Publication IV IEEE. Reprinted with permission.

Transcription

1 Publication IV Piippo, A. and Luomi, J. (6). Inductance harmonic in permanent magnet ynchronou motor and reduction of their effect in enorle control. In Proceeding of the XVII International Conference on Electric Machine (ICEM 6), no. 138, Chania, Greece, CD-ROM. 6 IEEE. Reprinted with permiion. Thi material i poted here with permiion of the IEEE. Such permiion of the IEEE doe not in any way imply IEEE endorement of any of the Helinki Univerity of Technology product or ervice. Internal or peronal ue of thi material i permitted. However, permiion to reprint/republih thi material for advertiing or promotional purpoe or for creating new collective work for reale or reditribution mut be obtained from the IEEE by writing to pub-permiion@ieee.org. By chooing to view thi material, you agree to all proviion of the copyright law protecting it.

2 138 1 Inductance Harmonic in Permanent Magnet Synchronou Motor and Reduction of Their Effect in Senorle Control Antti Piippo and Jorma Luomi Abtract The paper deal with peed and poition etimation of permanent magnet ynchronou motor having unwanted patial harmonic in the tator inductance and in the permanent magnet flux. The enorle control i baed on an adaptive oberver that i augmented with a pulating high-frequency ignal injection technique at low peed. The effect of harmonic on the peed and poition etimation are reduced by adding harmonic model of the inductance and the flux to the adaptive oberver and by modifying the injected high-frequency voltage ignal baed on the inductance variation. Simulation and laboratory experiment how that the etimation accuracy i improved and torque pulation are reduced. Index Term Permanent magnet ynchronou motor, enorle control, ignal injection, patial harmonic. I I. INTRODUCTION NFORMATION on the rotor poition i required for the vector control of a permanent magnet ynchronou motor (PMSM). In enorle control, the rotor peed and poition are obtained by mean of a uitable etimation algorithm. Thee etimation algorithm can be claified into fundamental-excitation method [1] [3] and ignal injection method [4] [6]. The former, baed on the back-emf of the motor, have good dynamic propertie, wherea the latter, baed on detecting magnetic aniotropy, are uited for etimation at low peed. To combine the advantage of the two approache, the etimation method can be changed from a fundamental-excitation method to either a high-frequency (HF) ignal injection method [7], [8] or a combined method [9], [1] at low peed. In permanent magnet ynchronou machine, unidealitie in the rotor geometry and in the magnetomotive force ditribution reult in a non-inuoidal variation of the back-emf and in patial harmonic in the tator inductance [11]. Thee unidealitie not only caue ripple in the electromagnetic torque [1], but alo lead to error in the rotor peed and poition etimation. The effect of inductance harmonic have earlier been modeled in rotating HF ignal injection method for induction Manucript received June 3, 6. Thi work wa financially upported by ABB Oy and Walter Ahltröm Foundation. The author are with the Power Electronic Laboratory, Helinki Univerity of Technology, P.O. Box 3, FI-15 TKK, Finland (phone: ; fax: ; antti.piippo@tkk.fi). machine [13] [15]. The aim of thi paper i to increae the accuracy of the rotor peed and poition etimation of PMSM when the patial variation of the tator inductance and flux linkage i noninuoidal. An adaptive oberver i ued throughout the whole peed region, and it i augmented with a pulating HF ignal injection method at low peed [1]. The adaptive oberver i extended with a model for the inductance and flux harmonic, and the HF voltage reference of the ignal injection method i modified for compenating the etimation error caued by the inductance harmonic. The performance of the method i invetigated by mean of imulation and experiment. II. PERMANENT