Poceeding of the IEEE, Vol. 9, No. 8, Aug., pp. 359-394 Senole Contol of Induction Moto Dive Joachim Holtz, Fellow, IEEE Electical Machine and Dive Goup, Univeity of Wuppetal 497 Wuppetal Gemany Abtact Contolled induction moto dive without mechanical peed eno at the moto haft have the attaction of low cot and high eliability. To eplace the eno, the infomation on the oto peed i extacted fom meaued tato voltage and cuent at the moto teminal. Vecto contolled dive equie etimating the magnitude and patial oientation of the fundamental magnetic flux wave in the tato o in the oto. Open loop etimato o cloed loop obeve ae ued fo thi pupoe. They diffe with epect to accuacy, obutne, and enitivity againt model paamete vaiation. Dynamic pefomance and teady-tate peed accuacy in the low peed ange can be achieved by exploiting paaitic effect of the machine. The oveview in thi pape ue ignal flow gaph of complex pace vecto quantitie to povide an inightful deciption of the ytem ued in enole contol of induction moto. Keywod: Induction moto, enole contol, vecto contol, complex tate vaiable, obeve, modelling, identification, adaptive tuning. INTRODUCTION AC dive baed on full digital contol have eached the tatu of a matue technology. The wold maket volume i about, million US$ with an annual gowth ate of 5%. Ongoing eeach ha concentated on the elimination of the peed eno at the machine haft without deteioating the dynamic pefomance of the dive contol ytem []. Speed etimation i an iue of paticula inteet with induction moto dive whee the mechanical peed of the oto i geneally diffeent fom the peed of the evolving magnetic field. The advantage of peed enole induction moto dive ae educed hadwae complexity and lowe cot, educed ize of the dive machine, elimination of the eno cable, bette noie immunity, inceaed eliability and le maintenance equiement. The opeation in hotile envionment motly equie a moto without peed eno. A vaiety of diffeent olution fo enole ac dive have been popoed in the pat few yea. Thei meit and limit ae eviewed baed on a uvey of the available liteatue. Fig. give a chematic oveview of the methodologie applied to peed enole contol. A baic appoach equie only a peed etimation algoithm to make a otational peed peed etimation paaitic popetie v/f contol oto model oto field oientation field angle etimation tato model vecto contol MRAS, obeve, Kalman filte tato field oientation Fig.. Method of enole peed contol eno obolete. The v/f contol pinciple adjut a contant volt-pe-hetz atio of the tato voltage by feedfowad contol. It eve to maintain the magnetic flux in the machine at a deied level. It implicity atifie only modeate dynamic equiement. High dynamic pefomance i achieved by field oientation, alo called vecto contol. The tato cuent ae injected at a well defined phae angle with epect to the patial oientation of the otating magnetic field, thu ovecoming the complex dynamic popetie of the induction moto. The patial location of the magnetic field, the field angle, i difficult to meaue. Thee ae vaiou type of model and algoithm ued fo it etimation a hown in the lowe potion of Fig.. Contol with field oientation may eithe efe to the oto field, o to the tato field, whee each method ha it own meit. Dicuing the vaiety of diffeent method fo enole contol equie an undetanding of the dynamic popetie of the induction moto which i teated in a fit intoductoy ection.. INDUCTION MACHINE DYNAMICS. An intoduction to pace vecto The ue of pace vecto a complex tate vaiable i an efficient method fo ac machine modelling []. The pace vec-
jim jim cuent denity ditibution A (α) a b axi ia Re phae a winding axi a Re α c axi (a) ymbolic epeentation Fig.. Stato winding with only phae a enegized (b) geneated cuent denitiy ditibution to appoach epeent the induction moto a a dynamic ytem of only thid ode, and pemit an inightful viualization of the machine and the upeimpoed contol tuctue by complex ignal flow gaph [3]. Such ignal flow gaph will be ued thoughout thi pape. The appoach implie that the patial ditibution along the aigap of the magnetic flux denity, the flux linkage and the cuent denitie (magnetomotive foce, mmf) ae inuoidal. Linea magnetic ae aumed while ion loe, lotting effect, deep ba and end effect ae neglected. To decibe the pace vecto concept, a thee-phae tato winding i conideed a hown in Fig. (a) in a ymbolic epeentation. The winding axi of phae a i aligned with the eal axi of the complex plane. To ceate a inuoidal flux denity ditibution, the tato mmf mut be a inuoidal function of the cicumfeential coodinate. The ditibuted phae winding of the machine model ae theefoe aumed to have inuoidal winding denitie. Each phae cuent then ceate a pecific inuoidal mmf ditibution, the amplitude of which i popotional to the epective cuent magnitude, while it patial oientation i detemined by the diection of the epective phae axi and the cuent polaity. Fo example, a poitive cuent a in tato phae a ceate a inuoidal cuent denity ditibution that lead the winding axi a by 9, having theefoe it maximum in the diection of the imaginay axi a hown in Fig. (b). The total mmf in the tato i obtained a the upepoition of the cuent denity ditibution of all thee phae. It i again a inuoidal ditibution, which i indicated in Fig. 3 by the vaying diamete of the conducto co ection, o, in an equivalent epeentation, by two half-moon haped egment. Amplitude and patial oientation of the total mmf depend on the epective magnitude of the phae cuent a, b and c. A the phae cuent vay with time, the geneated cuent denity pofile diplace in popotion, foming a otating cuent denity wave. The upepoition of the cuent denity pofile of the in- dividual phae can be epeented by the patial addition of the contibuting phae cuent. Fo thi pupoe, the phae cuent need to be tanfomed into pace vecto by impating them the patial oientation of the petaining phae axe. The eulting equation ( ) = a + b + 3 i a i a c () define the complex tato cuent pace vecto. Note that the thee tem on the ight-hand ide of () ae alo complex pace vecto. Thei magnitude ae detemined by the intantaneou value of the epective phae cuent, thepatial oientation by the diection of the epective winding axi. The fit tem in (), though complex, i eal-valued ince the winding axi of phae a i the eal axi of the efeence fame. It i nomally omitted in the notation of () to chaacteize the eal axi by the unity vecto = e j. A a complex quantity, the pace vecto. a epeent the inuoidal cuent denity ditibution geneated by the phae cuent a. b axi c axi jim cuent denity ditibution a b c Re a axi Fig. 3. Cuent denitiy ditibution eulting fom the phae cuent a, b and c
