Study on Bifurcation and Chaos in Boost Converter Based on Energy Balance Model
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1 ergy ad Power geerg, 009, doi:10.436/epe Published Ole August 009 ( Study o Bifurcatio ad Chaos Boost Coverter Based o ergy Balace Model Quam NI, Zhizhog J Departmet of Iformatio Coutermeasure, Air Force Radar Academy, AFRA, Wuha, Cha mail: qm1@tom.com Abstract: Based o boost coverter operatg discotuous mode, this paper proposes a eergy balace model (BM) for aalyzg bifurcatio ad chaos pheomea of capacitor eergy ad output voltage whe the coverter parameter is varyg. It is foud that the capacitor eergy ad output voltage dyamic behaviors exhibit the typical period-doublg route to chaos by creasg the feedback ga costat K of proportioal cotroller. he accurate positio of the first bifurcatio pot ad the iterative diagram of the capacitor eergy with every K ca be derived from BM. Fally, the uderlyg causes for bifurcatios ad chaos of a geeral class of olear systems such as power coverters are aalyzed from the eergy balace viewpot. Comparg with the discrete iterative model, BM is simple ad high accuracy. his model ca be easily developed o the olear study of the other coverters. Keywords: power coverter, olear, bifurcatio, chaos, eergy balace model 1 Itroductio he bifurcatio ad chaos pheomea appeared power system have becomg a focus subject at preset. It is foud that basic DC/DC coverters exhibit bifurcatio ad chaos pheomea as well as parallel-coected DC/DC coverters ad PFC system. here are o uified methods researchg olear power system. I the appeared literatures, the averaged model ad the samplg data model are usually adopted olear aalysis for DC/DC coverters [1][]. But the averaged model eglects the dyamic characteristic of the system at high frequecy, ad oly ca be used for aalyzg the dyamic behavior at low frequecy. his model has limitatios as follows. Firstly, the depedece o itial coditio for system dyamic behavior is eglected small sigal aalysis ad it ca't predict dyamic behavior whe the coverters work the saturated mode. Secodly, this model ca t predict the stability of system uder the fast-scale coditio [3]. Accordg to the periodic workg characteristic of power system, the samplg data modelg method build the relatioship be- twee state variable at preset samplg time stat ad state variable at ext samplg time stat. Based o this idea, four models amed stroboscopic map model, sychroous switchg map model, asychroous switchg map model ad geeral two-by-two switchg map model are give [] accordg to the differet samplg time stat. hese models are used to aalyze olear pheomea such as bifurcatio ad chaos. But some shortcomgs exist these models. For example, it is difficult to get accurate aalytic models as the duty ratio of cotrol pulses is the olear fuctio of state variables. If we eglect olear effect o system trasfer matrix, big errors ad much amout of calculatio will brg about. I order to improve the accuracy of simulatio results, this paper proposes a eergy balace model (BM) for boost coverter. Accordg to eergy balace prciple, BM was established ad the dyamic behavior of the capacitor eergy state was vestigated. From the bifurcatio diagrams of capacitor eergy state ad output voltage, BM preseted this paper is more accurate tha the stroboscopic map model preseted [4]. Furthermore,
2 39 Figure 1. Schematic circuit diagram of closed-loop boost coverter eergy bifurcatio mechaism for power coverters ca be foud from BM. Buildg ergy Balace Model he circuit diagram of the closed-loop boost coverter is show Figure 1. Suppose that boost coverter works DCM. he feedback ga costat of voltage amplifier is k ad error sigal u is amplified through the voltage amplifier. he amplified error voltage compares with saw-tooth wave sigal ad produces variable duty cycle cotrol pulse. he variable duty cycle deotes d Figure 1. he output voltage is regulated by chagg d whe put voltage ad output load is fluctuatg. I oe switchg period [,( 1) ], the coverter satisfies eergy balace coditio R C L, (1) where R L c deote the eergy supplied by put power, the eergy cosumed o resistace load, the storage eergy ductor ad the storage eergy capacitor respectively oe switchg period. I DCM coverter, the ductor curret always starts from zero, i.e., 0, so the eergy balace formula is L R c. () he curret flowg out from put power equals to the curret flowg to ductor a switchg period as show Figure. Accordg to Figure, we ca write So, we obta Figure. Iductor curret waveform il d d L L. (3) d ca be writte as d. (4) 1 i t dt i d d ( 1) () ( ) L L d L he eergy cosumed by load is. (5) ( 1) 1 uc, uc,( 1) R uc () t dt [ ] R R R. (6) [ c, c,( 1) ] RC he storage eergy capacitor oe switchg period is. (7) c c,( 1) c, Substitute (6), (7) to (), we obta 1 K 1 RC c,( 1) S c,. (8) Substitute (5) to (8), we ca build BM of boost coverter c,( 1) KSc, Ad, (9)
