Bifurcation and Chaotic Aspects in Peak Current Controlled Buck-Boost Converters

Transcription

1 Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan Bifurcaion and Chaoic Apec in Peak Curren Conrolled BuckBoo Converer DAN NEGOITESCU, DAN ASCU, VIOE POPESCU, COINA IVAN Deparmen of Applied Elecronic Poliehnica Univeriy of Timioara, Faculy of Elecronic and Telecommunicaion Bd. Vaile Parvan, 33 Timioara OMANIA Abrac: The paper inveigae he bifurcaion and chaoic behavior of a currenmode conrolled Buck Boo converer operaing in coninuou conducion mode (CCM). The analyi comprehend boh open loop and proporional cloed loop operaion cae. Claical averaged dynamic model canno predic he chaoic converer behavior which lead o ypical phenomenology like perioddoubling, quaiperiodic and chaoic operaion. The mo imporan dynamic behavior apec are revealed by imulaion, ogeher wih a rigorou mahemaical decripion ha exacly predic he converer operaion mode. KeyWord: Noninear dynamic, bifurcaion, chao, ubharmonic ocillaion, dcdc converer, buckboo converer, currenmode conrol, power elecronic Inroducion A i i known, currenmode conrol i one of he mo popular conrol mehod ued for achieving fa oupu regulaion in wiching converer []. The goal of hi conrol i o force he inducor curren o follow a reference value which i provided by an oupu feedback circui. The inner curren loop can become unable under cerain condiion. The baic phenomenology aociaed wih hi inner loop i he perioddoubling which i no deecable by he averaged dynamical model. The block diagram of a currenmode conrolled buckboo converer operaed in CCM i repreened in Fig.. The operaion of he inner curren loop can be briefly decribed a follow. Tranior curren, i, i ened and compared o a reference curren, generaing he onoff driving ignal for he wich S. The wich S i urned on by a clock a he beginning of each wiching cycle and he inducor curren increae unil i reache he value of. A hi poin he wich S i urned off and remain in hi ae unil he nex period begin. The conrol equaion for hi operaion mode can be derived from he fir opological ae when S i on, a follow: di I i v () ref n, on d dn where v,on i he inducor volage in he fir opological ae. From () he duy cycle reul a: V g S i S Q d I ref n n () v, on S D i The converer can be operaed in an open loop mode when he oupu feedback loop i no preen or in a cloed loop mode, when he oupu feedback loop i added. C COMP COCK K V ref V o OUTPUT FEEDBACK OOP Fig. Schemaic diagram of a currenmode conrolled buckboo converer Iue 7, Volume 7, July 8

2 Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan Open oop Operaion Mode A far a dynamic of he inner loop i concerned he converer can be operaed open loop. Becaue he oupu feedback loop i uually much lower and i purpoe i o adju he reference value in he even of inpu volage and load variaion, he omiion of he oupu volage loop hould no aler he high frequency dynamic of he inner curren loop. For inveigaing he chaoic behavior for a currenmode conrolled buckboo converer in open loop operaion mode, he following circui parameer were conidered: V g = V, =,mh, =,47Ω, f = khz, = 5Ω, C = 4,4μF. The circui operaion wa imulaed uing he CASPOC Simulaion eearch Sofware []. In Fig. are preened he inducive curren waveform, he phae porrai and he inducive curren harmonic pecrum in he cae of period operaion. I can be oberved ha he converer operae in a able and periodical mode due o he mall value of he bifurcaion parameer. By increaing he value of, he converer operaion mode can change. In Fig.3 are preened he ame waveform bu in he cae of period operaion. I can be oberved ha he ampliude of he curren harmonic a half of he wiching frequency i higher han he harmonic correponding o he wiching frequency. Alhough ubharmonic, hi operaion mode i ill able and periodical. The cae of quaiperiodic operaion i preened in Fig.4. Due o i nonperiodic naure, hi operaion mode lead o preence of noie, epecially a low frequencie a i can be oberved from he inducor curren pecrum in Fig.3(c). 45.m 5.m 4.m 35.m 3.m 5.m.m 5.m.m 5.m.u 5.u.u 5.u.u 5.u 3.u 35.u 4.u 45.u 5.u m m m m m 6.464m m 6.464m 6.464m Curren harmonic ampliude [A] Ampliudinea armonicei [A] Frecvena [Hz] x Frequency [Hz] (c) Fig. Open loop operaion wih period ( =.4A). Inducor curren waveform, Phae porrai, (c) Inducor curren harmonic pecrum Iue 7, Volume 7, July 8

