Power Electronics. Prof. Dr.-Ing. Joachim Böcker. Lecture Notes. Last Update Paderborn University

Transcription

1 Power Elecroc Prof. Dr.-Ig. Joachm Böcker ecure Noe a Updae Paderbor Uvery Deparme of Power Elecroc ad Elecrcal Drve hee lecure oe are prmarly eded o be a accompayg lecure docume for ude of he Uvery of Paderbor. I alo avalable for oher ereed reader va ere. I ay cae, oly he prvae, dvdual, o-commercal uage permed. Publhg, preadg, duplcag or ug hee oe or par of hem aoher bu he above meoed coex o permed. Excepo alway requre he explc approval of he auhor. he auhor apprecae h, relaed o make or adequace.

2 Coe Coe Agme of Power Elecroc Compoe ad Syem... 5 Swche Ideal Swche Realzao of Swche by Mea of Power Elecroc Devce... 3 DC-DC Coverer Buck Coverer Prcple of Operao Buck Coverer wh Capacor for Smoohg of he Oupu Volage Realao of he Crcury Dcouou Coduco Mode of he Buck Coverer Boudary Coduco Mode of he Buck Coverer Boo Coverer Prcple of Operao Boo Coverer wh Capacor for Smoohg of he Oupu Volage Realzao of he Crcury Dcouou Coduco Mode of he Boo Coverer Boudary Coduco Mode of he Boo Coverer B-Drecoal DC-DC Coverer Coverer for Boh Curre Polare Coverer for Boh Volage Polare Four-Quadra Coverer Commuao Subber Crcu wh Zeer Dode RCD Subber Crcu Packagg echology Dyamc Averagg Average Modelg of a Reace Average Modelg of Iducor ad Capacor Averagg Model of ear me-ivara Dffereal Equao Average Modelg of a Swch Sae-Space Averagg of Varable-Srucure Dffereal Equao Dyamc Averagg Model of Buck Coverer Corol of he Buck Coverer Feedforward Corol wh Coa Duy Cycle Sgle-oop Volage Corol P-Coroller PI-Coroller PID-Coroller Volage Corol wh Ier Curre Corol oop Ier Curre Corol Ouer Volage Corol oop Curre mao wh Cacaded Corol... 75

3 Coe Curre Hyere Corol Boudary Curre Mode Corol Peak Curre Mode Corol Pule Wdh Modulao Pule Wdh Modulao wh Couou-me Referece Value Pule Wdh Modulao wh Dcree-me Sepo Pule Wdh Modulao Coderg Varable Supply Volage Pule Wdh Modulao wh Feedback of he Oupu Volage Harmoc of Pule Wdh Modulao Harmoc a a Coa Sepo Harmoc wh Suodal Sepo... 9 Ierlockg me... Drver Power Supply of he Drver... 5 Four-Quadra Coverer Crcu opology Pule Wdh Modulao for he 4QC Four-Quadra Coverer a a Recfer for Sgle-Phae Grd Saoary Aaly Corol Noch fler Sarg-Up ad Shudow Procedure ad Requred Crcury Parallel- ad Sere Coeco of 4QC module raformer Recfer wh Boo Coverer PFC Recfer hree-phae Volage-Source Iverer Exerally Commuaed hyror Coverer Ceer-apped ad Brdge hyror Crcu Commuao e-sde Harmoc wo-way Coverer Cyclo Coverer Hgh-Volage Drec Curre ramo Curre-Fed oad-commuaed Iverer Curre-Fed Self-Commuaed Iverer Curre-Fed Self-Commuaed Iverer wh ur-off Devce Mul-evel Iverer... 73

4 Coe 4 5. hree-evel Iverer eraure... 74

5 Agme of Power Elecroc Compoe ad Syem 5 Agme of Power Elecroc Compoe ad Syem u u Fg. -: Coverer he ma ak of power elecroc he covero of oe form of elecrcal eergy o aoher. h proce volve he covero of volage ad curre erm of magude or RMS value, he chage of frequecy, ad he umber of phae. Power elecroc compoe dog uch covero are referred o a coverer. Coverer are ued may dffere power ad volage rage. he pecrum of coverg elecrcal power rage from a few mw o everal MW, he volage rage exed from a few vol o everal kv or eve kv ad eve curre rag rage from ma o ka. O he way from power geerao va drbuo o ulzao, a eadly creag rao of elecrcal eergy covered a lea by oe, ofe by everal power elecroc age. Some example of applcao of power elecroc yem: Power upple for elecroc home applace, offce, for elecommucao equpme ad for peroal compuer or compuer erver Coverer for moor drve o allow peed-varable operao of elecrcal drve e.g. mache ool, coveyor drve, dural drve, pump ad fa Elecrcal wd urbe Phoovolac yem Supplyg a gle-phae 6 / 3 Hz ralway grd from 3-phae, 5 Hz aoal power grd. Supply of elecrc arc furace. gheg corol Audo amplfer

6 Agme of Power Elecroc Compoe ad Syem 6 Ceral Geeraor/Sorage grd couplg ale grd Auoomou vehcle G ~ arge power pla G ~ Wd off-hore Hydro GW 5 MW Pump orage ocal Geeraor G ~ ~ G ~ A ~ ~ kw ~ G ~ ~ Combed hea ad power ~ Wd,5- MW GW ~ ~ ~ ~ 38 kv kv ~ Coumer Elekroc - 5 kv HVS SC/ AF,- 3 GW ghg ~ ~ kv hgh volage -3 kv 4 V ~ 5-6 MW medum volage ale grd SC/ AF low volage SC/ AF Small dural pla Home applace G ~ arge dural pla ~ ~ M ~ MW,7 kv Coveyace ~ ~ ~ M -3 kw 5 kv grd couplg 5 Hz ~ 6,6 Hz 5 - kw G ~ 4- MW ~ ~ M ~ Ralway yem - MVA Daa ceer, rouer Auomao ~ ~ 5 MW ~ kv ~ M G ~ ~ Solar ~ kw ~ - 5 W - kw ~ ~ ~ ~ Prvae Geeraor 3 - W W kw ~ kw - 5 kw H Fuel cell Solar ~ ~ ~ kw ~ ~ ~ W ~ ~ kw ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Fg. -: Power elecroc geerao, drbuo ad ulzao of elecrc eergy

7 Agme of Power Elecroc Compoe ad Syem 7 Defo of erm of power covero Covero o from drec volage 3 drec volage DC-DC coverer alerag volage gle or hree phae verer DC-AC coverer alerag volage gle or hree phae 3 recfer AC-DC coverer AC-AC coverer he dreco of power flow, mo cae, u-drecoal. However, ome cae ca be alo b-drecoal. Ofe he erm AC-AC coverer uderood uch a way ha oly he magude or RMS value of he volage or curre beg chaged, bu o he frequecy. I geeral, however, chage of frequecy alo ubjec of uch a covero. h ca be performed a a oe-age covero by o-called cyclo coverer or marx coverer. I may cae h doe, however, by wo covero age, va recfcao o a ermedae DC volage or DC curre ad he vero aga o AC. Such coverer are called DC-lk coverer. 3 Moly we are dealg wh volage-ource yem ad peak of volage covero. E.g., h he cae wh he eergy upply grd. I geeral, alo he curre-ource yem are poble. hough curre-ource grd are o commo, however, coverer wh ermedae DC curre lk are kow very well.. Impora apec covero of elecrcal eergy: Co fe me ad relably Qualy of volage ad curre e.g. volage accuracy rpple coe, harmoc of volage ad curre, corol dyamc ec. Effcecy oe loe are o oly a ue of eergy coumpo, bu alo becaue of geg rd of he hea Volume, wegh epy. for moble applcao

