A Simplified Steady-State Analysis of the PWM Zeta Converter

Transcription

1 Proeeng of the 3th WSEAS International Conferene on CICUITS A Simplified Steady-State Analyi of the PW Zeta Conerter ELENA NICULESCU *, INA IAA-PUCAU *, AIUS-CISTIAN NICULESCU **, IN PUCAU *** AN AIAN AIA **** * epartment of Eletroni and Intrumentation, * * epartment of ehatroni, Unierity of Craioa, Faulty of Automation, Computer and Eletroni Al. I. Cuza Street, No. 3, Craioa, ANIA, E-ail: eniuleu@eletroni.u.ro, dpuraru@eletroni.u.ro, * ** S.C. G Impex S..L., C. Branoeanu Street, No. 0, 0033 Craioa, ANIA, E-ail: puraruion@yahoo.om * ** * Energeti Shool Complex, Amaraa Street, No. 59, 0083 Craioa, ANIA, E-ail: mariamarian@yahoo.om Abtrat: - The paper preent a implified teady-tate analyi of the PW Zeta onerter for both ontinuou and ontinuou indutor urrent mode. Treating the onerter with eparate indutor a a partiular ae of the more general oupled-indutor ae, the analyi reult oer both the onerter erion: oupled - and nonoupled indutor onerter. The teady-tate propertie of onerter are obtained a low-entropy expreion that omprie the eential element neeary in the firt tage of an appliation deign that onit in topology omparion and/or in modelling proe. The analyi i baed on the equation written on the iruit for eah time interal orreponng to the tate of withe (tranitor and ode) and the low-ripple apaitor oltage aumption, and the urrent and oltage waeform of the onerter. Key-Word: - Steady-tate analyi, PW Zeta onerter, Continuou indutor urrent mode, iontinuou indutor urrent mode. Introdution Baially, the PW Zeta onerter i a tep up/down onerter of noninerting-polarity type and it an be deign to ahiee low-ripple output urrent either with oupled indutor or with eparate indutor []. A PW Zeta onerter i urrently ued in power fator orretion and oltage regulation deign [] [7]. The PW Zeta onerter topology oniting in two withe (a ingle tranitor and a ingle ode) and four reatie element (two indutor and two apaitor) exhibit many ommon feature with the PW Cuk and Sepi onerter whoe analyi ha urfaed in many form oer the year. The mot ommon operating mode of thee PW onerter are the ontinuou indutor urrent mode (CIC or CC) and ontinuou indutor urrent mode (IC or C), although thee are not the only poible operating mode in onerter topologie with more than two reatie element. The PW onerter are nonlinear dynami ytem with trutural hange oer an operation yle. Two network are repeatedly withed by the ation of the onerter withe in CC, while in C three network are perioally withed [8]. The purpoe of thi paper i to unify the analyi tep for thi onerter with oupled or nonoupled indutor and with both CC and C, and to gie implified expreion for their teady-tate propertie. It i aumed that all element in the onerter are ideal and the onerter topology with nonoupled indutor i treated a a partiular ae of the more general oupled-indutor ae. The teady-tate propertie of onerter are obtained a low-entropy expreion that omprie the eential element neeary in the firt tage of an appliation deign that onit in topology omparion. Alo, ome dynami aeraged model of PW onerter require the knowledge of thee teady-tate propertie [8]. A omplete teady-tate haraterization for oupled- or nonoupled indutor PW Zeta onerter with both CC and C i proided in thi paper, the onerter with nonoupled indutor being treated a a partiular ae of the more general oupled-indutor ae. The paper i organized a ISSN: ISBN:

2 Proeeng of the 3th WSEAS International Conferene on CICUITS follow. In Setion, the teady-tate analyi of PW Zeta onerter with CC i made. The expreion of aerage input and output urrent, output oltage and oltage aro energy torage apaitor, whih take the effet of indutor oupling into aount, are deried here. The ame teady-tate propertie of PW Zeta onerter with C and the boundary equation are deried in Setion 3. Setion 4 onlude the work. Nomenlature i L - the urrent through indutor L i - the ripple omponent of urrent i L I L0 - the initial and final ontion of urrent i L I L - the aerage alue of urrent i I L - the aerage alue of urrent i L i L - the urrent through indutor L i - the ripple omponent of urrent i L I L0 - the initial and final ontion of urrent i L I L - the aerage alue of urrent i I L - the aerage alue of urrent i L I - the aerage alue of output urrent i - the urrent through with i S I - the urrent through ode V - the C input oltage - the output oltage L - the oltage aro indutor L L - the oltage aro indutor L VC - the aerage oltage aro energy torage apaitor - the with-on duty yle - the ode-on duty yle - the with and ode off ratio 3 f - the withing frequeny L - the mutual indutane k - the oupling oeffiient of indutor n - the turn ratio - the load reitane η - the onerion effiieny of onerter - the C oltage onerion ratio in CC d - the C oltage onerion ratio in C Steady-State Analyi of the PW Zeta Conerter with CC Aume that the onerter operate in ontinuou ondution mode and that the two indutor are magnetially oupled together to redue the input urrent ripple and inreae the power denity. Thi analyi aim to find the expreion of C oltage onerion ratio, aerage indutor urrent, aerage urrent through with and ode, aerage oltage aro the energy torage apaitor C. The agram of the original PW Zeta onerter topology with oupled indutor i gien in Fig.. Fig.. The PW Zeta onerter topology with oupled indutor. The equialent iruit of two oupled indutor and Zeta topology are gien in Fig., and the waeform of indutor urrent and oltage are hown in Fig. 3, where. Fig.. Equialent iruit for two oupled indutor (a) and Zeta topology with oupled indutor (b). A it an be een from the indutor urrent waeform hown in Fig. 3b and d, the ripple omponent i repetiely i i uperpoed oer the C omponent I L0 repetiely I L0 that repreent the initial and final ontion of indutor urrent in L repetiely L : ISSN: ISBN:

3 Proeeng of the 3th WSEAS International Conferene on CICUITS i i + I () L L0 i L i + I L0 () In order to determine the aerage alue of the urrent omponent, we need to find / and / firtly. The input and output oltage a well a the oltage aro the energy torage apaitor C are uppoed without ripple. Baed on the equialent iruit of two oupled indutor in Fig. a, we hae L L + L (3) L L + L. (4) Soling for the unknown / and /, we find LL LL L L L L L L (5) L L L L (6) LL L LL L After ome algebra, the preiou equation an be written a L k L / n, (7) k L ( ) k n L ( k ) L L. (8) For 0 t T, ubtituting the indutor oltage L and L + VC V into (7) and (8) yield: k ( + VC V )/ n, (9) k L ( ) ( V + V V ) C ( k ) L k nv I I. (0) We proeed imilarly for ( ) T t T when L VC and L V, and we obtain: VC + k V / n, () k L V + k nv ( k ) L C. () Sine the total rie of i and repetiely i during 0 t T mut to be equal to the total fall of the ame urrent during T T T ( ) t T T 0 t T T t T ( ) T, we hae: 0 t T T t T (3) (4) The ubtitution of (9), (0) and (), () into (3) and (4) yield the teady-tate relationhip between the d input oltage and the aerage output oltage, and the aerage oltage aro the energy torage apaitor C : V C V and V. Fig.3. The waeform of urrent and oltage in PW Zeta onerter with CC From thee reult, the C oltage onerion ratio of oupled-indutor PW Zeta onerter with CC i obtained a V (5) and V V. (6) C ISSN: ISBN:

4 Proeeng of the 3th WSEAS International Conferene on CICUITS The aboe reult hae been obtained een for the eparate indutor ae. So, the magnetially oupling of indutor do not mofie the C oltage onerion ratio that only depend on the duty yle of PW onerter and alo the relationhip between the oltage V C, V and V I. n the other hand, the relationhip (6) how that the oltage aro the two indutor are equal during eah tint time interal that allow oupling the indutor in order to redue the input or output urrent ripple [], []. Uing thee new reult, (9), (0) and (), () beome: (7) L e V I (8) L e and V (9) L e V (0) L e where we denoted two effetie indutane: ( k ) L Le () k / n ( k ) L L e. () k n The equation (7) to (0) how that oupled indutor fed by the ame oltage waeform will exhibit a two equialent indutor, whih only depend on the phyial indutor truture. The ame equation gie the theoretial ontion to anel the indutor urrent ripple that i L or L e. So, the analyi reult from the oupled indutor Zeta onerter an be extended to nonoupled indutor Zeta onerter and ie era. Furthermore, two oupled indutor exhibit a two equialent indutor with the effetie indutane gien by () and (), and hown in Fig. b. Therefore, Fig. b ere exluiely for a lear repreentation of (7) and (0). The aerage alue of the indutor urrent ripple an be retly found from the waeform of the indutor urrent gien in the Fig. 3 a: T I L, (3) L k e m T I L. (4) L e k m In aboe relationhip, we denoted two parameter of ondution through the indutor a follow: e K m Lef / (5) K m L ef /. (6) Next, we determine the omponent I L0 and I L0 of indutor urrent. For thi purpoe, we ue the equation of the C urrent through the load and the effiieny a follow: V ( I L + I L0 ), (7) V η ( I L + I L0 + I L + I L0 ). (8) Uing the aumption of 00% effiieny, the following reult are obtained: ( ) I L0 (9) K m I L0. (30) K m We an now to find the aerage urrent of tranitor with and ode a follow: IS ( I L + I L ) (3) ( ) I ( )( I L + I L ). (3) ( ) The reult of thi implified analyi ompletely haraterize the teady-tate behaior of the oupled-indutor PW Zeta onerter in CC. For the nonoupled indutor ae, it i enough to et k at zero into the aboe reult. A onequene, Le L, L e L, Km K and K m K. 3 Steady-State Analyi of the PW Zeta Conerter with C For PW Zeta onerter a well a for a Cuk or Sepi PW onerter, a ontinuou ondution mode (C) mean that the third time interal of operation yle i nonzero, not that either indutor urrent i ontinuou. Coupled indutor tehnique an be applied to PW Zeta onerter with C to ahiee low-ripple output urrent []. Aume that the output urrent i now further dereaed to enable the onerter to enter into ontinuou ondution mode. The waeform of indutor urrent and oltage orreponng to thi operating mode are hown in Fig. 4. Three tint time interal appear there namely T, T and 3 T with for a ontant withing frequeny. For C, beide of expreion of aerage indutor urrent, aerage input and output urrent, aerage oltage aro the ISSN: ISBN:

5 Proeeng of the 3th WSEAS International Conferene on CICUITS energy torage apaitor C and aerage urrent tranitor and ode, we hae to determine the parameter that fixe the deay interal of indutor urrent and the C oltage onerion ratio impliitly. Conerning the waeform of indutor urrent, it an ee that the hape of the urrent in Fig. 4 are imilar to thoe hown in Fig. 3. nly the ale are fferent and the urrent I L0 and I L0 are equal and hae an inere ene: I L0 I L0 I L0. (33) The urrent through indutor an be expreed a i i + I (34) i L L L0 i I. (35) L0 For the firt two time interal T and T, (9), (0) and (), () keep their ality. For the third time interal ( ), the oltage aro 3 T L L indutor are zero: 0. Taking into aount that the aerage oltage aro the indutor oer a withing period are zero, the following relationhip reult: V C V. (36) The aboe relationhip yield the C oltage onerion ratio V d (37) and V C V. (38) The remark regarng the relationhip (38) that i the ame a (8) for the onerter with CC keep it ality for the onerter with C. The determination of the onerion ratio need the alue of parameter too. It i obiou that the ontion of ripple anellation from the indutor urrent found for CC remain unhanged for C. Alo, the two effetie indutane and two parameter of ondution through the indutor hold their expreion. Uing the waeform of indutor urrent, the formula of aerage alue of ripple omponent of indutor urrent an be written a: ( + ) ( + ) I L (39) L f K e m ( + ) V ( + ) I I L. (40) L ef K m The alulation of thee omponent of indutor urrent need to find the parameter firtly. In d order to find the expreion of the omponent I L0, we ue again the relationhip of the onerter effiieny, that i V I ηv I. (4) I S Fig.4. The waeform of the urrent and oltage in PW Zeta onerter with C For aumption of 00% effiieny, the aboe equation yield I L0. (4) K m Km A it an be een from the waeform of urrent in the tranitor and ode, hown a i and repetiely i in Fig. 4f, the total rie of i S during 0 t T hould be equal to the total fall in i T t +. Thi mean that we during ( ) T S ISSN: ISBN:

6 Proeeng of the 3th WSEAS International Conferene on CICUITS hae T + 0 t T. (43) T T t ( + ) T After ubtituting the orreponng lope of indutor urrent and uing (38) in (43), we find the C oltage onerion ratio in (37) one more. The aerage urrent of tranitor and ode reult a follow: d IS ( I L + I L ), (44) + I d ( I L + I L ). (45) + Now, we proeed to determine the formula for the parameter. For thi purpoe, we ue the C load urrent ombining (39), (40) with (45). A reult, the parameter i gien by the formula K em, (46) where K em L emf /. (47) The quantity K em repreent the parameter of ondution through an equialent indutor with the indutane L em Le // L e. For a omplete haraterization of the PW Zeta onerter operating, we find now the equation of the boundary between CC and C for the ouplendutor ae. It i well known that a onerter will hange the operating mode when the following equality i atified:. (48) Thi mean that the parameter ritial alue K em,rt ( ) K em reahed it. (49) The ame equation an be deried tarting from (33). All thee reult haraterize the PW Zeta onerter with eparate indutor, ae in whih we hae to et k at zero. 4 Conluion Simplified mathematial model for the teady-tate behaior of onerter with CC and C hae been obtained a reult of thi analyi. No fferene exit between thee model and thoe obtained by uing the tate-pae aeraging modelling method. Alo, thi implified analyi yield the ame reult a one baed on the largeignal PW with [8]. All the C propertie of onerter inlude the oupling indutor effet. A a reult of thi implified analyi, the C onerion ratio of PW Zeta onerter with oupled-indutor and operating in C ha the ame formula a in the nonoupled indutor ae. But, thee formulae are only formally the ame beaue of the parameter that take the oupling effet into aount, by mean of the parameter of ondution through an indutor with an equialent indutane, L em. eferene: [] J. W. Kolar, H. Sree, N. ohan and F. C. Zah, Noel Apet of an Appliation of Zero - ipple Tehnique to Bai Conerter Topologie, IEEE PESC 97 Pro., Vol., 997, pp [] A. Pere,. C. artin, and I. Barbi, Zeta Conerter Applied in Power Fator Corretion, in 994 Power Eletroni Speialit Conf. Pro., ol., 994, pp [3]. C. artin, Zeta Conerter perating in Continuou Condution ode Uing the Unity Power Fator Tehnique, The 6th Int. Conf. Power Eletroni and Variable Speed rie Pro., 996, pag. 7-. [4] C. Sudhakarababu and. Veerahary, Zeta Conerter for Power Fator Corretion and Voltage egulation, 004 TENCN Pro., ol. 4, 004, pp [5] J.-L., Lin, S.-P., Yang, and P.-W. Lin, Small- Signal Analyi and Controller eign for an Iolated Zeta Conerter with High Power Fator Corretion, Eletri Power Sytem eearh 76, 005 Eleier B.V., 005, pp [6] Fujitu Com., New produt: 4-Channel C/C Conerter IC with Synhronou etifiation B39A0, Find, ol. 0, no. 4, 00, pp [7] Fujitu Com., New produt: 5-Channel C/C Conerter IC with Synhronou etifiation B39A, Find, ol., no. 4, 004, pp [8] V. Vorperian, Simplified analyi of PW onerter uing model of PW with, Part I (CC) and II (C), IEEE Tran. on Aeropae Eletroni Sytem, Vol. 6, 990, pp ISSN: ISBN: