DC-DC Converters - Dynamic Model Design and Experimental Verification

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1 D-D erter - Dym Mdel De d Expermetl Verft Jh, Bet 005 k t publt tt fr publhed er APA: Jh, B D-D erter - Dym Mdel De d Expermetl Verft Deprtmet f Idutrl Eletrl Eeer d Autmt, ud Ittute f Tehly Geerl rht pyrht d mrl rht fr the publt mde eble the publ prtl re reted by the uthr d/r ther pyrht wer d t dt f e publt tht uer ree d bde by the lel requremet ted wth thee rht. Uer my dwld d prt e py f y publt frm the publ prtl fr the purpe f prte tudy r reerh. Yu my t further dtrbute the mterl r ue t fr y prft-mk tty r mmerl Yu my freely dtrbute the U detfy the publt the publ prtl Tke dw ply If yu belee tht th dumet brehe pyrht plee tt u prd detl, d we wll reme e t the wrk mmedtely d ette yur lm. UNDUNI VES I TY PO Bx7 00 ud

2 D-D erter - Dym Mdel De d Expermetl Verft Bet Jh Dtrl Dertt Idutrl Autmt Deprtmet f Idutrl Eletrl Eeer d Autmt

3 Deprtmet f Idutrl Eletrl Eeer d Autmt ud Uerty P.O. Bx 8 SE- 00 UND SWEDEN ISBN ODEN:UTEDX/TEIE-04/-94/004 Bet Jh, 004 Prted Swede by Med-Tryk ud Uerty ud, 004

4 Abtrt T bt hh perfrme trl f d-d erter, d mdel f the erter eeded. The ld uully ffet the dym d e wy t tke th t dert t rerd the ld prt f the erter. The ld fte the mt rble prt f th ytem. If the ld urret d the utput lte re meured there re d pblte t bt d mdel f the ld -le. Adpte trl the be ppled t mpre the trl. I pek urret-mde trl, the utput lte d the dutr urret re meured d utlzed fr trl. I the uthr lette the, lyt mdel were dered fr the e where the ld urret l meured d utlzed fr trl. The trl-t-utput trfer fut, the utput mpede, d the ud ueptblty were dered fr the buk, bt, d buk-bt erter perted tuu dut mde the e f rete ld. The ue f ld urret be ee hedul the e where the ld retr. G hedul be dered pel e f dpte trl. The mjrty f the reult the lette the were ldted by mpr the frequey repe predted by the lyt mdel d wthed lre-l mult mdel. I th the, ddtl reult re preeted fr the buk erter. Expermetl reult bted by me f etwrk lyzer erfy the dered trl-t-utput trfer fut d the ud ueptblty but t the utput mpede t lw frequee. I the expermetl buk erter there re try rete the dutr, trtr, d dde but thee try rete were t dered the lette the. A ew trfer fut fr the utput mpede dered where thee try rete re dered d t d reemet wth the expermetl reult l t lw frequee. If the urret t the utput ptr meured ddt t the utput lte d the dutr urret, the ld urret be lulted

5 the dfferee betwee the dutr d ptr urret the e f the buk erter. Hee, the meuremet f the ld urret be repled by meuremet f the ptr urret. If th pblty utlzed d the ptr urret meured by me f urret trfrmer, lwfrequey ree trdued the frequey repe rd t expermetl reult. The re fr th ree due t the hh-pflter hrtert f the urret trfrmer. A ew lyt mdel dered whh predt the ree.

6 Akwledemet Frt f ll, I wuld lke t thk my uperr Prfer Gutf Ol d Dr Mtz eell fr ude dur my wrk d fr mmet the murpt f th the. Thk l t Dr Per Krl fr red d mmet the murpt. Furthermre, I wuld lke t thk Jh Fält, Er Mrwe Sytem AB, d Mkel Appelber, Er Pwer Mdule AB, fr du but prtl prblem wth d-d erter d fr help me wth meuremet. I wuld l lke t thk the tff t the Deprtmet f Tehly, Uerty llee f Klmr, d the tff t the Deprtmet f Idutrl Eletrl Eeer d Autmt IEA, ud Uerty, wh he helped me my wy. Th wrk h bee flly upprted by The Kwlede Fudt, Uerty llee f Klmr, d Er Mrwe Sytem AB. Flly, I wuld lke t thk my fmly fr upprt me. Klmr, Nember 004 Bet Jh

7

8 Ntt Frequetly ued l d prmeter re preeted wth me d derpt the lt belw. Sl d prmeter tht ly pper e f the hpter re t luded the lt. The me f l t f lwer-e letter. Hweer, exept re mde fr the ubrpt prt f the me. The me f the l re l ued t dete ther d lue but ptl letter re ued th e. Hweer, the letter the ubrpt prt re t hed. The d lue me re t luded the lt. Nme Derpt pte f the ptr d Duty yle d ' d' d urret referee p ptr urret e Exterl rmp ued fr lpe mpet j urret jeted t the utput te Idutr urret ld d urret k f Iput lte feedfrwrd ee Set.3 k r Output lte feedfrwrd ee Set.3 Idute f the dutr m Slpe f the dutr urret whle the trtr m Slpe f the dutr urret whle the trtr ff m elte lpe f the exterl rmp, m M e M M e Slpe f the exterl rmp ete f the ld retr Aere try rete, D D' ff

9 Equlet Sere ete ES f the ptr d ete the dde whle t dut G f dutr-urret er ete the dutr ff Stry rete whle the trtr ff, ff d Stry rete whle the trtr, t t ete the trtr whle t T Swth perd Iput lte Output lte ref Vlte referee δ trl l f the trtr drer ω Hlf the wth frequey, ω π T Sl re fte dded t d prt d prt. The prt deted by u the ht-ymbl ^. A meted erler, the d prt deted by u ptl letter. T expltly dete tht l fut f tme, the rble t dded t the me,.e. l met. The mpled er f tuu-tme l deted by repl the rble t wth. The ple trfrm f tuu-tme l deted by repl the rble t wth. The Z-trfrm f dretetme l deted by repl the rble wth z. The tt t me extet he uh tht t mptble wth the e ued by dley 99.

10 tet HAPTE INTODUTION.... BAKGOUND.... MOTIVATION FO THE WOK ITEATUE EVIEW MAIN ONTIBUTIONS OUTINE OF THE THESIS PUBIATIONS...7 HAPTE MODES FO BUK ONVETES A SUMMAY OF THE IENTIATE THESIS...9. HAPTE SUVEY...9. STATE-SPAE AVEAGING UENT-MODE ONTO A NOVE MODE IMPOVED MODES APPOXIMATIONS OF OBTAINED EXPESSIONS USING OAD UENT FO ONTO SUMMAY AND ONUDING EMAKS...84 HAPTE 3 MODE INUDING STAY ESISTANES INTODUTION STATE-SPAE AVEAGING UENT-MODE ONTO USING OAD UENT FO ONTO SUMMAY AND ONUDING EMAKS APPENDIX...5 HAPTE 4 EXPEIMENTA VEIFIATION EXPEIMENTA ONVETE...7 x

11 4. ESUTS SUMMAY AND ONUDING EMAKS HAPTE 5 UENT-TANSFOME INFUENE INTODUTION UENT TANSFOME MODE INUDING A APAITO-UENT SENSO EXPEIMENTA VEIFIATION SUMMAY AND ONUDING EMAKS APPENDIX HAPTE 6 SUMMAY AND FUTUE WOK ESUTS IMPIATIONS OF THE ESUTS FUTUE WOK HAPTE 7 EFEENES... 8 x

