Obtaining the Current-Flux Relations of the Saturated PMSM by Signal Injection

Transcription

1 Obtaining the Current-Flux Relation of the Saturated PMSM by Signal Injection Pacal Cobe, Françoi Malrait, Philippe Martin and Pierre Rouchon Schneider Tohiba Inverter Europe, 7 Pacy-ur-Eure, France Eail: {pacal.cobe, francoi.alrait}@chneider-electric.co Centre Autoatique et Sytèe, MINES PariTech, PSL Reearch Univerity, 756 Pari, France Eail: {philippe.artin, pierre.rouchon}@ine-paritech.fr arxiv:75.465v [c.sy] May 7 Abtract Thi paper propoe a ethod baed on ignal injection to obtain the aturated current-flux relation of a PMSM fro locked-rotor experient. With repect to the claical ethod baed on tie integration, it ha the ain advantage of being copletely independent of the tator reitance; oreover, it i le enitive to voltage biae due to the power inverter, a the injected ignal ay be fairly large. I. Introduction Good odel are uually paraount to deign good control law. Thi i the cae for Peranent Magnet Synchronou Motor PMSM, epecially when enorle control i conidered. In thi ode of operation, neither the rotor poition nor it velocity i eaured, and the control law ut ake do with only current eaureent; a uitable odel i therefore eential to relate the current to the other variable. When operating above oderately low peed, i.e., above about % of the rated peed, odel neglecting agnetic aturation are uually accurate enough for control purpoe; but at low peed, agnetic cro-aturation ut be taken into account, in particular when high-frequency ignal injection i ued, and the ore o for otor with little geoetric aliency, ee e.g. [] [6]. However, otor anufacturer very eldo provide aturation data, which ean the aturated current-flux relation ut be experientally deterined. The claical ethod to get thee data i to integrate over tie the tie derivative of the flux a given by the tator odel of the otor, ee [7] for a detailed account. Unfortunately, thi ethod i very enitive to the value of the tator reitance R, which ay ignificantly vary during a long experient; it alo require a good knowledge of the actually ipreed potential voltage enor ay be required becaue of the not very well-known voltage drop in the power tage, and of coure of the reulting current which are in practice alway eaured. It i thu not eay to ipleent thi ethod on a coercial variable peed drive for ue in the field. The goal of thi paper i to propoe an alternative approach to obtain the aturated current-flux relation, which i copletely independent on the knowledge of the tator reitance R ; it i alo le enitive to the voltage drop of the power tage. It i baed on high-frequency ignal injection. It i baed on ignal injection, a technique that wa originally introduced for enorle control at low velocity [8], but that can alo be ued for identification [9]. The key idea, thank to a uitable analyi of the effect of ignal injection, i to recover the flux-current relation by integrating over path in the plane of direct and quadrature current; it generalize a cruder procedure ued in []. Notice that the required experiental data can be obtained with only experient where the rotor i locked in a known poition, which i reaonable for indutrial ue in the field the claical ethod alo work in iilar condition. The paper run a follow: ection II preent the tructure of the odel, baed on an energy approach; ection III applie the claical ethod to a tet PMSM; ection IV detail the propoed approach, and applie it to the ae tet PMSM. The following convention are ued: if x ij, ij being αβ or, i the vector with coordinate x i and x j, we write indifferently x ij and x i, x j T ; if f i a function of everal variable, k f denote it partial derivative with repect to the k th variable, and kl f = lk f the econd partial derivative with repect to the k th and l th variable. Latly, the rotation atrix of angle α i denoted by co α in α Rα := ; of coure, RαR in α co α T α =, where R T α i the tranpoe of Rα. II. Energy-baed odel of the PMSM To write a odel of the tar-connected aturated inuoidal PMSM in the frae, we follow the energybaed approach of [], []. All the otor pecific inforation i encoded in the calar agnetic energy function H φ, where φ i the flux linkage vector. H i independent of the electrical rotor angle θ by the auption of inuoidal winding, and independent of the electrical rotor velocity ω a in any conventional electroechanical device. The tate equation of the PMSM then read dφ = v R ı ωj φ dω = nφ T J ı T l J l n

