Ealuaton of Back-EMF Etmato fo Snol Contol of JPE -4- http://x.o.og/.63/jpe...4. Ealuaton of Back-EMF Etmato fo Snol Contol of Pmannt Magnt Synchonou Moto Kwang-Woon an Jung-Ik Ha Dpt. of Elctonc Eng., Mokpo Natonal Matm Unty, Mokpo, Koa Dpt. of Elctcal an Comput Eng., Soul Natonal Unty, Soul, Koa Abtact Th pap pnt a compaat tuy of poton nol contol chm ba on back-lctomot foc (back-emf) tmaton n pmannt magnt ynchonou moto (PMSM). Th chaacttc of th tmat back-emf gnal a analyz ung aou mathmatcal mol of a PMSM. Th tanf functon of th tmato, ba on th xtn EMF mol n th oto fnc fam, a to how th mlaty. Thy a thn u fo th analy of th ffct of both th moto paamt aaton an th oltag o u to nt nonlnaty on th accuacy of th back-emf tmaton. Th ffnc btwn a pha-lock-loop (P) typ tmato an a unbg ob typ tmato, gnally u fo xtactng oto p an poton nfomaton fom tmat back-emf gnal, a alo xamn. An xpmntal tuy wth a 5-W nto-pmannt-magnt machn ha bn pfom to alat th analy. Ky wo: Back-EMF tmato, Pha lock loop, Pmannt magnt ynchonou, Snol contol I. INTODUCTION Tatonally nol fo pmannt magnt ynchonou moto (PMSM) ha bn wly u n aou applcaton bcau of th aantagou fatu uch a nca lablty an uc cot. Vaou nol mtho ha bn popo. Thy can b claf nto two goup: hgh funcy gnal njcton (HFSI) []-[4] an back-lctomot foc (back-emf) ba mtho [5]-[6]. Th HFSI ba nol mtho can po latly xact oto poton at tantll an n low-p opatng gon (typcally l than 5% of th at p of a machn) at th xpn of aubl no an atonal ngy lo. Th back-emf ba nol mtho acu oto poton fom th tato oltag an cunt wthout ung atonal hgh funcy gnal njcton. Th back-emf ba mtho cannot po labl oto poton nfomaton n low-p gon bcau th magntu of th back-emf ca a p ca. How, t ha bn pot that back-emf ba nol mtho can b uccfully appl to many applcaton (uch a compo) wh mpl tatng contol u Manucpt c Jan. 3, ; May, commn fo publcaton by Aocat Eto Hyung-Mn yu. Coponng Autho: jungkha@nu.ac.k Tl: 8-88-76, Fax: 8-878-45, Soul Nat'l Unty Dpt. of Elctonc Eng., Mokpo Natonal Matm Unty, Koa Fg... Functonal block agam of th back-emf ba nol mtho. [6]. Th functonal lmnt of th back-emf ba nol mtho a compo of a mathmatcal mol, a back-emf tmato, an a p/poton tmato, a hown n Fg.. Th back-emf tmato mak u of tato comman oltag (), tato cunt (), an a mathmatcal mol of th PMSM to th back-emf gnal. Th oto p an poton a ob fom th tmat back-emf gnal by man of anoth tmato uch a a pha-lock-loop (P) typ tmato o a unbg typ tat flt. Bcau th oto p an poton a ob fom th tmat back-emf gnal, th accuacy of th back-emf tmato ha a ct nflunc on th pfomanc of th nol. Vaou back-emf tmato ha bn popo fo th nol contol of PMSM. A cunt mol-ba EMF tmato wa lop n [5]-[6]. How, applyng th mtho to an nto PMSM (IPMSM) cau untabl nol opaton, a th aumpton aopt n th mol