8 Agme of Power Elecroc Compoe ad Syem 8 Fg. -3: he deg proce of power elecroc yem

9 Swche 9 Swche he applcao of power elecroc o perform power covero wh mmum loe clearly elmae he uage of reve eleme or oher mlar eleme wh hgh dpave loe a lea o he crcal pah. Alo elecroc compoe uch a raor cao be operaed her lear rego, becaue h would alo cur hgh power loe. So we coder oly he compoe wh mmum loe. hee are eeally reacve eleme uch a: Capacor Iducor raformer Alo jo o hee eleme: Swche Swche are key compoe power elecroc becaue hey are he oly eleme ha ca corol he flow of curre ad volage elecvely eher acvely by gae pule, exerally drve or pavely a reul of exeral elecrcal behavor of load or ework.e. load or ource commuaed.. Ideal Swche u cloed ope u Fg. -: wo-pole wch he mple dealzed wch he wo-pole wch, ca aume wo ae. cloed: u, ay value ope:, u ay value. he power lo of he deal wch alway zero, becaue eher he volage or he curre zero, p u..

10 Swche Fg. -: Realzao of he hree-pole wch by wo wo-pole wche, operaed u u Fg. -3: Four-pole wch he four-pole wch ca be ealy decrbed wh he help of he wchg fuco: cae of upper wch poo.3 ca e of lower wch poo u u.4 he quae u ad ca be codered a arbrarly gve, a hey are mpreed o he wch. Vce vera, ha o he cae for u ad! Due o he wchg behavor, u ad may chage ep-lke. hu, he crcu coeced o pu ad oupu of he wch mu be capable of uch ep-lke chage. I parcular, geerally o allowed o coec a ducor ere o he pu crcu of he wch, or o coec a capacor parallel o he oupu ermal. Excepo of h geeral rule are oly permed f ca be aured ha he wchg aco ake place exacly a uch me a, whe he ducor curre or he capacor volage zero for ure, o ha a ep-lke chage doe o occur. Such raege may be appled he coex of of wchg whch o a ubjec of h bac coure.

11 Swche opology o allowed allowed opology Fg. -4: Bac opologe he power balace apple for he four-pole wch a p p u u.5 u u u Fg. -5: Equvale crcu dagram of he four-pole wch wh corollable ource

12 Swche. Realzao of Swche by Mea of Power Elecroc Devce he deal wche codered o far wll of coure o be realzed by mechacal coac bu by elecroc devce. he mo mpora caddae of wo-pole wche are led he able below. Ou of h l, he bpolar juco raor ad he ehaceme-ype feld effec raor are well kow alo from coveoal elecroc crcu. Uually, hee eleme are operaed a couou, more or le lear operao rage, for example amplfer crcu. Here, h o he cae. he raor are operaed oly a wched mode. ha mea: Eher he devce o acve a all o ha oly a very mall blockg curre reul, whch ca uually be egleced or drve o aurao o ha he reulg forward volage geg a mall a poble. h called auraed operao rage wh he bpolar raor. I he able below, he characerc curve of he devce are how, however, dealzed wh egleced forward volage he coducg ae. I mpora, however, o pay aeo o he volage ad curre polare he devce are capable of. Wh he deal wo-pole wch, o rerco have bee dcued o far. May devce ca oly coduc he curre oe dreco ad whad blockg volage oly revere dreco. Excepo are he MOS- FE whch ca coduc he curre forward ad revere dreco, ad he hyror whch capable boh of forward ad revere blockg volage. For may applcao, however, hee horcomg of he real devce are o erou hadcap, becaue hadlg of boh volage ad curre polare ofe o requred. If ha eceary ome cae, everal devce mu be combed order o cover all eceary pecfcao. Such combao are how he ecod able below. hree ad four-pole wche are he realzed from wo-pole wche accordgly.

13 Swche 3 able: Idealzed characerc of ome power elecroc devce Red: characerc of o-drve devce,.e. gae/corol pu o acve Gree: characerc wh acve gae/corol pu Dode u u Bpolar juco raor ad Iolaed Gae Bpolar raor IGB IGB u u Power MOSFE u Body-Dode chael coducg u body dode coducg Gae ur Off GO hyror, u IGC Iegraed Gae Corolled hyror u RB-IGC Revere blockg IGC u u

14 Swche 4 able: Realzao of he wo-pole wch Q: Realzao for oly oe curre ad oe volage polary Q: Realzao for oly oe curre polary ad boh volage polare ad vce vera 4Q: Realzao for all curre ad volage polare he cae o how he able ca be obaed from he led cae by vero Q po. volage po. curre u u Q eg. volage po. curre u u Q po./eg. volage po. curre u u Q po. volage po./eg. curre u u 4Q po./eg. volage po./eg. curre u u u

15 3 DC-DC Coverer 5 3 DC-DC Coverer he ak of DC-DC coverer o cover a avalable DC volage o aoher DC volage hgher or lower ha he orgal. Power rag of DC-DC coverer may vary o a very large cale. ha may be oly a few vol ad ome ma whe provdg a exra upply volage o ome devce o a pred crcu board. Oher applcao may be, however, he rage of ome kv ad ka, reulg ome MW rag, a may be he cae wh ralway yem. 3. Buck Coverer 3.. Prcple of Operao S u u u u Fg. 3-: Schemac dagram of he buck coverer Fr, boh pu ad oupu volage hould be codered a gve ad coa. h cae a he oupu, e.g. wh he couervolage of a moor EMF or of a baery o be charged. u U, U he wch S clocked wh he duy rao. u. 3. erm ued: e Coverer volage: a e D 3. Swch-o me Swch op Swch-off me Swch boom e a Swchg perod f Swchg frequecy

16 3 DC-DC Coverer 6 u U durg he o perod u 3.3 durg he off perod u u u u u u u u u Durg o-me Durg off-me Fg. 3-: Equvale crcu dagram durg o- ad off-perod Seady Sae Aaly he evoluo of he curre v. me deermed by he followg dffereal equao cf. Fg. 3-3: From h, follow durg he o-perod, ] : ad durrg he off-perod e, ]: u u U 3.4 [ e U U 3.5 [ U U U U e e e e 3.6 he curre hrough he ducor aoary or perodc, f h lead o:. 3.7 U U U e e 3.8 U U U e e U e U

17 3 DC-DC Coverer 7 U e D 3.9 U he duy cycle deerme mlarly a a raformer wdg rao he rao of he volage. U u U e a U U max U m Fg. 3-3: me-doma behavor of buck coverer durg eady ae operao A alerave road he coderao of average value: he curre doe o chage over he perod, f he averaged ducor volage u zero,.e. u, becaue from follow va egrao over a wchg perod : u 3. u d u. 3. Meh equao: u u U 3.