MAGNET MOTOR MODEL The PMSM i modeled in the d-q reference frame fixed to the rotor. The d axi i oriented along the permanent magnet flux, whoe angle in the tator reference frame i θ m in electrical radian. The tator voltage equation i u = R i + ɺ + ω J (1) m where u = [ ud uq ] T i the tator voltage, i = [ id iq ] T the tator current, = [ d q ] T the tator flux, R the tator reitance, ω the electrical angular peed of the rotor, and m The tator flux i 1 J = 1 = Li + () where L i the inductance matrix and i the permanent magnet flux. When the rotor poition angle θ m i varied, the permanent magnet flux linkage of the three tator winding phae conit of a fundamental component and odd harmonic. In the d-q reference frame, the permanent magnet flux ha a contant component on the d axi and harmonic of order multiple of ix. For implicity, the harmonic having order higher than ix are omitted in thi paper, reulting in the permanent magnet flux vector [11] + d6 co(6 θm ) = (3) q6 in(6 θm)

3 138 u iɶ i Voltage model Current error Flux model ˆ, u ˆ, i Error term F Adaptation mechanim Fig. 1. Block diagram of the adaptive oberver. ˆm ω where d 6 and q6 are the coefficient of the ixth-order harmonic on the d and q axe, repectively. The elf and mutual inductance of the tator winding phae have an average component and even harmonic when the rotor poition angle i varied. In the d-q reference frame, the inductance matrix conit of the direct-axi inductance L d and quadrature-axi inductance L q on the d and q axe, repectively, and harmonic of order multiple of ix. When the higher-order harmonic are omitted, the tator inductance matrix i [11], [16] Ld + L6 co(6 θm) L6 in(6 θm) L = L6 in(6 θm ) Lq L6 co(6 θm) (4) where L 6 i the amplitude of the ixth harmonic of the tator inductance. The electromagnetic torque i given by 3p T = i + ( L L ) i i e q d q d q 6 6 m d q L in(6 θ )( i i ) 4L co(6 θ ) i i m d q + i co(6 θ )( + 6 ) q m d 6 q6 id in(6 θm)( q6 + 6 d 6) where p i the number of pole pair. A. Adaptive Oberver III. OBSERVER STRUCTURE In the adaptive oberver, the rotor peed and poition etimation i baed on the etimation error between two different model (a reference model and an adaptive model). In the following, the tator flux i the etimated variable, and the etimate of the rotor peed i ued for adjuting the adaptive model [1]. The block diagram of the adaptive oberver i hown in Fig. 1. The oberver i formulated in the etimated rotor reference frame. The reference model i baed on (); thi flux model can be written a, i (5) ˆ = Li ˆ + ˆ (6) Etimated quantitie are marked by ˆ and meaured quantitie expreed in the etimated rotor reference frame are marked by. Originally, the tator inductance matrix i a diagonal matrix with the diagonal element L d and L q, and the permanent magnet flux ha only the contant component. They can be extended to include the harmonic uing the definition (3) and (4). The adaptive model i baed on (1) and (). It i here referred to a a voltage model, and defined by ˆɺ = u Rˆ ˆi ˆ ω J ˆ + λ ɶ i (7), u m, u where the etimate of the tator current and the etimation error of the tator current are ˆ ˆ 1 i = L ˆ, u ˆ ( ) (8) ɶ i = i ˆi (9) repectively. The feedback gain matrix λ i varied a a function of the rotor peed [17]. For the adaptation, an error term i contructed a F = C1ɶ (1) where ɶ = ˆ, i ˆ, u i the difference between the flux etimate, and C 1 = [ 1]. Hence the flux difference in the etimated q direction i ued. The etimate of the electrical angular peed of the rotor, ued in (7), i obtained by a PI peed adaptation mechanim ω = k F k F dt (11) ˆm p i where k p and k i are nonnegative gain. The etimate for the rotor poition i evaluated by integrating ˆm ω. The gain k p and k i of the adaptation mechanim are elected a [1] k p fo α fo, ki ˆ = α ˆ = (1) where the deign parameter α fo correpond to the approximate bandwidth of the adaptive oberver. B. Coupling HF Signal Injection to the Adaptive Oberver Since the adaptive oberver cannot perform well at low peed due to inaccuracie in meaurement and parameter etimate, an HF ignal injection method i ued to tabilize the oberver [1]. A carrier excitation ignal alternating at angular frequency ω c and having amplitude u c i uperimpoed on the d component of the tator voltage in the etimated rotor reference frame. The injected HF voltage ignal i thu 1 u c = uc co( ωct) (13) An alternating HF current repone i detected in the q direction. Demodulation and low-pa filtering reult in an error ignal K ɶ θm, which i approximately proportional to the poition etimation error ɶ θ ˆ m = θm θm, K being the ignal injection gain. The error ignal i ued for correcting the etimated rotor peed and poition by influencing the direction of the tator

4 138 3 flux etimate of the adaptive model. The algorithm i given by and ˆɺ = u Rˆ ˆi ( ˆ ω ω ) J ˆ + λ ɶ i (14), u m, u ω = γ + γ (15) p i dt where γ p and γ i are the gain of the PI mechanim driving the error ignal to zero. In accordance with [9], thee gain are elected a i αi α γ p =, γ i = (16) K 6K where α i i the approximate bandwidth of the PI mechanim. At low peed, the combined oberver relie both on the ignal injection method and on the adaptive oberver. The influence of the HF ignal injection i decreaed linearly with increaing peed, reaching zero at tranition peed ω [1]. At higher peed, the etimation i baed only on the adaptive oberver. IV. MODIFICATION OF THE SIGNAL INJECTION METHOD A. Poition Error Caued by Inductance Harmonic For the HF ignal injection method, it i poible to olve the poition etimation error caued by tator inductance harmonic. The motor model i expreed in term of the tator current by ubtituting () for the tator flux in the voltage equation (1): u = R i + Li ɺ + Li ɺ + ɺ + ω JLi + ω J (17) m m The PMSM model i then tranformed to the etimated rotor reference frame. The reulting tator current derivative in the etimated rotor reference frame i derived in the Appendix. Since the HF ignal injection method i applied only at low peed, the approximation = i ued. The reitive voltage drop i alo omitted when the HF ignal injection i conidered. The permanent magnet flux derivative i proportional to the rotor peed, and i thu alo omitted when i low. The current derivative i implified to ɺi L ɶ L J ɶ JL u (18) = ( θm + θm ) When the HF voltage ignal (13) i fed to the tator winding, the derivative of the q-axi current i obtained from (18). If the rotor poition and the poition error are conidered contant, the q-axi current can be olved by integration. When the ixth inductance harmonic i included in the tator inductance matrix a in (4), the reulting q-axi current i given by (19) below. The q-axi HF current i demodulated, low-pa filtered, and driven to zero in teady tate. Hence the rational expreion in (19) become zero. Setting the numerator of (19) equal to zero yield the rotor poition error L in(6 θ ) ɶ 6 m m = () ( Lq Ld ) L6 co(6 θm) θ The inductance harmonic thu caue a periodic error in the rotor poition etimate when the original HF ignal injection method i ued for the etimation. B. Error Compenation The poition error caued by inductance harmonic can be compenated by modifying the HF voltage ignal (13). It i required that at the zero rotor poition error, the ignal injection method receive the HF current ignal uc 1 i = in( ωct) Ldω c (1) Inerting thi current into (17) give the HF voltage ignal to be fed to the motor. The reitive voltage drop and the backemf are ignored. The reulting modified HF voltage ignal i u 1 + ( L6 / Ld ) co(6 θm) = u co( ω t) ( L6 / Ld )in(6 θm) c c c ωm 5( L6 / Ld )in(6 θm) + u c in( ω c t) ωc 1 5( L6 / Ld ) co(6 θm ) () Thu the HF voltage hould be augmented with a ignal whoe amplitude i a function of the rotor poition and proportional to the amplitude L 6 of the harmonic inductance in order to obtain a more accurate poition