Jounal of Pow Elctonc, Vol., No. 4, July of th IPMSM a not al n all opatng ang. To ol th poblm, th xtn EMF mol wa popo n [7]. Th xtn EMF mol nclu alncy tm, a wll a th back-emf, o that th mplfyng aumpton ma to th mol a not ncay. In [7] an [8], an xtn EMF mol n th tatonay fnc fam wa u fo nol contol. Th xtn EMF tmat n th tatonay fnc fam n an AC fom. Thfo, om pha lay btwn th actual an th tmat EMF ntabl, a th tmato a flt an flt ha an ntnc pha lay. Th xtn EMF mol n th oto fnc fam po th poton o nta of th oto poton. How, th pha lay nglgbl n th ca, bcau th xtn EMF n th oto fnc fam a DC gnal [9]. Mol-ba back-emf tmato a nt to moto paamt aaton, back-emf hamonc, an oltag o n th nt []-[]. Th hamonc u to th nonlnaty of th nt th man cau of gaaton of pfomanc n back-emf ba nol at low p; a mall gan fo a back-emf tmato u to xpan th low opatng ang []. Voltag o compnaton mtho w u n []-[] to uc th ngat ffct of th nt hamonc on a back-emf tmato. Th aton of th moto paamt cau by magntc atuaton an thmal chang alo ga th pfomanc of back-emf ba nol. Onln paamt ntfcaton an altnat to moto paamt aaton []-[3]. Th pap aluat th kn of back-emf tmato (a popotonal-ntgal (PI) typ tat flt [8], a tubanc ob typ tmato [9], [4], an a uc-o ob [5]) ba on th xtn EMF mol n th oto fnc fam. Th tanf functon of back-emf tmato a an th mlat among th back-emf tmato a montat ba on th tanf functon. Th ffct of paamt aaton an nt hamonc on th accuacy of back-emf tmaton a ntgat n tal ung th tanf functon an t bo-plot. Th analy xplan why th tanc an th -ax nuctanc aaton ha a gat nflunc on th back-emf tmaton accuacy than oth paamt. Th pha-lock-loop (P) typ tmato an th unbg typ tmato a commonly utlz to xtact th oto p an poton fom th tmat back-emf gnal. Th ffnc btwn th two tmato a alo xamn. To po th alty of th analy pfom n th pap, xpmntal ult obtan wth an IPMSM a po. II. MATHEMATICA MODE OF A PMSM FO BACK-EMF ESTIMATION Fg. how a pac cto agam fo a PMSM [9]. Th α-β an - fam pnt th tatonay an th oto Fg.. Spac cto agam of PMSM [9]. fnc fam, pctly. Th α ax copon to th magntc ax of th u pha an th ax algn wth th cton of th N pol of th oto. Th - fam an tmat fam u n nol cto contol ung th oto fnc fam. θ an θ a th actual an tmat oto poton, pctly. A. Mathmatcal Mol n th Statonay fnc Fam Th PMSM oltag uaton n th tatonay fnc fam : α p( ) coθ p n θ α β p p n θ coθ β nθ () λpm coθ wh: α, β tato oltag n th tatonay α - β fam; α, β tato cunt n th tatonay α - β fam; tato tanc; p ffntal opato; λ PM pmannt magnt flux lnkag; oto angula locty; θ oto poton;,, an a th - an -ax nuctanc. Th con tm on th ght-han of () th back-emf an t nclu th oto poton nfomaton. In th ca of a ufac mount PMSM (SPMSM), an a ntcal,. Thu uaton () can b mplf a follow: α p α α () β p β β wh: α n θ λ. (3) PM β coθ In a SPMSM, α an β, th back-emf gnal n th tatonay fam can b aly tmat ung a mpl tmatng tatgy an uaton (). How, fo th IPMSM, t ffcult to contuct th back-emf ob