Such ditibution i epeented in Fig. (b). In the econd tem of (), a = exp(jp/3) i a unity vecto that indicate the diection of the winding axi of phae b, and hence a b i the pace vecto that epeent the inuoidal cuent denity ditibution geneated by the phae cuent b. Likewie doe a c epeent the cuent denity ditibution geneated by c, with a = exp(j 4p/3) indicating the diection of the winding axi of phae c. Being a complex quantity, the tato cuent pace vecto in () epeent the inuoidal patial ditibution of the total mmf wave ceated inide the machine by the thee phae cuent that flow outide the machine. The mmf wave ha it maximum at an angula poition that lead the cuent pace vecto by 9 a illutated in Fig. 3. It amplitude i popotional to =. The caling facto /3 in () eflect the fact that the total cuent denity ditibution i obtained a the upepoition of the cuent denity ditibution of thee phae winding while the contibution of only two phae winding, paced 9 apat, would have the ame patial effect with the phae cuent popely adjuted. The facto /3 alo enue that the contibuting phae cuent a, b and c can be eadily econtucted a the pojection of on the epective phae axe, hence ia = Re{ i} ib = Re{ a i} () i = Re a i c jim { } Equation () hold on condition that zeo equence cuent do not exit. Thi i alway tue ince the winding ta point of an invete fed induction moto i neve connected [4]. At teady-tate opeation, the tato phae cuent fom a balanced, inuoidal thee-phae ytem which caue the tato mmf wave to otate at contant amplitude in ynchonim with the angula fequency w of the tato cuent. flux linkage ditibution Fig. 4. Flux denitiy ditibution eulting fom the tato cuent in Fig. 3 y Re The flux denity ditibution in the aigap i obtained by patial integation of the cuent denity wave. It i theefoe alo a inuoidal wave, and it lag the cuent denity wave by 9 a illutated in Fig. 4. It i convenient to chooe the flux linkage wave a a ytem vaiable intead of the flux denity wave a the fome contain added infomation on the winding geomety and the numbe of tun. By definition, a flux linkage ditibution ha the ame patial oientation a the petaining flux denity ditibution. The tato flux linkage ditibution in Fig. 4 i theefoe epeented by the pace vecto y. A otating flux denity wave induce voltage in the individual tato winding. Since the winding denitie ae inuoidal patial function, the induced voltage ae alo inuoidally ditibuted in pace. The ame i tue fo the eitive voltage dop in the winding. The total of both ditibuted voltage in all phae winding i epeented by the tato voltage pace vecto u, which i a complex vaiable. Againt thi, the phae voltage at the machine teminal ae dicete, cala quantitie. They define the tato voltage pace vecto ( ) u = u a + a u b + a u 3 c (3) in a ame way a the phae cuent define the tato cuent pace vecto in (). Note that cuent pace vecto ae defined in a diffeent way than flux linkage vecto: They ae alway 9 out of phae with epect to the maximum of the cuent denity ditibution they epeent, Fig. 3. Againt thi, flux linkage vecto ae alway aligned with the maximum of the epective flux linkage ditibution, Fig. 4. Thi i a convenient definition, pemitting to etablih a imple elationhip between both vecto, fo intance y = l, whee l i the thee-phae inductance of the tato winding. The thee-phae inductance of a ditibuted winding i.5 time the pe phae inductance of that vey winding [].. Machine equation To etablih the machine equation, all phyical quantitie ae conideed nomalized, and oto quantitie ae efeed to the tato, i. e. caled in magnitude by the tato to oto winding atio. A table of the bae quantitie ued fo nomalization i given in Appendix A. The nomalization include the conveion of machine of abitay numbe p of pole pai to the two-pole equivalent machine that i hown in the illutation. It ha been found convenient to nomalize time a t = w R t, whee w R i the ated tato fequency of the machine. A otating coodinate ytem i choen to etablih the voltage equation of the induction moto. Thi coodinate ytem otate at an angula tato velocity w k, whee the value of w k i left unpecified to be a geneal a poible. Of coue, when a pecific olution of the ytem equation i ought, the coodinate ytem mut be defined fit.
The tato voltage equation in the geneal k-coodinate ytem i tato winding u i k t oto winding dy u = i + + jω d ky τ (4) whee i the eitive voltage dop and i the tato eitance. The um of the lat two tem in (4) epeent the induced voltage, o back emf, of which dy /dt i the tationay tem that account fo the vaiation in time of the tato flux linkage a een fom the moving efeence fame. The econd tem jw k y i the motion-induced voltage that eult fom the vaying diplacement of the winding conducto with epect to the efeence fame. In the oto, thi diplacement i w k w, whee w i the angula mechanical velocity of the oto, and hence the oto voltage equation i dy = i + + j( ω ) d k ω y τ. (5) The left-hand ide how that the oto voltage um up to zeo in a quiel cage induction moto. Equation (4) and (5) epeent the electomagnetic ubytem of the machine a a econd ode dynamic ytem by two tate equation, howeve, in tem of fou tate vaiable:, y, i, y. Theefoe, two flux linkage equation y = l + l m i i (6) y = l m + l i i (7) ae needed to etablih completene. In (6) and (7), l i the tato inductance, l i the oto inductance, and l m i the mutual inductance between the tato and the oto winding; all inductance ae thee-phae inductance having.5 time the value of the epective phae inductance. Equation (4) and (5) ae eaily tanfomed to a diffeent efeence fame by jut ubtituting w k with the angula velocity of the epective fame. To tanfom the equation to the tationay efeence fame, fo intance, w k i ubtituted by zeo. The equation of the mechanical ubytem i dω τ m = T e T L (8) whee t m i the mechanical time contant, w i the angula mechanical velocity of the oto, T e i the electomagnetic toque and T L i the load toque. T e i computed fom the z- component of the vecto poduct of two tate vaiable, fo intance a T = y i = y i y i e z ab ba (9) when y = y a + j y b and = i a + j i b ae the elected tate vaiable, expeed by thei component in tationay coodinate. u w.3 Stato cuent and oto flux a elected tate vaiable Mot dive ytem have a cuent contol loop incopoated in thei contol tuctue. It i theefoe advantageou to elect the tato cuent vecto a one tate vaiable. The econd tate vaiable i then eithe the tato flux, o the oto flux linkage vecto, depending on the poblem at hand. Selecting the oto