THE BOOST CONVERTER REVISITED

Transcription

1 TH BOOST CONVT VSTD B. W. Wllams, T. C. m Deparmen f lecrnc and lecrcal ngneerng, Unersy f Srahclyde, Glasgw G XW, UK Absrac - The dc--dc bs cnerer s a sngleswch, sngle-nducr, swchng crcu used effcenly ransfrm energy frm ne dc lage leel a greaer lage leel f he same relae plary. Fr a specfc resse lad range, as he duy cycle decreases, he bs cnerer nducr eners a dscnnuus curren mde f peran - he upu lad curren hang decreased a defnable leel. Ths paper analyses he fac ha a furher reducn f lad curren, as he duy cycle decreases wards zer, wll always resul n he re-emergence f a cnnuus nducr curren cndn. Furher, a he her lad exreme, hgh-curren, prgressely fr ncreasng lad curren, sarng a lw duy cycle cndns, he mnmum nducr curren always ncreases frm a fxed nrmalsed curren leel, fr a specfc lad range. These and her hher unexplred bs cnerer prperes are analysed and erfed mahemacally and wh PSpce smulans. Keywrds - swched mde pwer supples, smps, bs cnerer, dc dc cnerers.. NTODUCTON A. Speces Dsrbun Mdels (SDMs) The hree cmmn nn-slaed, sngle-swch, snglenducr, dc--dc cnerers (swch mde pwer supples - smps) are he frward (r buc r sep-dwn) cnerer; he sep-up (r bs) cnerer; and he nerng sep-up/dwn (r buc-bs) cnerer. ach f hese cnerers peraes n he prncple f ang npu dc supply energy, emprarly srng ha energy n he magnec feld f an nducr, hen ha energy s dered he lad, whch can be a a penal and plary dfferen he npu energy surce emf lage. Ths paper s specfcally cncerned wh he bs cnerer shwn n fgure a and exends he analyses and undersandng f s basc radnally acceped perang mdes and prperes. The bacgrund hery s brefly presened esablsh exsng bs cnerer peranal nerprean and defne he necessary cnceps and parameers. The bs cnerer n s basc frm cnerers dc lage surce energy a hgher lage leel, usng swch mde (hard r sf) echnques (namely, he swch s eher cu-ff r fully n - sauraed). The bs cnerer s used sep up he lw upu lage f PV arrays a lage leel cmmensurae wh nerer ln lages suable fr ransfrmerless ac grd nerfacng []. Because f s cnnuus npu curren characersc he bs cnerer s used exensely fr exracng snusdal curren a cnrllable pwer facr frm wnd urbne ac generars []. As shwn n fgure b, peran and he upu lage are characersed by he swch n-me duy cycle, and analyss s cenred n he nducr curren and nducr lage, as shwn n fgures c and d. Specfcally, n seady-sae, analyss s based n he nducr sasfyng Faraday s equan d d Seady-sae herecal analyss assumes zer swch and dde lsses, an nfne upu capacance C, and zer surce and nducr ressance; hen he swch n-sae and ff-sae currens creaed Krchhff lps yeld T D (V.s) () whch afer rearrangng he las equaly, ges he radnal bs cnerer lage and curren ransfer funcn expressn (pu) () where, because f zer cnerer lsses, he upu pwer s equal he npu pwer, namely, 7

2 - ermed pwer narance. Ths expressn, equan (), assumes cnnuus curren n he nducr when he swch s ff, durng he perd D, such ha T, where he seadysae swchng frequency s nrmalsed swch n-perd s D f s T / /. and he quan () hghlghs ha he relanshp beween he npu and upu lages (and currens) s ndependen f crcu cmpnens, C, and he swchng frequency f s: beng dependan nly n he swch n-sae duy cycle, prded nducr curren flws hrughu he whle swch ff-sae perd, D, nrmalsed as D = D / τ = -. Ths nducr curren cndn s ermed cnnuus cnducn. A cnnuus curren requremen s hghlghed when cnsderng he crcu energy ransfer balance beween he npu and he upu: ½ / / (W) () ¾, = / ¼, ½, ¼ ½ ¾ (b) D D (c) (d) Fg.. Nn-slaed, sep-up, flybac cnerer (bs cnerer) where : (a) crcu dagram; (b) lage ransfer funcn dependence n duy cycle ; (c) waefrms fr cnnuus npu (nducr) curren; and (d) waefrms fr dscnnuus npu (nducr) curren. The las erm n () s he cnnuus pwer cnsumed by he lad, whle he secnd erm s he energy (whence pwer) ransferred frm he nducr he lad. The frs erm s he energy dered frm he surce delered hrugh he nducr, whle he nducr s als ransferrng energy he lad (when 7

3 he swch s ff). A smlar energy ransfer mechansm ccurs wh he sep-up ac auransfrmer. n usng he facr - n equan (), s assumed ha energy s ransferred he lad frm he supply, durng he whle perd he swch s ff, ha s, cnnuus nducr curren. Snce he nducr s n seres wh he npu, he npu curren erms are nerchangeable wh he crrespndng nducr curren erms, fr example, Assumng pwer frm he emf surce s equal pwer delered he lad, pwer narance, hen equan () reduces equan (), r n a rearranged frm (V) () n hs frm, can be seen ha he upu lage has a cmpnen due he npu dc supply, and a bs cmpnen prprnal he swch n-sae duy cycle, llusrang ha he upu lage magnude s equal r greaer han he npu lage (emf) magnude. Tradnal hery cnsders when he lad energy requremen falls belw ha leel necessang he nducr carry curren durng he whle perd when he swch s ff: ermed dscnnuus nducr curren. Three sequenal cycle sages (raher han w) nw ccur durng he neral τ: he perd when he swch cnducs, T, T nrmalsed as / he perd when he nducr and dde cnduc, D, nrmalsed as D D / he perd x s when he nducr and dde cease cnduc befre = τ; nrmalsed as x x / ( ) ½ ha s, afer arus subsuns: (6) Ne ha pwer narance s ald fr all nducr curren cndns, namely, bh cnnuus and dscnnuus currens durng seady-sae peran. Bu equan (6) s nly ald fr dscnnuus nducr curren, and s bundary wh cnnuus nducr curren peran. quan (6) s cnssen wh pwer narance, where, fr dscnnuus curren, equan () becmes ½ / (7) Operanal neres cenres n he bundary cndns beween cnnuus and dscnnuus nducr curren, where he basc lage and curren ransfer funcn n () remans ald. On he erge f cnnuus cnducn, any f he equans n (6) rearranged ge he bundary crcal upu curren crcal Usng ( ) (8) and equan (), he crcal bundary lad ressance crcal s gen by crcal crcal Ω (9) such ha T + D + x = τ r when nrmalsed + D + x =, as shwn n fgure d. Frm equan (), usng T (5) where when he swch s urned n hen ½ ½ Assumng pwer narance and subsung A fac n preusly specfcally saed n he leraure, s ha he las equaly n hs crcal ressance expressn s cmmn all hree snglenducr, sngle-swch smps (buc, bs and bucbs): crcal () whch, fr he bs cnerer, afer dfferenang he frs deny n equan (9) and equang zer, gng = ⅓, has a mnmum alue f 7

4 crcal 7 () when = ⅓ and =.5, whch crrespnds a maxmum crcal curren f crcal (A) 7 crcal () A her upu lages, when ⅓, lwer currens (larger lad ressance, ) crcal can be leraed befre he nse f dscnnuus nducr curren. Alernaely, equan () shws ha crcal ressance s nersely prprnal nducr rpple curren. Tha s, a large cnnuus nducr rpple curren and lw duy cycle undesrably enhance he early nse f nducr dscnnuus rpple curren. crcal. NDUCTO ONTNUOUS/DSCONTNUOUS CUNT BOUNDAY OPATONA ANAYSS MTHODS () The expressns n equan (6) are nly ald fr dscnnuus nducr curren, and n he bundary wh cnnuus cnducn, snce her deran s based n. Tw mehds are cmmnly emplyed analyse dscnnuus curren cnducn peran, equan (6), namely: The lage ransfer funcn s nrmalsed wh respec he maxmum dscnnuus nducr curren n erms f he npu curren he maxmum dscnnuus nducr curren n erms f he upu curren where n each case he npu lage and upu lage are, n urn, assumed cnsan. The lage ransfer funcn s nrmalsed n erms f he lad ressance. Ths nles he smps me cnsan /, (frmed by he crcu when he swch s ff), beng nrmalsed by he recprcal f he swchng frequency, namely, τ. A. Nrmalsan n erms f he maxmum dscnnuus nducr curren Of he fur pssble -V cmbnans ha can be analysed, he case specfcally cnsdered here, by way f example, s when he upu lage s assumed cnsan (as ppsed he npu lage, ) and s nrmalsed wh respec he upu curren, specfc case ( (as ppsed he npu curren, ). Ths, effecely lad ressance) s cnsdered snce laer represens he suan whch bes presens hher unexplred prperes. Cnsder equan (6), specfcally () whch ges / (5) A maxmum dscnnuus nducr curren (a he bundary beween cnnuus and dscnnuus nducr curren), he ransfer funcn, equan () s ald, and n subsun n equan (5) ges a cubc plynmal n (6) The maxmum aerage upu curren n he bundary, fr a cnsan upu lage, s fund by dfferenang (6) wh respec duy cycle, and equang zer. On subsung he cndnal resul = ⅓, hs yelds (as n equan ()) when and ½ 7 (7) Nrmalsng equan (5) wh respec he maxmum dscnnuus curren gen by (7), afer slang he lage ransfer ra, yelds 7

5 cnsan ½ ½ 7 / (8) The bundary equan beween cnnuus and dscnnuus nducr curren cnducn, relang he upu curren and duy cycle can be fund by subsung he ransfer funcn /, whch s ald n he bundary, n (8), whch yelds 7 crcal (9) crcal Hweer, fgure shws he mre praccal suan f hw, fr an ncreasng lad ressance, fr a gen fxed duy cycle, he upu lage ncreases, when he lad (hence npu) curren decreases nce enerng he dscnnuus cnducn regn. Because f pwer narance, he w rgh hand pls hae he same shape. The fur pls n fgure predc an nfne upu lage, bu a fne upu curren as he lad curren decreases: fr a cnsan duy cycle, dscnnuus nducr curren ccurs fr all currens belw he crcal leel.. ersus - fgure quans () (9) are presened n he penulmae clumn f Table, where he resuls f smlar analyss fr he her nrmalsed npu and upu lage and curren cndns, beng esablshed bacgrund, are als summarsed. The able als shws rearranged bundary cndns fr each arable (, /, and / ), n erms f he her arables, as n equan (9), fr example. These nrmalsed equans and her arus rearranged frms are pled n fgures,, and. ach fgure has fur dfferen pls, ne fr each f he pssble nrmalsed npu and upu lage and curren cmbnan.. ersus / - fgure [] Fgure shws hw, manan a cnsan upu lage, he duy cycle mus decrease as he lad lage ends ncrease nce he dscnnuus cnducn regn s enered. As wuld be expeced, ndependen f whch npu r upu parameer s cnrlled, dscnnuus nducr curren resuls n er chargng he upu capacr, hus he duy cycle mus be decreased reduce he energy ransferred he lad, s as manan he requred cnsan upu lage. Fgure c shws he nrmalsed cndn as per equan (7), namely a pea bundary cndn f = ⅓ a =.5. Because f pwer narance, he w pls n he rgh n fgure, hae he dencal shape, wh arables nerchanged apprpraely.. ersus / - fgure [], [] Fgure affrds a mre nfrmae represenan f he arus crcu equans. ach f he fur pars f fgure shw lage ransfer funcn aran wh duy cycle. ach pl herefre nles he basc lage ransfer ra cure, n he cnnuus curren regn: The man regn f cncern s when he nrmalsed curren arable s less han ne, n her wrds, when here s a pssbly f dscnnuus nducr curren. By way f example, each graph has pled w nrmalsed currens f less han ne, namely / and. Bh f hese lad curren cndns nrduce a regn where he ransfer funcn becmes lad curren dependen dscnnuus nducr curren. n he case where he npu lage s cnsan, fr lads based n he npu curren, he duy cycle bundary beween he w mdes ccurs a where / fr Fr lads based n he upu curren and cnsan npu lage,, he bundary beween he w nducr curren mdes ccurs a Fgure cneys smlar nfrman fgure. 75

6 ½ ½ - where / / bundary nersecn pns n he pls n fgures and.the prperes f he cubc plynmal n equan () bare clser examnan. Frm he Appendx, hree real rs exs (dscnnuus curren cndns) f beween he w duy cycle bundary pns. Agan, because f pwer narance hs pl s he same as when he upu lage s cnsan, fr aryng npu curren cndns. The remanng pl n fgure and s nser, s ms mpran snce represens he mde where he upu cndns are cnrlled, as s he usual mehd f usng he bs cnerer. A regn f dscnnuus nducr curren ccurs, fr duy cycle alues abe and belw whch peran reurns cnnuus cnducn [5]. The w bundary cndns fr are w f he rs f he cubc (see Appendx fr he general expressns fr rs f hs cubc) 7 () and he crrespndng lage ransfer funcn s as shwn n Table. Frm Table he lage ransfer funcn s ½ 7 / The lage ransfer funcn fr dscnnuus nducr curren s apprxmaely lnear wh duy cycle er a wde range: ½ / The pl shws ha dscnnuy cmmences a wh ½ when Anher ranal bundary slun s and - wh when ½ / / whch s rue fr / - he dscnnuus nducr curren cndn. The exac rs f equan () n erms f he ceffcens f he cubc are gen n he Appendx. The lage ransfer funcn can be expressed n erms f he duy cycle a he bundary f cnnuus nducr curren. Tha s, frm Table he hrd frmula clumn, fr cnsan and he upu curren nrmalsed 7 7 () Facrsng he las equaly, whch prduces a cubc plynmal, yelds / / / () The frs r cnfrms ha he bundary fr dscnnuus cnducn s / () Alhugh real rs exs he quadrac plynmal n equan (), fr allwable duy cycle alues <, bh are meanngless excep her han cnfrm he maxmum bundary cndn a, when / ½ and /. n all fur cases (npu/upu -V), he bundary cndns crrespnd he apprprae ercal lne 76

7 77 Table. Sep-up cnerer ransfer funcns wh cnsan npu lage,, and cnsan upu lage,, wh respec and cnnuus nducr curren, equan () cnsan cnsan dscnnuu s nducr curren equan (6) Nrmalsed wr / where 8 / where 7 / where 7 / where 8 = ½; = ; = ⅓; ½ = ½; change f arable 7 change f arable ½ 7 ½ cnducn bundary 7 7 cnducn bundary ½ ½ 7 ½ ½ cnducn bundary

8 78 5 ½ 5 ½ / /, ⅓, ½ cnnuus cnnuus cnsan dscnnuu 7 7 ½ / /, ½ cnnuus cnnuus cnsan dscnnuu 5 ½ 5 ½ ½ Fg.. Cnrl f duy cycle manan a cnsan upu lage. 7 7, ½ =½ ½, cnnuus cnnuus cnsan / dscnnuu =½ / cnnuus cnnuus cnsan dscnnuu s ½ ½, =½ =¼ =½ =¼

9 Fg.. Vlage aran fr cnsan duy cycle. cnnuus / duy cycle ½ dscnnuus / 8 / cnnuus dscnnuus ½, duy cycle ½ cnsan ½ - ½ - ½ + ½ - cnnuus ½ ½ dscnnuus / 8 duy cycle ½ ⅓,½ cnsan 7 / / dscnnuus ½ ½, cnnuus duy cycle ½ - ½ - ½ + ½ - / cnsan dscnnuus ½ ½ 7 /. cnnuus ¼.. Fg.. Oupu lage aran fr cnsan curren. 79

10 B. Nrmalsan n erms f lad ressance n he preus analyss, he ransfer funcn fr dscnnuus cnducn (and s bundary) s nrmalsed wh respec he maxmum dscnnuus curren f he frm: Mulplyng bh sdes by he lad ressance ges () (5) where 7 / (p.u.) Q (6) whch s he ra f w me cnsans he swch ff-perd crcu me cnsan / and he recprcal f he swchng frequency, namely,. Beng relae crcu Q, he symbl n (6) s he ra f energy delered dded by al energy sred per cycle a characersc n preusly bsered. Dscnnuus nducr curren [5]: The pea nducr curren, durng dscnnuus nducr curren peran s deermned slely by he duy cycle, accrdng equan (5). Tha s The aerage nducr curren fr dscnnuus nducr cnducn s shwn n Table. Cnsder he cenral deny expressn n equan (6), whch assumes (7) whch s based n he energy equan (7) ½ / On subsun f n equan (7), afer suable mulplcan by, hen =, ges (see equan (6)) ½ (8) slang he lage ransfer funcn, ges fr dscnnuus cnducn ½ (9) where he upu curren has been nrmalsed by he mnmum upu curren, when =, he erm / The lage ransfer funcn fr dscnnuus nducr curren n (9) alng wh he nrmal lage ransfer funcn n equan (), are pled n fgure 5. The cndn = ½, he erge f dscnnuus nducr curren a = ⅓ s shwn. Als shwn s he bundary cndns fr ncreased lad ressance, =, =.6. The magnfed ew nse shws ha cnnuus nducr curren peran re-emerges a a lw duy cycle f less han =.. Ths reemergence f cnnuus nducr curren a lw duy cycles ccurs fr > ½, and can be characersed mre rgrusly by nesgang he mnmum nducr curren characerscs. Cnnuus nducr curren: The ey crcu parameer s he mnmum nducr curren, gen by, whch fr cnnuus nducr curren, s (A) () Nrmalsan wh respec he mnmum lad curren, /, ges. 8

11 ½ () The lage ransfer funcn, equans () and (9), arus nducr currens (aerage mnmum, pea, equan ()), ec. are summarsed n Table. Crcal crcu cndns, namely he bundary beween cnnuus and dscnnuus nducr curren, ccur when he mnmum nducr curren equals zer, ha s, n equan (), equals zer: ½ () whence, n subsung he lage ransfer funcn, equan (), whch s ald n he bundary, yelds 7 / () as he dscnnuus curren cnducn bundary cndn. Ne he smlary equan (9). Such analyss dere hs equan appears n exs [5], bu analyss prgresses lle furher.. = bundary ½. - /(-) = = =.5.. bundary =.6 - = ½ =½ = bundary /(-) =. = = = ⅓ =.5 = ½ Fg.5. Vlage ransfer funcn aran wh duy cycle, shwng dscnnuus curren bundares. 8

12 8 Table. Sep-up cnerer ransfer funcns, lad ressance nrmalsed.. MATHMATCAY ANAYSS OF TH BOUNDAY CUBC POYNOMA Cnsder equan () rearranged n a mre general cubc plynmal frm where y c c () As a cubc plynmal, a leas ne real r exss ; T Cnerer Flybac-bs r sep-up dscnnuus cnnuus crcal 7 7, ½ D D D x x D - x,, ½ =½ D ½ mn 9 mn c D T T D T = ½ - hen herwse, / T c T f C ½ C C ½

13 fr. The effec f he cnsan erm c s prduce a Y-axs shf f he basc cubc funcn, hus n hs case, deermnng f ne r mre real rs exs. The duy cycle range f neres s and c (represenng pse upu curren). Alhugh prmarly neresed n he rs f hs cubc, hse rs are unquely asscaed wh he prperes f he lcal maxma and mnma. quang he frs dfferenal zer (and esng fr a maxma r mnma) yelds and, bh ndependen f c. The lcal mnma always ccurs a a alue f -c, he Y-axs nercep alue (when = ). The nflexn pn (secnd dfferenal equaed zer), whence a lcal maxma and mnma always exs. ges nflex The lcal maxma and mnma represen he alues a whch rs emerge and dsappear respecely, as he alue f c (he Y-axs nercep) shfs he cubc pl up he Y-axs (decreasng curren), as shwn n fgure 6. decreasng = 7 τ =Ω cubc d (-d)² - c,, , 7, 7, 7 c = /7 c =, d c ne real r X c c 7 ne real r X 7 hree real rs, X Fg.6. Bundary cndns fr hree real rs he cubc equan. Slable, unque bundary sluns exs fr when all hree real rs exs such ha w are cncden (real and equal). These cases are pled n fgure 6. quan () s equaed he fllwng general cubc wh w cncden rs, whch releases a slun degree f freedm. c (5) X crcal By expandng bh sdes and equang ceffcens, because f he released degree f freedm, a unque, able, quadrac slun resuls, yeldng: and fr = ⅓, c = /7 ( = ½). The r X = supprs he fac ha when nly ne real, pse r exss, ha r ccurs fr X. Smlar analyss n he lcal mnmum yelds ha f nly ne real negae r exss ha sngle r mus be less han zer, hence a sngle r slun s n he range X ; always uwh he range f neres,. The lcal mnma als yelds cncden rs, a =, when c = (, an upu pen crcu). xac sluns fr he rs f he cubc plynmal, n erms f s ceffcens, can be fund n he Appendx. r and crcal X 8

14 V. NTPTATON OF TH BOOST CONVT BOUNDAY CUBC POYNOMA nerprean f he cubc analyss resuls, s: f < ½ (c > /7, wh n real rs beween < < ), dscnnuus nducr curren des n ccur fr any duy cycle. Frm equan (6), cnnuus nducr curren resuls f 7 (6) As ncreases abe ½ ( < c < /7), as he lad ressance ncreases and he lad curren decreases, a regn abu = ⅓ spreads asymmercally (wards = and wards = ), where n ha range resuls n dscnnuus nducr curren. The mplcan f he crcal range spreadng n bh drecns abu = ⅓, s sgnfcan. Fr a gen fxed lad ressance, wh > ½, as he duy cycle decreases frm a maxmum, dscnnuus nducr curren resuls befre = ⅓, bu, as he duy cycle s reduced belw =⅓, furher wards zer, cnnuus nducr curren cnducn always re-emerges [5]. A crllary, a frs sgh cnradcry, s ha: f he duy cycle s mananed cnsan as he lad curren s decreased ( ncreased), nce dscnnuus nducr curren ccurs, cnnuus nducr curren peran des n re-emerge (f he duy cycle s mananed cnsan), as cnfrmed by any f he fur pls n fgure. n fgure nce a cnsan duy cycle cnur eners he shaded dscnnuus curren regn, he upu lage ncreases and he cnsan duy cycle cnur remans n he dscnnuus curren regn as he curren decreases zer. Only f he duy cycle s decreased (decreasng he energy beng ransferred he lad) can peran re-ener he cnnuus nducr curren regn. Therecally cnnuus nducr curren ccurs a =. The nrmalsed desgn mngram n fgure 7, llusrang he equans n Table wh =, fr whch dscnnuus cnducn ccurs accrdng equan (), llusraes he prperes f he cubc bundary equan (). Fr =, he bundary alues fr dscnnuus cnducn are. cr.6. n fgure 7, he nducr curren waefrm s cnnuus fr =.65, hen dscnnuus when he duy cycle s reduced =.. f he duy cycle s furher reduced, =.5, cnnuus cnducn s predced, as subsanaed by he PSpce pls n fgure 8c. Fgure 8, pars a c, shw PSpce pls fr hree lad cndns ( =, ½, and ). The frs, fgure 8a, s when =, and cnnuus nducr curren ccurs fr all, as predced. The pl n fgure 8b, shws peran n he erge f dscnnuus cnducn, when fr =⅓ he mnmum nducr curren jus reaches zer, as predc fr = ½. Fgure 8c shws he case fr =, when dscnnuus nducr curren resuls, bu reemerges a a lwer duy cycle. The reasn why cnnuus curren recmmences a lw duy cycles (<⅓) s relaed he amun and hw energy s ransferred he lad. Durng nrmal peran, when he swch s ff, w surces ransfer energy he lad, as shwn by equan (). ½ / The surce energy s prprnal lad curren, whle he energy frm he nducr s quadrac curren dependen, and he lad curren s prprnal lage. The upu lage decreases accrdng /- when nducr cnducn s cnnuus, bu s apprxmaely prprnal duy cycle (frm equan (9) and fgure 5), when dscnnuus. The energy frm he supply when he swch s ff s ha necessary manan he upu a s mnmum alue, equan (), whle he nducr energy prduces he bs lage abe,. A hgh duy cycles he necessary quadrac nducr energy reduces a he same rae as he upu energy fall rae whch s dependan n he duy cycle. A he bundary f dscnnuus nducr curren, he rughs f he nducr curren are unable reerse (heren ransferrng energy bac he dc supply), whch resuls n excess energy beng ransferred and reaned by he lad crcu. As he duy cycle s decreased belw hs leel, he lad energy rae decreases apprxmaely lnearly wh duy cycle, a a 8

15 faser rae han he nducr energy rae reducn. As he duy cycle decreases furher, he lad requremen s such necessae nducr energy asscaed wh cnnuus nducr curren. The nducr energy agan balances he bs lage accrdng he ransfer funcn /-. 6 Prgresse a larger percenage f energy mus be prded by he supply energy cmpnen, whch can nly suppr an upu lage. As ends zer he as majry f he lad energy s prded drecly by he dc supply; wh a cnnuus lad (hence nducr) curren, / frm he supply when = and he nducr rpple curren s zer. 5. nducr curren = = = =..6. me ¼ ½ ¾ ¼ ½ ¾ me c= = D.5 = cr =.8 ranslan.6. D maxmum X.9.6, x. ¼ ½ ¾ = cr = Fg.7. Sep-up cnerer perfrmance mngram fr =, gng dscnnuus nducr curren fr. cr.6. nducr me dman curren waefrms fr cn =.65 (cnnuus nducr curren) and ds =. (dscnnuus nducr curren). Capacr dscharge n swch-ff perd.7. 85

16 V. GNA ANAYSS OF TH BOOST CONVT Fgure 9 shws he mnmum nducr curren a he bundary f dscnnuus nducr curren, as gen by equan (), pled n hree dfferen dman cmbnans. Specfcally he pls are cmbnans f,, and he nrmalsed mnmum nducr curren: ½ (7) Tgeher he fur shwn pls reeal he underlyng mechansms f he bs cnerer. eersble cnerer peran has been assumed n equan (7), ha s he lage ransfer funcn, equan (), s always ald. Ths recgnses ha f he bs cnerer s reersble (exra swch and dde) hen he nducr curren can reerse and he ransfer funcn gen by equan () remans ald fr all, prded wh a passe lad. The frs pl, 8a, f mnmum nducr curren ersus lad,, shws he mnmum nducr curren reachng zer when he lad ressance reaches = ½ fr = ⅓, hereby cnfrmng he slun (5). The mnmum curren lcus s dered frm dfferenan f equan (7) d ½ d whence such ha The pl n fgure 9b shws mnmum nducr curren pled agans duy cycle fr dfferen lad cndns,. The lcus f he mnmum pssble nducr curren fr a gen lad s dered by dfferenang equan (7) wh respec and equang zer, gng whch n subsun bac n equan (7) yelds he lcus: mn (9) The pl (and equan (9) when equal zer) cnfrms he crcal nducance curren cndn ccurrng a = ½ fr = ⅓. Fgure 9b sheds sme lgh n he cnerer mechansms when <. As he lad curren ncreases, (ha s, lad ressance decreases - decreases) he lcus f mnmum nducr curren ncreases, he mnmum alue f whch ncreases as duy cycle decreases. A =, he mnmum pssble nducr curren f pu, /, ccurs a =, any furher decrease n lad ressance resuls n he mnmum nrmalsed nducr curren f pu cnnung ccur a =. The herecal lcus fr =, ha s an upu shr crcu s shwn, wh < beng shaded as an unbanable perang regn. Subsun f hs duy cycle cndn bac n equan (7) ges ½ (8) mn The sragh lne (equan ()) angens (f slpe ½ frm dfferenan f equan (7) wh respec ) represen he mnmum nducr curren aran fr a cnsan duy cycle as he lad, s changed. Ths pl s n readly nerpreed when < (hgh lad curren leels), fr ends predc angens such ha <. Obusly = s a resrcn, hence he mnmum lcus pl n fgure 9a s shwn dashed fr <. 86