12 hpter Itrdut Th the ered wth the mdel d trl f d-d erter wth urret-mde trl. Spel fu pled u ld urret meuremet fr trl. I the lette the, Jh 003, umber f mdel fr buk, bt d buk-bt erter were deelped d lyzed detl. ter expermet erfed the ury f me f the mdel d reeled ert tee ther mdel. Therefre the expermet he mtted further refemet f the mdel. Th the emphze the buk erter mdel. I th hpter, the bkrud f the prblem derbed, the mtt fr the wrk preeted d the trbut f the the re utled.. Bkrud D-D erter Fure. hw d-d erter blk bx. It ert d put lte, t, t d utput lte, t, wth mtude ther th the put lte Erk d Mkm, 000, Set.. The erter fte lude e r eerl trtr rder t trl the utput lte, u the trl l δ t. It derble tht the er be mde wth lw le the erter. Therefre, the trtr t perted t ler terl. Ited, t perted wth d the trl l bry. Whle the trtr, the lte r t lw whh me tht the pwer l the trtr lw. Whle the trtr ff, the urret thruh t lw

13 hpter. Itrdut ld t t D-d pwer erter d t δ t Fure.: A d-d erter. d the pwer l l lw. T bt lw le, retr re ded the erter. ptr d dutr re ued ted e delly they he le. The eletrl mpet be mbed d eted t eh ther dfferet wy, lled tple, eh e h dfferet prperte. The buk, bt, d buk-bt erter re three b erter tple. The buk erter h utput lte tht lwer th the put lte. The bt erter h utput lte tht hher th the put lte tedy tte. The buk-bt erter ble t he utput lte mtude tht hher r lwer th the put lte mtude. Fure. hw the buk erter wth tw trller. Here t umed tht ll mpet re del. The ld t f retr wth rete. The erter h lw-p utput flter t f dutr wth dute d ptr wth pte. Whle the trtr, the dutr urret, t, ree e the put lte hher th the utput lte the buk erter. A the trtr tured ff, the dde mut trt t dut e the dutr urret t tp flw tteuly. The lte r the dde zer whe t dut d the dutr urret wll deree. Fure.3 hw the wefrm f the trl l d the dutr urret. The erter uully deed tht the mtude f the rpple the utput lte beme mll. If the rpple ft, the dutr urret ree d deree lerly hw Fure.3. The lte r the dde equl

14 hpter. Itrdut 3 t ld t t p t t V ref Vlte trller t urret trller Drer δ t Fure.: The buk erter wth urret trller d lte trller. δ t 0 T dt T t t Fure.3: The wefrm f the trl l d the dutr urret. t t the put lte r equl t zer. The utput flter f the erter flter th lte wefrm d the mtude f the rpple the utput lte deped the flter de. If the dutr urret beme zer befre the trtr tured, t wll rem t zer utl the trtr tured e the dde ly dut e dret. If the erter perted tht the dutr urret zer dur me prt f the wth perd,

15 4 hpter. Itrdut t d t be perted dtuu dut mde. Otherwe, t perted tuu dut mde. The wth perd, T, f the erter determed by the trl l δ t, hw Fure.3. I th fure, the wth perd held tt. The ere utput lte trlled by h the wdth f the pule. I Fure.3, the fll ede trlled.e. whe the trtr huld tur ff. The duty yle, d t, rel lue the terl 0 t d t equl t the rt f the wdth f pule t the wth perd. The trl l δ t be bted frm d t by u pule wdth mdultr. The duty yle be ee drete-tme l. Stte-Spe Aer The erter t tme-rt ytem whle the trtr. Whle the trtr ff the erter t ther tme-rt ytem d f the dutr urret rehe zer, the erter t yet ther tme-rt ytem. If the trtr trlled derbed preuly, the erter be derbed wth betwee dfferet tme-rt ytem dur the wth perd. equetly, the erter be mdeled tme-rt ytem. Stte-pe er Mddlebrk d uk, 976 e methd t pprxmte th tme-rt ytem wth ler tuu-tme tme-rt ytem. Th methd ue the ttepe derpt f eh tme-rt ytem trt pt. Thee tte-pe derpt re the ered wth repet t ther durt the wth perd. The ered mdel ler d tme-rt d h the duty yle, d t, the trl l ted f δ t. Th mdel flly lerzed t the pert pt t bt mll-l mdel. Frm the mdel we wll extrt three mjr trfer fut: The trl-t-utput trfer fut derbe hw he the trl l ffet the utput lte. The utput mpede derbe hw he the ld urret ffet the utput lte. The ud ueptblty derbe hw he the put lte ffet the utput lte.

16 hpter. Itrdut 5 urret-mde trl Fure. hw the buk erter trlled by tw trl lp. The dutr urret fed bk t the urret trller the er lp d the utput lte fed bk t the lte trller the uter lp. Th trl methd lled urret-mde trl. The me urret trller ued ted f urret mdultr th the, ee Set.3. Aume tht the uter lp t preet. The ytem the led lp ytem e the dutr urret fed bk. If the uter lp dded, ew led lp bted. The trl l frm the uter lp t the referee l fr the urret trller. The three trfer fut meted be wll eerl be dfferet fr the ew led lp ytem. The urret trller trl the dutr urret. Th be mde dfferet wy. Oe wy t trl the pek lue f the dutr urret eh wth perd. dley 99 d T d Mddlebrk 995 he preeted tw mdel fr urret-mde trl. The lte trller tully exluded. The m dfferee betwee the tw mdel the mdel f the urret lp. Al-Mthfr d Hmmd 999 fud tht l the ud ueptblte predted by the tw mdel re dfferet. The ere lue f the dutr urret be trlled ted f the pek lue. Th trl methd uully lled ere urret-mde trl. The utput lte fed bk t the lte trller tht the utput lte kept er the lte referee l V ref ee Fure.. The lte trller trl the referee l f the urret trller, t. A lterte t let the lte trller trl the duty yle dretly. Th me tht the meuremet f the dutr urret d the urret trller re t eeded. Th trl methd lled lte-mde trl.. Mtt fr the Wrk My pet mut be dered the e where erter t be deed. Oe uh pet keep the utput lte the pefed lte terl. Here re me exmple f he tht deree the rt f the utput lte:

17 6 hpter. Itrdut he the prperte f me f the mpet the erter, e.. ree the pte f the ptr. he the erter tply. he t mre ded trller. Iree the umber f l tht re meured d ued by the trller. Eh e f thee he h e r eerl ddte uh : Hher t. Ireed weht d lume. wer relblty. wer effey ee P, Te, d u 999. Therefre, the he r he tht re mt utble deped t lre extet the erter peft t hd. erter be mpred better mpet re deelped d mre kwlede beme lble. Th mtte reerh the re f mpet, erter tple d trller fr exmple. T bt hh perfrme trl f ytem, d mdel f the ytem eeded. A mdel f ytem be dered by u the lw f phy d/r by u meuremet f the ytem,.e. ytem detft ju, 999. Whe the ytem hed dur the tme t ue, t dte t pply ytem detft tht be ued -le fr updt the mdel. The djuted mdel the ued t djut the prmeter f the trller, whh the eee f dpte trl Åtröm d Wttemrk, 995. A dpte trller perfrm better th dpte trller, whh mut be deed fr the wrt e. Oe dffulty wth dpte trl mk the detft uh tht the mdel djut uffetly ft dur ytem he wthut mk the detft ete t meuremet e. If the djutmet lw, the trller mut be deed t be utu. Th me tht there wll be ft mpremet mpred t -dpte trller. The djutmet, eerl, be mde fter f the umber f prmeter t be etmted ytem fewer. Oe wy t hee th t fx the prmeter whe lue re kw wth ret pre d ry ly lhtly. Ather wy t meure lrer umber f l the pre d the re fr th expled fllw. A wy t deree the umber f prmeter t be etmted t mply detfy prt f the ytem. T

