Average Small-Signal Modelling of the Power Stage of Power Factor Correctors with a Fast Output- Voltage Feedback Loop

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1 erage SmallSgnal dellng f he wer Sage f wer Facr rrecr wh a Fa Oupu lage Feedbac p Jaer Sebaán, Deg. amar, ara aría Hernand, guel dríguez and rur Fernández Unerdad de Oed. rup de Sema Elecrónc de lmenacón SE Edfc Deparamenal nº 3. ampu Unerar de eque. 33 jón. SIN eba@un.e brac If a wer Facr rrecr F degned wh a fa upulage feedbac lp, bh he eadyae and he dynamc behaur f h F ary n cmparn wh ha baned when upulage feedbac lw. Th a cnequence f he lage rpple ha appear n he cnrl gnal, whch mdfe he cnerer peran. Th lage rpple quanfed by defnng w parameer: relae mdule,, and phae angle,. Thee w parameer hae been aen n accun n h paper deelp a new aerage mallgnal mdel f he pwer age f F. The mdel crrepnd a frrder yem and mlar he ery wellnwn mdel baned fr F wh a lw upulage feedbac lp. The man dfference beween he w mdel, hweer, ha he ne crrepndng he F wh a fa upulage feedbac lp depend n he afremenned parameer, and, whch n urn depend n he gan f he errr amplfer a wce he lne frequency. Th n a cmmn uan: he pwer age mdel depend n he behaur f he errr amplfer he regular a a pecfc frequency a wce he lne frequency. h he cnrl and he lneupu ranfer funcn and he penlp upu mpedance f he F can be ealy deduced frm he new mdel hu baned. ll f hee funcn cncde wh he wellnwn mdel crrepndng he cae f a lw upulage feedbac lp when he cnrl gnal rpple dappear.e.,. Fnally, he new mdel erfed expermenally. I. INTODUTION The ue f an ace wer Facr rrecr F [, ] rually mandary n rder lm he harmnc cnen n he lne curren f mancnneced elecrnc equpmen. Fgure hw he general cheme f an ace F cnrlled by w feedbac lp, whch he m wdelyued crcury cnrl cnerer f h ype. The nner feedbac lp hwn n h fgure an npucurren feedbac lp, whle he uer ne an upulage feedbac lp. The curren lp mae he lne curren fllw a reference gnal whch baned by mulplyng a recfed nudal waefrm baned frm he lne lage by. Thu, he lne curren, g, a nud whe amplude deermned by he alue f. The andard degn f he lage feedbac lp mple lw rpple n n rder ad lne curren drn [, ]. Hweer, h degn caue lw bandwdh n he upulage feedbac lp, hu lmng he ranen repne f he F. Under hee cndn, he ranen repne f a F n fa enugh afy he requremen f me lad [, 3]. T achee a faer ranen repne, he g g bandwdh f he errr amplfer,, mu be relaely hgh [3, ], hu leadng cnderable lage rpple n he cnrl gnal, ee he waefrm hwn n Fg.. Thu, aumng ha ha been degned wh hgh bandwdh allwng nnneglgble lage rpple n a achee a fa repne, bh he ac [5] and dynamc behaur f he F change n relan ha crrepndng a degn wh lwbandwdh n. The udy f h dynamc behaur n fac he gal f he preen paper. The arng pn fr h udy he ac mdellng deelped n [5], whch reewed n he nex ecn. II. g g / EIEW OF THE STTI ODEIN OF OWE FTO OETOS WITH FST OUTUT OTE FEED OO The lage and curren a he npu f he pwer age hwn n Fg. can be wren a fllw: n, g n g, where he pea alue f g, he lne angular frequency, a cnan and he lage a he errr amplfer upu. Th lage can be rewren a fllw:, 3 ac g g wer age urren lp ac n, W E lage lp ref Hgh bandwdh Fg. : F wh a fa upulage feedbac lp. ad 9788/9/$5. 9 IEEE 998