17 nducr curren (A) 8 6 =.5 τ = ¾ =. ¼ ½ ¾ 8 6 mnmum nducr curren (A) =.5 =¾ = nducr lage ( ) (V) - τ =.5 = ¾ =.5 ¼ ½ ¾ =¾ = - =¾ ¾ ½ ¼ upu lage ( ) duy cycle (pu) 5 Fg.8a. Sep-up cnerer perfrmance, ==. [ = 5V, = Ω, = μh, τ = μs, =.5, ⅓, ¾] nducr curren (A) 8 6 =.5 τ = ¾ =. ¼ ½ ¾ 8 6 mnmum nducr curren (A) =.5 =¾ = ½ nducr lage ( ) (V) - τ =.5 = ¾ =.5 ¼ ½ ¾ =¾ = - =¾ ¾ ½ ¼ upu lage ( ) duy cycle (pu) 5 Fg.8b. Sep-up cnerer perfrmance, =½=.[ = 5V, = Ω, = μh, τ = μs, =.5, ⅓, ¾] 87

18 nducr curren (A) 6 =.5 τ = ¾ =. ¼ ½ ¾ 6 mnmum nducr curren (A) =.5 =¾ = = =.6 nducr lage ( ) (V) - τ = ¾ =.5 =.5 ¼ ½ ¾ dscnnuus =¾ = - =¾ ¾ ½ ¼ upu lage ( ) duy cycle (pu) 5 Fg.8c. Sep-up cnerer perfrmance, ==.[ = 5V, = Ω, = μh, τ = μs, =.5, ⅓, ¾] Furher nsgh s ganed n peran belw =, by pl 8c whch shws lad ressance pled agans duy cycle. The lcus f mnmum lad ressance fr a gen duy cycle s gen by dfferenang (7) wh respec duy cycle, when he equan s expressed n erms f, namely whch n dfferenan and equang zer yelds and n bac subsun ges () mn The regn f dscnnuus nducr curren fr > ½ s shwn shaded, and s ndcaed as reersble. Nce ha quadrac ype mnmum nducr curren cnurs are n baned when <. The bes way nerpre he regn fr < s examne he exreme lm, when =. n he lm, as he lad ressance ends zer, =, he mnmum nducr curren s resrced by he duy cycle. Ceran perang curren and duy cycle cndns are unbanable, as shwn n pl 8d. As he mnmum nducr curren ncreases (less rpple curren) he mnmum necessary duy cycle ncreases n rder manan he upu lage. Snce he rpple curren maxmum pea pea s fxed (bu prprnal duy cycle), hs means a lman n he mnmum rpple curren a hgh curren leels and lw duy cycles. The bundary fr frbdden peran n fgure 9d (and als shaded n fgure 9b) s gen by = n equan (7), ha s () The aerage nrmalsed nducr curren s gen when = n equan (7) r equan (). The lman area s beween < <, as shwn n fgure 9d, where he upper bunds s 88

19 () earrangng equan () ges he mnmum duy cycle fr <, fr whch he mnmum nrmalsed nducr curren s n resrced, as shwn n fgure 9d. fr () ad Mnmum nducr curren (p.u.) 5 6 mnmum lcus reersble ½ duy cycle c= mnmum lcus duy cycle.9 frbdde n (c) (b) (d) (a) Mnmum nducr curren (p.u.) Mnmum nducr curren (p.u.) reersble frward reersble frward frbdde n ½ mnmum lcus mnmum lcus duy cycle lad Fg.9. Sep-up cnerer characersc lad cures. Maxmum perd X fr zer nducr curren Fgure shws he bs cnerer characerscs fr dscnnuus nducr curren when perang a he maxmum lengh f me wh a dscnnuus curren cndn, X. Frm Table, he perd f zer nducr curren, s gen by X - and =½ The maxmum nn-cnducn perd, (afer elmnang he lage ransfer funcn, hen dfferenang and equang zer) s when 89