18 3 DC-DC Coverer 8 Average value aoary codo: u u U U 3.3 he average value of he wchg volage gve a u U u d e DU 3.4 From h, follow u U DU 3.5 or U D 3.6 U he average of he curre reul a d e d e e D 3.7 h lead fally o U D 3.8 U h cocluo gve u a equvale crcu of he averaged value D U DU U Fg. 3-4: Saoary averaged model of he buck coverer

19 3 DC-DC Coverer 9 he curre of he buck coverer ever coa; ead vare he form of ragle. h varao rpple of he curre reul a U D D U max m e a 3.9 he curre rpple maxmal wh duy cycle D. 5 herefore: U U max f 3. 4D D max max.5 Fg. 3-5: Curre rpple veru duy cycle D he curre rpple ca be flueced by he moohg ducor or he wchg perod, or hrough he wchg frequecy f /, repecvely. ypcal wchg frequece le he rage of oly ome Hz up o khz, depedg o he applcao feld ad power rag. Wh hgh wchg frequece or mall volage, uually MOSFE are employed ead of IGB. A mpora meaure for aeg he curre flucuao bede he peak-o-peak value he RMS coe of he devao whch wll be deoed by, I d. 3.

20 3 DC-DC Coverer From he above equao, we ca e.g, deerme he addoal loe caued by he curre flucuao a reor whch reul a RI P R 3.3 he rao of peak ad RMS value of a ragular hape, depede of he parcular form, alway gve by 3, reulg I 3 4D D 3 max 3.4 I 3 Fg. 3-6: RMS value of he curre flucuao 3.. Buck Coverer wh Capacor for Smoohg of he Oupu Volage S I U u C C u C u Fg. 3-7: Buck coverer wh a capacor o moohe he oupu volage If he oupu volage u cao be codered coa by elf, a capacor hould be employed for moohg he volage. he load modeled a a coa curre k, he capacor curre I 3.5

21 3 DC-DC Coverer C I. 3.6 I eady ae, he averaged curre mu be zero,, reulg he momeary volage value ca be calculaed va u C I 3.7 d I C d. 3.8 C C C I parcular, he volage rpple of ere. he egrao ca be ealy doe by geomerc coderao a he curre o be egraed of ragular form cf. Fg. 3-8: u C uc max uc m I d C 3.9 C 4C D D U uc 3.3 8C For mea of mplfcao he volage value rpple uc aumed o be mall compared wh he averaged capacor volage value u o ha he chaged fluece o hape of he curre eglgble. he maxmum volage flucuao reached a D. 5 : U uc max 3.3 3C

22 3 DC-DC Coverer U u u u e a max U u u chaged reaco of he curre due o he volage rpple I / m I u u C u C max u u C u C m Fg. 3-8: me behavor of buck coverer wh a moohg capacor 3..3 Realao of he Crcury echcally, he wch wh he buck coverer realzed by emcoducor uually a bpolar or Feld Effec raor ad a dode. However, h crcu opology wll o allow he full fucoaly of he deal wch a ca provde oly pove curre ad pove volage ee Fg. 3-9 below, compare wh Seco 3.3. h coverer ca operae oly oe quadra. Wh he deal wch uch rerco dd o apply.

23 3 DC-DC Coverer 3 u u u Fg. 3-9: Realzao of buck coverer wh raor ad dode 3..4 Dcouou Coduco Mode of he Buck Coverer he realzao of he deal wch hrough dode ad raor for he buck coverer reul he curre ad power flow oly oe dreco u-drecoal. If he average curre value mall, he he curre flucuao due o pulg may reul eve hg he zero value durg he off-me. he he curre expred ce he dode cao coduc he curre he revere dreco. he curre rema zero ul he raor wched o aga. A a reul, he curre flow how a dcouy durg perod a cf. Fg. 3-, Fg. 3-. h pheomeo called dcouou coduco mode DCM. he ormal operao codered up o ow called couou coduco mode CCM. he dcouou coduco mode chage alo he volage relaohp. A he dervao of he pu-oupu volage rao aumed couou curre flow, h relao here o loger vald: U e D 3.3 U e a For h cae, he relao bewee pu ad oupu volage ha o be calculaed eparaely.

24 3 DC-DC Coverer 4 U u U e a a U U U Fg. 3-: Dcouou coduco mode buck coverer u U U U u u U U U U Durg e Durg a Durg Fg. 3-: Equvale crcu dagram of buck coverer a varou wchg age durg dcouou mode of coduco a raor o-me: Coduco me of he dode: Neher he raor, or he dode are coduco durg e, a, a e D a D e a

25 3 DC-DC Coverer 5 he couou coduco mode characerzed by he codo: max D D 3.33 wh U 4 max 3.34 If h codo o fulflled, we have he cae of dcouou coduco mode. he goal ow o calculae he volage rao for he DCM cae. For ha purpoe, he curre flucuao aalyzed: Rg edge: e D U U U U 3.35 Fallg edge: a D U U ' 3.36 Solve for ' D : D U U D U U U U D 3.37 Calculao of he averaged curre durg DCM: max max ' a e U U D U U U D U U D D U U U U D D D 3.38 We ca ow olve he equao for he volage rao whch deped o he duy rao a well a o he averaged load curre: max max D D U U 3.39 Pleae oe ha cae of couou coduco mode, he volage rao 3.6 depede of he load curre. Fg. 3- how he volage rao for dcouou ad couou coduco mode veru load curre. I couou coduco mode, he oupu volage depede of he load curre ad deped oly o he duy cycle. Durg dcouou coduco mode, he oupu volage deped rogly o he load curre.

26 3 DC-DC Coverer 6 Fg. 3-: oad curve for he buck coverer wh CCM ad DCM operao area Uforuaely, he DCM corol characerc o ha mple a CCM whch make he corol deg more dffcul. However, he dcouou coduco mode offer eve a advaage: I DCM, he dode curre expre aurally a he ed of he fallg lope,.e. he dode o hard ured-off. ha avod or mmze he o-called revere recovery curre of he dode whch a mall curre peak he revere dreco ha remove he charge carrer ou of he juco. A he revere recovery curre a reao of he wchg loe, hee loe are mmzed DCM. However, he raor ur-off commuao ll a hard commuao Boudary Coduco Mode of he Buck Coverer he coverer ca alo be operaed uch a way ha he boom po of he curre hape ouche he zero le. I ha cae he average curre ju half of he peak curre whch alo equal o he curre rpple, ee Fg. 3-3, 3.4 ha operao mode called boudary coduco mode BCM. I Fg. 3- BCM characerzed by he red curve. Whe operag h mode, he wchg frequecy cao be kep coa. Accordg o 3.9, he frequecy wll deped o he averaged load curre,

27 3 DC-DC Coverer 7 f D D U U U U 3.4 U he advaage of he boudary coduco mode - mlar o DCM - ha he dode o hard ured-off, a he commuao occur ju whe he curre aurally expred. We peak of zero-curre wchg, ZCS. h welcome o reduce wchg loe. For he deal of corol deg ee Seco 6.5. U u U e a max m Fg. 3-3: Boudary coduco mode buck coverer

28 3 DC-DC Coverer 8 3. Boo Coverer 3.. Prcple of Operao S U u U Aumg coa volage: Fg. 3-4: Schemac dagram of he boo coverer u U, u U. 3.4 u U U a e max U U U m Fg. 3-5: me behavor of he boo coverer eady ae codo

29 3 DC-DC Coverer 9 Pleae oe ha he me erval e ad a are here aged dfferely o he wch poo compared wh he buck coverer. ha wll be uderood whe lookg a he realzao of he wch, ee Seco Defo of he duy cycle: I eady ae operao, hold e D 3.43 U D 3.44 U U DU D D U max m e Boo Coverer wh Capacor for Smoohg of he Oupu Volage S I U u C C u u C Fg. 3-6: Boo coverer wh a capacor o mooh he oupu volage Smoohg of he oupu volage wh capacor. Aume coa load curre, I eady ae, due o, hold he volage flucuao reul a: C I 3.46 I 3.47 I I D D D uc uc max uc m e 3.48 C C C

30 3 DC-DC Coverer 3 he volage flucuao codered mall wh repec o averaged volage o ha he effec o he behavor of he curre ca be egleced. u U a e max U u U Chaged behavor of curre due o volage rpple m I u u C u C max u u C m I C Fg. 3-7: me behavor of he boo coverer wh moohg capacor 3..3 Realzao of he Crcury Fg. 3-8 how he mplemeao of he boo coverer cog of a raor ad a dode. Alo h crcu opology ca oly provde pove volage ad curre. However, compare Seco 3.3.

31 3 DC-DC Coverer 3 U u U Fg. 3-8: Implemeao of he boo coverer wh raor ad dode 3..4 Dcouou Coduco Mode of he Boo Coverer u U U e a a Fg. 3-9: Dcouou coduco mode of he boo coverer

32 3 DC-DC Coverer 3 U u U U u U U u U U Durg e Durg a Durg a Fg. 3-: Equvale crcu dagram of he boo coverer a varou wchg age wh dcouou coduco mode he formula are derved mlar o he cae of buck coverer. Durg DCM, alo he dode of he boo coverer ured-off ofly, reulg low wchg loe Boudary Coduco Mode of he Boo Coverer he boudary coduco mode ca be appled a well a for he buck coverer, compare Seco h mode ca be employed o mmze he dode wchg loe alo here.

33 3 DC-DC Coverer B-Drecoal DC-DC Coverer he prevou elemeary crcu ca provde curre ad volage oly wh oly oe polary, reulg a u-drecoal power flow from he pu o he oupu. o revere he power flow by reverg eher he curre or he volage, he followg ehaceme ca be codered Coverer for Boh Curre Polare u u Fg. 3-: DC-DC coverer for boh curre polare wo-quadra coverer, realzao wh IGB ad dode he orgal buck coverer Fg. 3-9 ow equpped wh wo addoal raor ad wo addoal dode order o allow curre flow boh dreco. Pleae ee alo Seco.. he raor are drve complemeary: If coducg, mu be blocked ad vce vera. he opology of Fg. 3- or Fg. 3- omeme referred o a oem-poleopology. Depedg o he vewpo whch of volage u or u codered a pu or oupu, or depedg upo he dreco of power flow, he coverer behave lke a buck or a boo coverer. he problem wh he dcouou coduco mode doe o occur here. he polary of he volage, however, ll o reverble h crcu opology. hu, he coverer operae wo of he four poble curre-volage quadra. So he coverer may be called wo-quadra coverer.

34 3 DC-DC Coverer 34 u D D u Fg. 3-: DC-DC coverer for boh curre polare wo quadra coverer, realzao wh MOSFE ad dode eher bul- body dode or addoal exeral dode he crcu how Fg. 3- wh wo MOSFE eve ued, f o revere flow of curre ad power requred. Ulke IGB, MOSFE are capable of revere coduco whle howg ohmc characerc whou ay forward hrehold volage ha ypcal of dode or IGB. A a reul, he curre wll ake pah o hrough he dode bu wh oly a mall volage drop hrough he raor. ha why uch crcu are employed eve udrecoal applcao order o mmze loe, parcularly low-volage applcao. Depe of he MOSFE revere coduco, he a-parallel dode cao be removed. O he oe had, hey are uremovable ed o he MOSFE he emcoducor rucure a body dode. O he oher had, a lea D eceary o eure a free-wheelg pah cae of wchg delay durg he commuao ad durg hudow of he crcu. Someme ha approach referred o a ychroou recfcao hough he word oud rage he coex of DC-DC coverer. he ource of ha wordg are dode recfer, where he dode are paralleled wh MOSFE ha are fred ychroouly durg he ormal coduco erval of he dode parallel.

35 3 DC-DC Coverer Coverer for Boh Volage Polare u u Fg. 3-3: DC-DC coverer for boh volage polare aymmercal half-brdge Aumpo:, u 3.49 Swchg fuco { ; ; } 3.5 u u u u u Alo h coverer gover ou of 4 poble quadra. However, hee are dffere oe compared o he coverer of Seco he ame wo-quadra coverer could be ued a well, bu herefore o very pecfc. he word aymmercal half brde alo commo.

36 3 DC-DC Coverer Four-Quadra Coverer he four-quadra coverer 4QC ca provde boh curre ad volage polare a he oupu de. I Seco wll be dcued more deal. u u Fg. 3-4: Four-Quadra Coverer

37 4 Commuao 37 4 Commuao So far, wchover were dealzed, parcular wa aumed ha curre ad volage ca be wched aaeouly whou ay delay. ha o very realc. Eve f dealzed wchg behavor of wchg devce aumed, parac ducace ad capacace of he elecrcal coeco lead o a chaged wchg behavor. If hee ue would o be codered, he power elecroc devce may be damaged by exceedg he allowed rag. he udy of commuao problem carred ou ug he buck coverer a example. I h age of coderao, raor ad dode are ll aumed a dealzed wchg devce; however, he parac ducace D, are ow cluded he pah of dode ad raor. ypcally, hee ducace are he order of few H. I he equvale crcu how Fg. 4- already clear ha a curre he raor cao be wched off uddely, ay aemp o do o would reul a very hgh volage peak, exceedg he allowed volage rag, ad fally he damage of he raor. Addoal crcu roduced o proec he compoe from uch deruco are called a ubber crcu. I U D D u Fg. 4-: Buck coverer wh parac ducace ad capacace 4. Subber Crcu wh Zeer Dode A meaure o lm he exce volage acro he raor mply a Z-dode. I he followg, boh wch-o commuao ad wch-off commuao are examed. Durg a commuao, he curre hrough he oupu ducor of he buck coverer aumed o be approxmaely coa. I he equvale crcu herefore a coa curre I aumed. herefore he oupu ducor o how he pcure above. he wch-o commuao doe o caue ay rk o he compoe. I aumed ha he raor goe aaeouly o deal coducg mode from he blockg mode. he lope of ha ramp deermed by he drvg volage, called commuao volage, whch he pu volage U, ad he oal parac ducace, ee Fg. 4-3 ad Fg k D

38 4 Commuao 38 Durg he wch-off commuao, he volage acro he raor lmed by he Z-dode. Alhough alo he commuao bewee raor ad Z-dode could be peced more deal, hould be aumed ha he Z-dode ake over he raor curre whou delay. Z I U D D u Fg. 4-: Buck coverer wh parac ducace. Subber crcu for he raor wh Z-dode I U Z I U D D u U Z D D u Durg wch-o commuao Durg wch-off commuao Fg. 4-3: Equvale crcu dagram for he commuao

