Motion Sensorless Control of BLDC PM Motor with Offline FEM Info Assisted State Observer

Transcription

1 2, 2th Internatonal Conference on Optmzaton of Electrcal and Electronc Equpment, OPTIM 2 Moton Sensorless Control of BLDC PM Motor wth Offlne FEM Info Asssted State Observer Aln Ştrban, Ion Boldea, Fellow, IEEE, Gheorghe-Danel Andreescu, Senor Member, IEEE, Dorn Iles, Member, IEEE, and Frede Blaabjerg, Fellow, IEEE Dept. of Electrcal Machnes and Drves, Dept. of Automaton and Appled Informatcs, Unversty Poltehnca of Tmsoara, Tmsoara, Romana, e-mal: strban.aln@yahoo.com, boldea@lselnux.upt.ro, danel.andreescu@aut.upt.ro ebm-papst St. Georgen GmbH&Co. KG., 782 St. Georgen, Germany, e-mal: les@eee.org Insttute of Energy Technology, Aalborg Unversty, Aalborg, Denmark, e-mal: fbl@et.aau.dk Abstract Ths paper descrbes a new offlne FEM asssted poston and speed observer, for brushless dc (BLDC) PM motor drve sensorless control, based on the lne-to-lne PM flux lnkage estmaton. The zero-crossng of the lne-to-lne PM flux lnkage occurs rght n the mddle of two commutaton ponts, whch s used as a bass for the poston and speed observer. For performance applcatons, the poston between commutaton ponts s obtaned by comparng the estmated lne-to-lne PM flux lnkage wth the FEM calculated lne-to-lne PM flux lnkage. Even f the proposed observer reles on the fundamental model of the machne, a safe startng strategy under heavy load torque, called I-f control, s used, wth seamless transton to the proposed sensorless control. The I-f startng method allows lowspeed sensorless control, wthout knowng the ntal poston, and wthout machne parameters dentfcaton. Dgtal smulatons and expermental results are shown, demonstratng the relablty of the FEM asssted poston and speed observer for BLDC PM motor sensorless control operaton. Index Terms Brushless dc (BLDC) motor, sensorless control, startng method, varable speed range. F I. INTRODUCTION OR brushless dc permanent magnet (BLDCPM) motors, many sensorless control strateges for obtanng the rotor poston and speed have been proposed n the lterature [- 6]. One category, the back-emf sensng technque () [3-8], s a scheme estmatng the rotor poston ndrectly, by usng the zero-crossng pont detecton from the termnal voltage of the unenergzed phase wndng. It s the most popular sensorless control method, and has been mproved for wde range speed control [6]. Other sensorless methods are: back- EMF ntegraton technques (2) [9]; flux lnkage-based technque (3) [9]; and freewheelng dode conducton (4) []. All mentoned methods have advantages (easy mplementaton and low computatonal burden) as well as dsadvantages (need of an external hardware crcutry [, 2] and [4], error accumulaton problem at low speeds [2] and [3], poston error n transent state ([] and [4]). References [2] and [3] ntroduce a sensorless poston detecton technque based on a speed-ndependent poston functon for BLDC motors, whch proved good relablty n a wde speed range. Reference [4] proposes the calculaton of the commutaton nstants based on the slope varatons of the common dc current I MAX, whch s obtaned by takng the absolute values of two of the three phase currents. Ths last technque ntroduces unavodable delay n the swtchng status at hgher frequences. Ths paper descrbes a new offlne FEM asssted poston and speed observer for BLDC PM motor drve sensorless control, based on the lne-to-lne PM flux lnkage. Usng measured phase currents and lne-to-lne voltages that can be measured or calculated (by multplcaton of the dc bus voltage by swtchng status, whch are known n the controller), the lne-to-lne PM flux lnkage can be estmated. The zero-crossng of the lne-to-lne PM flux occurs rght n the mddle of two commutatons ponts. At constant or slowly varyng speed, the tme perod from one commutaton pont to zero-crossng, and