Ealuaton of Back-EMF Etmato fo Snol Contol of 3 ung uaton () owng to th unknown paamt θ, whch cau by th machn alncy. Th xtn EMF mol, pnt n [7], can mplfy th oltag uaton fo an IPMSM a follow: α p ( ) α nθ E (4) x β p β coθ wh E x th xtn EMF, an fn by (5): Ex [( ) λpm ] ( )( p ). (5) Th con tm on th ght-han of (4) copon to α an β. Th oto poton can b calculat ctly ung (6): α θ tan (6) β wh θ th tmat oto poton, an ê an α ê β a th back-emf gnal tmat n th tatonay fnc fam ung () o (3). How, ê α an ê a β AC gnal. Thfo, a pha lay xt btwn th actual an th tmat back-emf gnal u to th ntnc pha lay of th back-emf tmato. Th ffct ult n om oto poton tmaton o. A a ult, th pcal pha lay compnaton mtho gnally u. B. Mathmatcal Mol n th oto fnc Fam Th oltag uaton of th PMSM n th oto fnc fam gn by: p. (7) p λpm Th oltag uaton of th PMSM n th - fam a follow [9]: p n θ λpm p coθ (8) a p ( b ) c wh Δθ th poton o btwn th - an th - fnc fam, an tmat oto angula p, an ( ) n θ n θ coθ n θ coθ n θ ( ) n θ coθ n θ n θ n θ coθ ( ) n θ coθ co θ n θ n θ co θ n θ coθ (9) a () b.() c Euaton (8) too complcat to b uful n bulng an tmato. How, fo a SPMSM,, an thu uaton (8) can b mplf a follow: p n θ λpm co p θ () ( ). To mplfy th oltag uaton of an IPMSM n th - fnc fam, th xtn EMF can alo b appl to th oto fnc fam mol a follow [7], [9]: Fg. 3. Back-EMF tmato ung PI typ tat flt. p ( ) (3) p wh: n θ E. (4) x coθ Th con tm on th ght-han of (3) th back-emf. How, th back-emf n th - fnc fam nclu th oto poton o, ath than th oto poton. Th th ffnc btwn th tatonay an th oto fnc fam mol. Un th tay-tat conton, t pobl to gno th th tm on th ght-han of (3) bcau th o btwn an uffcntly mall an uaton (3) can b mplf ung (5): p. (5) p Th tmat oto poton o θ can b calculat ung (6): θ tan (6) wh ê an ê a th back-emf gnal tmat ung (5) n th - fnc fam. Whn ung th oto fnc fam mol, th tmat back-emf gnal a DC alu. Thfo, th pha lay btwn th actual an th tmat gnal nglgbl. Th th aantag of th oto fnc fam mol whn compa to th tatonay fnc fam mol. On th oth han, an atonal oto poton tmato u fo th oto fnc fam mol bcau th oto poton o (Δθ) tmat nta of th oto poton (θ ). In aton, th th tm n (3), gno n (5), may gnat a back-emf tmaton o n th tannt-tat conton, wh th o btwn an no long nglgbl. III. ANAYSIS OF THE BACK-EMF ESTIMATO Th back-emf gnal can b tmat ung th th α-β o th - fnc fam mol. Th pap focu on th analy of back-emf tmato ba on th xtn-emf mol n th - fnc fam, bcau th pha lay n th back-emf tmato latly mall whn compa to that n th α-β fnc fam mol.