39 4 Commuao 39 D I U k UZ U k I Z u U U D k e U k ka U UZ D k ke a Fg. 4-4: Commuao characerc durg wch-o ad wch-off wh Z-Dode a ubber oal ducace of he commuao crcu: k 4. D Durao of he wch-o ad wch-off commuao: Ik ke 4. U I k ka 4.3 UZ U A Fg. 4-4 how, he Z-dode eured regular behavor durg wch-off. he Zeer volage, however, mu be choe hgher ha he pu volage. Oherwe, he curre of he Z-dode would o be ured off. O he oher had, he Zeer volage mu be maller ha he maxmum raor blockg volage, leadg o he deg rerco

40 4 Commuao 4 U U Z U 4.4 CE max If he goal ha wch-o ad wch-off ake he ame amou of me, volage hould be choe a, he Zeer ke ka U Z U 4.5 h requre a allowed raor blockg volage of wce of he pu volage U a mmum. I pracce, U Z would ed o be choe maller. I ca alo be ee from Fg. 4-4 ha wch volage u devae form he dealzed hape a a reul of he commuao. h error ca be decrbed erm of volage-me area. Durg he wch-o commuao, he devao he error of he wch- commuao 4.6 e keu I k 4.7 D U UZ I D a ka k So, he average volage error oe wchg perod reul a f 4.8 e a u I Wh repec o he average value calculao we ca coclude ha he reduco of oupu volage caued by commuao ca be repreeed by a equvale ohmc reace. k k k R f 4.9 Moreover, alo he averaged pu curre chaged due o he commuao. Compared o deal wchg, he devao of he pu curre reul a kei kai f k fi 4. U U U Z I he cae U Z U h devao zero. Sce he volage-me area ha he ame phycal dmeo a a magec flux, he ame ymbol ued. I fac, h quay correpod o he magec flux of he parac ducace.

41 4 Commuao 4 R k DU U Fg. 4-5: Seady ae averaged model of he buck coverer oupu crcu wh a equvale reace repreeg he commuao No loe occur durg he wch-o commuao a log a he modelg doe o h level where devce-depede loe are ll o coderd. Durg he he wch-off commuao, however, he loe occur Z-Dode. he oal lo work W V UZ k UZ Ia ka I 4. U U Z h reul o averaged lo power ha due o he ubber crcu of P V UZ k I f 4. UZ U 4. RCD Subber Crcu R C C D I U D D u D Fg. 4-6: Buck coverer wh parac ducace ad RCD ubber crcu

42 4 Commuao 4 I C I C I U D D u U C u U C D D u a Durg ke b Durg ka c Durg ka Fg. 4-7: Equvale crcu dagram for he varou commuao age D I U k C I Qa Q a u U U D k e U k I C a ka ke ka a I D C k Fg. 4-8: Commuao characerc durg ur-o ad ur-off wh he RCD ubber he wch-o commuao o dffere from he prevou ubber vara of Seco 4.. he error of he volage-me area aga 4.3 e keu I k

43 4 Commuao 43 Whe urg he raor off, he raor curre aumed o commuae quckly o he pah D ad C. I he followg, wo phae have o be dguhed. A log a he volage u pove, he dode D blocked. h uao how by he equvale crcu dagram Fg. 4-7 b. he coa curre I ow charge he capacor C, a a reul he capacor volage creae learly ramp-we ad u drop accordgly. Sce he curre coa durg ha phae, here would o be ay volage drop acro. he me ake for volage u o h zero U C ka 4.4 I I h phae, comparg wh a deal wch, a volage-me error of CU a ka U 4.5 I reul. Afer he volage u ha chaged g, he dode D become coducg ad he equvale crcu dagram of Fg. 4-7 c apple. h reul a reoa crcu comprg of he capacor C ad he commuao ducace k D ha beg o ocllae wh he agular frequecy gve by k 4.6 k C he dode D, however, errup he ocllao a he curre C chage he g, wh whch he wch-off commuao compleed. h uao reached afer oe quarer of he ocllao perod, ka kc k I he reoa crcu, he rao of he ocllao amplude of ducor volage ad capacor curre deermed by he characerc mpedace k / C. he oupu volage u deermed from oal ducve volage acro k hrough he ducve volage dvder o ha he mmum value of volage u durg he ocllao ur ou o be u m D k D I I. 4.8 C C k k ogeher wh he form facor of he uodal ocllao of h phae a /, volage-me error reul

44 4 Commuao 44 D ka a I u m 4.9 he volage error, averaged over he ere wchg perod, he k a a e f I CU I u 4.. he reor R o drecly volved he commuao. he reace ca be choe relavely hgh for oly repoble o dcharge he capacor C durg he followg perod, whe he raor ured o aga o ha dcharged for he ex ur-off commuao. h way, he ored eergy of he capacor dpaed ad corbue o he loe. o deerme he ored eergy of he capacor, fr he charge curre deermed by egrag he curre, C I U C C I U C I I Q Q Q k k ka ka a a a 4. he capacor volage a he ed of wch-off commuao herefore: C I U Q C u k a C max 4. Ad he eergy C I U C E k C 4.3 reulg fally he averaged power loe of h ubber crcu a C I U Cf E P k C v 4.4. he requred rag of he raor blockg volage reul alo drecly form he from maxmum capacor volage a C I U Q C u u k a C CE max max. 4.5.

45 4 Commuao Packagg echology he parac ducace he commuao crcu are o oly deermed by he power elecroc compoe bu o a large exe by he paal coruco of he crcury whole. he geomerc area paed by he commuao meh of parcular mporace. he commuao meh defed a ha meh where he curre durg a commuao chagg. h meh area hould be a mall a poble ce he area correpod drecly o he parac ducace he crcu. og ercoeco bewee he eleme lead o hgh parac ducace whch hould herefore be avoded or mmzed. If he log ercoeco from he upply DC volage are evable, he a addoal capacor cloe o he wchg compoe recommeded a how he Fg. 4-9 below o ha h capacor by-pae he hgh frequecy curre compoe. I U C D D u Commuao meh Fg. 4-9: Commuao meh oday packagg echology make ue of mul-layer coruco. I he rage of mall power rag up o ome kw, ha doe wh mul-layer pred crcu board PCB. For hgh power, he layer are old copper or alumum plae, eparaed by olag fol. O oe had, he arrow layer how a raher low ducace compared wh gle wre. O he oher had, he layer provde a addoal capacace ha hghly welcome order o eable a commuao pah for he hgh-frequecy compoe of he commuao curre, compare Fg Fg. 4- how a ypcal coruco of coverer module cog of wo IGB ad a-parallel dode wh plaar layer a ued a a b-drecoal DC-DC coverer or a oe of he phae of a hree-phae verer. Cu-ou he meal layer ad fol eable he eceary elecrc coeco.