3 40 where K S 1 RC 1, A 1 1 L RC RC As show Figure 1, the duty cycle the th period is d d D K( u ), (10) S where D is steady duty cycle, is samplg value of s output voltage at the th period. After the coverter eters to steady state, the closed-loop system should satisfy u. D S For actual coverter, expressio 1 KS d A. (1) should satisfy the followg 0; DS K( u ) 0 d DS Ku ( );0 DS Ku ( ) 1. (13) 1; DS K( u ) 1 3 Results ad Discussios 3.1 Diagrams of Storage ergy Capacitor 1 c,( 1) c, C Based o (9) ad (13),we ca depict the sequece diagram of capacitor eergy for boost coverter operated. (11) From (9) ad (11), the steady duty cycle expressio is period as show Figure 3. he circuit parameters is t s, 16 V, 5 V, L 08 H, C F, R 1.5 he storage eergy other period oscillatio is show Figure 4. he les appeared Figure 3 ad Figure 4 defed as follows. Substitute d 1 to (9),we ca write K A (14) c,( 1) S c, Figure 3. ergy locus of capacitor period heore, le a is depicted from the above expressio. Likewise, substitute d 0 to (9), we obta K (15) c,( 1) S c, he le b is depicted from (15). Le a ad le b are two parallel les ad show storage eergy state capacitor uder two utmost coditios. For actual coverter, duty cycle is limited ad the eergy locus is also limited betwee a ad b If the eergy locus touch with le b,it shows that the duty cycle of correspodg cotrol pulse is zero, i.e., power switch has beg tured-off. Whe storage eergy capacitor is equilibrium, the equatio is expressed as the followg Figure 4. ergy locus of capacitor period 4, period 8 ad chaos. (16) c,( 1) c,
4 41 Le c is depicted from (16). I CCM ad critical mode, DCM, d d 1. d d 1, ad From (4), we obta the duty cycle of coverter operatg critical mode. d c (17) So, the storage eergy capacitor ca be depicted as le d. he correspodg equatio ca be writte as K A c,( 1) S c, ( ). (18) It is obvious that the coverter will eter to CCM while eergy locus lies above le eergy locus lies below le d. Le e d, ad DCM while deotes erece eergy, the equatio is 1 C. (19) Figure 3(b) is the elarge diagram of eergy locus period. As show Figure 3(b), eergy state trasforms betwee dot 1 ad dot while dot 1 lies the left upper part of le d ad dot lies the right lower part of le d. his meas that ductor curret o dot 1 works CCM, ad ductor curret o dot works DCM. Because there is o tersectio betwee eergy locus ad le b, it dicates that there are o skipped cycles period oscillatio. For the other periodic oscillatio behavior, the operatio characteristics of coverter are compreheded by the eergy locus diagrams as show Figure 4. For example, the operatio characteristics period 4 are described as follows. Iductor curret is cotuous the first switchg period; ad discotuous the secod switchg period accompayg with skipped cycles. Furthermore, ductor curret returs to be cotuous the third switchg period; ad discotuous without skipped cycles the fourth switchg period. Such switchg sequeces make output voltage trasform amog four values. he same coclusio will be obtaed by aalyzg other eergy locus Figure 4(a) ad Figure 4(b). However, from Figure 4(c) we ca fd that eergy locus is much complex ad has may tersectios with le b. Meawhile, may eergy states lie o the top of le d. It shows that the correspodg cotrol pulses sequeces is very complex. he output of coverter has etered to chaos.. I geeral, we ca acquire much operatio formatio of coverter from the diagram of capacitor eergy locus. he eergy locus of capacitor moves aroud the erece eergy (le e ). heore, the correspodg output voltage u is fluctuatg at 0. he eergy states o the top of le d shows that coverter operates CCM ad the eergy states at the bottom of le d shows that coverter operates DCM. he pots where eergy locus tersects with le b dicate that coverter operates with skipped cycles. 3. Bifurcatio Diagrams As show Figure 5, the bifurcatio diagram of the capacitor eergy is derived from (9) ad (13) whe the feedback ga costat k of error voltage amplifier is varyg. It is obvious that the capacitor eergy exhibit the route from period-doublg to chaos by creasg k. Assume that the coverter is steady state. 0 d 1. Substitutg d Ds K( u ) to (9) ad usg the followg expressios Figure 5. Bifurcatio diagram of capacitor storage eergy