2 dθ = ω, 3 where v i the ipreed potential vector, R the tator reitance, n the nuber of pole pair, T l the load torque and J := gradient of H ı, i.e., ; the tator current vector ı := H φ = H φ D, φ Q H φ D, φ Q i the. 4 The phyical control input i the potential vector v αβ := Rθv ipreed in the αβ frae, i.e., the fictitiou potential vector v rotated by Rθ. Siilarly, the eaured current vector i := Rθı ; in enorle control, thi i the only available eaureent. Taking advantage of the contruction yetrie of the PMSM the tator and the rotor of the PMSM are yetric with repect to a plane, ee [] for detail, we can oreover write H φ D, φ Q = H φ D, φ Q, 5 i.e., the agnetic energy function H of a PMSM i even with repect to the q-axi flux linkage φ Q. Notice that thank to the auption of inuoidal winding, all that i needed to cloe the odel 3 are the flux-current relation 4; thi i no longer the cae for non-inuoidal winding, ince the electro-agnetic torque in 3 explicitly depend on H. Notice alo that the iplet acceptable function i the quadratic for H φ := L D φd M + L Q φq, which yield the current-flux relation ı D = φd M L D ı Q = φq L Q and the electro-agnetic torque T e = n L D M φ Q + n L Q L D φ D φ Q ; in other word, the iplet acceptable agnetic energy function repreent the unaturated PMSM. The unaturated odel i uually ufficient for control above oderately low peed. Thi energy approach enjoy everal intereting feature: it naturally enforce the reciprocity condition ıd ı Q φ D φ Q [3], ince H φ D, φ Q = H φ D, φ Q it yield a valid expreion for the agnetic torque, even in the preence of agnetic aturation it jutifie the odeling of aturation in the fictitiou rotor frae for a tar-connected otor. Though thi point i uually taken for granted, it i not copletely obviou becaue of the nonlinearitie due to aturation that: i the tranforation fro = the phyical abc-frae to the fictitiou -frae behave well; ii the decoupling between the - and the -axe i till valid it require only a very baic knowledge of the otor internal layout finally, it i particularly aenable to an analyi of the effect of ignal injection. In all the experient, the rotor will be locked in a known poition, o that 3 reduce to dφ = R T θ lr v αβ R ı, 6 with θ lr contant and known. Notice that ince θ lr i known, we can conider that v := R T θ lr v αβ i the ipreed potential, and ı := R T θ lr i the available eaureent. III. Claical ethod The ot widely ued ethod to obtain the currentflux relation, ee [7], aue that an ipreed potential trajectory t v t i known, together with the reulting current trajectory t ı t. The correponding flux linkage i obtained by tie integrating 6, i.e., φ t = φ + v τ R ı τ dτ, 7 the unknown initial value φ being yet to deterine. A pair ı t, φ t i thu obtained for each tie t; provided the current trajectory t ı t cover a ufficient area of the current plane, thi yield the deired current-flux relation. The initial value φ i choen o a the flux linkage i zero when the current i zero. Thank to the filtering effect of the integration, the ethod i rather inenitive to eaureent noie. However, it i trongly affected by biae in: the ipreed potential. The power tage i uually an IGBT bridge couted with PWM; owing to voltage drop in the tranitor and dead tie, the actually ipreed potential oewhat differ fro the deired one, ee e.g. [4]. Thi i not a proble if voltage enor are available, a in a laboratory experient, but atter for ipleentation on indutrial drive, which are uually not equipped with uch enor the tator reitance R etiation; thi i the ain proble, ince the value of the reitance can ignificantly vary during a long experient. A. Experiental reult The ethod wa ued to obtain the current-flux curve of a 4 W PMSM rated paraeter in table I. The otor i fed by an.5 kw ATV7 ATV7HU5N4 inverter bridge driven by a dspace board DS5. The rotor wa firt aligned o that the αβ and frae coincide, and then locked with a echanical brake. To check the conitency of the reult, three trajectorie were ued:

3 Current ref. A Current ref. A Current ref. A Flux Wb 4 - Table I Rated paraeter of tet PMSM. Rated power Rated voltage RMS Rated current RMS Rated frequency Rated peed Rated torque 4 W 39.3 V.66 A 6 Hz 8 rp. N Nuber of pole pair n Stator reitance R 4.5 Ω D-axi inductance L D 43.5 H Q-axi inductance L Q 69.5 H Tie Tie Tie Figure. The three current trajectorie on D- and Q-axi ued in the experient. contant ı D, -periodic trapezoidal ı Q with aplitude 3 A, ee fig. top; the experient wa repeated for ı D := 3 A, A, A, A, A, A and 3 A contant ı Q, -periodic trapezoidal ı D with aplitude 3 A, ee fig. iddle; the experient wa repeated for ı Q := 3 A, A, A, A, A, A and 3 A proportional -periodic trapezoidal current on both axe with total aplitude 3 A, ee fig. botto. To enforce thee trajectorie, the current were controlled with Proportional-Integral controller on the D- and Q- axe daping ratio ξ := ; bandwih ω := 5 Hz. The tator reitance i evaluated by coputing the ratio between the voltage and the current during the phae when the current i contant; it i found to vary fro 4.5 A to 5.5 A over the whole experient. The flux linkage i then coputed according to 7. The enitivity to voltage biae i illutrated in fig. : when the tie integration i perfored over everal identical iilar pattern in our cae, the hape of the experiental Current A Figure. Experiental current-flux curve φ D ıd, M and φ Q, ı Q ; olid line: uncopenated voltage drop; dahed line: copenated voltage drop; dotted line: averaged curve. current-flux curve i altered, epecially when the current i all; with uncopenated inverter voltage drop and dead tie, the reult i very bad olid line; with copenation by a uitable odel, the loop in the currentflux curve i uch aller, but till preent dahed line; the final current-flux curve i then obtained by averaging all the flux linkage point obtained for a current point dotted line. The full 3D current-flux relation are diplayed in fig. 3. The agnetic aturation i clearly viible. A expected fro 5, which iplie H φ D, φ Q = H φ D, φ Q H φ D, φ Q = H φ D, φ Q, it can be een that φ D i even with repect to ı Q, and φ Q odd with repect to ı Q. IV. Propoed ethod A. Data acquiition with ignal injection Signal injection wa originally introduced for enorle control at low velocity [8], but it can alo be ued for identification [9]. We follow the quantitative analyi introduced in [6], [9] and tudied in detail in [5]; it i valid even in the preence of nonlinearitie due to agnetic aturation, and whatever the hape of the periodic injected ignal. Ipreing in 6 a potential vector of the for v αβ = v αβ + ṽαβ fωt, 8 where f a -periodic function and Ω i a large frequency, the analyi how that the actual flux linkage i φ = φ + Ω RT θ lr ṽαβ F Ωt + O Ω, 9 where F i the priitive of f with zero ean, and O i the big-o ybol of analyi; φ i the flux linkage without ignal injection, i.e., the olution of dφ = R T θ lr v αβ R ı.

4 ? Q Wb? D! M Wb can be extracted fro the actual eaureent the etiation uing.. t = Ω t = Ω t Ω t Ω τdτ τ Ω t Ω ı αβ τ F Ωt F Ωτdτ { Q A - - { D A a D-axi flux linkage in function of ı In other word, the virtual eaureent ha been ade available beide the actual eaureent. Notice that ince θ lr i known, we can conider that v := R T θ lr v αβ and ṽ := R T θ lr ṽαβ are the ipreed potential, and that ı := R T θ lr and ı := R T θ lr, are the available eaureent. It i then poible to experientally acquire the ix expreion Experientally, it i poible to apply any deired current ı by a uitable choice of v ; by ignal injection with at leat two independent vector ṽ and extraction of the correponding ĩ, it i then poible to obtain the coplete aliency atrix, hence H φ, H φ and H φ. Since the flux linkage φ i unknown, what i actually obtained i H ı, H ı and H ı, where H kl ı := kl H of the flux-current relation 4. ı and i the invere { D A - { Q A b Q-axi flux linkage in function of ı Figure 3. Current-flux relation obtained by the claical ethod. Plugging 9 into 4 and expanding then yield = + γ Ω αβ F Ωt + O Ω, where γ αβ := S θ lr, φ ṽαβ and Sθ, φ := Rθ H φ H φ H φ H R T θ. φ We call S θ lr, φ the aliency atrix ; indeed, it effectively depend on θ lr if the otor exhibit aliency, whether geoetric or induced by agnetic aturation. In other word, how that a all ripple of aplitude := γ Ω αβ, produced by the injected ignal ṽαβ fωt, i uperipoed to the current without ignal injection, produced by v αβ. A explained in [5], both and B. Obtaining the current-flux relation fro the H kl ı By the very definition of, we can write ı, or, ore explicitly, H ı D ı Q ı = H D ı D, ı Q, Q ı D, ı Q = H D ı D, ı Q, Q ı D, ı Q. = Differentiating thee relation with repect to ı D and ı Q, we find that H ı H ı D ı H ı H ı i the identity atrix, which iplie D ı D ı Q ı Q ı = D ı Q ı Q ı =: Hkl ı H kl ı H kl ı H kl ı Ldd ı L dq ı L dq ı L qq ı. We thu know the partial derivative of, or, equivalently, the integrable differential for d D ı = L dd ı dı D + L dq ı dı Q d Q ı = L dq ı dı D + L qq ı dı Q.