46 4 Commuao 46 Iolao U U Cu or Al + _ IGB module Hea k Fg. 4-: ow-ducace deg, here of a b-drecoal DC-DC coverer

47 5 Dyamc Averagg 47 5 Dyamc Averagg he pule frequecy compoe of curre ad volage are caued by he wchg operao of power elecroc yem ad are prcple evable. he coderao of hee curre ad volage flucuao a mpora ue of he power elecroc deg. Such vegao uually aume he wug- eady ae. However, alo mpora o aalyze he dyamc behavor of he power elecroc yem ade he eady ae, parcularly whe come o deg a coroller. For uch ak, omeme cumberome ad edou o coder he pule frequecy compoe of curre ad volage. he coroller hall o md hee pule frequecy compoe ad hall o ry o compeae for hee flucuao a hey are herely due o he power elecroc covero ad cao be avoded. So, for he aaly of he dyamc ad he followg corol deg derable o abrac form he dealed wchg behavor ad he pule frequecy flucuao. ha accomplhed by he dyamc averagg approach. For ha purpoe, a local me average of a varable x wh a me wdow of legh hall be defed a x / / x d 5. I hould be oed ha we make o parcular aumpo of he varable x ; may vary me arbrarly. If he quay, however, coa ocllao of frequece f k k /, hee frequecy compoe are cacelled ou by he averagg. If we elec frequecy compoe ad hgher harmoc are he elmaed by averagg., he pule 5. Average Modelg of a Reace By applyg he local averagg modelg for u R, 5. follow mmedaely he ame relaohp alo for averaged quae, u R Average Modelg of Iducor ad Capacor u 5.4 Averagg:

48 5 Dyamc Averagg 48 / / / d u d 5.5 / / / u 5.6 he dfferece of he curre o he lef-had de ca ow be wre a me dervave of he average value 3 : d d / / d u 5.7 he equece of applyg averagg ad dffereao hu erchageable. h become alo clear from abrac po of vew, f oe realze ha boh he dffereao ad averagg are lear me-vara operao, whch are allowed o be erchaged he equece of compuao.e. he operaor are commuave. A a reul, he dffereal equao of he ducor alo exacly vald for he averaged quae, whou applyg ay approxmao or eglgece: u 5.8 We ca, of coure, llurae h equao a a equvale crcu wh he curre ad volage u : u Fg. 5-: Averagg model equvale crcu repreeao of a ducor Smlarly for he capacor, Cu 5.9 he ame equao reul alo a a averaged model: Cu 5. 3 Pleae oe he cha rule for egral: d dx g x g x dg x dg x f z dz f g x f g x dx dx

49 5 Dyamc Averagg Averagg Model of ear me-ivara Dffereal Equao he procedure decrbed here ca alo be appled o all ype of lear me-vara dffereal equao, for example, o he -order marx dffereal equao, whch popularly ued yem ad corol heory, x A x Bu y Cx Du 5. Applyg he local averagg procedure, follow drecly whou ay eglgece, x A x Bu y C x D u Average Modelg of a Swch Wh he help of he above relaohp, we codered arbrary lear me-vara ework, bu o ye power elecroc crcu, a wche are o covered o far. We udy he four-pole wche wh he equao below, u u If he wchg fuco codered depede of curre ad volage, hee equao are lear wh repec o volage ad curre, however, he equao are o mevara. he average model o be aalyzed more deal. I order o geeralze he problem, he vegao hould ar wh a produc lke he averagg gve re o z z x y 5.5 / / x y d 5.6 Uder cera codo, he rgh-had de of he equao may be approxmaed by he produc of he mea value of x ad y,.e. z xy x y 5.7

50 5 Dyamc Averagg 5 he eceary codo hould be derved by a error aaly. e u ar wh he erm y x x y y x y x y x x y x y x y x 5.8 Applyg averagg over he varable, yeld / / d y x x y y x y x y x 5.9 / / d y x x z y x 5. he egral o he rgh-had de ca be emaed a follow: / / / / / / max d y x x d y x x z y x 5. Fally, max / / y x x z y x 5. h mea he accuracy of he approxmao y x z beer wh a mall varao of varable x. If he flucuao zero, whch equvale o. co X x, he he relaohp eve exacly rue. I he above approxmao he erm x ad y ca be erchaged. ha mea ha approxmao of he averaged produc by a produc of average applcable, f a lea oe of he cluded varable how a uffcely mall flucuao. If ha codo volaed,.e. f boh varable how coderable large flucuao, he approxmao 5.7 wll be poor. Example: he wchg volage of a buck coverer, u u 5.3 ca be approxmaed wh uffce accuracy by u u 5.4

51 5 Dyamc Averagg 5 f he feedg pu volage u coderably mooh,.e. hould exhb oly mall flucuao. However, he wchg fuco ha a wde rage of varao. So he above codo mu be afed by mall flucuao u. Alo, he pu curre of he buck coverer, 5.5 ca alo be approxmaed by he average value wh he aumpo ha here oly mall varao oupu curre : 5.6 he above codo ca be acheved by a mooheg ducor buck coverer. Excepo: he buck coverer operaed wh a mall ducor e.g. boudary coduco mode, whch aocaed wh large curre flucuao. If he coverer feed he ohmc reace drecly whou mooheg ducor he alo he above codo o me: he he oupu curre of pulag form ad herefore ha alo a large varao. Alo h cae he above approxmao o permed. However f he codo are fulflled he we ca ue averagg approxmao. h ca alo be expreed by a equvale crcu. Praccally, he equvale crcu for he curre value ad he chagg average value do o dffer. Neverhele, we mu pay aeo o he meag of he quae. he aaeou value model of wch fuco ake oly he dcree value ad, wherea he averagg model of wchg fuco whch equal o duy cycle ca ake ay value bewee ad. u u u Fg. 5-: Equvale crcu of he wch for aaeou value u u u Fg. 5-3: Equvale crcu of he wch for averaged quae

52 5 Dyamc Averagg Sae-Space Averagg of Varable-Srucure Dffereal Equao A ework wh wchg eleme doe o lead o a lear me-vara dffereal equao yem, bu ofe o a yem havg hee propere pecewe. e u ay ha he yem ca be decrbed wh a perod pecewe by a e of dffereal equao u B x A x k k for k k 5.7 he value of he ae vecor a he ed of oe me erval x k alo he al value of he ex ub-erval. Wh he ub-erval, he ae ae-pace marce k k B, A of wchg yem ca be ypcally aumed coa. However, mplfe he furher ep f we aume eve marce,b A varyg me. hu, u B x A x 5.8 Now we apply aga averagg o h equao: u B x A x x 5.9 O he lef-had de of he equao allowed o erchage he dffereao ad averagg compuao equece for he reao explaed above. Aga we have o fd ou, wheher he average of produc erm o he rgh-had de may be approxmaed by a produc of average. he compuao flow follow he cheme of prevou eco, oly he marx vecor produc are codered: / / A x A x A d x x 5.3 Or wrg h compoe-we, / / j j j j j j j j j d x x A x A x A 5.3 A upper boud ca be foud a follow: max / / / / / / / / x x d A d x x A d x x A x A x A j j j j j j j j j j j j j j j j j 5.3 So ha bee proved ha he error deped aga o he flucuao bad of he ae varable x j. he maller he varao wh he averagg erval, he more prece he

53 5 Dyamc Averagg 53 approxmao. he ame apple o he ecod par of he equao, he erm of dffereal equao where he varao of he pu varable u j ha mpora. Iofar he orgal dffereal equao allowed o be approxmaed by he dffereal equao of averaged quae: x A x B u 5.33 We ow aume ha he rucure of a yem wche back ad forh bewee oly wo cofgurao A, B ada, B. he cofgurao A, B aumed o be acve durg he duy cycle d / ad A, B are acve durg he remag poro d of he oal wchg perod. he he averaged yem marce are, d A d d A 5.34 A d B d d B 5.35 B Smlar a he orgal pu vecor u, alo he duy cycle d ow a pu o he yem. However, d appear dfferely a mulplcao wh he ae vecor, x A d x B d u 5.36 Now, he yem o loger lear. I order o apply he mehod of yem heory ad corol egeerg for lear yem, we apply a learzao a he operao po. e x, u d be a operag po,.e. hold,, Furhermore, A d x B d u 5.37 x x x u u u d d d 5.38 are defed a devao from he operag po. h reul : d d x x Bd d u u x x A 5.39 For mall devao, h ca be approxmaed by fr-order erm, 5.4 A B x d d x A d d x x B d d u u d d A d x B d A d u d x B d d u d