etimate. An HF voltage ignal i uperimpoed on the q component of the tator voltage in addition to the d component in the etimated rotor reference frame. V. RESULTS The propoed method wa invetigated by mean of imulation and laboratory experiment. The block diagram of the control ytem compriing cacaded peed and current control loop i hown in Fig.. PI-type peed control with active damping i ued. The current component reference i d, ref and i q, ref are calculated according to maximum torque per current control [18]. The current control i implemented a PI-type control in the etimated rotor reference frame, where the crocoupling term and the back-emf are decoupled [19]. The dclink voltage of the converter i meaured, and a imple current feedforward compenation for dead time and power device voltage drop i applied []. The experimental etup i illutrated in Fig. 3. A ix-pole interior-magnet PMSM (. kw, 15 r) i fed by a frequency converter that i controlled by a dspace DS113 PPC/DSP board. The motor data are given in Table I. Me- L in(6 θ ) + ɶ θ ( L L ) L ɶ θ co(6 θ ) u i = q 6 m m q d 6 m m c Ld Lq + L6 co(6 θm)( Lq Ld ) L ω 6 c in( ω t) c (19)

5 138 4, ref Speed contr. θˆm ˆm ω i, ref Curr. contr Adaptive oberver Error ignal u u, ref c + + dq αβ i dq αβ PMSM Fig.. Block diagram of the control ytem. Block Speed contr. include both the peed controller and the minimization of the current amplitude. Freq. converter PMSM Servo Speed for monitoring Computer with DS113 Freq. converter Fig. 3. Experimental etup. Mechanical load i provided by a ervo drive. chanical load i provided by a PMSM ervo drive. An incremental encoder i ued for monitoring the actual rotor peed and poition. The nominal dc-link voltage i 54 V, and the witching frequency and the ampling frequency are both 5 khz. The HF carrier excitation ignal ha a frequency of 833 Hz and a zero-peed amplitude of 4 V, and the amplitude become zero at the tranition peed ω =.13 p.u. The electromagnetic torque i limited to Nm, which i 1.57 time the nominal torque T N. Other parameter of the control ytem are given in Table II. The MATLAB/Simulink environment wa ued for the imulation. Fig. 4 how the rotor poition error a a function of the rotor poition at no-load in teady tate. The rotor peed reference wa a low a.15 p.u in the imulation. The imulation reult obtained uing the original HF voltage ignal (13) correpond well to the error evaluated analytically uing (). When the modified HF voltage ignal () i ued, the etimation error i reduced ignificantly. Although the frequency of the HF voltage ignal i above the current controller bandwidth of 4 Hz, the current control trie to compenate the HF current. To reduce thi compenation, the etimated current i ued for feedback in the proportional part of the current controller intead of the meaured current. However, the current controller caue a mall periodic error in the etimated poition, which can be een in Fig. 4. Almot zero error could be achieved by filtering the current or the fundamental voltage reference, but the filtering would degrade the current control performance. Fig. 5 how experimental reult at the nominal load torque and contant peed in the motoring mode. Only the adaptive 1 TABLE II CONTROL SYSTEM PARAMETERS Current controller bandwidth π 4 rad/ Speed controller bandwidth π 5 rad/ Speed adaptation bandwidth α fo π 5 rad/ Bandwidth α at zero peed π 5 rad/ i TABLE I MOTOR DATA Nominal power. kw Nominal voltage 37 V Nominal current 4.3 A Nominal frequency 75 Hz Nominal peed 1 5 r Nominal torque T N 14 Nm Number of pole pair p 3 Stator reitance R 3.59 Ω Direct-axi inductance L d 36 mh Quadrature-axi inductance L q 51 mh Inductance harmonic amplitude L mh Permanent magnet flux.545 V Flux harmonic coefficient d 6 1. mv Flux harmonic coefficient q6 1.4 mv Total moment of inertia.15 kgm Fig. 4. Rotor poition etimation error a a function of the rotor poition: analytical (olid), imulated with