2 where he dc cmpnen f, ac ac cmpnen, he pea alue f ac and phae lag angle Fg.. The relae alue f he lage rpple n defned a fllw:. 5 a cnequence f hang lage rpple n he cnrl gnal,, all he ac elecrcal quane f he F ary n relan he crrepndng he andard degn cae [5],.e., wh a cnrl gnal wh n rpple. Fgure hw me lne waefrm crrepndng a arey f degn cndn. h fgure hw, he lne curren ha apprecable drn, whch a cnequence f he lage rpple n. I huld be ned ha nly rpple f wce he lne frequency aumed n, whch a lgcal cnequence f he behaur f bh he upu cell and he errr amplfer,, a frequence greaer han wce he lne frequency. Therefre, h lage rpple can be defned by mean f nly w parameer: magnude,, and phae lag angle,. he lage rpple magnude can be relaed hrugh 5, hen, and cmpleely defne he ae f he cnrl arable,. The nfluence f, and n F peran n he eadyae can be ummarzed n he equalen crcu hwn n Fg. 3 [5]. The alue f he dependen curren urce repreened n h fgure are gen n Table, where hee alue hae been expreed a funcn f and and f he pwer age arable.e.,, and. The alue f hee curren urce can al be expreed a funcn f he g ne g g g g g g g g g Fg. : ne curren fr dfferen alue f and. g g g3 dc / / ad & parac cnrl arable dered cnrl arable Fg. 3: Seady Sae mdel f a F wh a fa upulage feedbac lp. Inpu urce Oupu urce g g 3 n c n c3 n dc n cnrl arable, dered, parac and parac, a gen n Table. When, he alue f curren urce g3 and becme zer n bh e f expren and he alue f he remanng curren urce becme he f he andard F wh a lwbandwdh upulage feedbac lp.e., wh n rpple n h feedbac lp. bh able hw, curren urce g, g3, and are ac urce. Ther frequence are he lne frequency, hree me he lne frequency, wce he lne frequency and fur me lne he lne frequency, repecely. Hweer, dc a dc curren urce, whe eadyae alue cncde wh he lad curren. n mpran degn parameer f any F he upu lage rpple. In he cae f F wh a fa upulage feedbac lp, h rpple manly generaed by he curren urce. The cnrbun f he her ac curren urce,, h rpple alm neglgble [5], a Fg. hw. In h fgure, he relae upu lage rpple f wce he lne [ n c ] n n Table : alue f he curren urce repreened n Fg. 3 a funcn f,,, and. Inpu urce Oupu urce g g 3 dc n c c3 n n c n Table : alue f he curren urce repreened n Fg. 3 a funcn f,,, and. 9788/9/$5. 9 IEEE 999

3 frequency, r due, cmpared wh ha f fur me he lne frequency, r due. The alue f r neglgble cmpared wh r fr alm all cndn. Only n he cae f hgh alue f near and 9º de he alue f r becme gnfcan and een hgher han r. Hweer, h cae f n praccal nere becaue cann be ealy mplemened and generae hgh drn n he lne curren [5]. l frm [5], he cmpnen f wce he lne frequency n he upu lage wll be: n c. 6 n Frm h expren, he magnude f wll be: n p. 7 n reer, he ealuan f he gan magnude and phae f he upulage feedbac lp a he rpple frequency yeld:, 8 p n arcan, 9 c where and are he magnude and he phae angle f a wce he lne frequency.e.,. Thee equan allw u deermne he gan f he errr amplfer a frm he dered alue f and. menned preuly, he cnrl arable,, ha w cmpnen: dc alue,, whch he dered cnrl arable, and rpple, ac, whch he parac cnrl arable ee Fg. 5. Thee w cnrl arable deermne w parallel feedbac lp: he dered ne and he parac ne. The frmer he le feedbac lp preen n he andard degn wh a lwrepne lp, wherea he laer nly preen n he cae f a farepne lp. The parac lp furher dded n w lp, ne crrepndng he magnude f he lage rpple and he her crrepndng phae. r [%] r / / / / / / r / / Fg. : alue f he upulage rpple a wce and fur me he lne frequency. wer age and cnrl III. EE SSIN ODEIN OF OWE FTO OETO WITH FST OUTUTOTE FEED OO The aerage mallgnal mdel f he pwer age can be baned fllwng a mlar prce ha prped n [6], hugh ang n accun n h cae he nfluence f he aran f and n he aran f dc. n he cae f any aerage mdel, he aerage upu lage ha n lage rpple. Hweer, he acual upu lage rpple mu be aen n accun cmpue he rpple n he cnrl gnal,. menned n he preu ecn, h rpple ha a rng nfluence n he eadyae behaur f he F and wll al mdfy he F dynamc repne. The nan ued n h ecn decrbe he dfferen quane he uual ne,.e., capal leer hae been ued decrbe eadyae quane, wherea lwercae leer wh ha hae been emplyed fr he perurban f he ame quane. The eadyae frm f he alue f dc can be ealy baned frm Table and Table : Idc n, where I dc,,, adc, and are he eadyae alue f dc,,, adc, and, repecely. erurbng he alue f dc n Table and ang n accun, we can ban: dc dc dc dc dc î ˆ dc ˆ ˆ ˆ ˆ, dc dc dc n dc ac Oupulagerpple lp parac D upulage lp dered n n E dc ref dc ref, Fg. 5: lc dagram f he crcu hwn n Fg.., 3, 9788/9/$5. 9 IEEE