the tme perod from zero-crossng to the next commutaton pont are equal to each other. Ths s used as a bass for the poston and speed observer. Snce the shape of lne-to-lne PM flux lnkage s dentcal at all speeds, t provdes rather precse commutaton ponts. For BLDC PM motors where the objectve s to acheve quas-square current waveforms, only the poston of commutaton ponts s necessary. But, for specal purpose control strateges (e.g. advanced angle control for the feld weakenng operaton), the poston between commutaton ponts s obtaned by comparng the estmated lne-to-lne PM flux lnkage wth the FEM calculated one (or the one measured under no-load operaton). As the proposed observer reles on the fundamental model of the machne, a safe startng strategy under heavy load torque s stll needed. For the startng method from standstll to a certan low speed, varous methods were proposed: nductance varaton method [] or the algn and go method wth the peak currents lmted durng the startng perod [6]. In ths paper, a safe, robust sensorless start-up method, called I-f control [7], wthout ntal poston detecton, s used. Seamless transton from I-f control to the proposed sensorless control, based on the lne-to-lne PM flux lnkage s acheved. Dgtal smulatons and expermental results demonstrate the relablty of the offlne FEM asssted poston and speed observer for BLDC PM motor sensorless control operaton. The paper s organzed as follows: Secton II: BLDC PM Machne model, Secton III: I-F control startng method, Secton IV: FEM asssted poston and speed observer, Secton V: Transton strateges, Secton VI: Smulaton results, Secton VII: Expermental results, Secton VIII: Concluson //$26. '2 IEEE 32 Authorzed lcensed use lmted to: Gheorghe Andreescu. Downloaded on August 3,2 at 2:6:7 UTC from IEEE Xplore. Restrctons apply.

2 II. BLDC PM MACHINE MODEL Fg. a shows the cross-secton of the nteror permanent magnet brushless dc motor wth 8 poles and (6+6) nonunform slots wth concentrated wndngs, chosen as a case study [8]. The BLDC PM specfcaton data and machne parameters are summarzed n the Appendx. (It has been shown [9] that, wth segmental skewng the full torque pulsatons wth trapezodal shape currents s less than 3%, despte of the IPM character of the rotor). B C B C A A A A Fg.. BLDC PM motor cross-secton. Generally, BLDC PM motors are drven by a three-phase nverter wth what s called sx-step commutaton. Each phase s conductng 2 electrcal degrees. Therefore, for ths knd of machne, only two of the three phases are conductng at any tme, leavng the thrd (floatng) phase, open. Fg. 2 shows an equvalent crcut of the three-phase BLDC PM motor drve. /2V DC /2V DC T T 3 D D 3 V a a V b V c T 4 D 4 b T 6 D 6 T D c T 2 D 2 a V ab b V bc V ca c C B C B R s V bn L bb Fg. 2. Schematc nverter system dagram. R s L E a aaeb n Each phase n an ac motor can be descrbed by a frst order dfferental equaton. The general equatons of the phase voltages can be represented n the matrx form, as: ( θ,,, ) ( θ,,, ) ( θ,,, ) Va a Ψ A r a b c Vn d Vb = Rs b + ΨB r a b c Vn dt + Vc c ΨC r a b c V n where V x s the phase voltage, R s s the phase resstance, x s the phase current, θ r s the rotor poston, Ψ x s the total flux lnkage of the wndng, and V n s the voltage between motor and nverter neutral ponts. The flux lnkage n a phase ncludes both self and mutual flux lnkages, and may be defned n the followng form: Ψ A Laa( θr, Lab( θr, Lac( θr, a λpma( θr) Ψ B Lba( θr, Lbb( θr, Lbc( θr, = b + λpmb( θr) Ψ L ( θ, ) L ( θ, ) L ( θ, ) λ ( θ ) C ca r a cb r b cc r c c PMc r V an Ec V cn L cc R s () (2) where, the term, L xx (θ r, x ) x, represents the self flux lnkage of phase x; the term, L xy (θ r, y ) y stands for the mutual flux lnkage between phases x and y; and the term, λ PMx (θ r ) s the phase flux lnkage n phase x, due to the permanent magnets. The