A Steady-state Analysis of PWM SEPIC Converter

Transcription

1 Proeeng of he h WSEAS nernaional Conferene on CCUTS, ouliagmeni, Ahen, Greee, July -, 6 (pp7-) A Seady-ae Analyi of PWM SEPC Converer EENA NCUESCU, DONA-MOAA PUCAU M. C. NCUESCU Deparmen of Eleroni nrumenaion, Meharoni Univeriy of Craiova Al.. Cuza Sree, No. 3, Craiova, O-585 OMANA Abra: - The paper preen a unified eady-ae analyi of PWM SEPC onverer wih oupled induor eparae induor for boh oninuou oninuou onduion mode. The reul of hi implified analyi allow o deermine he operaing poin of onverer furher he mall-ignal low-frequeny parameer of linear model, o deign he onverer oo. The expreion of all eady-ae properie of onverer ake he effe of induor oupling ino aoun. The onverer wih eparae induor i reaed a a pariular ae of he more general oupled-induor ae. Key-Word: - Coupled non-oupled induor PWM SEPC Converer, Seady-ae analyi nroduion A i i well known, he PWM onverer are nonlinear dynami yem wih ruural hange over an operaion yle: wo for he oninuou onduion mode (CCM) hree for he oninuou onduion mode (DCM). Coupled induor ehnique an be applied o PWM SEPC onverer operaing eiher wih CCM or DCM o ahieve low-ripple inpu urren [], [], [3]. The paper provide a omplee eady-ae haraerizaion for oupled-induor PWM SEPC onverer wih boh CCM DCM. The onverer wih eparae induor i reaed a a pariular ae of he more general oupled-induor ae. n Seion, he eady-ae analyi of PWM SEPC onverer wih CCM i made. The expreion of average inpu oupu urren, oupu volage volage aro energy orage apaior, whih ake he effe of induor oupling ino aoun, are derived here. The ame eady-ae properie of PWM SEPC onverer wih DCM are derived in Seion. The boundary equaion i given in Seion. Simplified mahemaial model for he eady-ae behavior of onverer wih CCM DCM have been obained a reul of hi analyi. The analyi i baed on he equaion wrien on he irui for eah ime inerval orreponng o ae of wihe (ranior ode) he lowripple apaior volage, he waveform of elerial quaniie orreponng o operaing mode of he onverer. The analyi modelling of SEPC onverer ha urfaed in many form over he year [] [5]. The purpoe of hi paper i o make a implified analyi of oupled-induor SEPC onverer avoing magnei heory. Nomenlaure i = urren hrough i = ripple omponen of i = iniial final onion of i D = average value of i = average value of i i = urren hrough i = ripple omponen of i = iniial final onion of i D = average value of i = average value of i O = average oupu urren i S = urren hrough wih i D = urren hrough ode = d inpu volage v = oupu volage O v = volage aro v = volage aro C = average volage aro C D = wih-on duy yle D = ode-on duy yle D 3 = wih ode off raio f = wihing frequeny = induane of filering induor

2 Proeeng of he h WSEAS nernaional Conferene on CCUTS, ouliagmeni, Ahen, Greee, July -, 6 (pp7-) = induane of filering induor M = muual induane k = oupling oeffiien n = urn raio C = apaiy of energy orage apaior C C = apaiy of oupu filering apaior C = load reiane η = onverion effiieny of onverer M = d volage onverion raio Seady-ae Analyi of SEPC Converer wih CCM The agram of onvenional PWM SEPC onverer wih oupled induor i given in Fig. a. The opology hown in Fig. b i anoher ruure of PWM SEPC onverer wih oupled induor idenial operaing behaviour a ha in Fig. a. Thi one learly highligh he PWM wih he poibiliy o ue he PWM wih model wih all i benefi [6]. A ompared o he onvenional SEPC opology, he onrol of he wih ha o be realized poenial free wih repe o he poenial referene in he eond opology. Aume ha he onverer operae in oninuou onduion mode ha he wo induor are magneially oupled ogeher o redue he inpu urren ripple inreae he power deniy. The equivalen irui of wo oupled induor i given in Fig. a, he waveform of induor urren volage are hown in Fig. 3, where D D. Fig. Two SEPC PWM onverer opologie wih oupled induor The arge of hi analyi i ha o find he expreion of d volage onverion raio, average induor urren, average urren hrough wih ode, average volage aro he energy orage apaior C. A i an be een from he induor urren waveform hown in Fig. 3a d, he ripple omponen i repeively i i uperpoed over he d omponen repeively ha repreen he iniial final onion of induor urren in repeively : i = i + () i = i + () Fig. Equivalen irui for: a. wo oupled induor; b. SEPC opology wih oupled induor n order o deermine he average value of he urren omponen, we need o find / d / d firly. The inpu oupu volage a well a he volage aro he energy orage apaior C are uppoed wihou ripple. Baed on he equivalen irui of wo oupled induor in Fig. a, we have v = + M d d (3) v = M +. d d (4) Solving for he unknown / d / d, we find v vm d (5) d v M v M (6) M M Afer ome algebra, he previou equaion an be wrien a v kv / n = (7) d k d v ( ) k nv ( k ) = (8) M

3 Proeeng of he h WSEAS nernaional Conferene on CCUTS, ouliagmeni, Ahen, Greee, July -, 6 (pp7-) a. v D T D T T b.. d. e. f. g. i i D v i i D i S D T Fig.3 The waveform of urren volage in PWM SEPC onverer wih CCM e DT e i D For DT, ubiuing he induor volage v = v = C ino (7) (8) yield: kc / n = (9) d k ( ) k n =. () d C ( k ) We proeed imilarly for D T T when v = C v, we obain: C + k / n = () d k d ( ) + k ( ) n C ( k ) O O. () Sine he oal rie of i repeively i during DT mu o be equal o he oal fall of he ame urren during D T T, we have: D T ( D ) = T d d (3) D T DT T D T ( D ) = T d d. (4) D T DT T The ubiuion of (9), () (), () ino (3) (4) yield he eady-ae relaionhip beween he d inpu volage he average oupu volage, he average volage aro he energy orage apaior C : D C = O =. D From hee reul, he d volage onverion raio of oupled-induor PWM SEPC onverer wih CCM i obained a D M (5) D C =. (6) The above reul have been obained even for he eparae induor ae. So, he magneially oupling of induor do no mofie he d volage onverion raio ha only depend on he duy yle of PWM onverer alo he relaionhip beween he volage C,. On he oher h, he relaionhip (6) how ha he volage aro he wo induor are equal during eah in ime inerval ha allow o ouple he induor in order o redue he inpu or oupu urren ripple [], []. Uing hee new reul, (9), () (), () beome: = (7) d e

4 Proeeng of he h WSEAS nernaional Conferene on CCUTS, ouliagmeni, Ahen, Greee, July -, 6 (pp7-) = (8) d e (9) d e O () d e where we denoed wo effeive induane: ( k ) e = () k / n ( k ) e =. () kn The equaion (7) o () how ha oupled induor fed by he ame volage waveform will exhibi a wo equivalen induor, whih only depend on he phyial induor ruure. The ame equaion give he heoreial onion o anel he induor urren ripple ha i or e. So, he analyi reul from he oupled induor SEPC onverer an be exended o nonoupled induor SEPC onverer vie vera. Furhermore, wo oupled induor exhibi a wo equivalen induor wih he effeive induane given by () (), hown in Fig. b. Therefore, Fig. b erve exluively for a lear repreenaion of (7) (). The average value of he induor urren ripple an be rely found from he waveform of he induor urren given in he Fig. 3 a: DT D D (3) k e m DT D D (4) e k m where we denoed wo parameer of onduion hrough he induor a follow: K = f / (5) m e m e K = f /. (6) Nex, we deermine he omponen of induor urren. For hi purpoe, we ue he equaion of he d urren hrough he load he effiieny a follow: = D = ( D )( D + + D + ) (7) = η ( D + ). (8) Uing he aumpion of % effiieny, he following reul are obained: e D ( ) D = D K m (9) D =. D K m (3) We an now o find he average urren of ranior wih ode a follow: D M S = D ( + ) ( D ) (3) D M D = ( D )( + ). ( D ) (3) The reul of hi implified analyi ompleely haraerie he eady-ae behaviour of he oupled-induor PWM SEPC onverer in oninuou onduion mode. For he eparae induor ae, i i enough o e k a zero ino he above reul. A onequene, e, e, K m K K m K. 3 Seady-ae Analyi of SEPC Converer wih DCM For SEPC PWM onverer a well a for Cuk PWM onverer, a oninuou onduion mode (DCM) mean ha he hird ime inerval of operaion yle i nonzero, no ha eiher induor urren i oninuou. Coupled induor ehnique an be applied o PWM SEPC onverer wih DCM o ahieve low-ripple inpu urren []. Aume ha he oupu urren i now furher dereaed o enable he onverer o ener ino oninuou onduion mode. The waveform of induor urren volage orreponng o hi operaing mode are hown in Fig. 4. Three in ime inerval appear here, namely D T, D T D 3 T wih D + D + D3 = for a onan wihing frequeny. For DCM, beide of expreion of average induor urren, average inpu oupu urren, average volage aro he energy orage apaior C average urren ranior ode, we have o deermine he parameer D ha fixe he deay inerval of induor urren he d volage onverion raio impliily. Conerning he waveform of induor urren, i an ee ha he hape of he urren in Fig. 4 are imilar o hoe hown in Fig. 3. Only he ale are fferen he urren have an invere ene: are equal =. (33) The urren hrough induor an be expreed a

5 Proeeng of he h WSEAS nernaional Conferene on CCUTS, ouliagmeni, Ahen, Greee, July -, 6 (pp7-) i = i + (34) i = i. (35) a. v D T D T b. d.. i i D 3 T T D v e. - f. g. i D T Fig.4 The waveform of he urren volage in PWM SEPC onverer wih DCM e i D i S D T e i D For he fir wo ime inerval D T D T, (9), () (), () keep heir valiy. For he hird ime inerval ( D 3 T ), he volage aro induor are zero: v = v =. Taking ino aoun ha he average volage aro he induor over a wihing period are zero, he following relaionhip reul: D C = =. (36) D The above relaionhip yield he d volage onverion raio D M = O = (37) D C =. (38) The remark regarng he relaionhip (38) ha i he ame a (8) for he onverer wih CCM keep i valiy for he onverer wih DCM. The deerminaion of he onverion raio M need he value of parameer D oo. i obviou ha he onion of ripple anellaion from he induor urren found for CCM remain unhanged for DCM. Alo, he wo effeive induane wo parameer of onduion hrough he induor hold heir expreion. Uing he waveform of induor urren, he formula of average value of ripple omponen of induor urren an be wrien a: D ( D + D ) D ( D + D ) D (39) f K e m ( D + D ) D ( D + D ) D D. (4) e f K m The alulaion of hee omponen of induor urren need o find he parameer D firly. n order o find he expreion of he omponen, we ue again he relaionhip of he onverer effiieny, ha i O = η ( D + ). (4) For aumpion of % effiieny, he above equaion yield D D D =. (4) K m Km A i an be een from he waveform of urren in he ranior ode, hown a i S repeively i D in Fig. 4g, he oal rie of i S during DT hould be equal o he oal fall in i D during D T ( D + D ) T. Thi mean ha we have

6 Proeeng of he h WSEAS nernaional Conferene on CCUTS, ouliagmeni, Ahen, Greee, July -, 6 (pp7-) DT + d d DT d d D T = (43) D T ( D + D ) T afer ubiuing he orreponng lope of induor urren uing (38), we find he d volage onverion raio in (37) one more. The average urren of ranior ode reul a follow: S D M = ( D + D ) = (44) D + D D M D = ( D + D ) =. (45) D + D Now, we proeed o deermine he formula for he parameer D. For hi purpoe, we ue he d load urren ombining (39), (4) wih (45). A reul, he parameer D i given by he formula D = K em (46) where he quaniy K em = em f / (47) repreen he parameer of onduion hrough an equivalen induor wih he induane = //. em e e eady-ae urren uh a he average inpu oupu urren, he average urren of ranior ode are found. For he d volage onverion raio he average volage aro he energy orage apaior, we found he ame formula a in he eparae induor ae beaue we negleed he equivalen erie reiane of apaior. The eady-ae analyi developed here upplie he formula of equivalen induane in he ouplenduor ae ha are enailed by a dynami analyi baed on he mall-ignal equivalen irui of onverer. Thi implified analyi yield he ame reul a one baed on he large-ignal PWM wih [6]. Formally, he d onverion raio of PWM SEPC onverer wih oupled-induor operaing in DCM ha he ame formula a in he eparaeinduor ae. Bu, he parameer D ake he oupling effe ino aoun, by mean of he parameer of onduion hrough an equivalen induor wih he induane em. The average inpu oupu urren, he average urren of ranior ode a well a he boundary equaion inlude he oupling effe oo. So, he eady-ae operaing poin he parameer of mall-ignal low-frequeny model of onverer depend on he oupling parameer. 4 Boundary beween he Coninuou Dioninuou Conduion Mode For a omplee haraerizaion of he PWM SEPC onverer operaing, we find now he equaion of he boundary beween CCM DCM for he ouplenduor ae. i well known ha a onverer will hange he operaing mode when he following equaliy i aified: D = D. (48) Thi mean ha he parameer K em reahed i riial value K ( D ) emr. (49) The ame equaion an be derived aring from (33). All hee reul haraerize he PWM SEPC onverer wih eparae induor, ae in whih we have o e k a zero. 5 Conluion A eady-ae analyi ha ompleely haraerize he behavior of PWM SEPC onverer wih oupled or eparae induor operaing in CCM or DCM wa made in hi paper. Taking he effe of induor oupling ino aoun, he expreion of all eferene: [] J. W. Kolar, H. Sree, N. Mohan F. C. Zah, Novel Ape of an Appliaion of Zero - ipple Tehnique o Bai Converer Topologie, EEE PESC 97, ol., pp [] J. Chen C. Chang, Analyi Deign of SEPC Converer in Boundary Conduion Mode for Univeral-line Power Faor Correion Appliaion, in Pro. EEE PESC,. [3] S. Ben-Yaakov, D. Adar G. ahav, A SPCE ompaible Behavioral Model of SEPC Converer, EEE PESC 96, ol., pp [4] W. M. Moua, Modeling performane evaluaion of a DC/DC SEPC onverer, APEC 95, ol., pp [5] D. Adar, G. ahav S. Ben-Yaakov, A unified behavioral average model of SEPC onverer wih oupled induor, EEE PESC 97, ol., pp [6]. orperian, Simplified analyi of PWM onverer uing model of PWM wih, Par (CCM) (DCM), EEE Tran. on Aeropae Eleroni Syem, ol. 6, 99, pp