20 fr 7 () lad curren. The capacr rpple lage due hs cnsan dscharge s gen by whch yelds fr T C d T C C X (5) 7 Tha s D (6) 6 C C pu (8) when, durng dscnnuus nducr curren, a cnsan upu lage resuls ½ (7) These equans recnfrmed he erge f dscnnuus nducr curren cndn, when =½, = ⅓, x =, and / = ½. Operan a he lnges nducr dscnnuus curren duran pn resuls n nly n a cnsan upu lage, equan (7) bu als n a cnsan aerage nrmalsed nducr curren gen by ½ ½ 9 pu whle he mnmum nducr curren s zer fr all > ½, he pea nducr curren s These nducr currens are shwn n fgure, alng wh he bundary case = ½ fr cnnuus nducr curren cnducn. V. DSCONTNUOUS CAPACTO CUNT DUNG TH SWTCH OFF POD Oupu rpple lage s dmnaed by he capacr rpple lage, whch s relaed he upu capacr chargng curren aran equalen seres ressance equalen seres nducance Generally fr he bs cnerer, when he swch s n, he upu capacr prdes he enre cnsan Ths equan assumes ha he capacr nly dscharges when he swch s n. Under lwer duy cycle cndns he capacr s prne prde lad curren when he swch s ff, when he nducr curren falls a leel whch s nsuffcen prde all he lad curren requremen. Such a cndn ccurs wh dscnnuus nducr curren, ha s, as shwn n fgure (whch s fgure 9c reprduced fr clary). Ths bundary cndn s defned by equan (), ha s (9) The capacr rpple lage ncreases abe ha gen by equan (8) and s gen by ½ C C (5) The lage ransfer funcn s gen by equan (9). n fac a mre resrce bundary han equan (9) exss, whch s characersed by equang he mnmum nducr curren he lad curren such ha c =, ha s. Frm Table, fr cnnuus nducr curren he bundary s gen by ½ whch, as shwn n Table, prduces a regn defned by (5) 9

21 D X upu lage 8 6 f >½ X D decreasng Nrmalsed nducr currens 8 f <½ 9 9, fr 7 fr all fr decreasng Fg.. cus f maxmum dscnnuus curren characerscs. Ths bundary s relaed by he duy cycle he bundary specfed by equan (9), bh equans beng shwn n fgure. Prded dscnnuus nducr curren des n ccur, he lage ransfer funcn gen by equan () remans ald. The upu rpple lage magnude s gen by C ½ - C ½ - C C (5) 9

22 The frs equaly s fr when he rpple lage ncreases, as he capacr s recharged mmedaely afer he swch s urned ff. The secnd equaly cmprses he w cmpnens when he capacr lage decreases, namely when he swch s n and durng he laer par f he swch ff perd nce he nducr curren falls belw he lad curren leel ad 5 quan reersbl 6 ½ duy cycle quan ½ mnmum c= lcus quan (8) quan (9) Fg.. Zer capacr curren characerscs n swch ff-sae. V. CONCUSONS. quang yelds earrangng ges he nrmalsed cubc equan ½ ½ Subsung X / 6 X and slng he resulan quadrac yelds The dscnnuus nducr curren prperes and characerscs f he bs cnerer hae been nesgaed. Specfcally, he re-emergence f cnnuus nducr curren a lw duy cycles has been quanfed. Ths prpery neer ccurs f he lad s ared and he duy cycle s fxed. The mnmum nducr curren prperes a hgh curren hae als been nesgaed. Oupu capacr rpple lage has als been quanfed fr bh dscnnuus nducr curren and a mre lely cndn where he nducr curren falls belw he lad curren leel. V. APPNDX: XACT OOTS OF A CUBC QUATON N TMS OF TS COFFCNTS Frm Table, a he bundary f cnnuus cnducn, equan (), X ¼ 7 whence fr X / 6 X ¼ 7 6 ¼ 7 ¼ 6 ¼ 7 7 Frm he dscrmnan, fr mre han ne real r 7 fr whch whence a 7 Ne ha a negae dscrmnan s he crrec cndn fr mre han ne real r [6]. and frm Table, equan () The slun equan (), fr he duy cycle n erms f he nrmalsed maxmum crcal nducr 9

23 curren ( le / ) s (5) The remanng w rs, are ½ j, (5) whch are all real rs f. The slun equan (), fr he duy cycle n erms f he nrmalsed lad ressance facr, s fund by subsung 7 n equans (5) and (5), such ha mre han ne real r resuls f 7, ha s 7. FNCS [] Kjaer, S.B.; Pedersen, J.K.; Blaabjerg, F. 'A reew f sngle-phase grd-cnneced nerers fr phlac mdules', ndusry Applcans, Transacns n, 5, Vl., ssue: 5, pp.: 9-6, DO:.9/TA [] Al-Saffar, M.A.; smal,.h.; Sabzal, A.J.. 'negraed Buc Bs Quadrac Buc PFC ecfer fr Unersal npu Applcans., Pwer lecrncs, Transacns n, 9, Vl., ssue:, pp , DO:.9/TP.9. [] Mhan. N., e al., Pwer lecrncs, J Wley & Sns,, Chaper 7. [] Bausere.., e al., Pwer lecrncs, dc-dc cnersn, Sprnger-Verlag, 99, Chaper 6. [5] rcsn..w., Fundamenals f Pwer lecrncs, Sprnger,, Chaper 5. [6] hp:// 9