronc crcuts, such as state-space aeragng, can only represent the detals of the behaour of the system to a certan harmonc order [3]. The behaour of an electrcal crcut can be characterzed n ts steady-state, f any, or n the transent state. In ts steady state, an electrcal crcut can exhbt one of the followng four behaours [4]: () pont stablty, () cycle stablty, () nstablty (but saturated), and () chaotc stablty. In pont stablty, the crcut currents and oltages settle down to a constant alue. In ths case the crcut s called stable and representaton of the system n phase-space s a sngle pont. Most crcuts are desgned to operate n ths mode. In cycle stablty, the crcut states repeat themseles as perodc functons of tme wth a sngle perod of T, perod T and ts multples, or some dsproportonate perod. An oscllator crcut s perhaps the most used example of ths type of behaour. In saturated nstablty, oltages and currents derge untl bounded by an external factor, e.g., lmted oltage of the power supply. Some crcuts wth ery specfc functons, for example Schmtt trggers, oltage clppers, and flp-flops use ths mode of operaton. In chaotc stablty, the dynamcal system s dergent but ts trajectory s bounded. Ths fourth class s called chaotc behaour and that trajectory s called strange attractor, whch arses n many power electronc conerters, such as buck conerter, boost conerter, and the rpple regulator crcut (a buck conerter wth constant reference oltage nstead of a PWM feedback control) [5]. Exstence of chaos n power electronc crcuts has receed great attenton durng last two decades. Due to ther smpler structure, the most studed power electronc crcuts are DC/DC conerters. Chaotc behaour of buck conerters has been studed n [6]-[9]. A method for controllng chaos n the buck conerter based on pole-placement s suggested n [1]. Boost conerters are consdered n [11]-[13]. Ths paper presents a study of the chaotc behaour of the buck conerter. In Secton II a bref ntroducton to chaos s presented. Secton III dscusses the buck conerter and ts mathematcal modelng. Three methods for smulaton of the buck conerter, consstng of the exact soluton, numercal ntegraton, and smulaton n PSCAD/EMTDC program are presented n Secton IV and results are compared. yapuno exponents are defned and calculated n Secton V. Some fnal remarks n Secton VI conclude the report. II. REVIEW OF CHAOTIC DYNAMICS Chaotc operaton s the fourth class of stablty of a dy /7/$ IEEE

2 namcal system. A contnuous system goerned by a set of at least three frst-order, nonlnear, dfferental equatons wth no external nput (autonomous), or of lower order but wth an external nput such as tme (non-autonomous), can exhbt chaotc behaour [14]. The sgnals resultng from a chaotc system, although aperodc, are bounded. The behaour of a system s referred to as chaotc f the trajectory of ts states possesses three propertes. Frst, t should show hgh senstty to the ntal condtons. Een the smallest changes can lead to ery large dfferences n the trajectory, although the chaotc system s goerned by a set of completely determnstc equatons and een n the absence of nose. The second property of chaos s the underlyng process of foldng. Whle trajectores do not ntersect, they are lmted to a certan area the strange attractor. The thrd characterstc of chaos s mxng, whch means trajectores, regardless of the ntal condtons, wll eentually reach eerywhere n the phase-space. A more formal defnton s that for any two open nterals of non-zero length, a alue from one nteral maps to another pont n the other nteral after a suffcent number of teratons [15, p. 52], [16]. Two types of dagrams are frequently used n the study of chaotc systems: the phase-space dagram and the bfurcaton dagram [17]. The phase-space dagram, whch s an n-dmensonal dagram wth n beng the number of states of the autonomous system, shows the state trajectory of the system. For a stable system, the phase-space dagram s just a sngle pont. For a perodc system, t s a closed trajectory. For an unbounded unstable system, the phase-space dagram s dergent, whle for a chaotc system, although the phasespace dagram s dergent, the trajectory s bounded. Such a trajectory s non-ntersectng [16]. Note that, n general, any projecton of the strange attractor on a sub-space below ts embeddng dmenson becomes ntersectng. A bfurcaton dagram s a sual summary of successon of perod-doublngs. In a bfurcaton dagram, the bfurcaton parameter s plotted on the abscssa and the states of the system are plotted on the ordnate. Crcut parameters [8], [11] or feedback loop parameters [17] can be chosen as the bfurcaton parameter. There are seeral methods to characterze chaos. The largest yapuno exponent and the nformaton dmenson are among them [16]. The largest yapuno exponent