dependence of machne nductances on currents has been thoroughly treated n [9]; here, however, for control desgn, approxmated constant values are consdered (see Appendx). The phase flux lnkages λ PMx (θ r ), are modeled through Fourer seres, based on FEM-analyss of the BLDC IPM motor: λ θ = A cos k p θ ( ) ( ) PMa r k r k =,3, 2 PMb ( r ) = Ak cos k p r 3 k =,3, 2 PMc ( r ) = Ak cos k p r + 3 k =,3, λ θ θ λ θ θ π π where p s the number of pole pars, and the terms A k, are the frst three coeffcents of Fourer seres (n Appendx, Table II). The back-emf equaton s gven by dλpmx dλpmx dθ e r x = = = kex ωm (4) dt dθr dt where k ex stands for back-emf constants, ω m s the mechancal rotor speed. Substtutng (2) and (4) nto (), the phase voltage equatons become: Va a Laa Lab Lac a ea Vn d Vb = Rs b + Lba Lbb Lbc b eb Vn dt + + () Vc b Lca Lcb Lcc b e c V n Snce the neutral pont of the motor s not avalable, the lne-to-lne voltage equatons are used as follows: Vab a Laa Lab Lac a eab d Vbc = Rs K b + K Lba Lbb Lbc b + ebc dt Vca c Lca Lcb Lcc c e ca where K s the matrx transformaton from phase quanttes to lne-to-lne quanttes, and has the followng form, K = From (6) and knowng the fact that the lne-to-lne back EMF s the lne-to-lne PM flux lnkage dervatve, the lne-tolne PM flux lnkages can be evaluated n the followng form: λ t PMab Vab a λpmbc = Vbc Rs K b dτ λ PMca Vca c Laa Lab Lac a + K Lba Lbb Lbc b Lca Lcb L cc c Drect measurement of the phase currents and lne-to-lne voltages, can allow the estmaton of the lne-to-lne PM flux (3) (6) (7) (8) 322 Authorzed lcensed use lmted to: Gheorghe Andreescu. Downloaded on August 3,2 at 2:6:7 UTC from IEEE Xplore. Restrctons apply.

3 lnkages. The lne-to-lne voltages can also be calculated, by multplcaton of the dc bus voltage and swtchng status. From these lne-to-lne PM flux lnkages, the poston of the rotor can be estmated as explaned n the next sectons. III. I-F CONTROL STARTING METHOD I-f control method conssts n rampng the stator current frequency whle the current reference s kept constant by the current controller [7]. In ths paper, durng the experments, n order to have a fnte acceleraton durng start-up, the frequency tme-varaton form was chosen as: fmax ( cos ( ω f t) ), ωf t π f = 2 (9) fmax, ωf > t π ωf tf = π 2 π ff tf = π ff = () 2 t f where t f s the tme perod of the reference frequency ramp (t f =.2 [sec], n our case). The frequency s used to calculate a current reference angle θ er, whch s the feedforward poston-angle correspondng to the mposed synchronous reference frame: IV. θ () er = 2 π f dt FEM ASSISTED POSITION AND SPEED OBSERVER A. Commutaton Strategy and Poston Observer Fg. 3 shows the back-emfs, phase currents, lne-to-lne PM flux lnkages of the BLDC PM motor n [8], and the detecton of the zero crossngs of the PM lne-to-lne flux lnkages. Back-EMFs [V] Phase Currents [A] Lne-to-lne PM flux lnkages [Wb] Zero-crossng detecton of the lne PM flux lnakges a b c x -3 Mode Mode 2 Mode 3 Mode 4 Mode Mode 6 λ PMbc λ PMca e a e a e b e b e c e c tme [s] Fg. 3. Bass of the sensorless operaton of the BLDC PM: back-emfs; phase currents; lne-to-lne PM flux lnkages; zero-crossng detecton of the lne-to-lne PM flux lnkages. As can be seen n Fg. 3, the zero-crossngs of the lne-tolne PM flux lnkages occur rght n the mddle of two commutatons ponts. At constant or slowly varyng speed, the tme perod from one commutaton pont to zero-crossng, and the tme perod from zero-crossng to the next commutaton pont are equal to each other. Ths s used as a bass for the mplementaton of the poston and speed sensorless observer. In each mode (sector), only two of the three phases are conductng at any tme, leavng the thrd floatng phase open. The phase voltage n the floatng phase s equal wth the Back-EMF. Only one lne-to-lne PM flux lnkage (from Eq. (8)) s used