18 hpter. Itrdut 7 detfy th ubytem, t put d utput l mut be meured. If lrer umber f l the pre re meured, t my be pble t dde the pre t dfferet mller prt. Nte tht the tme fr the mpl d mputt re t dered th du. The ld uully ffet the dym d e wy t tke th t dert t rerd the ld prt f the erter. If meuremet f the ld urret, ld t, ee Fure. trdued, t pble t der the ld e prt t be detfed. The utput lte the rerded the put l d the ld urret the utput l f th prt. If dpte trl t be trdued, utble frt tep my be t ly detfy the ld. Ofte th the mt rble prt f the erter. Th frt tep my be uffet t bt trller tht meet the perfrme peft. A ed tep, detft f the ret f the erter my further mpre trl. mputtl tme e pre t py. Th ed tep my be mre expee th ther lut t mpre the perfrme f the led lp ytem. Th du mtte the reerh detft f the ld. A meted be, the utput lte d the ld urret huld be meured t bt ft ld detft. There re eerl pper tht uet tht the ld urret huld be meured d utlzed fr trl f the erter d they hw wht prperte re bted. Tw f thee pper re meted here. I thee tw pper, the utput lte d the dutr urret re umed t be meured bede the ld urret. edl d Skl 986 hw tht the tret the utput lte due t tep he the ld be muh redued. They ll the ue f the meured ld urret feedfrwrd. Fr deft f feedfrwrd, ee Åtröm d Hälud 995, Set 7.3. edl d Skl l hw tht the trl-t-utput trfer fut de t he whe th feedfrwrd trdued. The d f the trl-t-utput trfer fut rmlly deped the ld. Ht d Brje 993 ue the meured ld urret t mke the trl-t-utput trfer fut rt fr dfferet ld t d fr the bt erter. Ht d Brje thu hw tht the trl-t-utput trfer fut he whe the ue f meured ld urret trdued. The trl Ht d Brje ue tur ut t be extly the me the e edl d Skl prpe fr the bt erter. T ummrze, edl d Skl hw tht the trl-t-utput trfer fut de t he whe the ue f meured ld urret trdued whle Ht d Brje hw tht t de he. It thu eem

19 8 hpter. Itrdut t be trdt. Se the utput lte d the ld urret re umed t be meured the tw pper, the ly my be eted t detft f the ld me wy. Therefre, t mtted t ette th pble et d trdt befre the wrk wth detft f the ld trt..3 terture eew The umber f referee w lmted the preu et t mke the derpt ler. I th et, exteded reew f wrk relted t th the mde. Sme mdel fr utrlled d-d erter re frt reewed. Mdel fr urret-mde trl re the dered. Next, me wrk relted t the effet ld he the dym f the erter re reewed. Flly, eerl wrk where the meured ld urret ued fr trl re reewed. Mdel fr D-D erter wthut trller There re eerl methd tht be ued t bt ler tuutme tme-rt mdel f d-d erter. Stte-pe er Mddlebrk d uk 976 d Mthell 988, rut er Weter d Mddlebrk 973 d Vrper 990, d the urretjeted pprh lque d Frd 977 d Klk, edl d Skl 99 re me f them. If thee methd re ppled t erter tht perted tuu dut mde, the reult mdel re urte bth t lw d hh frequee. The frequee re here relted t the terl d t hlf the wth frequey,.e. hh frequee re lwer th hlf the wth frequey. Hweer, f thee methd re ppled t erter tht perted dtuu dut mde, the reult mdel re ly urte t lw frequee. Su et l. 00 preet mdfed methd t bt mdel fr erter perted dtuu dut mde tht re urte bth t lw d hh frequee. Tymerk 99, 994 ue tme-ry ytem thery t dere mdel fr the frequey fut d thee mdel re lmed t be ext fr ll frequee,.e. l hher th hlf the wth frequey. The trl-tutput frequey fut dered the e where the erter perted tuu dut mde r dtuu dut mde. The mdel re mre mplted th the preuly meted mdel.

20 hpter. Itrdut 9 Whe the trl-t-utput frequey fut f erter ued t de trller, the frequey terl d t hlf the wth frequey the mt teret. The preuly meted mdel my therefre be uffet whe de trller. Mdel fr urret-mde trl A lre umber f tuu-tme mdel fr urret-mde trl he bee preeted dur the yer. Sme f thee mdel re teded t be urte l t hh frequee. The mdel preeted by dley 99, T d Mddlebrk 995, d Tymerk d 993 re deed t be urte frm d t hlf the wth frequey. Tymerk d 993 preet tte-pe mdel whle the dley d T mdel ue the PWM wth mdel Vrper, 990. Tymerk 994 dere mdel fr the frequey fut frm trl l t utput lte d t lmed t be ext fr ll frequee. The m dfferee betwee the dley d T mdel the mdel f the urret lp. The urret lp be meured by u dtl mdultr h d ee, 984 r l tehque T d Mddlebrk, 995 but the reult re t the me. The urret lp rd t the dley mdel predt the meuremet reult bted by ue f dtl mdultr whle the T mdel predt the reult bted whe the l tehque ued. The l tehque preferred rd t T d Mddlebrk 995. Hweer, d K 999 lm tht the l tehque t rret fr meur the urret lp d tht dtl mdultr huld be ued ted. A dtl mdultr dd ft dely t the pule wdth mdultr. Myer d K 00 preet mdel fr the urret lp tht lude the effet f dely the pule wdth mdultr. If the dely et t zer th mdel, the predted urret lp the me the e predted by the dley mdel. emrk The dley mdel lude feedfrwrd. dley 99 preet pprxmte expre fr the urret lp fr the buk erter d the feedfrwrd re red the dert f th expre. Th mke the pprxmt errr ft t lw frequee f the erter perte le t dtuu dut mde dley, 990b. If the feedfrwrd re t red, the fllw pprxmte expre bted:

21 0 hpter. Itrdut T 0.5. ' H e. m D T The term -0.5 dded mpred t the expre preeted by dley. Hee,. t tht muh mre mplted. The pprxmt errr f. ft, ee whe the erter perte le t dtuu dut mde. Nte tht the etty t try rete the dutr, trtr, d dde fr the three mjr trfer fut re dereed whe the urret lp led e the dutr urret trlled. Hweer, th t the e fr the urret lp whh h the me etty the pe lp trfer fut. Effet f d erter Dym The ld uully ffet the dym f the erter. I Set., the ld w dered t be prt f the erter. A lterte t exlude the ld frm the erter d ue the Thee equlet rut mdel f the utput f the erter. Oe wy t eure reble tblty mr t defe mpede peft f the ld. Fe et l. 00 du frbdde re fr the mpede rt Z Z where Z the utput mpede f the erter wthut ld d Z the mpede f the ld. Sudhff et l. 000 preet frbdde re tht mller th ll preuly preeted frbdde re d, hee, the ertee redued. h et l. 00 preet methd t de the uter lte trller urret-mde trl the e where the ld dym re ukw. The de mde uh tht the trlled ulded erter h utble tblty mr d lw utput mpede f rete hrtert. I the de, tt urret k tully umed t be eted t the utput t be ble t bt dered pert pt. Hee, the erter ly ulded the mll-l ee. Vr d 99 try t dere trl tehque uh tht the utput mpede zer. Hweer, there hrtm ther prf f the tblty f the er led lp ytem. The lp f th ytem fllw Equt 9:

22 hpter. Itrdut Z ' ek G.. Z ' Z '' ek They lm tht the ytem tble e the mtude f G le th uty. Hweer, e Z ' ek d Z '' ek re mplex umber, G jω my be reter th uty fr me ω. The trl tehque utlze pte feedbk f the dutr urret. Sme f the wrk preeted the ext ubet l lyze hw the ld ffet the dym f the erter. U d urret fr trl Seerl wrk where the meured ld urret ued fr trl re reewed th ubet. A meted Set., edl d Skl 986 hw hw the meured ld urret huld be ued t redue the tret the utput lte whe tep he the ld ur. They ly der the e where urret-mde trl ued. Shem d Mthell 989 lyze the prped ue f ld urret further the e f the buk erter. Hweer, there hrtm ther wrk ee Set.7. Klk, edl d Skl 99, Set.4 preet trl tehque whh tre t equlze the utput d put pwer f the erter. T d th, the ld urret, utput lte, put urret, d put lte mut be meured. Sut et l. 003 dere eerl mll-l mdel fr the e where the meured ld urret ued fr trl. They ue utermted twprt mdel f the erter trt pt. The utput prt f th mdel t f the Thee equlet rut. A meted Set., Ht d Brje 993 ue the meured ld urret t mdfy the urret-mde trl fr the bt erter uh tht the d beme depedet f the ld. Th trl tehque exteded by Ht d Bryeh 994 uh tht the prmeter the uter lte trller re dpted t dfferet ld urret d utput lte leel. Vr d 989 exted lte mde trl f the buk erter by lud er lp where the meured ld urret utlzed uh tht the utput mpede zer. They d t ette the tblty f the led ld urret lp. A prelmry ett mde by the uthr f th the hw tht the mtude f the ld urret lp ek

23 hpter. Itrdut theretlly lwer th uty f the ld pte rete. Fr ther type f ld the led ld urret lp my t be tble. Oe ddte wth the trl tehque tht the derte f the ld urret mut be ued. Th t eery the trl tehque prped by edl d Skl 986 whh ue urret-mde trl trt pt. Kemru et l. 00 l exted lte mde trl f the buk erter. The trl tehque utlze the ld urret whh meured by me f urret trfrmer. A urret trfrmer h hh-p-flter hrtert d by h urret trfrmer wth lw metz dute, the rer frequey f the urret er hh. I th e, the utput f the urret er pprxmtely prprtl t the derte f the ld urret. Hee, the trl tehque mlr t the e preeted by Vr d 989. Hweer, the rer frequey f the urret er d the mplft f the utput l f the urret er re he uh tht the utput lte qured errr mmzed. rr et l. 995 de fuzzy-l trller fr the buk erter. The trller utlze meuremet f the utput lte d the ld urret. Zh et l. 004 preet el trl heme fr lte reultr mdule where meuremet f the utput lte d the ld urret re utlzed fr trl. Vlte reultr mdule re ued pwer upple fr mrprer. A meted preuly, edl d Skl 986 exted urret-mde trl by utlz l meuremet f the ld urret. Shem d Mthell 989 hw tht, the e f the buk erter, equlet trl be bted f the urret t the utput ptr meured ted f the dutr d ld urret. Hee, ly the utput lte d the ptr urret mut be meured. Id et l. 998 preet el trl de pprh fr ere urret-mde trlled buk erter. Furthermre, they prpe tht the ptr urret meured ted f the dutr urret e eerl dte be ed. They hw tht the utput mpede ftly redued d tht the dym f the erter rem lmt uhed wth ld rt. They l hw tht the effet f try rete the dutr the led-lp trfer fut mr. They hw tht the try rete the dutr trdue zer the utput mpede trfer fut t lw frequey. Id d M 999 l utlze meured ptr urret ted f dutr urret the trller.

24 hpter. Itrdut 3.4 M trbut Th the tut f the lette the, Jh 003. Therefre, the m trbut f the lette the re frt preeted d the the m trbut f th the re preeted. The prtl mprte f me f the trbut dued. The ette The The m trbut f the lette the re preeted th ubet. Sme f the prperte tht be bted u meured ld urret fr trl re lyzed. The ly ly mde fr the e where urretmde trl ued. A urte mdel ued the e where the ld ler retr.. The ly frm tht lw utput mpede be bted.. The ly hw tht the e where the ld urret ure,.e. the ld urret depedet f the utput lte, the fllw prperte re bted: The ue f meured ld urret fr trl feedfrwrd. The trl-t-utput trfer fut de t he whe th feedfrwrd trdued. If t hd hed, t wuld t he bee feedfrwrd. 3. The ly hw tht the e where the ld ler retr, the fllw prperte re bted: The trl-t-utput trfer fut he whe the meured ld urret trdued fr trl. The erter beme utble whe the meured ld urret trdued fr trl. The trl-t-utput trfer fut be lmt rt fr dfferet ler rete ld f the meured ld urret ued fr trl. Th epelly the e fr the buk erter. The ue f meured ld urret fr trl t feedfrwrd. It ted be ee hedul, whh be dered pel e f dpte trl Åtröm d Wttemrk, 995, hpter 9.

25 4 hpter. Itrdut It l hw tht the tw publhed mdel fr urret-mde trl, dley 99 d T d Mddlebrk 995, e urte expre fr the trl-t-utput trfer fut d the utput mpede but t fr the ud ueptblty. A el mdel fr the ud ueptblty preeted d t ued t mpre the dley d T mdel. The el mdel me e urte t lw frequee but the mpremet re mde uh wy tht th hrtm t trferred t the mpred mdel. The mpred mdel re urte. Aurte tuu-tme expre fr the trl-t-utput trfer fut, the utput mpede, d the ud ueptblty re dered fr d-d erter tht meet the fllw peft: The erter tply buk, bt r buk-bt. The erter perted tuu dut mde. urret-mde trl wth tt wth frequey d pekurret mmd ued. The ld ler retr. Th The The m trbut f th the re preeted th ubet. After the lette the w publhed, expermet were mde t erfy me f the preeted mdel. Thee expermet erfed the ury f me f the mdel d reeled me tee me ther mdel. Therefre the expermet he mtted further refemet f the mdel. edl d Skl 986 prpe trl lw fr the buk, bt d bukbt erter. I the lette the, thee trl lw were dered mprtt pel e. The prped trl lw fr the buk erter ery mple t mplemet d my therefre be the mt teret t ue prte. Therefre, th the emphz mdel fr the buk erter. The m trbut f th the re:. If the frequey fut predted by the trfer fut dered the lette the re mpred wth expermetl reult bted by me f etwrk lyzer, t fud tht there lre dfferee e e. The trfer fut fr the utput mpede de t predt the expermetl reult t lw frequee the e where the meured ld urret utlzed fr trl. The re fr the dfferee tht the try rete the dutr, trtr, d dde re t