and the nformaton dmenson for the studed buck conerter are 4 and 2.21, respectely [18]. III. THE SECOND-ORDER BUCK CONVERTER A buck conerter s a step-down power electronc conerter that conerts an unregulated DC oltage to a lower DC oltage regulated by means of closed-loop feedback operaton. The crcut dagram of the buck conerter s shown n Fg. 1. E S D Hgh: ON ow: OFF ramp con C A out T R V U V V ref Fg. 1. Schematc of the second-order buck conerter wth smplfed control crcutry. A. Behaour of the Crcut There are two swtches n a second-order buck conerter. One swtch s uncontrolled (dode D) and the other one (S) s controlled by the feedback controller. At any tme, only one of these two swtches s n the ON state. A capactor C s connected n parallel wth the load to help mantanng a relately constant load oltage. The seres nductor s used as an energy-storng dece. Durng the ON state of S, energy from the source E s stored n. When S s open, the nductor delers the stored energy to the load R. The feedback loop tres to keep the load oltage, out, constant. The load oltage s measured and passed to the subtractor block to form the error sgnal, con, whch s = A V (1) con ( ) where A s the amplfcaton factor. Ths sgnal s then compared wth a saw-tooth ramp sgnal wth a mnmum of V, maxmum of V U and perod of T, defned as = V V V mod( t T,1 (2) ramp out ref ( ) ) U If the magntude of the saw-tooth sgnal s greater than that of the error sgnal con, S s turned ON, otherwse S remans OFF. Ths means that the swtch state changes wheneer con = ramp s satsfed. Normally, the load oltage s passed through a low-pass flter (an ntegrator, whch can be realzed as a shunt RC crcut) before beng fed to the subtractor to reduce ts rpple. Ths flter s neglected here for smplcty. B. Model of the Crcut At each nstant, the system state s determned by the two state arables (capactor oltage) and (nductor current) as well as the state of swtch S. The buck conerter can be consdered as two crcuts multplexed n tme. The dfferental equatons for and are d 1 1 = dt RC C (3) d 1 ζ () = t E dt where ζ s the control sgnal and s 1 when the swtch s ON 1146

3 and s when the swtch s OFF. The crcut s smulated usng the parameters shown n Table I. Input oltage E s used as the bfurcaton parameter and s ared between 15 and 4 V. TABE I: BUCK CONVERTER PARAMETERS USED FOR SIMUATION Crcut R (Ω) (mh) C (µf) Parameters Controller V U (V) V (V) T (µs) V ref (V) A Parameters The dfferental equatons (3) of the crcut are soled n the next secton by three methods: the exact closed-form soluton, numercal ntegraton, and PSCAD smulaton. IV. SIMUATION OF THE BUCK CONVERTER Behaour of the closed-loop buck conerter s analyzed usng three methods. Frst, the pecewse closed-form soluton of the system equatons s presented [7]. The extreme senstty of the crcut s the man ncente for lookng for the exact soluton of the crcut, so that the round-off error does not propagate from one step to another and the most accurate results can be obtaned. The system equatons are also soled by numercal ntegraton and by the commercal smulaton emtp-type program PSCAD/EMTDC. In all cases, crcut elements are assumed to be deal. A. Closed-Form Soluton In ths method, (3) s soled for (t) and (t) wth a constant ζ and the closed-from soluton s obtaned. The swtchng happens wheneer the followng boundary condton s satsfed con( tc ) = ramp ( tc ) (4) where t c s the swtchng tme. Then, the equaton s soled usng Newton-Raphson method wth a maxmum allowable error of 1-1 to fnd the exact swtchng tme. For E equal to 24, 28, 32, and 33 V, phase-space plots show perod-1, -2, -4, and chaotc behaour of the system as shown n Fg. 2. Fgure 3 shows tme-doman waeforms for chaotc operaton for E = 33 V. It can be clearly seen that t s possble for con to skp some cycles (no swtchng n a cycle) as well as to ntersect the ramp oltage more than once n a cycle (multple swtchngs n a cycle). Bfurcaton dagram s plotted for nput oltage E swept from 15 to 4 V (Fg. 4) and s obtaned by recordng the oltage at the end of each perod. It clearly shows the successon of perod doublngs. The separaton between perod-doublng ponts decreases wth the number of perods. The rato of successe bfurcaton parameters approaches the Fegenbaum number, Ths number, whch s beleed to be transcendental but not yet proed to be so, also arses n many physcal systems before they enter the chaotc regme. The abrupt transton from the perod-doublng to chaotc regon s related to the E = 24 V E = 32 V E = 28 V E = 33 V Fg. 2. Phase-space dagram of the buck conerter showng perod-1 (E = 24 V), perod-2 (E = 28 V), perod-4 (E = 32 V), and chaotc (E = 33 V) waeforms obtaned form the exact soluton. V ramp, V con (V) con ramp t (s) t (s) Fg. 3. Chaotc operaton of the buck