n each mode: ths s the PM flux lnkage between the two actve phases. Based on ths dea, n each mode, the proper estmated lne-to-lne PM flux lnkage can be used to detect the zero-crossng, and then a delay of 3 s appled to swtch-on the current n the correspondng phases (next commutaton pont). After commutaton s produced, a new couple of phases wll be conductng, and the correspondng lne-to-lne PM flux lnkage wll gve the nformaton for the next commutaton. Table I shows the proper lne-to-lne PM flux lnkage at each mode. TABLE I. LINE-TO-LINE PM FLUX LINKAGES AT EACH MODE Mode and 4 Mode 2 and Mode 3 and 6 Postve speed λ PMca λ PMbc The lne-to-lne PM flux lnkage estmator (Fg. 4) based on the voltage model n stator reference frame, employs an equvalent ntegrator n close-loop, wth a speed-adaptve proportonal controller to compensate the dc offset and phasedelay. The transfer functon of ths equvalent ntegrator s: T H( s) = ; T = (2) + st ω where ω s the corner frequency. For BLDC PM motor general drves, where the objectve s just to acheve quas-square current waveforms, knowng the poston of commutaton ponts suffces. But, for specal purpose drves wth advance angle control for the feld weakenng operaton, the poston between commutaton ponts can be obtaned by comparng the estmated lne-tolne PM flux lnkage wth a look-up table, whch contans the poston versus characterzed lne-to-lne PM flux lnkage. Ths PM flux lnkage can be calculated by usng FEMmethod [8], f the motor was desgned usng ths method, or measured by runnng the machne as a generator wth open phases at constant speed. The lne voltage sgnals are recorded and by ntegratng these voltages, the lne-to-lne PM flux lnkages can be calculated. The poston between commutaton ponts can be estmated because the correspondng lne-to-lne PM flux lnkage at each mode s ether ncreasng or decreasng. 323 Authorzed lcensed use lmted to: Gheorghe Andreescu. Downloaded on August 3,2 at 2:6:7 UTC from IEEE Xplore. Restrctons apply.

4 I abc R s V ab, V bc,v ca Κ I abc [L] X Κ ω ω r λ PM lneto-lne Fg. 4. Lne-to-lne PM flux estmator (K s from Eq. (7)). λpm(θer) θer Lookup Table B. Speed Estmaton Wth no sensors avalable, the speed must be calculated (for speed feedback) usng the nformaton from the commutaton controller. The mechancal model of the drve s approxmated as: θer θer d ω r p J ω r p J Te dt = + TL TL θˆer (3) where θ er s the rotor poston, ω r s the rotor electrcal speed and the load torque T L was assumed constant. The electromagnetc torque T e s regarded as known nput and s calculated by the followng equaton: ea a + eb b + ec c Te = = kea a + keb b + kec c (4) ωm From the mechancal model an observer can be constructed: ˆ θ ˆ er θ er k d ˆ ω ˆ ˆ r p J ω r p J = Te k2 ε dt + + () Tˆ ˆ k L T L 3 The PLL observer error s: ( ˆ er ˆ r ) ( ˆ er ) ( ˆ r ) ( ˆ er ) ( ˆ r ) ε = sn θ θ = sn θ cos θ cos θ sn θ (6) The PLL speed observer block dagram s shown n Fg.. Ths observer estmates the rotor speed and the load torque. However, the load torque estmaton s poor. TˆL Tˆe p/j ˆr ω θˆr J k = S, k2 = SP2, k3 = P (7) p where: S = p + p 2 + p 3, SP 2 = p p 2 + p 2 p 3 + p 3 p, P = p p 2 p 3. Gan values are lmted only by nose consderaton. If the nose level s small, t s possble to select all three poles real, negatve. However, for better dynamcs, two of them can be selected complex conjugates wth not too large magnary parts. V. TRANSITION STRATEGIES A. Transton from I-f Control to FEM Asssted Poston and Speed Observer I-f control method s used only for start-up and low speeds. When the reference frequency exceeds a certan level (f mn ), the system automatcally ntates the transton to the sensorless control operaton of the BLDC, based on FEM asssted poston and speed observer (Fg. 6. In order to acheve a smooth transton, the angle used for control has to er go