26 hpter. Itrdut 5 dered the lette the. A ew trfer fut fr the utput mpede dered where thee try rete re dered.. Expermetl reult re preeted fr the e where the ld retr d thee reult erfy the frequey fut predted by the trlt-utput trfer fut, the trfer fut fr the ud ueptblty, d the ew trfer fut fr the utput mpede. 3. edl d Skl 986 uet tht the ld urret t meured dretly. Ited, the urret t the utput ptr, p t, meured d the ld urret the be lulted the dfferee betwee the dutr urret d ptr urret ee Fure.. I the expermet meted tem d be, the ptr urret meured by me f urret hut. Th lut h eerl drwbk uh reed mtude f the rpple the utput lte d dereed effey. edl d Skl 986 uet tht urret trfrmer ued t meure the ptr urret. Hweer, the urret trfrmer h hh-p-flter hrtert d thee trdue ree the buk erter t lw frequey. A ew mdel dered fr the e where urret trfrmer ued t meure the ptr urret. Expermetl reult re preeted l fr th e d they erfy the frequey fut predted by the ew mdel. Prtl Imprte f the New Mdel The lette the d th the ree the udertd f the prperte f the erter d hw t dere mdel. Furthermre, the fllw ew trfer fut re preeted:. The el mdel fr ud ueptblty fr urret-mde trl.. The three mjr trfer fut the e where the ld urret meured wth urret hut d utlzed fr trl. 3. The me trfer fut tem exept tht the ld urret meured dretly by me f urret trfrmer. It erfed bth by mult d expermet tht thee trfer fut re urte the expermetl erft ly mde fr the buk erter.

27 6 hpter. Itrdut The el mdel tem be mly f dem teret e the errr the predt f the ud ueptblty the dley d T mdel fte re mll d ly pper t hh frequee. The trfer fut tem be my be mre teret t ue prte e the he f the urret thruh the ld retr re uted fr. Th t mde Shem d Mthell 989 ee Set.7. Hweer, e f the buk erter the urret thruh the ld retr mly ffet the prperte f the erter t lw frequee. Th me tht the de f the uter lte trller de t he muh f the ew mdel ued ted e the f the trller uully ery hh t lw frequee. The mt teret trfer fut t ue prte re prbbly the e tem 3 be whh be ued the e where the ptr urret meured wth urret trfrmer. The re fr th tht the ree he the phe ure ftly. The ree frequey ree f the rer frequey f the urret trfrmer ree ee Set 5.4. Therefre, the hher the rer frequey f the urret trfrmer, the mre the de f the lte trller ffeted. The utput mpede trfer fut tem d 3 be re mpred by der the try rete the dutr, trtr, d dde the mdel. Th mpre the predt f the utput mpede t lw frequee. Th mpremet my t be mprtt prte e the lte trller uully mke the utput mpede lw t lw frequee. A exept my be the de f lte reultr mdule where t utble t bt utput mpede whh tt fr ll frequee Zh et l., 004. The deft f utput mpede th e the mpede the Thee equlet rut. Oe my k f prte t mprtt t ue mdel whh lude hh frequey exte. I the e f the buk erter t my be mprtt e the rer frequey be deed t be hh, e.. e teth f the wth frequey. Ard t the mll-l mdel fr the bt d buk-bt erter, thee erter re t mmum-phe ytem. Therefre, the rer frequey mut uully be deed t be rther lw. equetly, the hh-frequey prperte f the bt d buk-bt erter my t be mprtt t der prte whe de the lte trller.

28 hpter. Itrdut 7.5 Outle f the The I hpter, bref reew f the lette the preeted. The ew trfer fut fr the utput mpede dered hpter 3. The frequey fut predted by me f the trfer fut preeted hpter d hpter 3 re mpred wth expermetl reult hpter 4. The e where urret trfrmer ued t meure the ptr urret lyzed hpter 5. A ummry d future wrk re preeted hpter 6..6 Publt The uthr h publhed the fllw feree pper:. Jh, B. d eell, M. 000, Pblte f bt mll-l mdel f D-t-D pwer erter by me f ytem detft, IEEE Itertl Telemmut Eery feree, pp , Phex, Arz, USA, Jh, B. 00, Aly f D-D erter wth urretmde trl d rete ld whe u ld urret meuremet fr trl, IEEE Pwer Eletr Spelt feree, l., pp. 65-7, r, Autrl, Jh, B. 00b, A mpr d mpremet f tw tuu-tme mdel fr urret-mde trl, IEEE Itertl Telemmut Eery feree, pp , Mtrel, d, 00. Pper ether luded the lette the r th the. Pper d 3 t prt f the lette the. Errt fr the three pper re preeted hpter 9 f the lette the.

29

30 hpter Mdel fr Buk erter A Summry f the ette The The ext hpter re prtly bed the reult preeted the lette the by Jh 003. Therefre, t utble t frt mke ummry f the lette the d th mde th hpter. Hweer, e ly the buk erter dered the ext hpter, few reult fr the bt d buk-bt erter re preeted here.. hpter Surey I Set., the pert f the buk erter derbed. The methd f tte-pe er expled. The mdel bted by pply th methd t the buk erter preeted d mpred wth reult frm mult f wthed mdel. urret-mde trl expled Set.3. The dley d T mdel re reewed d mpred. The mdel bted by pply thee mdel t the buk erter re preeted d mpred wth mult reult. The reult f the mpr re expled. The el mdel fr the ud ueptblty preeted Set.4. The expre bted by pply th mdel t the buk erter preeted d mpred wth the rrepd e Set.3. I Set.5, the dley d T mdel re mpred by u the reult Set.4. I Set.6, me pprxmt f the mdel fr urret-mde trl preeted the preu et re hwed. 9

31 0 hpter. A Summry f the ette The Sme prperte tht be bted whe u ld urret meuremet t trl the erter re lyzed Set.7. The reult f th ly re mpred wth mult reult. A ummry d lud remrk re preeted Set.8.. Stte-Spe Aer The erter be derbed wth betwee dfferet tmert ytem dur eh wth perd d ubequetly tmert ytem. There re eerl methd tht pprxmte th tme-rt ytem wth ler tuu-tme tme-rt ytem. Stte-pe er Mddlebrk d uk, 976 ued here. The utle f th et fllw. The rut d pert f the buk erter re frt preeted. The methd f tte-pe er the expled. The trl-t-utput trfer fut, the utput mpede, d the ud ueptblty be bted by pply th methd t the buk erter d the reult re preeted. Flly, wthed mult mdel f the buk erter preeted. It hw hw the frequey fut f the erter re bted frm th mult mdel. The frequey fut re preeted d mpred wth the three trfer fut fr the buk erter. Opert f the Buk erter The rut d pert f the buk erter re preeted th ubet. Numeru tt re trdued. The mpet f erter re t del d me f thee delte be dered mdel. Oly e -delty dered the lette the. The ptr mdeled del ptr ere wth del retr wth rete. The rete lled the Equlet Sere ete ES f the ptr. Fure. hw the rut f the buk erter where the ES f the ptr luded. The wefrm f the l the rut re hw Fure. d they re bted frm mult. Stedy tte rehed d, therefre, the trl l, δ t, t f pule wth tt wdth. The tme terl where the trl l δ t hh re lled t d the e where δ t lw re lled t ff. The wth perd, T,

32 hpter. A Summry f the ette The tr t t ld t t Drer dde t dde t t ES t t p t t δ t Fure.: The rut f the buk erter. the tme betwee tw uee pte flk f δ t d hee equl t the um f t d t ff. The rt f t t T lled the duty yle r the duty rt d t deted by d t. The duty yle tt tedy tte d equl t D the d lue f d t. Dur t the trtr perte the tte d dur t ff the trtr perte the ff tte. The lte r the dde, dde t, equl t the put lte, t, dur t d equl t zer dur t ff. The put lte held tt t V dur the mult. The dde lte fltered by the - lwp utput flter. The rer frequey f th flter he t be muh lwer th the wth frequey t bt mll mtude f the rpple the utput lte, t. equetly, the utput lte pprxmtely equl t the me lue f the dde lte d lwer th t. The lte r the dutr, t, equl t the dfferee betwee dde t d t. Dur eh tme terl, the lpe f t lmt tt e t lmt tt. The dutr urret equl t the trtr urret, tr t, dur t d equl t the dde urret, dde t, dur t ff. The ptr urret, p t, equl t the dfferee betwee t d the ld urret, ld t. The me lue f p t zer tedy tte. The mtude f the rpple t lrer th the mtude f the rpple the lte r the del ptr, t, due t the ptr ES. Tble. hw the prmeter lue ued the mult. Thee re l ued by dley 99. The wth frequey, f, equl t 50 khz the ere f T.