4 dc n n, 5 n p I p n n, 7 dc c c n. 6 p I p n n, 8 The dc cmpnen f he upu lage relaed dc hrugh he mpedance f he cell: dc dc, 7 and herefre: ˆ dc îdc. 8 reer, he aran f and f.e., ˆ and ˆ are relaed he upu lage rpple. T udy he aran f hee quane, curren mu al be perurbed. I huld be ned ha an ac quany and herefre bh magnude and phae angle mu be perurbed. The alue f gen n Table can be rewren a fllw: p n ϑ, 9 p n, n n ϑ arcan arcan. c c Frm 9, can be rewren a fllw: n ϑ arcan. c The alue f can be fund frm afer ung me rgnmerc relanhp: ar c c. 3 Tang n accun, he eadyae expren crrepndng and 3 can be ealy baned: Ip n, ar c n n c, 5 erurbng he alue f p and ang n accun, we can ban:, 6 î p p ˆ p ˆ p ˆ p ˆ pc ˆ p p p I p n n, 9 I p n n n n n, 3 I p n n c n n. 3 c n n Smlarly, perurbng 3 and ang n accun 5, we ban: ˆ ˆ ˆ, 3 c c c c, Tang n accun 5, 33 and 3 can be rewren a fllw: c n c n I huld be ned ha eablhe he relanhp beween he phae f.e., ϑ and he phae f he rpple.e.,. Smlarly, he magnude f.e., p and he magnude f he rpple.e., mu be relaed. Thu, he magnude f he upu lage rpple wll be: 9788/9/$5. 9 IEEE

5 and herefre: p p, 37 îp ˆ p, 38 and ang n accun 8, we fnally ban:, 39 ˆ p ˆ. The relanhp baned beween he perurbed quane can be ummarzed n he blc dagram hwn n Fg. 6 and Table 3. aran n he lad curren,, hae al been ncluded n h dagram. I huld be ned ha he pwer p dc dc dc 9 p p D upulage lp dered wer age, mulpler cnrl crcury and parac lp dc Fg. 6: aral ranfer funcn baned frm drecly perurbng he man equan. age, he cnrl crcury he nner curren feedbac lp and he analg mulpler and he parac lp caued by he wce lnefrequency lage rpple hae been drawn eparaely frm he dc upulage feedbac lp, whch he dered feedbac lp. Thu, he mdel repreened n Fg. 6 he cmplee cledlp mdel f he F wh a fa upulage feedbac lp. Th blc dagram can be reduced he ne gen n Fg. 7 afer a edu prce f Z Z g wer age, mulpler cnrl crcury and parac lp D upulage lp dered Fg. 7: Fnal blc dagram. µ n σ 3 n µ 3 n n µ n n n g dc σ µ Table : an ranfer funcn hwn n Fg. 7 and me f her degn parameer. n c 3 n n n n 5 n 6 n n n c 7 n n 8 n n n 9 n n c n 3 c n Table 3: an paral ranfer funcn n he blc dagram hwn n Fg /9/$5. 9 IEEE