cuent vecto a a tate vaiable i not vey pactical, ince the oto cuent cannot be meaued in a quiel cage oto. Synchonou coodinate ae choen to epeent the machine equation, ω k = ω. Selecting the tato cuent and the oto flux linkage vecto a tate vaiable lead to the following ytem equation, obtained fom (4) though (7): τ ' σ t σ ' jt' σ di k + i ωτ ωτ τ σ' i u d = j ( j ) y + τ y σ σ (a) dy τ + y ω ω τ τ = j( d ) y + lm (b) The coefficient in () ae the tanient tato time contant τ σ '= σl / σ and the oto time contant t = l /, whee σl i the total leakage inductance, σ = l m /l l i the total leakage facto, σ = + k i an equivalent eitance, and k = l m /l i the coupling facto of the oto. The elected coodinate ytem otate at the electical angula tato velocity w of the tato, and hence in ynchonim with the evolving flux denity and cuent denity wave in the teady-tate. All pace vecto will theefoe aume a fixed poition in thi efeence fame a long a the teadytate pevail. The gaphic intepetation of (8) to () i the ignal flow diagam Fig. 5. Thi gaph exhibit two fundamental winding tuctue in it uppe potion, epeenting the winding ytem in the tato and the oto, and thei mutual magnetic coupling. Such fundamental tuctue ae typical fo any ac l m T e k T L jt w w Fig. 5. Induction moto ignal flow gaph; tate vaiable: tato cuent vecto, oto flux vecto; epeentation in ynchonou coodinate t t m y w
u aaa tato winding u i t σ ' machine winding. The popetie of uch tuctue hall be explained with efeence to the model of the tato winding in the uppe left of Fig. 5. Hee, the time contant of the fit ode delay element i τ σ '. The ame time contant eappea a facto jτ σ ' in the local feedback path aound the fit ode delay element, uch that the epective tate vaiable, hee, get multiplied by jω τ σ '. The eulting ignal jω τ σ ', if multiplied by σ, i the motion-induced voltage that i geneated by the otation of the winding with epect to the elected efeence fame. While the facto ω epeent the angula velocity of the otation, the ign of the local feedback ignal, which i minu in thi example, indicate the diection of otation: The tato winding otate anti-clockwie at w in a ynchonou efeence fame. The tato winding i chaacteized by the mall tanient time contant τ σ ', being detemined by the leakage inductance and the winding eitance both in the tato and the oto. The dynamic of the oto flux ae govened by the lage oto time contant τ if the oto i excited by the tato cuent vecto, Fig. 5. The oto flux eact on the tato winding though the oto induced voltage u i k t k = j τ ( ) oto winding Fig. 6. Induction moto at zeo tato fequency, ignal flow gaph in tationay coodinate ωτ y () in which the component jω y pedominate ove y /τ unle the peed i vey low. A typical value of the nomalized oto time contant i τ = 8, equivalent to 5 m, while y i cloe to unity in the bae peed ange. The electomagnetic toque a the input ignal to the mechanical ubytem i expeed by the elected tate vaiable and deived fom (6), (7) and (9) a Te = k y () z.4 Speed etimation at vey low tato fequency The dynamic model of the induction moto i ued to invetigate the pecial cae of opeation at vey low tato fequency, ω. The tato efeence fame i ued fo thi pupoe. The angula velocity of thi efeence fame i zeo and hence ω in () i eplaced by zeo. The eulting ignal lm w t jt y flow diagam i hown in Fig. 6. At vey low tato fequency, the mechanical angula velocity ω depend pedominantly on the load toque. Paticulaly, if the machine i fed by a voltage u at zeo tato fequency, can the mechanical peed be detected without a peed eno? The ignal that can be exploited fo peed etimation ae the tato voltage vecto u and the meaued tato cuent. To invetigate thi quetion, the tanfe function of the oto winding y lm = τ + jωτ i (3) i conideed, whee y ~ and i ~ ae the Laplace tanfom of the pace vecto y and, epectively. Equation (3) can be diectly veified fom the ignal flow gaph Fig. 6. The ignal that act fom the oto back to the tato in Fig. 6 i popotional to (jωτ )y. It Laplace tanfom i obtained with efeence to (3): u i k k jωτ = ( jωτ ) y = lm i. (4) σ στ στ τ+ jωτ A ω appoache zeo, the feeding voltage vecto u appoache zeo fequency when obeved in the tationay efeence fame. A a conequence, all teady-tate ignal tend to aume zeo fequency, and the Laplace vaiable. Hence we have fom (4) u i k lim = lmi σ στ. (5) The ight-hand ide of (5) i independent of ω, indicating that, at zeo tato fequency, the mechanical angula velocity ω of the oto doe not exet an influence on the tato quantitie. Paticulaly, they do not eflect on the tato cuent a the impotant meauable quantity fo peed identification. It i concluded, theefoe, that the mechanical peed of the oto i not obevable at ω =. The ituation i diffeent when opeating cloe to zeo tato fequency. The afoementioned teady-tate ignal ae now low fequency ac ignal which get modified in phae angle and magnitude when paing though the τ -delay element on the ight-hand ide of Fig. 6. Hence, the cancelation of the numeato and the denominato in (4) i not pefect. Paticulaly at highe peed i a voltage of ubtantial magnitude induced fom the oto field into the tato winding. It influence on meauable quantitie at the machine teminal can be detected: the oto tate vaiable ae then obevable. The angula velocity of the evolving field mut have a minimum nonzeo value to enue that the induced voltage in the tato winding i ufficiently high, thu educing the influence of paamete mimatch and noie to an acceptable level. The inability to acquie the peed of induction machine below thi level contitute a baic limitation fo thoe etimation model that diectly o indiectly utilize the induced
Te T er 4 3 t diect on-line tating teady tate t Dw w 5 % 5 at ated peed at % ated peed..4.6 (a) Lage-ignal epone: diect on-line tating compaed with the teady-tate chaacteitic w w R m t Fig. 7. Dynamic behavio of the uncontolled induction moto (b) Small-ignal epone: peed ocillation following a tep change of the tato fequency voltage. Thi include all type of model that eflect the effect of flux linkage with the fundamental magnetic field. Speed etimation at vey low tato fequency i poible, howeve, if othe phenomena like atuation induced aniotopie, the dicete ditibution of oto ba, o oto aliency ae exploited. Such method bea a pomie fo peed identification at vey low peed including utained opeation at zeo tato fequency. Detail ae dicued in Section 8. Othe than the mechanical peed, the patial oientation of the fundamental flux linkage with the machine winding, i. e. the angula oientation of the pace vecto y o y, i not impoible to identify at low and even at zeo electical excitation fequency if enabling condition exit. Stable and peitent opeation at zeo tato fequency can be theefoe achieved at high dynamic pefomance, povided the component of the dive ytem ae modelled with atifying accuacy..5 Dynamic behavio of the uncontolled machine The ignal flow gaph Fig. 5 epeent the induction moto a a dynamic ytem of 3d ode. The ytem i nonlinea ince both the electomagnetic toque T e and the oto induced voltage ae computed a poduct of two tate vaiable, y and i, and w and y, epectively. It eigenbehavio i chaacteized by ocillatoy component of vaying fequencie ω* gadient limite t g v/f cuve * u * u ag( u *) PWM Fig. 8. Contant volt pe hetz contol ac main cuent limite ~ ~ 3~ M u which make the ytem difficult to contol. To illutate the poblem, a lage-ignal epone i diplayed in Fig. 7(a), howing the toque-peed chaacteitic at diect-on-line tating of a non-enegized machine. Lage deviation fom the coeponding teady-tate chaacteitic can be obeved. Duing the dynamic acceleation poce, the toque initially ocillate between it teady-tate beakdown value and the nominal geneating toque T er. The initial ocillation ae pedominantly geneated fom the electomagnetic inteaction between the two winding ytem in the uppe potion of Fig. 5, while the ubequent limit cycle aound the final teady-tate point at w = w R i moe an electomechanical poce. The nonlinea popetie of the induction moto ae eflected in it epone to mall-ignal excitation. Fig. 7(b) how diffeent damping chaacteitic and eigenfequencie when a % inceae of tato fequency i commanded fom two diffeent peed value. A detailed tudy of induction moto dynamic i epoted in [5]. 3. CONSTANT VOLTS-PER-HERTZ CONTROL 3. Low cot and obut dive One way of dealing with the complex and nonlinea dynamic of induction machine in adjutable peed dive i avoiding excitation at thei eigenfequencie. To thi aim, a gadient limite educe the bandwidth of the tato fequency command ignal a hown in Fig. 8. The band-limited tato fequency ignal then geneate the tato voltage efeence magnitude u * while it integal detemine the phae angle ag(u *). The v/f chaacteitic in Fig. 8 i deived fom (4), neglecting the eitive tato voltage dop and, in view of bandlimited excitation, auming teady-tate opeation, dy /dt. Thi yield u = jω y (6) o u /w = cont. (o v/f = cont.) when the tato flux i maintained at it nominal value in the bae peed ange. Field
w* peed contolle w * p p p contolle w u u ' l y k k w t ' y jt' w w R p R p Equ. 9 J y y k T e t m machine w Fig. 9. Dive contol ytem fo modeate dynamic equiement T L weakening i obtained by maintaining u = u max = cont. while inceaing the tato fequency beyond it nominal value. At vey low tato fequency i a peet minimum value of the tato voltage pogammed to account fo the eitive tato voltage dop. The ignal u * and ag(u *) thu obtained contitute the efeence vecto u * of the tato voltage, which in tun contol a pulewidth modulato (PWM) to geneate the witching equence of the invete. Oveload potection i achieved by imply inhibiting the fiing ignal of the emiconducto device if the machine cuent exceed a pemitted maximum value. Since v/f -contolled dive opeate puely a feedfowad ytem, the mechanical peed w diffe fom the efeence peed w * when the machine i loaded. The diffeence i the lip fequency, equal to the electical fequency w of the oto cuent. The maximum peed eo i detemined by the nominal lip, which i 3-5% of nominal peed fo low powe machine, and le at highe powe. A load cuent dependent lip compenation cheme can be employed to educe the peed eo [6]. Contant volt-pe-hetz contol enue obutne at the expene of educed dynamic pefomance, which i adequate fo application like pump and fan dive, and toleable fo othe application if cot i an iue. A typical value fo toque ie time i m. The abence of cloed loop contol and the etiction to low dynamic pefomance make v/f-contolled dive vey obut. They opeate table even in the citical low peed ange whee vecto contol fail to maintain tability (Section 7.). Alo fo vey high peed application like centifuge and ginde i open loop contol an advantage: The cuent contol ytem of cloed loop cheme tend to detabilize when opeated at field weakening up to 5 to time the nominal fequency of 5 o 6 Hz. The amplitude of the motion-induced voltage jω τ σ ' in the tato, Fig. 5, become vey high at thoe high value of the tato fequency ω. Hee, the complex coefficient jω intoduce an undeied voltage component in quadatue to any manipulated change of the tato voltage vecto that the cuent contolle command. The phae diplacement in the motion-induced voltage impai the tability. The paticula attaction of v/f contolled dive i thei extemely imple contol tuctue which favo an implementation by a few highly integated electonic component. Thee cot-aving apect ae pecifically impotant fo application at low powe below 5 kw. At highe powe, the powe component themelve dominate the ytem cot, pemitting the implementation of moe ophiticated contol method. Thee eve to ovecome the majo diadvantage of v/f contol: the educed dynamic pefomance. Even o, the cot advantage make v/f contol vey attactive fo low powe application, while thei obutne favo it ue at high powe when a fat epone i not equied. In total, uch ytem contibute a ubtantial hae of the maket fo enole ac dive. 3. Dive fo modeate dynamic pefomance An impoved dynamic pefomance of v/f-contolled dive can be achieved by an adequate deign of the contol tuctue. The ignal flow gaph Fig. 9 give an example [7]. The machine dynamic ae epeented hee in tem of the tate vaiable y and y. The ytem equation ae deived in the tationay efeence fame, letting ω k = in equation (4) though (7). The eult i dy = u y k y d ( τ σl ) (7a) dy t' y j wt' y d ky τ + = +, (7b) whee τ = στ = σ l / i a tanient oto time contant, and k i the coupling facto of the tato. The coeponding ignal flow gaph of the machine model i highlighted by the haded aea on the ight-hand ide of Fig. 9. The gaph how that the tato flux vecto i geneated a the integal of u., whee
= k σl ( ) y y. (8) The nomalized time contant of the integato i unity. The key quantity of thi contol concept i the active tato cuent p, computed in tationay coodinate a u* o i ip = = i i u* a coϑ + b inϑ (9) fom the meaued othogonal tato cuent component a and b in tationay coodinate, whee = a + jb and ϑ i the phae angle of the tato voltage efeence vecto u * = u *. e jϑ, a contol input vaiable. The active tato cuent p i popotional to the toque. Accodingly, it efeence value i * p i geneated a the output of the peed contolle. Speed etimation i baed on the tato fequency ignal ω a obtained fom the p -contolle, and on the active tato cuent p, which i popotional the oto fequency. The nominal value p R of the active tato cuent poduce nominal lip at oto fequency ω R, thu = ω R /p R. p. The etimated peed i then ω = ω ω ()wheehe hatch mak a an etimated vaiable. An inne loop contol the active tato cuent p, with it efeence ignal limited to pevent oveloading the invete and to avoid pull-out of the induction machine if the load toque i exceive. Fig. 9 how that an extenal. -ignal compenate eliminate the intenal eitive voltage dop of the machine. Thi make the tajectoy of the tato flux vecto independent of the tato cuent and the load. It povide a favoable dynamic behavio of the dive ytem and eliminate the need fo the conventional acceleation limite (Fig. 8) in the peed efeence channel. A toque ie time aound m can be achieved, [7], which matche the dynamic pefomance of a thyito convete contolled dc dive. 4. MACHINE MODELS Machine Model ae ued to etimate the moto haft peed, and, in high-pefomance dive with field oiented contol, to identify the time-vaying angula poition of the flux vecto. In addition, the magnitude of the flux vecto i etimated fo field contol. Diffeent machine model ae employed fo thi pupoe, depending on the poblem at hand. A machine model i implemented in the contolling micopoceo by olving the diffeential equation of the machine in eal-time, while uing meaued ignal fom the dive ytem a the focing function. The accuacy of a model depend on the degee of coincidence that can be obtained between the model and the modelled ytem. Coincidence hould pevail both in tem of tuctue and paamete. While the exiting analyi method pemit etablihing appopiate model tuctue fo induction machine, the paamete of uch model ae not alway in good ageement with the coeponding machine data. Paamete may ignificantly change with tempeatue, o with the opeating point of the machine. On the othe hand, the enitivity of a model to paamete mimatch may diffe, depending on the epective paamete, and the paticula vaiable that i etimated by the model. Diffeential equation and ignal flow gaph ae ued in thi pape to epeent the dynamic of an induction moto and it vaiou model ued fo tate etimation. The chaacteizing paamete epeent exact value when decibing the machine itelf; they epeent etimated value fo machine model. Fo bette legibility, the model paamete ae motly not pecifically maked () a etimated value. Suitable model fo field angle etimation ae the model of the tato winding, Fig., and the model of the oto winding hown in Fig. below. Each model ha it meit and dawback. 4. The oto model The oto model i deived fom the diffeential equation of the oto winding. It can be eithe implemented in tato coodinate, o in field coodinate. The oto model in tato coodinate i obtained fom (b) in a taightfowad manne by letting ω =. dy τ + y ωτ τ = j d y + lm () Fig. how the ignal flow gaph. The meaued value of the tato cuent vecto, and of the otational peed ω ae the input ignal to the model. The output ignal i the oto flux linkage vecto y (S), maked by the upecipt (S) a being efeed to in tato coodinate. The agument ag(y ) of the oto flux linkage vecto i the oto field angle δ. The magnitude y i equied a a feedback ignal fo flux contol. The two ignal ae obtained a the olution of (S) w (S) y = y coδ + jy in = yα + jyβ oto winding l m t jt y (S) Fig.. Roto model in tato coodinate δ atan x y x + y () d y
integato F t u u z t t y k y low pa t i i z σl (a) ignal flow gaph y σ ag (F) p t low pa integato ω (b) Bode diagam Fig.. Stato model in tationay coodinate; the ideal integato i ubtituted by a low pa filte whee the ubcipt α and β mak the epective component in tato coodinate. The eult i yβ δ = actan, y = yα + y y β (3) α The oto field angle δ mak the angula oientation of the oto flux vecto. It i alway efeed to in tato coodinate. The function (3) ae modeled at the output of the ignal flow gaph Fig.. In a pactical implementation, thee function can be condened into two numeic table that ae ead fom the micocontolle pogam. The accuacy of the oto model depend on the coect etting of the model paamete in (). It i paticulaly oto time contant τ that detemine the accuacy of the etimated field angle, the mot citical vaiable in a vecto contolled dive. The othe model paamete i the mutual inductance l m. It act a a gain facto a een in Fig. and doe not affect the field angle. It doe have an influence on the magnitude of the flux linkage vecto, which i le citical. 4. The tato model The tato model i ued to etimate the tato flux linkage vecto, o the oto flux linkage vecto, without equiing a peed ignal. It i theefoe a pefeed machine model fo enole peed contol application. The tato model i deived by integating the tato voltage equation (4) in tato coodinate, w k =, fom which ( ) y = u (4) i obtained. Equation (6) and (7) ae ued to detemine the oto flux linkage vecto fom (4): y = ( ( u i) σli)= ( y y ) (5) k k σ The equation how that the oto flux linkage i baically the diffeence between the tato flux linkage and the leakage flux y. One of the two model equation (4) o (5) can be ued to etimate the epective flux linkage vecto, fom which the petaining field angle, and the magnitude of the flux linkage i obtained. The ignal flow diagam Fig. (a) illutate oto flux etimation accoding to (5). The tato model (4), o (5), i difficult to apply in pactice ince an eo in the acquied ignal u and, and offet and dift effect in the integating hadwae will accumulate a thee i no feedback fom the integato output to it input. All thee ditubance, which ae geneally unknown, ae epeented by two ditubance vecto u z (t) and i z (t) in Fig. (a). The eulting unwaway of the output ignal i a fundamental poblem of an open integation. A negative, low gain feedback i theefoe added which tabilize the integato and pevent it output fom inceaing without bound. The feedback ignal convet the integato into a fit ode delay having a low cone fequency /t, and the tato model (4) and (5) become τ and dy + y = τ ( u i), y = ( y σ l k i) (6) τ d y τ + y = u i σ l k di (7) epectively. The Bode diagam Fig. (b) how that the fit ode delay, o low pa filte, behave a an integato fo fequencie much highe than the cone fequency. It i obviou that the model become inaccuate when the fequency educe to value aound the cone fequency. The gain i then educed and, moe impotantly, the 9 phae hift of the integato i lot. Thi caue an inceaing eo in the etimated field angle a the tato fequency educe.
y = y d + j w y l m k Te Fig.. Induction moto ignal flow gaph at foced tato cuent. The dotted line epeent zeo ignal at oto field oientation. The deciive paamete of the tato model i the tato eitance. The eitance of the winding mateial inceae with tempeatue and can vay in a : ange. A paamete eo in affect the ignal in Fig.. Thi ignal dominate the integato input when the magnitude of u educe at low peed. Reveely, it ha little effect on the integato input at highe peed a the nominal value of i low. The value ange between. -.5 p.u., whee the lowe value apply to high powe machine. To ummaize, the tato model i ufficiently obut and accuate at highe tato fequency. Two baic deficiencie let thi model degade a the peed educe: The integation poblem, and the enitivity of the model to tato eitance mimatch. Depending on the accuacy that can be achieved in a pactical implementation, the lowe limit of table opeation i eached when the tato fequency i aound - 3 Hz. 5. ROTOR FIELD ORIENTATION Contol with field oientation, alo efeed to a vecto contol, implicate poceing the cuent ignal in a pecific ynchonou coodinate ytem. Roto field oientation ue a efeence fame aligned with the oto flux linkage vecto. It i one of the two baic ubcategoie of vecto contol hown in Fig.. 5. Pinciple of oto field oientation A fat cuent contol ytem i uually employed to foce the tato mmf ditibution to a deied location and intenity in pace, independent of the machine dynamic. The cuent ignal ae time-vaying when poceed in tato coodinate. The contol ytem then poduce an undeiable velocity eo even in the teady-tate. It i theefoe pefeed to implement the cuent contol in ynchonou coodinate. All ytem vaiable then aume contant value at teady-tate and zeo teady-tate eo can be achieved. The bandwidth of the cuent contol ytem i baically detemined by the tanient tato time contant τ σ ', unle the witching fequency of the PWM invete i lowe than t w T L jt t m w about khz. The othe two time contant of the machine (Fig. 5), the oto time contant τ and the mechanical time contant τ m, ae much lage in compaion. The cuent contol theefoe eject all ditubance that the dynamic eigenbehavio of the machine might poduce, thu eliminating the influence of the tato dynamic. The dynamic ode educe in conequence, the ytem being only chaacteized by the complex oto equation (b) and the cala equation (8) of the mechanical ubytem. Equation (b) and (8) fom a econd ode ytem. Refeing to ynchonou coodinate, ω k = ω, the oto equation (b) i ewitten a dy τ + y ωτ τ = j d y + lm, (8) whee ω i the angula fequency of the induced oto voltage. The eulting ignal flow gaph Fig. how that the tato cuent vecto act a an independent focing function on the eidual dynamic ytem. It value i commanded by the complex efeence ignal * of the cuent contol loop. To achieve dynamically decoupled contol of the now deciive ytem vaiable T e and y, a paticula ynchonou coodinate ytem i defined, having it eal axi aligned with the oto flux vecto [8]. Thi efeence fame i the oto field oiented dq-coodinate ytem. Hee, the imaginay oto flux component, o q-component y q, i zeo by definition, and the ignal maked by dotted line in Fig. aume zeo value. To etablih oto field oientation, the q-component of the oto flux vecto mut be foced to zeo. Hence the q-component of the input ignal of the τ -delay in Fig. mut be alo zeo. The balance at the input umming point of the τ -delay thu define the condition fo oto field oientation lmq i =ωτy d, (9) which i put into effect by adjuting ω appopiately. If condition (9) i enfoced, the ignal flow diagam of the moto aume the familia dynamic tuctue of a dc machine, Fig. 3. The electomagnetic toque T e i now popotional to the foced value of the q-axi cuent i q and hence independently contollable. Alo the oto flux i independently contolled by the d-axi cuent i d, which i kept at it nominal, contant value in the bae peed ange. The ma- flux command toque command i d i q l m t k y T e T L machine Fig. 3. Signal flow gaph of the induction moto at oto field oientation t m w
u t t l m t tato model l k oto model chine dynamic ae theefoe educed to the dynamic of the mechanical ubytem which i of fit ode. The contol concept alo eliminate the nonlineaitie of the ytem, and inhibit it inheent tendency to ocillate duing tanient, illutated in Fig. 7. 5. Model efeence adaptive ytem baed on the oto flux The model efeence appoach (MRAS) make ue of the edundancy of two machine model of diffeent tuctue that etimate the ame tate vaiable on the bai of diffeent et of input vaiable [9]. Both model ae efeed to in the tationay efeence fame. The tato model (6) in the uppe potion of Fig. 4 eve a a efeence model. It output i the etimated oto flux vecto ŷ S. The upecipt S indicate that ŷ oiginate fom the tato model. The oto model i deived fom (b), whee ω i et to zeo fo tato coodinate dy τ + y ωτ τ = j d y + lm. (3) Thi model etimate the oto flux fom the meaued tato cuent and fom a tuning ignal, in Fig. 4. The tuning ignal i obtained though a popotional-integal (PI) contolle fom a cala eo ignal e = ŷ S ŷ R z = ŷ S ŷ R in α, which i popotional the angula diplacement α between the two etimated flux vecto. A the eo ignal e get minimized by the PI contolle, the tuning ignal appoache the actual peed of the moto. The oto model a the adjutable model then align it output vecto ŷ R with the output vecto ŷ S of the efeence model. The accuacy and dift poblem at low peed, inheent to the open integation in the efeence model, ae alleviated by uing a delay element intead of an integato in the tato model in Fig. 4. Thi eliminate an accumulation of the dift eo. It alo make the integation ineffective in the fequency ange aound and below /τ, and neceitate the addition of an equivalent bandwidth limite in the input of the adjutable oto model. Below the cutoff fequency ω R /τ S ŷ R ŷ e Fig. 4. Model efeence adaptive ytem fo peed etimation; efeence vaiable: oto flux vecto t jt ŷ - 3 Hz, peed etimation become neceaily inaccuate. A eveal of peed though zeo in the coue of a tanient poce i nevethele poible, if uch poce i fat enough not to pemit the output of the τ -delay element to aume eoneou