33 hpter. A Summry f the ette The δ t t dde t t t tr t dde t ld t p t t ES t t t ff t t ff x 0-5 t Fure.: The wefrm f the l tedy tte fr buk erter. The ut f the lte Vlt d the ut f the urret Ampere.

34 hpter. A Summry f the ette The 3 Tble.: The prmeter lue ued the mult f the buk erter. Prmeter Vlue 37.5 µh 400 µf V 4 mω Ω V D T 0 µ The Methd f Stte-Spe Aer I th ubet, the methd f tte-pe er expled. The trl-t-utput trfer fut, the utput mpede, d the ud ueptblty be bted by pply th methd t the erter d the reult re preeted. The erter be derbed wth betwee dfferet tmert ytem d ubequetly tme-rt ytem. The tte-pe derpt f the dfferet tme-rt ytem re ued trt pt the methd f tte-pe er. der the buk erter Fure.. Whle the trtr, the lte r the dde equl t the put lte. The rut Fure.3 therefre be ued mdel f the buk erter dur t. I the fure, urret ure dded. It jet the urret j t t the utput te f the erter. Th urret put l d eeded t determe the utput mpede. Whle the trtr ff, the lte r the dde equl t zer d the rut Fure.4 be ued mdel. Aume tht the tte-pe derpt f the rut Fure.3 d Fure.4 re dx t A x t Bu t dt. y t x t Eu t

35 4 hpter. A Summry f the ette The t j t t t t Fure.3: The rut f the buk erter dur t. t j t t t Fure.4: The rut f the buk erter dur t ff. d dx t A x t B u t dt. y t x t Eu t repetely, where t x t, t.3 t u t, j t.4

36 hpter. A Summry f the ette The 5 y t t..5. d. re tw dfferet ler tme-rt ytem. I tte-pe er, thee tw ytem re frt ered wth repet t ther durt the wth perd: dx t dt y t d t A d t A x t d t B d t B d t d t x t d t E d t E u u t.6 t.6 pprxmt f the tme-rt ytem d ew rble me huld frmlly he bee ued. T lmt the umber f rble me, th t mde. The duty yle, d t, ddtl put l.6. A ew put etr therefre defed: u t u' t..7 d t Th t mde trdtl preett f tte-pe er, where the trl l d t kept eprte frm the dturbe l t d j t. Hweer, ytem thery, ll trl l d dturbe l re put put etr. Se the duty yle be dered t be drete-tme l wth mpl terl T, e t expet the ytem.6 t be ld fr frequee hher th hlf the wth frequey. The ed tep tte-pe er lerzt f the ler tme-rt ytem.6. The det frm pert pt re defed fllw: x t X x t,.8 u' t U' u ' t,.9 y t Y y t..0

37 6 hpter. A Summry f the ette The ptl letter dete the pert-pt d, tedy-tte lue d the ht-ymbl ^ dete perturbt l. The reult f the lerzt lerzed, mll-l ytem: dx t A'x t B'u' t dt y t 'x t E'u' t.. Bede the mdel., d mdel be bted frm.6 by ett x t, u ' t, y t, d d x t dt t zer. If the ple trfrm f. lulted, eerl trfer fut be extrted. By pply tte-pe er t the three treted erter the fllw reult re bted fr the three mjr trfer fut. The trl-t-utput trfer fut, the utput mpede, d the ud ueptblty fr the buk erter re Z ut V,. d,.3 j D..4 T be pred frm trdu ew rble me, the ple trfrm f l deted by the me me the l, e.. { t }, ee f th t frmlly rret tt. The pert-pt lue f d t deted D d D' D..5 Nte tht the demtr the three trfer fut.-.4 re the me. Therefre, the pt f the tw ple re the me fr the three trfer fut. Th l the e fr the zer ued by the ES f utput ptr. The trfer fut fr the utput mpede h

38 hpter. A Summry f the ette The 7 ddtl zer t the r. Th me tht the utput mpede ted t zer the frequey ted t zer. Fr the bt erter, the trl-t-utput trfer fut, the utput mpede, d the ud ueptblty re fud Jh 003, Set.6, Equt The rrepd reult fr the bukbt erter re fud Jh 003, Set.9, Equt Smult Mdel d eult I th ubet, wthed mult mdel f the buk erter preeted. It hw hw the frequey fut f the erter re bted frm th mult mdel. The frequey fut re preeted d mpred wth the three trfer fut fr the buk erter. We multe buk erter by u the ftwre MATAB/SIMUINK lud Pwer Sytem Blket. Fure.5 hw the mplete mult mdel. The buk erter ubytem. It mult mdel hw Fure.6 d t h three put l d three utput l. The put l, j, d delt re the put lte, jeted urret, d trl l, repetely. j multpled by - t bt dret f the jeted urret tht ree wth the e defed Fure.3. The put l delt trl the trtr. A trllble wth emulte the dde. The ere f delt ued t trl th wth e the dde huld dut whe the trtr t dut. T be ble t trt mult, dummy retr luded the mdel. The rete f th retr et t MΩ d t effet the mult reult elble. The utput l, ld, d re meuremet f the utput lte, ld urret, d dutr urret, repetely. The put d utput l f the erter re eted hw Fure.5, t bt the frequey fut f the erter. The put lte,, the um f t d lue, V, d t lue ht. The jeted urret, j, d the duty yle, d, re mplemeted rrepd wy. The d lue f j,.e. Ij, equl t zer ll the mult. Oly e l eertr t the tme tted. The pule wdth mdult PWM mke ue f w-tth l, e.. the e hw Fure.7. The perd f the l equl t T,.e. the wth perd. Whe the l wtth beme reter th the duty yle, d, the S-lth reet. A rely blk ued t eerte reet l

39 8 hpter. A Summry f the ette The V l mtude le ht Ij jht j S Q delt j ld delt Buk erter ld D d!q dht wtth 0.5 Fure.5: The mplete mult mdel. l - m Trtr m Dde emultr dummy ld l - NOT - 3 delt j Fure.6: The buk erter ubytem. tht ether 0 r. A pule eertr et the S-lth. The et l yhrzed wth the wtth l tht the S-lth et eh tme the wtth l e frm t 0. The utput l f the S-lth, delt, pule tr d the wdth f eh pule determed by the duty yle l d. der the utput l f the erter whh repreet the utput lte. Th l h Furer mpet wth frequey equl t the frequey f the l frm the te l eertr. There ext ther Furer mpet the utput lte Erk d Mkm,