3 Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan 7.m 65.m 6.m 55.m 5.m 45.m 4.m 35.m 3.m 5.m.m 5.m 665.5m 6.78m 57.78m 5.78m 47.78m 4.78m 37.78m 3.78m 7.78m.78m 7.78m.m 5.m 4.u 45.u 4.u 45.u 4.u 45.u 43.u 435.u 44.u 445.u 45.u (c) Frecvena [Hz] x Frequency [Hz] 4.78m Fig.3 Open loop ubharmonic operaion wih period ( =.6A). Inducor curren waveform, Phae porrai, (c) Inducor curren harmonic pecrum Curren harmonic ampliude [A] Ampliudinea armonicei [A] m 95.m 9.m 85.m 8.m 75.m 7.m 65.m 6.m 55.m 5.m 45.m 4.m 35.m 3.m 5.m.m 5.m.m 5.m 8.u 8.u 8.u 83.u 84.u 85.u 86.u 87.u 88.u 89.u 9.u m m m m m m m m m m m m m m m 3.934m m m Curren harmonic ampliude [A] (c) Frequency [Hz] x 4 Fig.4 Open loop quaiperiodic operaion ( =.95A). Inducor curren waveform, Phae porrai, (c) Inducor curren harmonic pecrum Iue 7, Volume 7, July 8

4 Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan m 6.m 4.m.m 7.u 7.u 74.u 76.u 78.u 8.u 8.u 84.u 86.u 88.u 9.u m 94.77m m 84.77m m 74.77m m 64.77m m 54.77m m 44.77m m m Curren harmonic ampliude [A] (c) Frequency [Hz] x 4 (d) Fig.5 Open loop chaoic operaion ( =.3A). Inducor curren waveform, Phae porrai, (c) Inducor curren harmonic pecrum, (d) Poincaré ecion In Fig.5 he chaoic operaion mode waveform are preened. Thi operaion mode can be alo oberved from he irregular form of he Poincaré ecion in Fig.5(d) and i mu be avoided in he all applicaion involving wiching power upplie. By olving he ae equaion ha decribe he dynamic of a econd order dcdc converer wih CCM operaion, he ieraive map can be obained in he following form [3]: x( n ) f ( x( n), d) (3) where: f f g f ( x, d) x v g (4) f f g In he cae of he inveigaed buckboo converer, he analyical approximae expreion for he f and g funcion involved in he ieraive map, were deermined by he help of MATHEMATICA program. Inducor reiance,, wa aken ino accoun, reuling in [4]: Iue 7, Volume 7, July 8

5 Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan 3 6 d ( d) g 6 3dT dt 3 6 C From he buckboo circui opology and from he inducor curren waveform he duycycle value reul: iref in dn (6) Vg The above relaion wa obained aking in conideraion a linear hape of curren waveform during he fir opological ae. Thi approximaion lead only o an error of.% in he duycycle value. Baed on he ieraive map he heoreical bifurcaion diagram preened in Fig.6 wa obained in MATAB. A he curren reference value increae i can be oberved he occurrence of bifurcaion wih perioddoubling, quaiperiodic operaion and chaoic operaion. f f f f g d d d ( d) C C d d d ( d) C C C dt d ( d) dt ( d) C ( d) 6 3dT dt ( d) T Fig.6 Bifurcaion diagram of a currenmode conrolled buckboo converer wih open loop operaion The fir bifurcaion can be preciely deermined uing he characeriic muliplier. Thee can be ( d) T ( d) T ( d) T ( d) T C C C C C C C (5) ( d) C C obained a oluion of he characeriic equaion, he reul being preened in Table. Table Characeriic Muliplier,4,835;,689,5,979;,6838,5,994;,6839,5,34;,653,55,978;,698,5,9364;,736,54,876;,746,6,7985;,7638 I can be oberved ha for mall value he characeriic muliplier abolue value are le han, denoing a able period operaion. A value increae, one of he characeriic muliplier i moving oward. A an approximae value of.5a for one of he characeriic muliplier equal, indicaing a bifurcaion wih perioddoubling. From hi poin forward he converer ill operae in a able mode bu wih period operaion. The converer behavior can be alo oberved from he graphical repreenaion of he highe yapunov exponen veru he curren reference, performed in MATAB and preened in Fig.7. Nex, he bifurcaion diagram for he inveigaed buckboo converer wa obained hrough imulaion in CASPOC. Excep for ome mall dicrepancie caued by he approximaion ued in he ieraive map i can be oberved ha he imulaed reul, preened in Fig.8, are in accordance wih he heoreical expecaion Iue 7, Volume 7, July 8