smoothly from θ to ˆer θ : where: ( k) k ˆ θ = θ + θ (8) < ; for f f mn f fmn k = ; for fmn f f fmax fmn ; for f > fmax er er max (9) When the reference frequency reaches f max, the control swtches to FEM asssted poston and speed observer moton sensorless control (k=). Because the I-f feedforward poston-angle and the estmated poston are kept between and 2 π, based on the fact that Δθ sn ( Δθ) for small Δθ, (3) s mplemented n the followng form: θer ; for f < f mn ˆ = er + k sn ( er er ); for fmn f fmax ˆ θer ; for f > f max θ θ θ θ (2) k 3 k 2 k ε sn Δθ θˆer Fg.. Equvalent PLL state-observer structure The observer gans are determned by pole placement method. Let p, p 2, and p 3 be the observer poles, allocated such that the observer s fast and stable. In ths case, the observer gans are: Fg. 6. Control strategy: ) I-f control; 2) FEM asssted poston and speed observer moton sensorless control; 3) I-f control. 324 Authorzed lcensed use lmted to: Gheorghe Andreescu. Downloaded on August 3,2 at 2:6:7 UTC from IEEE Xplore. Restrctons apply.

5 B. Transton from FEM Asssted Poston and Speed Observer to I-f Control The transton from FEM asssted poston and speed observer moton sensorless control, to I-f control, s done automatcally by the system when the reference speed s below a certan level (Fg. 6. When the above condton s met, the control system generates an angle ( θ er - feedforward poston angle): f abs (2) er= dt= 2 f dt θ ω π 2 π f mn back θˆer u x θ er Fg. 7. Transton strategy from FEM asssted poston and speed observer moton sensorless control to I-f method. The transton to I-f control s performed by settng the ntal condton of the feedforward poston angle θ er, equal to the last value of the estmated rotor poston θ er (Fg. 7). VI. SIMULATION RESULTS The FEM asssted poston and speed observer for a brushless dc PM motor drve sensorless control, have been frst verfed through dgtal smulaton. A BLDC PM wth the characterstcs summarzed n the Appendx, was mplemented n Matlab/Smulnk, usng the general equatons of a brushless dc PM motor. In the mplementaton, the phase commutaton phenomenon was consdered [2]. In order to obtan accurate and effcent smulaton [2], the back-emf and the nductances (IPM rotor) were modeled through Fourer seres, and through look-up tables, based on FEM-method calculaton of the BLDC IPM motor, respectvely. Fg. 8 llustrates the performance of the proposed FEM asssted poston and speed observer moton sensorless control at RPM. The phase currents are controlled by a PI controller. The commutaton ponts from Fg. 8, represent n fact, the zero-crossng detecton of the lne-to-lne PM flux lnkages, delayed wth 3. Phase A Commutaton ponts Fg. 8. (contnue. Lne-to-lne PM flux lnkages [Wb] Poston [deg.] Poston error [deg.] Speed [rpm] x -3 estmated from FEM estmated encoder estmated speed actual speed f) Fg. 8. Smulaton results of moton-sensorless control of BLDC PM motor wth the proposed poston and speed FEM asssted observer at RPM, at. Nm load: phase A current; commutaton ponts; lne-to-lne PM flux lnkage; electrcal poston; e) poston error; f) speed. The numercal smulatons demonstrate the relablty of ths observer. Next, expermental results are presented, n order to valdate the FEM asssted poston and speed observer moton sensorless control n real-tme applcatons. VII. EXPERIMENTAL RESULTS The expermental setup conssts of BLDC IPM motor, wth the parameters as n smulatons (Appendx), loaded wth an dentcal nteror permanent magnet synchronous generator wth constant load. The dspace CLP3 platform s used to control the entre drve system (Fg. 9). Encoder BLDC PM motor 3-phase nverter & drver crcut IPMSG Current measurement devce Fg. 9. Expermental set-up. e) 32 Authorzed lcensed use lmted to: Gheorghe Andreescu. Downloaded on August 3,2 at 2:6:7 UTC from IEEE Xplore. Restrctons apply.