original HF voltage ignal (dahed), and imulated with modified HF voltage ignal (dotted). oberver contribute to the peed etimation at the peed =. p.u. Reult obtained without and with the ixth harmonic in the flux vector (3) and in the inductance matrix (4) are depicted. The peed variation at the ixth harmonic of the tator frequency i reduced ignificantly when the model for the inductance and flux variation i added to the oberver. It i to be noted that electromagnetic torque harmonic exit even without any poition error. Experimental reult howing peed reference tep at low peed are hown in Fig. 6. The HF ignal injection i in ue at the peed. p.u. When the model for the harmonic i not ued, pulation at the ixth harmonic of the tator frequency can be een in the rotor peed and it etimate. When the ixth harmonic i added to the inductance and flux model and the modified HF voltage ignal i ued, the peed pulation i reduced and the poition etimation accuracy i improved. Al-

6 t () Fig. 5. Experimental reult at nominal load torque and contant peed operation: electrical angular peed (olid), it etimate (dahed), and it reference (dotted). Firt ubplot how the operation without model for inductance and flux variation, and econd ubplot how operation with model for inductance and flux variation. though the pulation at the ixth harmonic i reduced, error at other frequencie are till preent, for example, due to current meaurement inaccuracie. Fig. 7 depict experimental reult of a low peed reveral at the nominal load torque. The poition etimation accuracy at low peed i improved due to the modified HF voltage ignal, and the pulation of the peed etimate i reduced. VI. CONCLUSION The rotor peed and poition of a PMSM can be etimated in a wide peed range by mean of an adaptive oberver that i augmented with an HF ignal injection technique at low peed. The effect of inductance and flux harmonic on the peed an poition etimation can be reduced by imple modification in the algorithm. Model for the patial harmonic are added to the tator inductance and the permanent magnet flux, and a poition-dependent component i added to the injected HF voltage ignal. In the example given in the paper, the ixth harmonic of the inductance and the flux are taken into account and, conequently, the ixth harmonic in the etimation error, torque and peed are reduced. APPENDIX The tator voltage and current tranformed from the true rotor reference frame into the etimated rotor reference frame are u = Tu (3a) i = Ti (3b) repectively, where T = co( ɶ θm ) I + in( ɶ θm) J i the coordinate tranformation matrix. The vector u and i are olved from (3) and ubtituted in (17). Solving the tator voltage in the etimated rotor reference frame give d u i TL T i TLT ɺ i dt = R + ( ) + 1 ωm ωm + T ɺ + TJLT i + TJ After applying the product rule for differentiation and differ- (4) t () (a) t () (b) Fig. 6. Experimental reult howing peed reference tep at zero load torque: (a) original HF voltage ignal and no model for inductance and flux variation; (b) modified HF voltage ignal and model for inductance and flux variation. Firt ubplot how electrical angular peed (olid), it etimate (dahed), and it reference (dotted). Second ubplot how etimated electromagnetic torque (olid) and load torque reference (dotted). Lat ubplot how etimation error of rotor poition in electrical degree. entiating T, the current derivative can be olved. The derivative of the tator inductance matrix i L ɺ = ω (5) 6 m JL h where the ixth-harmonic contribution to the tator inductance matrix i co(6 θm) in(6 θm ) L h = L6 in(6 θm) co(6 θm ) (6) When the rotor poition error i aumed mall and hence co( ɶ θm) 1, in( ɶ θm ) ɶ θ, and ɶ m θm, the current derivative in the etimated rotor reference frame i:

7 t () (a) t () (b) Fig. 7. Experimental reult howing low peed reveral at nominal load torque: (a) original HF voltage ignal and no model for inductance and flux variation; (b) modified HF voltage ignal and model for inductance and flux variation. Explanation of the curve are a in Fig. 6. ɺi L u ɶ L Ju ɶ JL u 1 = θm + θm 1 RL i + R ɶ θml Ji R ɶ θmjl i + ɶ ωmji + 6ωmL JLhi 6ω ɶ mθml JLhJi + 6ω ɶ mθmjl JLhi ωml JLi + ω ɶ mθml JLJi ω ɶ mθmjl JLi L ɺ ɶ θmjl ɺ ω 1 ml J ω ɶ 1 mθm JL J (7) enor, IEEE Tran. Ind. Applicat., vol. 8, no. 6, pp , Nov./Dec 199. [3] S. Bolognani, R. Oboe, and M. Zigliotto, Senorle full-digital PMSM drive with EKF etimation of peed and rotor poition, IEEE Tran. Ind. Electron., vol. 46, no. 1, pp , Feb [4] M. Schroedl, Senorle control of AC machine at low peed and tandtill baed on the INFORM method, in Conf. Rec. IEEE-IAS Annu. Meeting, vol. 1, San Diego, CA, Oct. 1996, pp [5] P. L. Janen and R. D. Lorenz, Tranducerle poition and velocity etimation in induction and alient AC machine, IEEE Tran. Ind. Applicat., vol. 31, no., pp. 4 47, March/April [6] M. Linke, R. Kennel, and J. Holtz, Senorle poition control of permanent magnet ynchronou machine without limitation at zero peed, in Proc. IEEE IECON, vol. 1, Sevilla, Spain, Nov., pp [7] E. Robeichl, M. Schroedl, and M. Krammer, Poition-enorle biaxial poition control with indutrial PM motor drive baed on INFORM and back EMF model, in Proc. IEEE IECON, vol. 1, Sevilla, Spain, Nov., pp [8] M. Turini, R. Petrella, and F. Parailiti, Senorle control of an IPM ynchronou motor for city-cooter application, in Conf. Rec. IEEE- IAS Annu. Meeting, vol. 3, Salt Lake City, UT, Oct. 3, pp [9] A. Piippo, M. Hinkkanen, and J. Luomi, Senorle control of PMSM drive uing a combination of voltage model and HF ignal injection, in Conf. Rec. IEEE-IAS Annu. Meeting, vol., Seattle, WA, Oct. 4, pp [1] A. Piippo and J. Luomi, Adaptive oberver combined with HF ignal injection for enorle control of PMSM drive, in Proc. IEEE IEMDC 5, San Antonio, TX, May 5, pp [11] T. S. Low, K. J. Teng, T. H. Lee, K. W. Lim, and K. S. Lock, Strategy for the intantaneou torque control of permanent-magnet bruhle DC drive, IEE Proc. B, Elect. Power Appl., vol. 137, no. 6, pp , Nov [1] J. Holtz and L. Springob, Identification and compenation of torque ripple in high-preciion permanent magnet motor drive, IEEE Tran. Ind. Electron., vol. 43, no., pp. 39 3, Apr [13] M. W. Degner and R. D. Lorenz, Uing multiple aliencie for the etimation of flux, poition, and velocity in AC machine, IEEE Tran. Ind. Applicat., vol. 34, no. 5, pp , Sept./Oct [14] J. Holtz, Senorle poition control of induction motor an emerging technology, IEEE Tran. Ind. Electron., vol. 45, no. 6, pp , Dec [15] N. Teke, G. M. Aher, M. Sumner, and K. J. Bradley, Suppreion of aturation aliency effect for the enorle poition control of induction motor drive under loaded condition, IEEE Tran. Ind. Electron., vol. 47, no. 5, pp , Oct.. [16] A. Madani, J. Barbot, F. Colamartino, and D. Marchand, An oberver for the etimation of the inductance harmonic in a permanent-magnet ynchronou machine, in Proc. IEEE CCA 97, Hartford, CT, Oct. 1997, pp [17] A. Piippo, M. Hinkkanen, and J. Luomi, Analyi of an adaptive oberver for enorle control of PMSM drive, in Proc. IEEE IECON 5, Raleigh, NC, Nov. 5, pp [18] T. Jahn, G. Kliman, and T. Neumann, Interior permanent-magnet ynchronou motor for adjutable-peed drive, IEEE Tran. Ind. Applicat., vol., no. 4, pp , July/Aug [19] F. Briz del Blanco, M. W. Degner, and R. D. Lorenz, Dynamic analyi of current regulator for AC motor uing complex vector, IEEE Tran. Ind. Applicat., vol. 35, no. 6, pp , Nov./Dec [] J. K. Pederen, F. Blaabjerg, J. W. Jenen, and P. Thogeren, An ideal PWM-VSI inverter with feedforward and feedback compenation, in Proc. EPE 93, vol. 5, Brighton, UK, Sept. 1993, pp REFERENCES [1] R. Wu and G. R. Slemon, A permanent magnet motor drive without a haft enor, IEEE Tran. Ind. Applicat., vol. 7, no. 5, pp , Sept./Oct [] R. B. Sepe and J. H. Lang, Real-time oberver-baed (adaptive) control of a permanent-magnet ynchronou motor without mechanical