54 5 Dyamc Averagg 54 where A d d B d d A A B B Α B 5.4 A fal reul: A x B u d x A d x B d u 5.4 h repreeao called dyamc mall-gal model. 5.6 Dyamc Averagg Model of Buck Coverer I he couou mode of operao, he dyamc behavor of he buck coverer ca be gve by he followg dffereal equao, d u u 5.43 Depedg o he vewpo wchg fuco v. raor duy cycle or d ued. o emphaze ha he duy cycle ow varyg me, he lower cae leer are ued, d 5.44 Alhough he averagg model ha abraced from he curre rpple, he curre rpple ca be calculaed from averaged value aferward, d d u 5.45 U u u R Fg. 5-4: Buck coverer wh reve load

55 5 Dyamc Averagg 55 d U du u R Fg. 5-5: Dyamc averagg model of he buck coverer wh reve load couou coduco mode Fg. 5-6: rae repoe of he buck coverer wh d = coa Comparo of he averagg model wh he aaeou value model I couou coduco mode, he averaged value of he curre follow he -order dffereal equao: d u u 5.46 I dcouou coduco mode, he coverer decrbe by he algebrac equao d u u 5.47 u he rao from couou o dcouou mode ad vce vera correpod o he rucural chage of he yem. No oly he parameer are chagg, bu he order of dffereal equao chage from o! he rao wll ake place o reachg he boudary codo, d d u max d, u 5.48 Such a behavor ca be expreed by a ae graph.

56 5 Dyamc Averagg 56 u d u dyamcal yem d, max u d, max u u u d d u u u ac yem Fg. 5-7: Hybrd ae graph a a dyamc averagg model of buck coverer wh rucural chage bewee CCM ad DCM

57 6 Corol of he Buck Coverer 57 6 Corol of he Buck Coverer Power elecroc crcu uually eed corol order o accomplh he dedcaed ak. he ak of a buck coverer uually o provde a oupu volage u o upply power o a load. Ideally he oupu volage hould be maaed coa, regardle of a flucuag load curre regardle of a flucuag pu volage u Source u u C u oad Fg. 6-: Buck coverer wh moohg capacor, feedg ource, ad load hee varou feable corol ad regulao cheme are dcued below. 6. Feedforward Corol wh Coa Duy Cycle For very mple applcao may uffce o adju he duy cycle of he coverer o a eceary eady ae value for dered volage value u : u D 6. U here o eed o meaure ay quay ad he duy cycle deermed va he referece volage value ad he aumed omal pu volage U. h kd of ope-loop corol very eay o mpleme. However he dadvaage are a follow: here o compeao poble regardg a chagg pu volage value. If he pu volage devae from he omal value, reul a oupu volage error. No compeao of here volage drop a due o commuao, by volage drop acro he emcoducor or ohmc eral reace. No fluece o he dyamc behavor.

58 6 Corol of he Buck Coverer 58 Source u u C u oad u corol PWM Fg. 6-: Ope-loop corol of he buck coverer wh coa duy cycle o udy he dyamc behavor of he coverer we ar wh average modelg. he average equao of ducor ad capacor are he aplace doma. he averagg model of wch u u 6. Cu C 6.3 u u 6.4 cao be readly raformed o he aplace doma. he produc me doma would reul o a covoluo operao frequecy doma. h way hould o be ued. Average modelg of he pule wdh modulao aume ha he demaded referece value really he ame a he mea value of he realzed wchg commad: 6.5 h way, he pule wdh modulao doe o occur he averaged model. he eady ae duy cycle ued a feedforward corol ga, ee Fg. 6-3: G f 6.6 U hee above relaohp lead o he followg dyamc yem corol block dagram:

59 6 Corol of he Buck Coverer 59 u u G f u u C u C G l Coroller Pla Fg. 6-3: Dyamc average model of buck coverer wh mple corol he rafer behavor of h corol yem ca ow be uderood from he block dagram or from he coderao of mpedace. u u u C C 6.7 u u C 6.8 C C u C u u 6.9 u 6. C C he ope-loop rafer fuco ha pole a j wh, C. he yem herefore mgh ed o ocllao due o he C compoe. he behavour of he load o ye bee codered here. I cae he load aumed o be modelled a a equvale ohmc reace R,.e. Gl u u 6. R he pla rafer fuco G p chage accordgly a follow: C u u u u 6. R u C u u R 6.3

60 6 Corol of he Buck Coverer 6 u G u u C R p 6.4 or a ormalzed form, u Gp u d 6.5 wh dampg facor ud characerc mpedace, Z d, R Z C. Fg. 6-4: rae repoe of he buck coverer oupu volage, corol wh a coa duy cycle he ope-loop corol hu how more or le rog ocllag behavor. he dampg maly due o he load. However alo eral reace o ye codered he model of he ducor may alo corbue o dampg. Ay excao of he yem, a chage of he e po u, he upply volage u, ad alo chage of he load, epecally he ur-o ad ur-off of he load, wll exce ocllao. Moreover, eve he dampg facor wll chage wh chagg load.

61 6 Corol of he Buck Coverer 6 h mple ype of corol may be employed cae of lle dyamc requreme o he yem. For more ophcaed requreme, h mple realzao o uffce. 6. Sgle-oop Volage Corol I order o avod a eady ae corol error of he oupu volage, eceary o meaure he volage ad o compeae for devao. Addoal goal o mprove he dyamc behavour,.e. o mprove he elg me ad durbace behavour by employg a cloedloop corol. Fg. 6-5 how he approach of a gle-loop volage corol. he eady ae feedforward block from he prevou eco may be reaed. he volage coroller he ha o compeae oly he re of he devao. I order o meaure he volage, a eor eceary. I low-volage applcao, power elecroc ad corol elecroc are ofe realzed o-olaed o he ame groud o ha mple aalog crcury e.g. a reve volage dvder or operaoal amplfer may be mple ad uffce oluo order o meaurg he volage. However, wh volage of ome V ad above, uually eceary o provde a galvac olao bewee power elecroc ad corol elecroc. h requre olaed meaureme raducer, whch are much more complex ad expeve. I h cae, alo he commad gal o he drver mu be raferred wh eparaed poeal ee Seco.. Source u u C C u oad Galvac olao f eceary u u Volage corol G c PWM G f Feedforward Fg. 6-5: Sgle-loop volage corol