4 Jounal of Pow Elctonc, Vol., No. 4, July A. Back-EMF Etmato Ung PI Typ Stat Flt Th PI typ back-emf tmato pnt n [8] can alo b mplmnt n th - fnc fam mol a hown n Fg. 3. In Fg. 3,,, an a th nomnal moto paamt, k p an k a th popotonal an ntgal gan of th tat flt, pctly, an a tato cunt cto, whch can b xp a follow: j (7) wh an a th tato cunt n th - fnc fam. î,, an Ê copon to th tmat tato cunt cto, th comman oltag cto, an th tmat back-emf cto n th - fnc fam, pctly, an can b xp n an uaton mla to (7). Th tmat back-emf n Fg. 3 gn by (8): E ( k p k )( ) ( k ) k E p j j j ( ) (8) wh th oltag cto n th - fnc fam. Th gan of th PI typ back-emf tmato a lct by ung th pol-zo cancllaton mtho a follow: k p, k (9) t t wh t th banwth of th back-emf tmato. By ubttutng (9) nto (8), th followng obtan: t E t E t t ( j j( ) ) { j. () By aumng that th o n th moto paamt, th oltag cto, an th tmat p a uffcntly mall, th tanf functon of th back-emf tmato, hown n Fg. 3, a follow: Ê t. () E t Fom (), t cla that th chaacttc th PI typ back-emf tmato a th am a tho of a ft-o low-pa flt. B. Back-EMF Etmato Ung a Dtubanc Ob Fg. 4 how a back-emf tmato ung a tubanc ob [9]. A ffntal opato nclu n Fg. 4. To mnmz th ngat ffct of th ffntal opaton, th Fg. 4. Back-EMF tmato ung tubanc ob. back-emf tmato ung th tubanc ob mplmnt ung both a low-pa flt an a hgh-pa flt a follow: E ( j ) t t t t. () Th tmat EMF ung th tubanc ob gn by (3): E t E t t { j t (3) ( ( j j ) ). Fom () an (3), t nt that th back-emf tmato ung th tubanc ob th am a th PI typ back-emf tmato. Thfo, th banwth of th low-pa flt n Fg. 4 alo tmn th banwth of th tanf functon fom th actual back-emf to th tmat back-emf. C. Back-EMF Etmato Ung a uc O Ob By aumng that th amplng funcy uffcntly hgh an that th back-emf contant ung a amplng po, th ynamc uaton of a PMSM ba on th xtn EMF mol a gn a follow: t (4)
Ealuaton of Back-EMF Etmato fo Snol Contol of 5. y (5) an nw y, whch a th tato cunt n th - fnc fam, a mauabl. Nw output aabl a fn ung th mauabl tat aabl a follow:. (6) Ung (6), th uc ob can b contuct a follow: (7) (8) wh an a th ob gan. To mo th ffntal tm n (7) an (8), nw tat aabl, an, a fn a follow: (9). (3) Ung th nw tat aabl, an, th ob can b gn a follow: (3). (3) To mplmnt th ob fom (3) an (3), th tmat oto angula p an th comman oltag a u, nta of th actual oto angula p an oltag. Fg. 5 how a back-emf ( ê ) tmato gn ung (9) an (3). Th alu of ê can b alo tmat ung a mla mtho, a hown n Fg. 5. Th gan of th uc ob houl b lct a: t, (33) thn th tmat back-emf gn by (34): { E E t t t t j j j. (34) Thu, th th kn of back-emf tmato ba on th xtn EMF mol n th - fnc fam ha th am opatng chaacttc, although th back-emf tmato ha a ffnt tuctu. D. Analy of th Back-EMF Etmaton Eo Th back-emf tmato u a mathmatcal mol of a PMSM an th comman oltag. Thfo, th accuacy of th back-emf tmato ctly affct by th moto paamt aaton an th oltag o u to nt nonlnat. To analyz th ffct of paamt aaton an oltag o on th accuacy of th back-emf tmato, th nomnal moto paamt mut b xamn, an th oto angula p an th comman oltag mut b tmat a follow:,, (35),, (36) wh Δ, Δ, an Δ a th o btwn th nomnal an th actual moto paamt, Δ th tmat oto p o, an Δ an Δ t t t t t t a th oltag o u to nt nonlnat. Ung (34), (35) an (36), th followng obtan: (37) t t t t t t (38) wh,. Un th tay tat conton, Δ uffcntly mall that Δ an Δ can b mplf a follow: (39). (4) Δ an Δ a functon of Δ, Δ, an Δ. Δ an Δ a th oltag o u to nt nonlnat. Thy cont manly of 6-th o hamonc an th funcy popotonal to th oto p. Fom uaton (37) to (4), t Fg. 5. Back-EMF tmato ung th uc o ob.