conerter obtaned by exact soluton wth E = 33 V (shown are con and ramp). sharp, sngular ponts n the phase-space dagram of the conerter. B. Numercal Integraton The equatons n (3) are already n the sutable form for computer mplementaton of numercal ntegraton of the statespace representaton. Both Euler s and trapezodal methods are mplemented wth a small tme-step of 1 µs. The results are shown n Fg. 5. The dscrepancy obsered between the results of ths method and the exact soluton s due to the extense round-off errors, whch are magnfed not only by the senstty of the crcut to ntal condtons, but also by the dscretzaton of tme t c, n contrast to the preous approach where Newton-Raphson method s used to fnd the almost exact t c. That s, a flow has been conerted to a map. C. PSCAD/EMTDC Smulaton The model s also mplemented n PSCAD/EMTDC electromagnetc transent smulaton program [19]. Fgure 6 shows the conerter model. The results are used to erfy those of numercal ntegraton method and to nestgate the effects of lmted accuracy used for t c. Takng adantage of the nterpolaton block n PSCAD [2], t c s found wth an accuracy of.1% of tme step. Phase-space dagrams for four nput oltages alues (24, 1147

4 (a) (b).2 Fg. 4. Bfurcaton dagram obtaned by samplng the output oltage at the end of each cycle E = 24 V E = 32 V E = 28 V 28, 32, and 33V) are shown n Fg. 7, whch are qute smlar to those of the exact soluton n Fg. 2. Ths s because of the proper selecton of tme-step as well as approxmatng the wtchng nstant by nterpolaton. Fgure 8 shows tme-doman oltage waeform an nput oltage of E = 33 V. V. THE ARGEST YAPUNOV EXPONENT yapuno exponent s a quanttate measure of the senste dependence of a dynamcal system on the ntal condtons [21]. It shows the rate of dergence of the system trajectores correspondng to close ntal condtons. The E = 33 V Fg. 5. Phase-space dagram of buck conerter showng perod-1 (E = 24 V), perod-2 (E = 28 V), perod-4 (E = 32 V), and chaotc (E = 33 V) waeforms obtaned form numercal ntegraton. Note jtters for E = 24 and 32 V. Fg. 6. PSCAD model of the buck conerter (c) (d) Fg. 7. Phase-space dagrams (output oltage on x-axs, nductor current on y- axs) of the PSCAD run for (a) E = 24 V showng perodc operaton, (b) E = 28 V showng perod-2, (c) E = 32 V showng perod-4, and (d) E = 33 V showng chaotc operaton. number of yapuno exponents for a system s equal to the dmenson of ts phase space. Normally the largest exponent s used, because t determnes the horzon of predctablty of the system. In ths sense, the nerse of the largest yapuno exponent s called yapuno tme, whch defnes the characterstc foldng tme of the system. The concept of yapuno exponents can be consdered as the nonlnear counterpart of egenalues for lnear systems. As t shows the rate of separaton of nfntesmally close trajectores, one can predct the behaour of the system based on the sgn of the yapuno exponent. A negate yapuno exponent s characterstc of dsspate (non-conserate) systems, whch exhbt pont stablty. The more negate the exponent, the faster the stablty. An exponent of shows the extremely fast conergence, and hence stablty. A yapuno exponent of zero s characterstc of a cycle-stable system. In ths case, the orbts mantan ther separaton. A poste yapuno exponent, on the other hand, mples that nearby ponts, no matter how close, wll fnally derge to an arbtrary separaton. Ths happens n the case of nstable as well as chaotc system. The dstncton between these two s made by usng the set of yapuno exponents. The largest yapuno exponent s defned as λ max = lm ( ) 1 lm ln t t δx δx() δx where δx(t) shows the perturbaton of the system. To oercome the problems n applyng the aboe equaton to power electronc crcuts [22], an approxmate method has been suggested by Müller [23]. Ths method s used for the buck conerter and λ max s calculated from ( t t ) ( t ) (5) 1 δx λ max = ln (6) T δx 1148

5 Vout Vramp Vcont Fg. 8. Plot of output oltage, ramp generator output, and the control oltage s. tme for the chaotc operaton wth E = 33 V. Whle for E = 24 V, the maxmum yapuno exponent s λ max = (practcally zero) that ndcates a stable system, for chaotc regon, E = 33 V, λ max = 8, whch s a poste number, n agreement wth [18]. VI. CONCUSIONS In ths paper, the buck conerter and ts operaton n the chaotc regme s studed usng tme-doman, phase-space, and bfurcaton dagrams, as well as yapuno exponents. The chaotc nature of crcut operaton ntensfes the need for precse determnaton of the swtchng nstances. Therefore, three methods (analytcal soluton, numercal ntegraton, and smulaton n the PSCAD/EMTDC program) are used to study the crcut and fnd the most sutable combnaton of smplcty of mplementaton and accuracy of results. 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