62 6 Corol of he Buck Coverer 6 G f u u u G c u u C u C G l coroller pla Fg. 6-6: Average modellg of buck coverer wh volage corol ad feed-forward corol If he upply volage u o coa, he corol loop exhb a o-lear behavour due o he mulplcao. By meaureme of he upply volage, a exac learzao poble a how he Fg. 6-7 ee alo Seco 8.3. However, ha eed a furher meaureme equpme alo for he upply volage u. A a reul, he coroller oupu ow drecly he demaded volage coverer ead of he duy cycle. I h repreeao, he eady-ae feedforward ga mply G f 6.6 G f u u u u u G c u u u C G l u C u Coroller Pla Fg. 6-7: Average modellg of buck coverer wh volage corol ad feed-forward corol a well a exac learzao of he fluece of he pu volage he fluece of u he corol loop ow compeaed; he mulpler ad dvder ca be omed he followg mplfed model, ee Fg By cloer examao, h argume oly approxmaely rue, ce he coroller ofe operae a a ampled dgal yem ha ca repod o chage of he pu volage oly wh a me delay. So he fluece of a chagg pu volage compeaed oly mperfecly. he remag error of uch compeao, bu alo oher durbg fluece reulg from he commuao or volage drop accordg o umodelled eral reace hould be

63 6 Corol of he Buck Coverer 63 ummarzed a overall volage error u d a how Fg I word of corol egeerg, h o be codered a durbace pu. G f u u d u G c u u u C u C G l u u G f u u d u G c u u u u G p u Fg. 6-8: Smplfed average model of he buck coverer wh volage coroller ad feedforward aco he behavour of he load o be acceped aga a a ohmc reace: G l 6.7 R 6.. P-Coroller A P-ype coroller hould be red a volage coroller: u G K u G c K 6.8 c u u G f u u u u h reul a rafer behavor a follow, d u d 6.9 u u K u u u u d d d 6.

64 6 Corol of he Buck Coverer 64 u u K K d u d 6. u u u d d 6. where he overall cloed-loop rafer fuco c c c d K d K K d K 6.3 ad he durbace rafer fuco c c c d d K 6.4 he yem behavor aga ha of a ocllag yem of d order, where characerc agular frequecy ad dampg facor are ow K c, K d d c 6.5 I hould be oed ha he abolue dampg he real par of he pole ad he decay me coa doe o dffer from uregulaed cae, d d c c 6.6 Icreag he coroller ga of he P-Coroller decreae he eady ae corol error, K d 6.7, bu doe o mprove he dyamc of he cloed loop: he abolue dampg o chaged, merely he ocllao frequecy creaed hrough h corol.

65 6 Corol of he Buck Coverer 65 Im Pole of he cloedloop yem Pole of he opeloop yem c K d d c c Re Fg. 6-9: Dplaceme of he ope-loop pole by a P-coroller 6.. PI-Coroller A ecod ral ca be doe wh a PI-ype coroller G c K K 6.8 A egral coroller aco ha, of coure, he advaage o avod eady-ae corol error. However, oe hould o expec o mprove he dyamc behavor compared wh a P-ype coroller. Ulke a P-coroller, a PI-coroller corbue alway o a phae lag, reulg reduced ably ad dyamc performace. he dampg provded by he ohmc behavor of he load hu eve more mpora a wh he mple P-coroller. he overall rafer fuco ad durbace rafer fuco would reul : G f Gc Gp, G G c p Gp d G G c p 6.9 K d K 3 K K d K K 6.3

66 6 Corol of he Buck Coverer 66 d d K d 3 K K 6.3 A you ca ee, a deal eady ae behavor guaraeed, becaue, d. 6.3 he maual coroller deg va pole placeme geg challegg becaue of he 3rdorder polyomal. Much clearer he deg frequecy doma wh he help of frequecy repoe curve Bode plo or oe may ue eracve ool e.g. oool from Malab wh whch oe ca hadle mulaeouly he frequecy characerc ad he pole zero locao of he yem. Wh he applcao of uer erave ool, modfcao ca be doe more eracvely. Oe reul how Fg. 6-. he deg wa carred ou, for example, for a dampg value d.5 of he ucorolled yem. Fally, he corol parameer are choe a, K he dagram how boh he locao of he reulg pole of cloed loop relao o he arg poo of he pole ad zero of he ope loop ope-loop pole, blue:, coroller pole ad zero, red:, ad alo he magude ad he phae repoe of he loop rafer fuco j G j G j 6.34 c are lluraed. he deg wa doe o ha all he hree pole of he cloed-loop rafer fuco how abou he ame real par of,335. he abolue dampg, a explaed above, deed maller ha ha of he ope loop d, 5. Furhermore wa red o fd a rade-off bewee uffce dampg ad hgh croover frequecy c. he croover frequecy.e. where hold j le a c.39 p c 6.35 wh a phae marg of h mall phae marg correpod wh a reoace magfcao of 8 db. he badwdh of he rafer fuco up o -3 db drop abou. b.

67 6 Corol of he Buck Coverer 67 3 Roo ocu Edor C 4 Ope-oop Bode Edor C 3 Magude db - - Imag Ax -3-4 G.M.: If Freq: If Sable loop Phae deg Real Ax P.M.: 36.5 deg Freq:.39 rad/ec Frequecy rad/ec Fg. 6-: Roo locu ad Bode plo for buck coverer wh PI-Coroller Bode Dagram Magude db Phae deg / Bode plo of he overall rafer fuco j / - Frequecy rad/ec 6..3 PID-Coroller A coroller wh dffereal aco would creae he phae reulg beer dampg, epecally o mprove he dampg f he yem elf oly weakly damped. Coroller wh D-aco, however, are rarely appled power elecroc. he reao le he

68 6 Corol of he Buck Coverer 68 wchg operao of power elecroc yem. A dffereao par he coroller would amplfy he hgh frequecy compoe, parcularly alo he wchg harmoc he corol varable, whch o dered from he vewpo of corollg he dyamc average. he flerg of wchg harmoc would roduce delay, whch coflc wh he ee of dffereao. For hee reao PID-coroller are o codered here. 6.3 Volage Corol wh Ier Curre Corol oop he corol of he oupu volage ug a eral curre corol loop a very powerful echque ad uually ouperform he gle-loop volage corol. However, h requre a addoal raducer o meaure he ducor curre. Now, he volage coroller geerae a referece value for he curre coroller. he exac learzao by meaurg he upply volage ee Seco 7.3 hould alo be ued here. Source u u C u oad u u volage coroller G cu curre coroller G c u u PWM feedforward pah G f u Fg. 6-: Buck coverer wh cacaded corol G f u u d G l u u G cu G c u u u u C C u Fg. 6-: Dyamc average model of buck coverer wh cacaded corol

69 6 Corol of he Buck Coverer Ier Curre Corol Fr, he er curre corol o be deged. For mplfcao he oupu volage aumed o vary oly lowly wh repec o he me cale of he curre corol loop. he volage feedback va u ake ogeher wh he coverer volage error u d a well a he feedforward aco u a a ew equvale durbace u u u 6.36 d d u wh codered a equvale durbace pu o he curre corol loop. he reul he followg mplfed block dagram of curre corol loop. u d G c u Fg. 6-3: Smplfed rucure of he curre corol loop Sce he corol cheme hould eure a zero eady ae corol error, a PI-coroller employed here, G c K K 6.37 he coroller ga K o dmeole, bu ha a u V/A. he overall cloed-loop rafer fuco reul a K K K 6.38 he characerc value of h d-order yem are K, d K 6.39 hee characerc value ca be e freely by approprae coroller parameer. If ad d are gve a pecfcao of he dered curre corol behavor, he he coroller parameer are e accordg o he coroller deg rule d, K d 6.4