value. Howeve, if the dive i opeated cloe to zeo tato fequency fo a longe peiod of time, the etimated flux goe atay and peed etimation i lot. The peed contol ytem upeimpoed to the peed etimato i hown in Fig. 5. The etimated peed ignal i upplied by the model efeence adaptive ytem Fig. 4. The peed contolle in Fig. 5 geneate a oto fequency ignal, which contol the tato cuent magnitude y i = + ω l τ, (3) and the cuent phae angle δ = ω τ + actan( ω d τ). (3) Equation (3) and (3) ae deived fom (9) and fom the teady-tate olution i d = y /l m of () in field coodinate, whee y q, and hence y d = y, i aumed ince field oientation exit. It i a paticula aet of thi appoach that the accuate oientation of the injected cuent vecto i maintained even if the model value of τ diffe fom the actual oto time contant of the machine. The eaon i that the ame, even eoneou value of τ i ued both in the oto model and in the contol algoithm (3) and (3) of the peed contol cheme Fig. 5. If the tuning contolle in Fig. 4 maintain zeo eo, the contol cheme exactly eplicate the ame dynamic elationhip between the tato cuent vecto and the oto flux vecto that exit in the actual moto, even in the peence of a oto time contant eo [9]. Howeve, the accuacy of peed etimation, eflected in the feedback ignal to the peed contolle, doe depend on the eo in τ. The peed eo may be even highe than with thoe method that eti- ŷ w* peed cont. w w field tato coodinate main Fig. 5. Speed and cuent contol yten fo MRAS etimato; CR PWM: cuent egulated pulewidth modulato * e jd d * CR PWM ~ ~ M 3~ u
u l tato model lm k S u i e oto model R u i Fig. 6. Model efeence adaptive ytem fo peed etimation; efeence vaiable: oto induced voltage mate the oto fequency ω and ue () to compute the peed: = ω. The eaon i that the tato fequency ω i a contol input to the ytem and theefoe accuately known. Even if in () i eoneou, it nominal contibution to i mall ( - 5% of ω R ). Thu, an eo in doe not affect vey much, unle the peed i vey low. A moe evee ouce of inaccuacy i a poible mimatch of the efeence model paamete, paticulaly of the tato eitance. Good dynamic pefomance of the ytem i epoted by Schaude above Hz tato fequency [9]. 5.3 Model efeence adaptive ytem baed on the induced voltage The model efeence adaptive appoach, if baed on the oto induced voltage vecto athe than the oto flux linkage vecto, offe an altenative to avoid the poblem involved with * i d i d peed contolle w * i q * i - contolle d w k i d * w l i q i q + w l i d * i - contolle q t kq B i q t jt field tato coodinate t ' * u d * u q e jd e -jd k A d * u w ŷ PWM open integation []. In tato coodinate, the oto induced voltage i the deivative of the oto flux linkage vecto. Hence diffeentiating (5) yield dy d = i u l k i σ, (33) which i a quantity that povide infomation on the oto flux vecto fom the teminal voltage and cuent, without the need to pefom an integation. Uing (33) a the efeence model leave equation () dy τ = y ωτ τ +j d y + lm, (34) to define the coeponding adjutable model. The ignal flow gaph of the complete ytem i hown in Fig. 6. The open integation i cicumvented in thi appoach and, othe than in the MRAC ytem baed on the oto flux, thee i no low pa filte that ceate a bandwidth limit. Howeve, the deivative of the tato cuent vecto mut be computed to evaluate (33). If the witching hamonic ae poceed a pat of u, thee mut be alo contained in (and in d /dt a well) a the hamonic component mut cancel on the ight of (33). 5.4 Feedfowad contol of tato voltage In the appoach of Okuyama et al. [], the tato voltage ae deived fom a teady-tate machine model and ued a the baic efeence ignal to contol the machine. Theefoe, though it model, it i the machine itelf that let the invete duplicate the voltage which pevail at it teminal in a given opeating point. Thi poce can be chaacteized a elf-contol. The component of the voltage efeence ignal ae deived in field coodinate fom () unde the aumption of teadytate condition, d/, fom which y d = l m i d follow. Uing uing the appoximation ω ω we obtain main Fig. 7. Feedfowad contol of tato voltage, oto flux oientation; k = σ y d /k, k = l m /τ y d d ~ ~ M 3~ u ud = id ωσ liq (35a) uq = iq +ω lid (35b) The d-axi cuent i d i eplaced by it efeence value i d *. The eulting feedfowad ignal ae epeented by the equation maked by the haded fame in Fig. 7. The ignal depend on machine paamete, which ceate the need fo eo compenation by upeimpoed contol loop. An i d -contolle enue pimaily the eo coection of u d, thu govening the machine flux. The ignal i q *, which epeent the toque efeence, i obtained a the output of the peed contolle. The etimated peed i computed fom () a the diffeence of the tato fequency ω
* i d i d * w A i q + w l i d * k w to k q * u d contol ytem B t ' * u q machine k t T L Fig. 8. Compenation channel (thick line at A and B) fo the enole peed contol ytem Fig. 7; k = /k σ y d -channel (thick line at A and B) fo the enole peed contol ytem Fig. 7; k = /k y d t ' and the etimated oto fequency ; the latte i popotional to, and theefoe deived fom, the toque poducing cuent i q. Since the toque inceae when the velocity of the evolving field inceae, ω and, in conequence, the field angle δ can be deived fom the i q -contolle. Although the ytem thu decibed i equipped with contolle fo both tato cuent component, i d and i q, the intenal co-coupling between the input vaiable and the tate vaiable of the machine i not eliminated unde dynamic condition; the deied decoupled machine tuctue of Fig. 3 i not etablihed. The eaon i that the poition of the otating efeence fame, defined by the field angle d, i not detemined by the oto flux vecto y. It i govened by the q- cuent eo intead, which, though the i q -contolle, acceleate o deceleate the efeence fame. To invetigate the ituation, the dynamic behavio of the machine i modeled uing the ignal flow gaph y * Fig. 5. Only mall deviation fom a tate of coect field oientation and coect flux magnitude contol ae conideed. A educed ignal flow gaph Fig. 8 w* i theeby obtained in which the d-axi oto flux i conideed contant, denoted a y d. A nonzeo value of the q-axi oto flux y q indicate a mialignment of the