5 Meaured q-axi current A To recover ı D = I, we integrate thee for on path κ κ in the current plane. Thi yield I κ D I κ = = κ L dd I κ κ κ ζ di D dζ ζdζ κ d D dζ I ζ dζ + L dq I ζ di Q ζdζ, κ dζ and a iilar expreion for Q I κ. Therefore, for each κ, a couple I κ, D I κ i obtained; provided the current path κ I κ cover a ufficient area of the current plane, thi yield the deired currentflux relation. The initial value I κ a i choen o a the flux linkage i zero when the current i zero. Notice the ethod i copletely iune to error in the knowledge of the tator reitance R, ince it never explicitly ue it value. Moreover, it i le enitive than the claical ethod to iperfection of the power inverter becaue the bae potential v doe not need to be known, wherea the injected potential v can be fairly large. C. Experiental reult The propoed approach wa applied with the ae experiental condition a in ection III-A, with iilar deired current path. Neverthele, the current were not enforced by controller not to diturb the ignal injection. A a conequence the actual current path lightly differ fro the deired one, ee fig. 4; thi i not a proble ince the true value of ı are known. The current path were dicretized with tep ı of about. A; for each current point ı, the injected ignal wa a 5 long quare ignal of frequency Ω := 5 Hz and aplitude ũ := 4 V, lowly rotating at frequency f i := Hz, u coπfi t := ũ. inπf i t The correponding current ripple, naely ı = ũ H ı coπf i t + H ı inπf i t, Ω H ı coπf i t + H ı inπf i t then directly yield H ı, H ı and H ı. The interet of lowly rotating the injected ignal i to provide any eaureent point ĩ, hence an accurate deterination of H ı, H ı and H ı. The flux linkage along a current path i then obtained by dicretizing the integral in, D I κ D I κ κ L dd I ζ I D ζ + L dq I ζ I Q ζ, ζ=κ + where I ζ := I ζ I ζ. The current-flux curve o obtained are diplayed in fig. 5. The conitency Meaured d-axi current A Figure 4. Actual ı path. can be checked by coputing the flux linkage difference at point where the current path interect, which hould be zero in theory: the axiu relative error i.3 % for the D-axi flux and.9 % for the Q-axi. Moreover, a with the claical ethod, φ D i even with repect to ı Q, and φ Q odd with repect to ı Q. Notice the ethod provide the Heian atrix of the energy function in function of ı H H H H or φ, i.e., the partial derivative of the current-flux relation. It i a ueful piece of inforation for fitting a paraetric odel to the current-flux relation; indeed, it i nuerically better to fit the derivative of a function than the function itelf. It alo provide it invere atrix, Ldd L dq, L dq L qq which i the inductance atrix. A expected fro 5, which iplie H φ D, φ Q = H φ D, φ Q φ D, φ Q = H H H φ D, φ Q φ D, φ Q = H φ D, φ Q, the parity/iparity relation are experientally atified, both for the Heian atrix and it invere. Thi i illutrated for the inductance atrix in in fig. 6; the cro- aturation i even ore viible than in fig. 5. D. Coparion with the claical ethod Fig. 7 copare oe current-flux curve obtained by the claical and the propoed ethod. The two ethod yield iilar curve, with neverthele oe all,

6 ? Q Wb? D! M Wb H D H H! 4! H { Q A - - { D A a D-axi flux linkage in function of ı -.? D Wb ? Q Wb a Ldd a a function of φ Q { D A - { Q A b Q-axi flux linkage in function of ı -.? D Wb ? Q Wb b Lqq a a function of φ.. Figure 5. Current-flux relation obtained by the propoed ethod. difference. Soe of the difference could be explained by experiental error, epecially biae in the claical ethod. Another poibility i that, owing to hyterei, the flux linkage coputed by ignal injection i yteatically aller than the flux linkage coputed by the claical ethod; thi wa noticed for the Synchronou Reluctance Motor in [], and raie the quetion of which odel hould be ued for enorle control at low velocity V. Concluion We have propoed a ethod baed on ignal injection to obtain the aturated current-flux relation of a PMSM fro locked-rotor experient. With repect to the claical ethod baed on tie integration, it ha the ain advantage of being copletely independent of the tator reitance; oreover, it i le enitive to voltage biae - -.? D Wb c Lqq a a function of φ.? Q Wb Figure 6. Coefficient of inductance atrix obtained by the propoed ethod..