6 Jounal of Pow Elctonc, Vol., No. 4, July Fg. 7. P typ p an poton tmato [9]. (a) an fom to th tmat ax back-emf, a hown n Fg. 6. Th combnatonal ca of moto paamt aton a con n th bo plot. To allow btt tmaton of th back-emf, th gan of th tanf functon fom th actual back-emf to th tmat back-emf houl b clo to B an that of th tanf functon fom to houl b ca a much a pobl. In Fg. 6, t can b n that th accuacy of th back-emf tmato mo nt to tato tanc o than to -ax nuctanc o. IV. SPEED AND POSITION ESTIMATOS (b) Fg. 6. Bo plot of th tanf functon (a) fom th actual ax back-emf to th tmat ax back-emf an (b) fom to th tmat ax back-emf ( 5.8Ω,.6H, t Hz). can b n that th tmat EMF a affct by Δ an Δ a th low-pa flt. Thu th ffct of Δ an Δ on th tmat EMF ca a nca, bcau th 6th o hamonc a attnuat by th low-pa flt. On th oth han, th ffct of Δ on th tmat EMF nca a nca, bcau Δ multpl by, a hown n th con tm on th ght of (39) an (4). In aton, th con tm of (39) an (4) a not attnuat by th low-pa flt, a thy a DC gnal. Thu, th accuacy of th tmat EMF bcom mo nt to Δ at hgh p, an t bcom mo nt to oltag o at low p. To mpo th accuacy of th back-emf tmato at low p, th banwth of th back-emf tmato t houl b ca o a to futh flt th unabl oltag hamonc. Da tm compnaton can b con a an altnat mtho to mpo th pfomanc of th back-emf tmato at low p. To analyz th ffct of Δ an Δ on th tmat EMF, con th bo plot of th tanf functon fom th actual ax back-emf to th tmat ax back-emf A. P Typ Sp an Poton Etmato Th P typ p an poton tmato hown n Fg. 7 gnally u to acu th tmat oto p an poton fom th tmat oto poton o. Th PI () n Fg. 7, whch gnally u, gn a follow: G K K (4) contoll fo G p an th tanf functon fom θ to θ gn by (4): θ Kp K. (4) θ K p K By aumng that th nomnato of (4) th am a th chaacttc uaton of th tana con-o ytm, K p an K can b lct ba on th ampng ato ζ an th unamp natual funcy n. Th tannt pon of th P typ tmato can b mpo by ang a oubl ntgal tm nto G () a follow [9]: G K K K (43) 3 Th tanf functon fom θ an θ gn by (44): θ K K K3. (44) 3 θ K K K 3 K, K, an K 3 can alo b lct by ung th ampng ato (ζ) an th unamp natual funcy ( n ) [9]. Fg. 8 how th bo plot of (4) an (44) wh ζ an n a t to an 5 a/, pctly. Bcau th pha lay ca, a hown n Fg. 8, whn th oubl ntgal tm a, t natual that th pfomanc of th oubl ntgal typ tmato mpo n th tannt tat. B. unbg Ob Typ Sp an Poton Etmato A unbg ob typ p an poton tmato can alo b u fo th tmaton of oto p an poton,
Ealuaton of Back-EMF Etmato fo Snol Contol of 7 wh J an B a th coffcnt of th nta an cou fcton, pctly, an Ĵ an B a th nomnal paamt. Th gan of th tmato n (45) can b lct uch that th chaacttc uaton of (45) ha th am oot a th followng [7]: 3 K 3β, K 3J β, K3 Jβ (46) wh β th oot of th chaacttc uaton. To contuct a unbg ob typ p an poton tmato, th mchancal paamt J an B a u, wha a P typ tmato o not u th u of mchancal paamt. It pobl to obtan zo pha lag wth