field oiented efeence fame. It i now aumed that the mechanical peed ω change by a udden inceae of the load toque T L. The ubequent deceae of ω inceae ω and hence poduce a negative dy q / at ignal the input of the τ -delay. Simultaneouly i the q-axi component k /. σ ω y d of the oto induced voltage inceaed, which i the back-emf that act on the tato. The conequence i that i q ie, delayed by the tanient tato time contant τ σ ', which etoe dy q / to it oiginal zeo value afte the delay. Befoe thi eadjutment take y i q l m w T e k t y d t w t m y q w place, though, y q ha aleady aumed a pemanent nonzeo value, and field oientation i lot. A imila effect occu on a change of ω * which intantaneouly affect dy q /, while thi ditubance i compenated only afte a delay of τ σ ' by the feedfowad adjutment of u q * though ω. Both undeied petubation ae eliminated by the addition of a ignal popotional to di q / to the tato fequency input of the machine contolle. Thi compenation channel i maked A in Fig. 7 and Fig. 8. Still, the mechanim of maintaining field oientation need futhe impovement. In the dynamic tuctue Fig. 5, the ignal jωτ y, which eentially contibute to backemf vecto, influence upon the tato cuent deivative. A mialignment between the efeence fame and the oto flux vecto poduce a nonzeo y q value, giving ie to a back-emf component that change i d. Since the feedfowad contol of u d * i detemined by (35a) on the aumption of exiting field alignment, uch deviation will invoke a coecting ignal fom the i d -contolle. Thi ignal i ued to influence, though a gain contant k q, upon the quadatue voltage u q * (channel B in Fig. 7 and Fig. 8) and hence on i q a well, cauing the i q -contolle to acceleate o deceleate the efeence fame to eetablih accuate field alignment. Toque ie time of thi cheme i epoted aound 5 m; peed accuacy i within ± % above 3% ated peed and ± pm at 45 pm []. w w flux contolle peed cont. i - contolle q N d field tato coodinate * i d * i q D l m t d i * (S) Fig. 9. Senole peed contol baed on diect i q etimation and oto field oientation. CRPWM: Cuent egulated pulewidth modulato; N: Numeato, D: Denominato y * i q ŷ CR PWM y etimato ac main e jd ~ ~ * i q d y * u M 3~ u
y i q D x + y N y y y u i t τ t ' y *(S) y * t e jδ δ u One can ee fom (36) that the facto û i / on the ight equal the oto flux vecto y ~, which vaiable i now ubtituted by it efeence value y ~ * : τ y y * = + i + τ u τ +. (38) Thi expeion i the equivalent of the pue integal of û i, on condition that y ~ = y~ *. A tanfomation to the time domain yield two diffeential equation τ dy d + y = τ i u i τ', (39) Fig.. Roto flux etimato fo the tuctue in Fig. 9; N: Numeato, D: Denumeato 5.5 Roto field oientation with impoved tato model A enole oto field oientation cheme baed on the tato model i decibed by Ohtani []. The uppe potion of Fig. 9 how the claical tuctue in which the contolle fo peed and oto flux geneate the cuent efeence vecto * in field coodinate. Thi ignal i tanfomed to tato coodinate and poceed by a et of fat cuent contolle. A poible mialignment of the efeence fame i detected a the diffeence of the meaued q-axi cuent fom it efeence value i q *. Thi eo ignal feed a PI contolle, the output of which i the etimated mechanical peed. It i added to an etimated value ω of the oto fequency, obtained with efeence to the condition fo oto field oientation (9), but computed fom the efeence value i q * and y *. The eaon i that the meaued value i q i contaminated by invete hamonic, while the etimated oto flux linkage vecto ŷ i eoneou at low peed. The integation of ω povide the field angle δ. The tato model i ued to etimate the oto flux vecto y. The dift poblem of an open integation at low fequency ae avoided by a band-limited integation by mean of a fit-ode delay. Thi entail a evee lo of gain in y at low tato fequency, while the etimated field angle lag conideably behind the actual poition of the oto field. The Bode plot in Fig. (b) demontate thee effect. An impovement i bought about by the following conideation. The tanfe function of an integato i y + = i = u u τ i τ + (36) whee y ~ and û i ae the Laplace tanfom of the epective pace vecto, and u i i the oto induced voltage in the tato winding (). The tem in the ight i expanded by a faction of unity value. Thi expeion i then decompoed a τ y y y = + i + + u u τ τ i = +. (37) whee u i i expeed by the meaued value of the teminal voltage and cuent efeing to (4), (6) and (7), and dy *( S τ + y = y ). (4) It i pecifically maked hee by a upecipt that y *(S) i efeed to in tato coodinate and hence i an ac vaiable, the ame a the othe vaiable. The ignal flow gaph Fig. how that the oto flux vecto i yntheized by the two component y and y, accoding to (39) and (4). The high gain facto t in the uppe channel let y dominate the etimated oto flux vecto ŷ at highe fequencie. A the tato fequency educe, the amplitude of u educe and ŷ get inceaingly detemined by the ignal y fom the lowe channel. Since y * i the input vaiable of thi channel, the etimated value of ŷ i then eplaced by it efeence value y * in a mooth tanition. Finally, we have ŷ y * at low fequencie which deactivate the oto flux contolle in effect. Howeve, the field angle d a the agument of the oto flux vecto i till unde contol though the peed contolle and the i q -contolle, although the accuacy of d educe. Field oientation i finally lot at vey low tato fequency. Only the fequency of the tato cuent i contolled. The cuent ae then foced into the machine without efeence to the oto field. Thi povide obutne and cetain tability, although not dynamic pefomance. In fact, the q-axi cuent i q i diectly deived in Fig. a the cuent component in quadatue with what i conideed the etimated oto flux vecto y i z iq =, (4) y independently of whethe thi vecto i coectly etimated. Equation (4) i viualized in the lowe left potion of the ignal flow diagam Fig.. A the peed inceae again, oto flux etimation become moe accuate and cloed loop oto flux contol i eumed. The coect value of the field angle i eadjuted a the q-axi cuent, though (4), now elate to the coect oto flux vecto. The i q -contolle then adjut the etimated peed, and