6 Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan Fig.7 The highe yapunov exponen veru Fig.8 Bifurcaion diagram obained by CASPOC imulaion of a currenmode conrolled buckboo converer wih open loop operaion, for γ=,7. The ame behavior ype (cacade of perioddoubling unil a chaoic operaion) can be obained for oher value of he γ=t /C parameer. In Fig.9 are preened he bifurcaion diagram obained by CASPOC imulaion in he cae of γ=.454 and repecively γ=.6. 3 Cloed oop Operaion Mode In he preence of he oupu volage feedback loop, wih K being he feedback gain, he converer operae in a cloed loop mode. The converer model require an addiional equaion in order o decribe he relaionhip beween he oupu volage and he reference curren. The bifurcaion parameer i choen o be he volage feedback loop gain K. In hee condiion he curren reference value i no longer a fixed one. Fig.9 Bifurcaion diagram obained by CASPOC imulaion of a currenmode conrolled buckboo converer wih open loop operaion, γ=,454, γ=,6. Becaue of he linear proporional feedback, he equaion for he curren reference i: i ref I ref d n C vcne Vref (7) In equaion (7), V ref i he eadyae reference oupu volage and i he eadyae reference curren which can be choen a he econdary bifurcaion parameer. The inducor curren waveform, for cloed loop operaion of he buckboo converer under currenmode conrol, i repreened in Fig.. In a imilar way o he open loop cae, he converer behavior reul from he bifurcaion diagram, obained from CASPOC imulaion and preened in Fig Iue 7, Volume 7, July 8

7 Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan 5.m 45.m 4.m 35.m 3.m 5.m.m 5.m.m 5.m 5.u 55.u 6.u 65.u 7.u 75.u 8.u 85.u 9.u 95.u 3.u Fig. Inducor curren waveform in cloed loop operaion mode K Fig. Bifurcaion diagram obained by CASPOC imulaion of a currenmode conrolled buckboo converer wih cloed loop operaion. I can be oberved ha unil he feedback gain value K of.3, he converer operae wih period. A around hi value he fir bifurcaion occur, hi lead o he perioddoubling phenomenon. Furhermore by increaing he value of he volage loop gain, he perioddoubling procee are repeaing unil he converer operae in a chaoic way. By increaing he value of he bifurcaion parameer over he value of., on he chaoic behavior will be overimpoed he onlimi colliion becaue of he diconinuou operaion mode on cerain ime inerval. The waveform for he inducor curren and phae porrai from CASPOC imulaion, correponding o he period, period, period 4 and chaoic operaion are preened in Fig Iue 7, Volume 7, July 8

8 5.m WSEAS TANSACTIONS on CICUITS AND SYSTEMS Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan 5.m 45.m 5.689m m m 4.m m m 35.m 47.69m 3.m m m 5.m m.m m 37.69m 5.m 97.69m.m 77.69m 57.69m 37.69m 45.u 455.u 46.u 465.u 47.u 475.u 48.u 485.u 49.u 495.u 5.u 7.69m 7.69m m 45.m 54.94m m m 4.m 44.94m 35.m m 3.m 5.m.m 5.m 34.94m 74.94m 4.94m.m 74.94m 5.m 4.94m 35.u 355.u 36.u 365.u 37.u 375.u 38.u 385.u 39.u 395.u 4.u 74.94m 74.94m m 55.m m m 5.m 53.49m 45.m m 4.m 43.49m 35.m 3.m 5.m (c) m 33.49m 73.49m.m 3.49m 5.m 73.49m.m 3.49m 5.m 73.49m 5.u 55.u 6.u 65.u 7.u 75.u 8.u 85.u 9.u 95.u 3.u 3.49m 3.49m m 55.m 5.m m 55.m 5.m 45.m 45.m 4.m 4.m 35.m 3.m 5.m.m 5.m.m 5.m (d) 35.m 3.m 5.m.m 5.m.m 5.m 5.u 7.u 9.u 3.u 33.u 35.u 37.u 39.u 4.u 43.u 45.u u Fig. Currenmode conrolled buckboo converer, wih cloed loop operaion, CASPOC imulaion waveform (inducor curren and phae porrai). Period operaion, K=,, Period operaion, K=,5, (c) Period 4 operaion, K=,9, (d) Chaoic operaion, K=,6, Iue 7, Volume 7, July 8