40 hpter. A Summry f the ette The wtth, d t x 0-4 delt t x 0-4 Fure.7: The wefrm f the l pule-wdth mdultr. 000, Set 7.. The utput f ler ytem uh.,.3, r.4 ly t f e Furer mpet f the put udl. The frequey f th mpet the me the frequey f the put udl. T be ble t mpre the mult reult wth the lerzed mdel, ly the Furer mpet the utput lte wth frequey equl t the frequey f the l frm the te l eertr dered. A etwrk lyzer l jut der th Furer mpet Erk d Mkm, 000, Set 8.5. T elute frequey fut t pef frequey, the te l eertr et t eerte udl wth th frequey. The pef frequey l et the Furer ly blk. The reult f the Furer ly the mtude d the phe f the mpet. The ly repeted dur the mult d the reult re ewed by u the llpe blk. At the trt f eh mult, the reult f the Furer ly he derbly e the dutr urret

41 30 hpter. A Summry f the ette The 40 0 Phe de; Mtude db Frequey Hz Fure.8: The trl-t-utput trfer fut f buk erter. Sld le: the lyt mdel. X: the mult reult. d the ptr lte re fr frm the fl d lue. The mult tpped whe the he the reult f the Furer ly elble. The frequey fut t pef frequey be expreed t mtude d t phe. The mtude equl t the rt f the mtude f the utput lte, bted frm the Furer ly, t the mtude f the l frm the te l eertr. The phe equl t the phe f the utput lte, bted frm the Furer ly, e the phe f the l frm the te l eertr zer. A mult duted fr eh e f the frequee tht eluted. Th predure repeted three tme e there re three dfferet l eertr. The frequey fut predted by the lerzed mdel f the buk erter re w mpred wth mult reult. The prmeter lue hw Tble. re ued the lerzed mdel d the mult

42 hpter. A Summry f the ette The Phe de; Mtude db Frequey Hz Fure.9: The utput mpede f buk erter. Sld le: the lyt mdel. X: the mult reult. mdel. Fure.8, Fure.9, d Fure.0 hw the Bde plt fr the trl-t-utput trfer fut., utput mpede.3, d ud ueptblty.4, repetely. Smult reult re l hw. The fure hw tht the predted frequey fut ree lely wth the mult reult. Nte tht the frequey xe re lmted t hlf the wth frequey 5 khz. Fr the bt d buk-bt erter the lu the me,.e. the predted frequey fut ree lely wth the mult reult..3 urret-mde trl I th et we der mdel fr erter wth urret-mde trl. The tw mdel preeted by dley 99 d T d Mddlebrk 995 re mpred wth mult reult.

43 3 hpter. A Summry f the ette The 0-0 Phe de; Mtude db Frequey Hz Fure.0: The ud ueptblty f buk erter. Sld le: the lyt mdel. X: the mult reult. urret-mde trl l lled urret prrmmed trl d urret-jeted trl. Derpt f urret-mde trl be fud e.. Klk, edl d Skl 99, hpter 5, Erk d Mkm 000, hpter, d Mthell 988, hpter 6. The utle f th et fllw. The pert f urret-mde trl frt expled. The dley d T mdel re the reewed d mpred. The trl-t-utput trfer fut, the utput mpede, d the ud ueptblty be bted by pply thee tw mdel t the buk erter wth urret-mde trl d the reult re preeted. The mult mdel exteded d mult reult re mpred wth the preeted trfer fut. The reult f the mpr re l expled.

44 hpter. A Summry f the ette The 33 ref t Vlte trller t urret trller δ t erter t t Fure.: urret-mde trl. ref t Vlte trller δ t erter t t Fure.: Vlte-mde trl. Opert f urret-mde trl The pert d mplemett f urret-mde trl re dued th ubet. I urret-mde trl, tw trl lp re ued edl d Skl, 986. See Fure.. The er lp ft d trl the dutr urret, t. The uter lp lwer d trl the utput lte, t. The dutr urret fed bk the urret trller the er lp whle the utput lte fed bk the lte trller the uter lp. The lte trller h the referee l ref t. The lte trller tre t et t equl t ref t by h t trl l, t. Th l ubequetly ued the referee l fr the urret trller. The urret trller m t ett t equl t t ee by h t trl l, δ t, whh the put trl l f the erter. Thu, urret-mde trl pplt f de trl Gdw, Grebe d Sld, 00, Set 0.7.

45 34 hpter. A Summry f the ette The Swtth d pule eertr S Q δ t ref t t t Fure.3: A lte trller lte-mde trl. Swtth d pule eertr e t S Q δ t t t Fure.4: A urret trller urret-mde trl. I the e f urret-mde trl, the trl l f the lte trller l d the trl l f the urret trller dtl bry. I the e f lte-mde trl, ee Fure., the trl l f the lte trller dtl u δ t. There urret trller d the dutr urret de t eed t be meured. A lte trller hw Fure.3. The frt left prt uully lte-errr mplfer d t utput l, t, l. The ed rht prt f the trller pule wdth mdultr mpre th mdultr wth the

46 hpter. A Summry f the ette The 35 rut hw Fure.5. The duty yle, d t, deped lerly the trl l t. Vlte-mde trl l lled duty rt trl. A typl urret trller urret-mde trl mplemeted hw Fure.4. The pek dutr urret trlled d the trl methd therefre lled pek urret-mde trl. Th the mt mm type f urret-mde trl d the wrd pek fte left ut. If the trller Fure.4 d Fure.3 re mpred, the urret trller eem t t ly f mdultr. Klk, edl d Skl 99, hpter 5 ue the me urret mdultr ted f urret trller. Aere urret-mde trl ther type f urret-mde trl Klk, edl d Skl, 99, hpter 5. The frt prt f the urret trller ere urret-mde trl urret-errr mplfer. It my th e t be utble t ll the urret trller urret mdultr e e my der t t f mre th mdultr mpre th urret trller wth the lte trller lte-mde trl. The me urret trller my therefre be ee mre eerl d t ued here. The mdultr ee lre r mll prt f the urret trller. The pert f the urret trller pek urret-mde trl hw Fure.4 wll w be expled. Fr mmet ume tht the w-tth l, e t, t preet. The perd f the l frm the pule eertr equl t T d the l et the S-lth. Eh tme th ur, the trtr tured d the dutr urret, t, trt t ree hw Fure.5. Whe t beme reter th the l t, the S-lth reet d t the deree utl ew et pule eerted. Th the me fut the pule wdth mdultr Fure.5 d Fure.3 exept the dutr urret, t, reple the w-tth l. mpre the wefrm hw Fure.5 d Fure.7. The l t the referee l f the urret trller. The urret trller tre t et t equl t t the ee tht t the pek lue f t tht f teret. I ere urret-mde trl, t the ere lue f t tht f teret. The urret trller pek urret-mde trl ft e t me t et the pek lue f t equl t t dretly. The er led lp ytem pek urret-mde trl therefre be ee urret ure. T be mptble wth the deft mde by dley 99, the wtth l, e t, frm w lled the exterl rmp. The feedbk f t ue tblty Erk d Mkm,

47 36 hpter. A Summry f the ette The δ t 0 T dt T t e t, t t Fure.5: The wefrm f the l urret trller pek urret-mde trl. t 000, Set.. The trl f the dutr urret utble f the tedy-tte duty yle, D, reter th 0.5. It utble the ee tht the duty yle, d t, eer rehe tt leel ee f t tt. Hweer, t tble the ee tht the pek lue f t equl t t. It pble t bt tblty l the e where D reter th 0.5 f lpe mpet utlzed. Wth lpe mpet, t mpred wth the um f t d exterl rmp, e t. The lpe f the um reter th the lpe f t le. The hrtert lue α w defed M M e α,.6 M M where M e the lpe f e t, M the lpe f t whle the trtr d M the lpe f t whle the trtr ff. Ne f M e, M, d M ete wth thee deft. M e mut be he uh tht α < t bt tblty. The dley d T Mdel Appled t the Buk erter I th ubet, the tw mdel preeted by dley 99 d T d Mddlebrk 995 re reewed d mpred. The trl-t-utput e

