Saliency Modeling in Radial Flux Permanent Magnet Synchronous Machines

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1 NORPIE 4, Tronheim, Norway Saliency Moeling in Raial Flux Permanent Magnet Synchronou Machine Abtract Senorle control of Permanent Magnet Synchronou Machine i popular for everal reaon: cot aving an ytem reliability. The bai for mot low an zero pee enorle control i that there exit a ifference in the irect an uaerature inuctance of the machine. Thi ifference i alo reefe to a aliency. In thi paper the ifferent contributor to the reulting aliency i moele. Thi type of moeling i ueful for everal reaon: increae unertaning of the funamental of enorle control, incorporating the combine effect of leakage an main inuctance variation, incorporating the effect of loaing of the machine an finally it give valuable inight when the aliency ratio i attempte increae by machine eign. S Sigur Ovrebo (igur@martmotor.no) Smartmotor AS, Stikletaveien 1 N-741 Tronheim, Norway Inex Term Senorle control, aliency moeling, PMSM I. INTRODUCTION enorle control of electric machine ha been a reearch topic for more than two ecae. The objective for thi work i to eliminate the nee for a poition enor by uing the machine a the poition enor. The poition enor, cabling an connector have been a ource of failure for motor control application. For mall rive the enor contribute conierable to the overall cot. The bai for mot low an zero pee enorle control [1-5] i the preent of a ifference in the an inuctance. Thi ifference i referre to a aliency. There are everal ource of aliencie in PM machine: rotor inherent aliency, aturation bae aliency (yoke, teeth) [3], rotor tator teeth harmonic [1], lamination irection bae aliency [7], ey current bae aliency [8], rotor eccentricity bae aliency. In thi paper a ingle aliency moel i evelope from the rotor inherent aliency an the aturation inuce aliency. The term aliency can be unertoo when coniering a alient pole ynchronou machine tructure (Figure 1). In thi tructure the rotor ha an inherent aliency ue to the hape of the rotor. In the -axi the air-gap i mall while there i a large air-gap in the -axi. The -axi inuctance will in thi cae be larger then the inuctance in the -axi. The injection bae enorle control cheme [1-5] extract rotor poition information from the ifference in the inuctance. Prof. Roy Nilen, Prof. Robert Nilen Norwegian Univerity of Science an Technology, 7491 Tronheim, Norway axi axi Figure 1 Salient pole ynchronou machine II. SAIENCY MODEING In Senorle Control of PMSM the rotor inherent aliency or the aturation bae aliency i uually ue a the bai for the aliency. The preente aliency moel i implifie a only thee ource of aliency are inclue. In orer to illutrate the aturation bae aliency a implifie moel of a 3 kw IPMSM machine wa imulate in FEMab. The relative permeability function (from upplier) wa inclue in the FEM imulation. In Figure the machine i unloae an only the flux from the permanent magnet contribute to the fiel. Three region of the machine can be een to have low relative permeability value: the permanent magnet (in rotor), the tator teeth (in the main flux path) an the tator yoke (in the main flux path). The tator teeth are more aturate compare to the portion of the tator yoke that i aturate. axi axi Yoke aturation Tooth aturation Figure Relative permeability, IPMSM (8-pole) FEM imulation

2 NORPIE 4, Tronheim, Norway The moeling of aliency i one by moeling the inuctance in a -rotor oriente reference frame. The inuctance i eparate in two part: the leakage inuctance an the main inuctance. In thi way the moel can be ue to incorporate the aliency ene by carrier ignal that i concentrate in the leakage flux path or main flux path (or both). The enorle cheme [1-5] will in general ene the reulting inuctance but the high freuency carrier flux itribution etermine the component of main- or leakage flux. In the firt tage of the moeling the machine will be moele at no loa (only flux from the permanent magnet). The loa epenency i inclue in the lat ection of the moeling. Figure 3 illutrate why the main an leakage inuctance houl be moele eparately: the main flux an the leakage flux are 9 electrical egree patially hifte. In orer to implify the moeling ome implification are mae: Only funamental component are coniere The flux from the magnet an the wining are aume inuoial itribute over the air gap. The itribute wining are implifie by repreenting them a a two axi moel. Saturation effect are aume to be inuoial itribute (uperpoition applie). Relative permeability aume to wary linear with flux enity Thee implification may eem crue. Some lamination material may have extremely non linear relation between relative permeability an flux enity. The machine in the tet etup ha lamination from Congent (M35-35). The relative permeability i erive from the BH curve from the upplier. Figure 4 how the relative permeability for the lamination (meaurement error in the 1 an Hz curve). From the 5 Hz curve an approximate linear relation can be obtaine in the region.4 to 1.3 T. The non linearity in the lower region i relate to the firt magnetization curve of the material. When the machine moel i coniere thi effect i neglecte a the lamination have been expoe to a β I ψ Figure 4 Relative permeability for 5,1, an 4 Hz excitation freuencie (Congent M35-35 lamination-upplier ata). magnetic fiel an the time contant for the material to return to it original ate i very large. A. No loa moel In the no loa moel only the flux from the permanent magnet are coniere. The inuctance will be moele in -rotor reference frame an the main an leakage inuctance will be moele eparately. 1) Main inuctance In the firt tage the machine i coniere a a uniform tructure with no magnet or aturation. main= (1) main= () The ource of aliency for a no loa conition i hown in Figure 5. Thee effect are inclue in the moel a: main= r t y (3) - main inuctance (ymmetrical machine) r - rotor eign (low relative permeability in magnet an inuce current in magnet) - tator tooth aturation t - yoke aturation y eakage flux region ξ α The yoke aturation i aume to have minimal impact on the -axi inuctance. If high freuency carrier i ue there may be inuce current in the magnet ue to the carrier flux. The effect of inuce current in the magnet i reuce -axi inuctance. Thi effect i inclue in the moel by the term r (combine effect of low permeability an inuce current). The inuce current i highly epenent on the carrier freuency [8]. Figure 3 Current itribution, current an flux pace vector