th u of a unbg ob typ tmato wth accuat machn paamt [8]. How, th tmato nt to th nta paamt o an t tuctu mo complx than that of a P typ tmato. Alo, th P typ tmato can flt hgh funcy no nclu n th tmat poton o, bcau t funcy pon th am a that of th low-pa flt, a hown n Fg. 8. Fg. 8. Bo plot of P typ tmato. Fg. 9. unbg ob typ p an poton tmato. [8], [7]. a hown n Fg. 9 [8],[7]. Th tanf functon of th unbg ob typ poton tmato, hown n Fg. 9, gn by (45): 3 θ J ( B JK) ( BK K ) K3 (45) 3 θ J ( B JK ) ( BK K ) K 3 V. EXPEIMENTA ESUTS To fy th ffctn of th analy of th back-emf tmato, xpmnt w pfom ung a 5-W IPMSM coupl to a pmannt-magnt DC (PMDC) loa moto, a hown n Fg.. Th paamt of th tt IPMSM a lt n Tabl I. Th DC-lnk of ach nt fo th tt IPMSM an th PMDC moto a connct togth o that atonal upmnt fo pocng th gnat ngy fom th PMDC moto not u. Th back-emf ba nol algothm a mplmnt on a Txa Intumnt TMS3F8335 floatng-pont gtal gnal poco (DSP). Th wtchng funcy of th nt khz an th a-tm 3 μ. Th amplng po m fo th p contol an.m fo th cunt contol, th nol p an th poton tmaton. Th banwth of th cunt contoll Hz. An nco wth a oluton of,4 pul p oluton (PP) u to monto th actual oto poton. Fg.. Expmntal tt tup.
8 Jounal of Pow Elctonc, Vol., No. 4, July TABE I 5-W IPMSM NOMINA PAAMETES Ba p λpm Pol 3 pm 5.8 Ω.6 H.65 H.59 Wb 6 A. Th Statonay fnc Fam Mol Ca To xamn th chaacttc of th back-emf tmato ung th tatonay fnc fam mol, xpmnt w pfom ung th PI typ back-emf tmato pnt n [8]. Fg. an Fg. how th xpmntal ult whn th banwth of th PI typ back-emf tmato chon to b 3Hz. A contant loa tou of.55 N m appl ung nol opaton. Th tmat oto poton n Fg. ctly calculat ung (6). In Fg., t can b n that om pha lay xt btwn th mau an th tmat oto poton. Th bcau th tmat back-emf gnal a AC gnal, a hown n Fg.. Th pha lay n th back-emf tmato unaoabl. Th pha lay btwn th actual an th tmat back-emf nca a th oto p nca. If th banwth of th back-emf tmato nca o a to uc th pha lay, th back-emf tmato bcom nt to nt no. Thu, an atonal pha lay compnaton mtho u fo th back-emf tmato n th tatonay fnc fam mol. Fg.. Etmat cunt an xtn EMF at oto p pm. Fg.. Stay-tat tmat poton at oto p pm. B. Th oto fnc Fam Mol Ca Fg. 3 an Fg. 4 how th tannt an tay-tat pon whn th PI typ back-emf tmato ung th oto fnc fam mol, hown n Fg. 3, u. Th banwth of th back-emf tmato chon to b Hz. Th oto p an poton a tmat though th P typ p an poton tmato hown n Fg. 7. Th PI contoll n th P typ tmato t to ζ an n 5 a/. Th PI contoll of (4) an (43) w u to pouc th gnal hown n Fg. 3 an Fg. 4, pctly. To xamn th tannt-tat pon accong to th tuctu of th PI contoll n th P typ tmato, th loa tou wa chang fom 5% (.37 N m) to % (.73 N m) whl th moto wa unnng at pm. Fom Fg. an Fg. 4, t can b n that th tannt-tat pfomanc of (43) btt than that of (4). Th copon wth th analy n Fg. 8. Whn ung th oto fnc fam mol, th tay-tat o btwn actual an th tmat oto poton mall whn