6 V dc ω r ˆr ω k pw, k w s s k p, k V dc Duty cycle drecton COMMUTATION STRATEGY ˆer θ CP S a, S b, S c : swtchng status PWM INVERTER a b Vab Vbc BLDC Motor s drecton - z Duty cycle MAX c ˆr ω Speed Estmaton (Fg. ) CP θˆer Lne-to-lne PM flux estmator (Fg. 4) ω r V ca Fg.. Block dagram of the proposed sensorless drve (CP - commutaton ponts). Fg. shows an overall system block dagram of the proposed sensorless drve. In ths experment, the lne-to-lne voltages are measured, but they also can be calculated by multplcaton of the dc bus voltage by swtchng status, whch are known n the controller. The use of the measured lne-to-lne voltages nstead of the calculated ones was done n order to acheve better stablty and dynamc responses durng the sensorless operaton. The performance of the proposed moton sensorless control system s nvestgated n four cases: constant speed, speed varatons, start-up wth I-f method and transton to the FEM asssted poston and speed observer, and back when the reference speed s below a certan level. A. Constant Speed Operaton The sensorless operaton of the BLDC motor wth the proposed FEM asssted poston and speed observer at constant speed (Fg. rpm, loaded at. Nm), shows a rather satsfactory agreement between encoder poston and estmated poston, certfyng good estmaton results. In both cases there are some dfferences between the estmated and measured speeds. Phase A Lne-to-lne PM flux lnkages [Wb] - - x -3 estmated - from FEM Fg.. (contnue. Poston [deg.] Poston error [deg.] Speed [rpm] estmated encoder estmated speed measured speed 9 e) Fg.. Expermental results of moton-sensorless control of BLDC PM motor wth the proposed poston and speed FEM asssted observer at RPM, at. Nm load: phase A current; lne-to-lne PM flux lnkage; electrcal poston; poston error; e) spee. B. Dynamcs Fg. 2 shows f the FEM asssted poston and speed observer s relable under acceleraton and deceleraton, and that the load s proportonal to speed. Durng these transent tests, the agreement between encoder poston and estmated poston s rather satsfactory. The error between these two postons may be mnmzed by a more carefully tunng of the poston and speed observer. From Fg. 2 t s obvous that the speed estmated wth the PLL observer (Fg. ) s n agreement wth the measured speed, n both steady state and transents. 326 Authorzed lcensed use lmted to: Gheorghe Andreescu. Downloaded on August 3,2 at 2:6:7 UTC from IEEE Xplore. Restrctons apply.