9 Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan 4 Simulaion reul A i wa menioned in he above ecion, he buckboo converer behavior wa imulaed wih he CASPOC Simulaion eearch ofware. Nex, in Fig.5 i preened he cloed loop operaion imulaion circui for he buckboo converer. SCOPE OUT V D.m C 4.4u 5 V D.m C 4.4u OUT SCOPE7 SCOPE TIME SIGNA TIME ime.5 DC.5 AC K F Phae. d I() CUENT SUB CUENT Fig.3 CASPOC imulaion circui of a currenmode conrolled buckboo converer open loop operaed. SCOPE7 S FF Q FF TIME SIGNA TIME ime.5 DC.5 AC K F Phae. d GAI. CUENT I() ADD. GAI SUB CUENT Fig.4 CASPOC imulaion circui of a currenmode conrolled buckboo converer open loop operaed ued for obaining he bifurcaion diagram. S FF Q FF SCOPE OUT D V.m C 4.4u.47 VOTAGE V()V(ou) SUB VOTAGE 7.6 GAI. VCON CUENT TIME SIGNA TIME ime.5 DC.5 AC K F Phae. d I() SUB.384 COMP FF COMP Q S D Fig.5 CASPOC imulaion circui of a currenmode conrolled buckboo converer in cloed loop operaion ued for obaining he bifurcaion diagram. For he open loop operaion he imulaion circui from Fig.3 wa ued. In Fig.4 i preened he open loop operaion imulaion circui ued for obaining he bifurcaion diagram. The inducor curren harmonic pecrum wa repreened in MATAB uing he imulaion reul of he circui from Fig Iue 7, Volume 7, July 8

10 Dan Negoiecu, Dan acu, Viorel Popecu, Corina Ivan The bifurcaion diagram for open loop operaion wa obained in MATAB uing he imulaion reul of he circui from Fig.4. The ieraive map wa deermined wih a MATHEMATICA program and he characeriic muliplier were deermined in MATAB. In he cloed loop operaion cae, he bifurcaion diagram wa deermined in MATAB uing he imulaion reul of he circui from Fig.5. 5 Concluion In cloed loop operaion he bifurcaion phenomenon i baically he ame a in he open loop cae wih perioddoubling and border colliion inerplaying o organize he bifurcaion paern. Similar raniion in he appearance of he bifurcaion diagram can be oberved in boh cae. A load ime conan increae he diance o chao along he axi i progreively horened, wih a reduced number of bifurcaion afer he border colliion. The ieraive map developed accuraely predic he converer behaviour. From he preened cae of he buckboo converer and from oher example conained in he lieraure, [9], [], [], i can be revealed ha bifurcaion wih perioddoubling and chaoic behavior are characeriic o all currenmode conrol operaed converer, regardle of he preence of he oupu volage feedback loop. Syem, (ISCAS'), Geneva Swizerland, June, pp. I [7] Hamill, D. C., Jefferie, D. J., Subharmonic and chao in a conrolled wichedmode power converer, IEEE Tran. on Circui and Syem, vol. 35, no. 8, Aug. 988, pp [8] e, C. K., ecen Developmen in he Sudy of Nonlinear Phenomena in Power Elecronic Circui, IEEE Circui and Syem Sociey Newleer, March Iue,, pp [9] Cafagna, D., Grai, G., Complex Dynamic Phenomena in Power Converer: Bifurcaion Analyi and Chaoic Behavior, WSEAS TANSACTIONS on CICUITS AND SYSTEMS, Iue 4, Volume 3, June 4, ISSN9734, pp [] HongZhong,., HaoZhong, C., ChengMin, W., A novel approach for deermining he hopf bifurcaion poin of dynamic volage abiliy in power yem, WSEAS TANSACTIONS on CICUITS AND SYSTEMS, Iue 7, Volume 4, July 5, ISSN9734, pp [] Dragan, D., Conrolling Chao in DC/DC Converer uing OGrebogiYorke and Pyraga Mehod, WSEAS TANSACTIONS on CICUITS AND SYSTEMS, Iue 6, Volume 5, June 6, ISSN9734, pp eference: [] Erickon,. W., Makimović, D., Fundamenal of Power Elecronic, Second Ediion, Kluwer Academic Publiher,. [] CASPOC eference Manual, Simulaion eearch, 999. [3] e, C. K., Complex Behavior of Swiching Power Converer, CC Pre C, 4. [4] Ivan, C.M., Conribuion regarding he analyi and modelling of power wiching converer, Ph.D. Thei, Poliehnica Univeriy of Timioara, 8. [5] Hamill, D. C., Deane, J. H. B., Aon, P. J., Some applicaion of chao in power converer, IEE Colloquium: Updae on new power elecronic echnique, ref. no. 997/9, ondon, May 997, pp. 5/5/5. [6] e, C. K., ai, Y. M., Conrol of Bifurcaion in Currenprogrammed DC/DC Converer: A eexaminaion of Slope Compenaion, IEEE Inernaional Sympoium on Circui and Iue 7, Volume 7, July 8