48 hpter. A Summry f the ette The 37 trfer fut, the utput mpede, d the ud ueptblty be bted by pply thee tw mdel t the buk erter wth urretmde trl d the reult re preeted. A lre umber f tuu-tme mdel fr urret-mde trl he bee preeted dur the yer. The dley d T mdel re deed t be urte frm d t hlf the wth frequey. They dffer mly the mdel f the urret lp. Whe urte mdel f urret-mde trl h bee eeded, the e preeted by dley 99 fte h bee he. A exmple f th d K 999, where the he w betwee the dley d T mdel. I d K 999, the T mdel dered upet d me ther uthr he l expreed th p. The dley d T mdel lude hh-frequey exte t be urte up t hlf the wth frequey. Thee hh-frequey exte re bed urte trl-t-urret trfer fut. We wll frt mke bref reew f hw th trfer fut bted. I Fure.5, e t dded t t but the me fut bted f e t ubtrted frm t d th ued Fure.6. Fure.6 hw the wefrm f the l t, t e t, d tw dfferet er f the dutr urret. The frt er ld le hw the dutr urret wefrm tedy tte,.e. the e where there re perturbt f t. The ed er dhed le hw the dutr urret wefrm the e where there tep perturbt t hw Fure.6. The trtr umed t tur t the pt t T, where teer. The trtr wll the tur ff t the pt t D T tedy tte. It umed tht the he the put lte d the utput lte re elble tht the lpe f the dutr urret be dered tt. The perturbt f t d t re hw Fure.6b d Fure.6, repetely. A pprxmt f t hw Fure.6d. If we erh fr lerzed mdel, the ue f the wefrm Fure.6d ted f the wefrm Fure.6 de t ue errr e the relte errr the terl f t the ftely mll. The l t ffet the wefrm f the dutr urret. If the he t re mll, the lue f t mprtt ly mll urrud f the pt t D T. If the he f t thee urrud re lw, ly mpled er f t r t eeded rder t bt urte mdel. A mpled er f t hw

49 38 hpter. A Summry f the ette The t t- e t t ^ t T k-d T kd T kd t ^ t T k-d T kd b T kd t T k-d ^ t pp. T kd T kd t ^ T k-d T kd d T kd t ^ k- k e k k- k f k Fure.6: Dfferet er f the urret urret trller.

50 hpter. A Summry f the ette The 39 Fure.6e d t deted. T be pred frm trdu ew rble me, the mpled drete-tme er f tuu-tme l deted by the me me the tuu-tme l, whh t frmlly rret tt. A mpled er f the pprxmte t hw Fure.6f d t deted. A expre tht derbe hw deped be dered. The Z-trfrm f th expre e drete-tme trfer fut. By der tht the pprxmte perturbed dutr urret hw Fure.6d retruted frm by u zerrder-hld rut the fllw urte tuu-tme trl-t-urret trfer fut bted: Fh T T T α e e, e T α.7 where α defed.6. Oe, te tht the dert f.7 mde wth the umpt tht the he the put d utput lte re elble. Bth the dley d T mdel re ufed mdel,.e. they be ppled t dfferet type f erter tple. The blk drm Fure.7 ued t mpre the dley d T mdel. Bth re mlll mdel d, therefre, the lerzed mdel f the erter luded Fure.7. The mdel f the urret trller t f x blk. The blk wll be expled lter th ubet. F m the trfer fut f the mdultr. he the put d utput lte ffet the trl d thee effet re mdeled wth the feedfrwrd k f d k r. Nte tht the put d utput lte re t fed frwrd Fure.. The re why they re eeded Fure.7 tht there re Furer mpet m the l { } mpred t the l t Fure. ee Set.. It t jut the d mpet tht m. The put d utput lte ffet the lpe f the dutr urret eh wth perd, whh mprtt ftr the urret trller. The ue f d the mll-l mdel f the urret trller therefre mplemet tht the wefrm f t better kw. The feedfrwrd re the dley d T mdel lulted wy tht mke the mplft f the led lp ytem rret t d. I the ext

51 40 hpter. A Summry f the ette The ^ erter ^ ^ k f d ^ F m k r H e ^ ^ Fure.7: A mll-l mdel f the urret trller d the erter. ubet, t wll be hw tht th wy trdue mdel errr t hh frequee. H e ued t lude the hh-frequey exte the dley mdel d t lulted t be T H e..8 T e The dley mdel utlze pprxmt f.8. The pprxmt: e T ωq ω Q z z ω ω,.9 where Q z,.0 π

52 hpter. A Summry f the ette The 4 π ω,. T ued bth by dley d T t reple the expetl fut. The pprxmt errr f.9 zer t d d hlf the wth frequey, ω. The pprxmte H e d F h re H e,. ω Q ω z Fh ω Q ω,.3 where Q π m D' 0.5,.4 m M e..5 M The hh-frequey exte the T mdel bted by lud ple F m. T d Mddlebrk 995 preet the fllw mdel fr the buk erter: F m M e D' D V T ω p,.6 k f DD' T,.7

53 4 hpter. A Summry f the ette The k 0,.8 r H e,.9 Ω,.30 ω ω p..3 Q I prtl urret-mde trller, the dutr urret meured d trfrmed t lte l. Vlte l l repreet the trl l d the exterl rmp l. dley mdel th by lud,, the dutr urret feedbk lp. T de t mdel th d therefre et t.30. The fllw rble re ued the dley mdel:,.3 S M,.33 S f M,.34 S M..35 e dley 990 preet the fllw mdel fr the buk erter: e F m Fm,.36 T m S T S S e DT D k f,.37 T kr,.38

54 hpter. A Summry f the ette The 43. z e Q H ω ω.39 By pply the dley mdel t the buk erter the fllw reult re bted fr the three mjr trfer fut:, de.40, de F Z h ut.4, ' de D D m D T.4 where, 0.5 ' D m T F de h.43 d F h defed.3. By pply the T mdel t the buk erter the fllw reult re bted fr the three mjr trfer fut:, de.44, de F Z h ut.45

55 44 hpter. A Summry f the ette The T D D m D' πω.46, de where de T πω F m D' 0.5, h.47 d F h defed.3. The demtr the T mdel,.47, lmt the me the e the dley mdel,.43. The dfferee fte ft fr erter tht re ued prte. The trl-t-utput trfer fut predted by the dley d T mdel re therefre pprxmtely the me e the umertr.40 d.44 re extly the me. The me true fr the utput mpede e the umertr.4 d.45 re extly the me. Hweer, the umertr.4 d.46 re t the me. The ud ueptblty predted by the T mdel lude extr zer mpred t the dley mdel. If the three trfer fut bted frm the dley mdel,.40-.4, re mpred wth the trfer fut fr the pe lp ytem,.-.4, t fud tht there re three ple ted f tw ple. Furthermre, the ES zer rem but the zer t the r fr the utput mpede repled by tw zer determed by F h. Th me tht the utput mpede de t ted t zer the frequey ted t zer. The trl-t-utput trfer fut, the utput mpede, d the ud ueptblty bted by pply the dley mdel t the bt erter re fud Jh 003, Set 3.6, Equt The rrepd reult fr the buk-bt erter re fud Jh 003, Set 3.7, Equt