5 4 u c,, (0) C. (1) C We ca write c, c,( 1) KS c, A[ D S K( C )] C. () It is obvious that () ca be expaded with aylor s series at the equilibrium pot cq,. If high-level items of aylor s series are eglected, the equatio ca be writte as where (3) c,( 1) c. AD K. c,( 1) S cq, K S c, C I the rage of small sigals, ca determe system stability. Whe 1 1, system will be steady. he positio of the first bifurcatio pot ca be obtaed at 1. K c C (1 KS) (4) AD Bifurcatio diagrams of output voltage simulated by three models are show as Figure 6. he diagram of Figure 6(a) is depicted by iterative 500 times of lear equatio with every K. Figure 6(b) is derived from stroboscopic map model [4]. Figure 6(c) is the bifurcatio diagram derived from BM. Comparg three diagrams, the routes from bifurcatio to chaos are similar. But the accurate positios of the first bifurcatio pot ad output voltage are differet. As show Figure 4, the bifurcatio diagram Figure 4(c) is closer to bifurcatio diagram Figure 4(a). heore, BM has high accuracy aalyzg bifurcatio ad chaos pheomea. Based o the above aalysis, BM still belogs to classificatio of stroboscopic map model although each modelg method is differet. he disadvatage of BM S Figure 6. Bifurcatio diagrams of output voltage simulated by three models is that it ca ot aalyze multiple pulses pheomea oe switchg cycle. However, multiple pulses pheomea ca be elimated by a trigger which is added to PWM modulator (see Figure 1). heore, BM is a uified model aalyzg olear pheomea of coverter. 4 he Reaso for Bifurcatio ad Chaos Coverters Based o ergy Balace Viewpot he mechaisms of bifurcatio ad chaos are so complex that there is ot a uified criterio to idetify them. he types of bifurcatios are various, e.g., period-doublg bifurcatio, saddle-ode bifurcatio, fork bifurcatio, Hopf bifurcatio ad border collisio bifurcatio [5]. I particular, border collisio bifurcatio usually appears piecewise smooth system. For ormal bifurcatio, the mechaism of bifurcatio accords with bifurcatio theory, i.e., bifurcatio happes whe eigevalues of Jacob matrix of switchg map model traverse uit circle. Furthermore, bifurcatio styles ca be distguished from traverse directio. However, border collisio bifurcatio ca ot be verified by eigevalues. Border collisio bifurcatio will happe if some pots o periodic orbits collide with the boudary. May research results show that saturatio of duty cycle power coverters results border collisio bifurcatio.
6 43 he geeral method for studyg mechaisms of bifurcatio is to seek breakthrough state space. As show Figure 5, eergy state of capacitor also exhibits bifurcatio ad chaos. heore, this paper firstly presets the mechaisms of bifurcatio ad chaos from eergy balace viewpot. From (9), we coclude that the preset eergy state is defed by former eergy state ad duty cycle. Whe duty cycle keeps steady, capacitor eergy ca keep balace every switchg period. But this is ot the case. Whe capacitor eergy ca ot keep balace every switchg period, switchg time will crease to two switchg cycles, four switchg cycles, et al. heore, period ad period 4 bifurcatios will happe. I geeral, imbalace of capacitor eergy oe switchg cycle results bifurcatio ad chaos. 5 Coclusios BM still belogs to stroboscopic map model. But it has more accurate tha stroboscopic map model. Comparg to iterative map model, the physical sigificace of BM is more distct. From eergy locus diagrams, we ca achieve abudat formatio about coverters. Because eergy balace theorem is uiversal rules the world, BM ca be geeralized to study the others olear power coverter as a uified model. RFRNCS [1] Middlebrook R D ad Cuk S, A geeral uified approach to modelg switchg-coverter power stage, I Power lectro. Spec. Cof. Rec., pp , 1976 [] Mario di Berardo ad Fracesco Vasca, Discrete-time for the aalysis of bifurcatios ad chaos DC/DC coverters, I rasactios o Circuit System Ⅰ, Vol. 47, No., pp , 000. [3] Ma Xikui, Li Mg, Dai Dog, ZhaoHao, et al., Reviews of research o complex behavior of power electroic circuits ad systems, rasactios of Cha lectrotechical Society, Vol. 1, No. 1, pp. 1-11, 006. [4] C. K. se, Flip bifurcatio ad chaos the three-state boost switchg regulators, I rasactios o Circuit System Ⅰ, Vol. No. 1, pp. 16-3, [5] C. K. se, Y. M. Lai, ad H. H. C. Iu, Hopf bifurcatio ad chaos a free-rug curret-cotrolled Cuk switchg regulator, I rasactios o Circuit System Ⅰ, Vol. 47, No. 4, pp , 000. [6] Guohui Yua, Soumitro Baerjee, dward Ott, et al., Bordercollisio bifurcatios the buck coverter, I rasactios o Circuit System Ⅰ, Vol. 45, No. 7, pp , [7] H. H. C. Iu ad C. K. se, Bifurcatio behavior parallelcoected buck coverters, I rasactios o Circuit System Ⅰ, Vol. 48, No., pp , 001. [8] Zhag Bo, Li Pg, ad Qi Qu, Method for aalyzg ad modelg bifurcatio ad chaos DC/DC coverters, Peoceedgs of CS, Vol., No. 11, pp , 00. [9] Liu Weizeg, Zhag Hao, ad Ma Xikui, Aalysis of termittet bifurcatios ad chaos pheomea boost PFC coverters bystroboscopic map, Proceedgs of CS, Vol. 5, No. 1, pp , 005. [10] Qu Yg ad Zhag Bo, he precise mathematical discrete model of buck coverter DCM ad its aalysis for bifurcatio stability, Acta lectroica Sica, Vol. 30, No. 8, pp , 00 [11] Ma Xikui, Liu Weizeg, ad Zhag Hao, Aalysis of fast-scale bifurcatios ad chaos pheomea boost PFC coverter, Proceedgs of CS, Vol. 5, No. 5, pp , 005.
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