7 Phase A Speed [rpm] estmated speed measured speed Lne-to-lne PM flux lnkages [Wb] Poston error [deg.] Speed [rpm] x -3 estmated from FEM estmated speed measured speed reference speed Fg. 2. Expermental results of moton-sensorless control of BLDC PM motor wth the proposed poston and speed FEM asssted observer for acceleraton from to RPM and deceleraton from to RPM: phase A current; lne-to-lne PM flux lnkage; poston error; speed. C. Transton from I-F Control to FEM Asssted Poston and Speed Observer BLDC PM Moton-Sensorless Control In ths paper, for a smooth start-up, the frequency tmevaraton form was chosen as n (9), up to Hz ( rpm). The control strategy automatcally swtches from I-f control method to the FEM asssted poston and speed observer moton sensorless control, when the reference frequency s between 7 and Hz (Fg. 3). Phase A Poston [deg.] estmated encoder Fg. 3. (contnue. Speed error [rpm] Poston [deg.] Fg. 3. Transton from I-f control method to FEM asssted poston and speed observer: phase A current; poston; speed; and speed error I-f estmated w Start transton End transton Fg. 4. I-f poston, estmated rotor poston, and the angle used for control [deg.], n transton to FEM asssted poston and speed observer. It s obvous from Fg. 4 that when the system automatcally swtches between the two control methods, the angle used for control swtches smoothly (Eq. (2)) from I-f feedforward poston to estmated poston D. Transton from FEM Asssted Poston and Speed Observer BLDC PM Moton-Sensorless Control to I-F Control When the reference frequency s below a certan level (2 Hz), the control strategy automatcally swtches from FEM asssted poston and speed observer moton sensorless control, to I-f control (Fg. ) wth currents frequency equal wth the reference frequency. A seamless transton from the estmated poston to I-f feedforward poston s vsble n Fg. 6 by usng the soluton from Fg. 7. Phase A Poston [deg.] estmated encoder Fg.. (contnue. 327 Authorzed lcensed use lmted to: Gheorghe Andreescu. Downloaded on August 3,2 at 2:6:7 UTC from IEEE Xplore. Restrctons apply.

8 Speed [rpm] Poston [deg.] estmated speed measured speed reference speed Fg.. Transton FEM asssted poston and speed observer to I-f control method: phase A current; poston; and speed To I-f I-f estmated Fg. 6. I-f poston and estmated rotor poston [deg.] n transton to I-f control. VIII. CONCLUSION The paper has ntroduced a FEM asssted poston and speed observer for brushless dc PM motor drve sensorless control, based on the lne-to-lne PM flux lnkage. The lne-to-lne PM flux lnkage can be estmated from measured phase currents and measured (or calculate lneto-lne voltages. The zero-crossng of the lne-to-lne PM flux occurs rght n the mddle of two commutatons ponts. Ths s used as a bass for the poston and speed observer. Ths sensorless control method s used together wth the I- f sensorless control, for start-up and for low speed control, wth seamless transtons between them. Dgtal smulatons and expermental results demonstrate that the FEM asssted poston and speed observer for BLDC PM motor sensorless control operaton, provdes rather precse commutaton ponts, even durng speed or load transent states. APPENDIX BLDC PM parameters: 2 p = 8 rotor poles, (6+6) slots, rated torque T emax =.4 [Nm], rated speed n n = [rpm], dc bus voltage V DC = 2 [V], very low coggng torque T cogg <% T emax, phase resstance R s = [mω], nerta: J= [kg m 2 ], constant values for nductances: L s = L aa = L bb = L cc = [uh], M = L ab = L ac = L ba = L bc = L ca = L cb = 7. [uh] [8]. TABLE II. FOURIER SERIES COEFFICIENTS A A 3 A [Wb] [Wb] [Wb] ACKNOWLEDGEMENT Ths work was partally supported by EU-FP7 EE-VERT Project (Grant agreement no. SCP7-GA EE- VERT). REFERENCES [] J.P. Johnson, M. Ehsan, and Y. Güzelgünler, Revew of Sensorless Methods for Brushless DC, IAS Annual Meetng 999, vol., pp.43-, October 999. [2] I. Boldea and S.A. 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Ştrban, L. Tutelea, D. Iles-Klumpner, and I. Boldea, FEM analyss of concentrated cols nonunform slot (6+6/8) IPMSM fed wth trapezodal current, OPTIM 28, pp. 4-2, May 28. [2] W. Hong, W. Lee, and B.-K. Lee, Dynamc Smulaton of Brushless DC Motor Drves Consderng Phase Commutaton for Automotve Applcatons, IEMDC 7, vol. 2, pp , May 27. [2] O.A. Mohammed, S. Lu, and Z. Lu, A Phase Varable Model of Brushless DC Motors Based on Fnte Element Analyss and Its Couplng wth External Crcuts, IEEE Trans. on Magnetcs, vol. 4, no., pp.76-79, May Authorzed lcensed use lmted to: Gheorghe Andreescu. Downloaded on August 3,2 at 2:6:7 UTC from IEEE Xplore. Restrctons apply.