compa to that n th tatonay fnc fam mol. Th bcau th tmat back-emf n th oto fnc fam mol a DC gnal. Thfo, th tay-tat pha lay n th back-emf tmato nglgbl. Expmnt on th tubanc ob typ back-emf tmato hown n Fg. 4 an th uc-o ob typ back-emf tmato hown n Fg. 5 how th am ult a Fg. 3 an Fg. 4 un th am conton. Th ffct of moto paamt o on th back-emf Fg. 3. Poton tmat wth PI-typ P at oto p pm. tmaton o can b monto a th tmat oto poton o, bcau th oto poton tmat fom th tmat back-emf. Fg. 5, Fg. 6, an Fg. 7 how th tmat oto poton o whn th nomnal tato tanc an th nomnal - ax nuctanc ay fom 7% to 3% of th nomnal alu (lt n Tabl I) whl th moto unnng at 5%, %, %, an 3% of th at p (3 pm) wth a % loa (.73 N m). Fom th fgu, t can b n that th ffnc btwn th maxmum an th mnmum alu of th tmat oto poton o ca a th oto p nca. Th ult occu bcau th oltag o u to nt nonlnat a flt though th low-pa flt nclu n th back-emf tmato. Alo, t can b ob that th tmat oto poton o a mo nt to th nomnal -ax nuctanc o. Th conc wth th analy pnt n th pap.
Ealuaton of Back-EMF Etmato fo Snol Contol of 9 Fg. 4. Poton tmat wth la-lag compnato-typ P at oto p pm. () Fg. 5. Etmat poton o u nomnal tato tanc aaton at at loa wth contant p ( (a) 5%, (b) %, (c) %, an () 3% of at p). (a) (a) (b) (b) (c) (c)
Jounal of Pow Elctonc, Vol., No. 4, July () Fg. 6. Etmat poton o u nomnal -ax nuctanc aaton at at loa wth contant p ( (a) 5%, (b) %, (c) %, an () 3% of at p). () Fg. 7. Etmat poton o u nomnal -ax nuctanc aaton at at loa wth contant p ( (a) 5%, (b) %, (c) %, an () 3% of at p). VI. CONCUSIONS (a) (b) Th pap ha analyz al back-emf tmato fo th nol contol of a PMSM an f th ffctn of th analy though xpmntal tu. Th followng pont ummaz th wok pnt n th pap. ) Whn ung th tatonay fnc fam mol, a pha lay btwn th actual an th tmat back-emf xt, bcau th back-emf n th tatonay fnc fam a AC gnal. On th oth han, th pha lay nglgbl whn ung th oto fnc fam mol, bcau th tmat EMF a DC gnal. ) Th a th kn of back-emf tmato ba on th oto fnc fam mol, whch nclu th PI typ, th tubanc ob typ, an th uc ob typ tmato. Thy all ha th am tanf functon an th am opatng chaacttc. 3) Th ffct of th moto paamt o an th oltag o u to nt nonlnat on th back-emf tmaton o w analyz an f though xpmnt. Th oltag o a flt by th low-pa flt nclu n th back-emf tmato. Th back-emf tmaton o u to oltag o ca a th oto p nca. To uc th back-emf tmaton o at low p, th banwth of th back-emf tmato houl b ca o a a-tm compnato can b u. Th back-emf tmato o mo nt to th -ax nuctanc o bcau th oltag o u to th -ax nuctanc o nca a th oto p nca an thy a not flt though th low-pa flt nclu n th back-emf tmato. Thu an atonal -ax nuctanc o compnaton mtho u fo tabl opaton of th nol contol at hgh p. Bo plot of th tanf functon fom th actual to th tmat back-emf how that th tato tanc o can alo ca th accuacy of th tmat back-emf. (c)
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