6 manpulang he equan ha relae he perurbed arable. The alue f he man funcn hwn n h fgure are gen n Table. h able hw, he ranfer funcn beween he cnrl and he dc upu lage.e.,, he penlp upu mpedance.e., Z and he frequency f he pwer age ple.e., depend n he ac alue f and.e., / and hrugh he dmennle parameer µ ee Table. Smlarly, he dmennle parameer σ deermne he ranfer funcn beween npu and upu lage. 5 µ, / / / / Fg. 8: µ a a funcn f and σ, / / / / /g [d] / Fg. 9: σ a a funcn f and Sandard degn 5 / g g. f [Hz] / / Fg. : agnude f fr a arey f dfferen degn. In all hee cae,.75, r. and he pwer prceed he ame. The alue f µ fr dfferen alue f and are gen n Fg. 8. h fgure hw, µ becme when he relae rpple zer; n h cae, he mdel hu baned cncde wh he mdel baned n [6]. Smlarly, he alue f σ fr dfferen alue f and are gen n Fg. 9. l n h cae, σ becme when he relae rpple zer, he mdel baned here cncdng wh ha baned n [6]. Fnally, Fg. hw he magnude f j n he cae f eeral degn wh a fa upulage feedbac lp all fr.75. I huld be ned ha j he ranfer funcn beween he upu lage and he derable cnrl arable,. In all cae, he F ha been calculaed fr he ame upu lage rpple % and fr he ame pwer prceed. gen n Table, h ranfer funcn alway a frrder ne. Fgure hw ha gnfcanly nfluenced by he alue f and. I. EXEIENT ESUTS The expermenal erfcan f he prped mdel cann be carred u n pen lp, becaue he effec f he upu lage rpple he parac feedbac lp n he cnrl gnal wuld dappear n ha cae. Therefre, a regular mu be ued cle he feedbac lp. We hae chen he regular hwn n Fg., he dc gan f whch m and whe cuff angular frequency p. Fgure hw he expermenal and herecal reul crrepndng a lad ep n a prype b F wh a fa upulage feedbac lp. The degn parameer f he regular hae been mdfed n rder ban dfferen alue f and and hence f and, frm alue crrepndng neglgble rpple n.6 her crrepndng a cnderable rpple n.67. In all cae, he aerage alue f he expermenal reul f he reul predced ung he mdel deelped n h paper ery well. hee expermenal reul hw, h mdel crrecly decrbe n nly he behaur f a F wh a lwrepne lage m ref ref p m p Fg. : egular mplemened n he expermenal prype. 9788/9/$5. 9 IEEE 3

7 /d m/d p 6.8 rad/, m,, 8.3º,.6, 8.º Therecal Expermenal /d m/d p 683. rad/, m, 5.7º,.6, 9.º Therecal Expermenal /d m/d p 68.3 rad/, m, 5.7º,.67, 7.8º becme he wellnwn ne fr F wh a lw upulage feedbac lp, namely ha gen n [6]. NOWEDENT Th wr ha been uppred by he Spanh nry f Educan and Scence under rjec TE76697/I and ran EFEENES []. J. cher and.. Segerwald, "n acdc cnerer wh hgh qualy npu waefrm", IEEE Tran. Ind. ppl., l. 9, n., 983, pp []. H. Dxn, "Hgh pwer facr preregular fr fflne pwer upple", Unrde wer Supply Degn emnar, 99, pp I I6. [3]. Spazz,. aaell, and. e, ehd mpre dynamc repne f pwerfacr preregular: an erew, 6 h Eurpean nference n wer Elecrnc and pplcan, 995, pp []. Fernández, J. Sebaán,. J. llega,.. Hernand and D.. amar, Dynamc m f a werfacr reregular, IEEE Tran. n Ind. Elecrn., l. 5, n., February 5, pp [5] J. Sebaán, D.. amar,. Fernández and.. Hernand, Seadyae analy f pwer facr crrecr wh a fa upulage feedbac lp, IEEE E 9. [6].. dley, erage mallgnal analy f he b pwer facr crrecr crcu, E Semnar rceedng 989, pp. 8. Therecal Expermenal Fg. : Expermenal erfcan. feedbac lp uch a he mdel deelped n [6], bu al ha f a F wh a farepne lage feedbac lp, whch n crrecly decrbed ung he mdel deelped n [6] baned neglecng he rpple n.. ONUSIONS When a F degned wh an upulage feedbac lp whch relaely fa, he mdel f he pwer age change n cmparn wh ha baned wh a lw upulage feedbac lp. The rean fr h he nfluence f bh he mdule and he phae angle f he lage rpple ha preen n he cnrl gnal,. The dynamc udy f a F wh fa upulage feedbac lp can be carred u ung w parameer: he mdule f he relae lage rpple n,, and phae angle,. I eady ae alue and deermne he aerage mallgnal mdel f he pwer age, whch a frrder yem, beng bh he cnrl ranfer funcn,, and he penlp upu mpedance, Z, funcn f and hrugh he parameer µ Table and Fg. 8. The alue f µ becme when.e., when here n lage rpple n and he deelped mdel 9788/9/$5. 9 IEEE