7 Flux Wb Flux Wb Current A a ı D D ıd, ıq M for ı Q = A, ı Q = A, ı Q = A Current A b ı Q Q ı D, ıq for ı D = A, ıd = A, ı D = A Figure 7. Coparion of current-flux curve obtained with the claical dahed line and propoed ethod olid line. due to the power inverter, a the injected ignal ay be fairly large. Beide, the ethod provide the inductance atrix a a function of the current or the flux linkage, which i an intereting piece of inforation by itelf, and can alo be ued to fit a paraetric odel to the currentflux relation; indeed, it i nuerically better to fit the derivative of a function than the function itelf. Reference [] D. D. Reigoa, P. Garcia, D. Raca, F. Briz, and R. D. Lorenz, Meaureent and adaptive decoupling of cro-aturation effect and econdary aliencie in enorle controlled IPM ynchronou achine, IEEE Tranaction on Indutry Application, vol. 44, no. 6, pp , Nov. 8. [] Y. Li, Z. Zhu, D. Howe, C. Bingha, and D. Stone, Iproved rotor-poition etiation by ignal injection in bruhle AC otor, accounting for cro-coupling agnetic aturation, IEEE Tranaction on Indutry Application, vol. 45, pp , 9. [3] P. Sergeant, F. De Belie, and J. Melkebeek, Effect of rotor geoetry and agnetic aturation in enorle control of PM ynchronou achine, IEEE Tran. Magnetic, vol. 45, no. 3, pp , 9. [4] N. Bianchi, E. Fornaiero, and S. Bolognani, Effect of tator and rotor aturation on enorle rotor poition detection, IEEE Tranaction on Indutry Application, vol. 49, no. 3, pp , May 3. [5] J.-H. Jang, S.-K. Sul, J.-I. Ha, K. Ide, and M. Sawaura, Senorle drive of urface-ounted peranent-agnet otor by high-frequency ignal injection baed on agnetic aliency, IEEE Tran. Indutry Application, vol. 39, pp. 3 39, 3. [6] A. K. Jebai, F. Malrait, P. Martin, and P. Rouchon, Senorle poition etiation and control of peranentagnet ynchronou otor uing a aturation odel, International Journal of Control, vol. 89, no. 3, pp , 6. [7] G. Stuberger, B. Polajzer, B. Stuberger, M. Toan, and D. Dolinar, Evaluation of experiental ethod for deterining the agnetically nonlinear characteritic of electroagnetic device, IEEE Tranaction on Magnetic, vol. 4, no., pp , Oct 5. [8] P. Janen and R. Lorenz, Tranducerle poition and velocity etiation in induction and alient AC achine, IEEE Tran. Indutry Application, vol. 3, pp. 4 47, 995. [9] A. Jebai, F. Malrait, P. Martin, and P. Rouchon, Etiation of aturation of peranent-agnet ynchronou otor through an energy-baed odel, in IEEE International Electric Machine & Drive Conference,, pp [] P. Cobe, F. Malrait, P. Martin, and P. Rouchon, Modeling and identification of ynchronou reluctance otor, in IEEE International Electric Machine & Drive Conference, 7. [] A. Jebai, P. Cobe, F. Malrait, P. Martin, and P. Rouchon, Energy-baed odeling of electric otor, in IEEE Conference on Deciion and Control, 4, pp [] P. Cobe, A. K. Jebai, F. Malrait, P. Martin, and P. Rouchon, Energy-baed odeling of AC otor, ArXiv e-print, 6, arxiv:69.85 [ath.oc]. [3] J. Melkebeek and J. Wille, Reciprocity relation for the utual inductance between orthogonal axi winding in aturated alient-pole achine, Indutry Application, IEEE Tranaction on, vol. 6, no., pp. 7 4, Jan 99. [4] A. R. Weber and G. Steiner, An accurate identification and copenation ethod for nonlinear inverter characteritic for ac otor drive, in IEEE International Intruentation and Meaureent Technology Conference Proceeding, May, pp [5] P. Cobe, A. K. Jebai, F. Malrait, P. Martin, and P. Rouchon, Adding virtual eaureent by ignal injection, in Aerican Control Conference, 6, pp