3 NORPIE 4, Tronheim, Norway β tator teeth aturation Stator eakage Flux permanent magnet θ α tator yoke aturation Rotor Figure 6 eakage flux path 3) Reulting no loa moel In the reulting no loa moel the leakage an main inuctance are umme up in each axi: Figure 5 Source of aliency no loa ) eakage inuctance If we conier a flux vector (pace vector) the leakage flux i locate 9 electrical egree patially hifte compare to the main flux itribution. When the - inuctance are moele the reference ytem i referre to the main flux. The effect on the leakage inuctance are there fore hifte 9 electrical egree. The bai for the leakage inuctance moel i alo a uniform tructure with no aturation: leakage = (4) leakage = (5) The leakage flux path are illutrate in Figure 6. Three region may lea to moeling of the inuctance: tator teeth, tator yoke an rotor urface. Variation (or aturation) in the rotor urface i neglecte in thi moel a the urface of the tet machine i uniform an the aturation level i moet. If we conier a tator flux vector oriente in the magnet -axi the leakage flux path for thi main vector can be influence by yoke aturation (aturation ue to flux from the permanent magnet). Thi effect i inclue in the moel a a reuction of the -axi leakage inuctance by a factor y. If a tator current vector i oriente in the -axi the leakage inuctance may be influence by tooth aturation (aturation ue to flux from the permanent magnet). Thi effect i inclue in the moel a a reuction of the -axi leakage inuctance by a factor : t leakage = t (7) leakage = y (6) - leakage inuctance (ymmetrical machine) y - yoke aturation effect on leakage inuctance t - tooth aturation effect on leakage inuctance = main + leakage = + = main + leakage = + t The ifference between an r t y y (8) (9) i the bai for aliency in the machine. The poition epenent part of the inuctance can be illutrate by expreing the inuctance on vector form: j 1e θ =Σ + (1) where: θ - rotor poition + Σ = + Σ = 1 = t + y + r + t + y 1 = r t y t y (11) (1) If the main inuctance variation i ue a the reference the variation in the leakage inuctance influence the magnitue of poition epenent inuctance. If the leakage yoke aturation term ( ) in (1) i larger than the leakage tooth aturation y t term ( ) the leakage term a to the main inuctance aliency. On the other han if t > y the leakage term oppoe the poition epenent term from the main inuctance.

4 NORPIE 4, Tronheim, Norway B. oa epenency Several factor may change from an unloae to a loae machine: flux an aturation level, flux an aturation orientation an aturation region. In thi moel the orientation of the reulting tator flux i coniere in the firt tage. In the next tage aturation from the loa current i inclue. In orer to implify the moel the control optimal control of the PMSM i not coniere. The control trategy * of the machine i aume to be i = an i = i. oa epenent eakage Flux Stator Saturate Region θ Ψ I Rotor Figure 7 oa epenency on tator flux Ψ m The tator flux increae when the machine i loae (Figure 7). The reulting flux vector i hifte by a term θ ue to the loaing of the machine. θ a in ψ i M + ( i ) (13) The hift in the tator flux reult in a hift of the tator yoke an teeth aturation region. In aition to the main flux effect the leakage effect mut alo be inclue. Figure 8 illutrate typical aturation region ue to the leakage flux from the loa current. Thi type of aturation will reult in a loa epenent term in the -axi leakage inuctance ( l ): = (14) leakage t l = + t l (15) The patial hift in the tator flux i incorporate in the moel by hifting the aturation bae aliency by the angle θ : ' jθ j( θ+θ) =Σ + e + 3e (16) Where the upper inex inicate that there may be a change in the aturation level ue to the loaing (a the tator flux increae) an the term in (16) can be expree a: ' ' ' + Σ = ' ' (17) ' 1 r t y + Σ = ' ' t y l r = (18) ' ' ' ' t + y + t + y l 3 = (19) Figure 8 Saturation ue to leakage flux from loa current In thi moel the aturation i aume inuoial itribute an linear. The ifferent term may be combine to form a reulting ingle aliency: ' j 4e ξ =Σ + () ξ = a in( θ ) 3 tan( ) + co( θ ) 4 ( ) co( θ) (in( θ) ) (1) = + + () The reulting moel in (), (1) an () ecribe the relation between the ifferent ource of aliency in the IPMSM an loa current in the -axi. III. EXPERIMENTA WORK A 3 kw raial flux IPMSM (motor ata in Table 1) wa ue in the experimental work. The machine wa eigne for high funamental flux (5-6 Hz) operation uring fiel weakening. Table 1 IPMSM parameter P N [kw] U N [V] I N [A] n N [rpm] R [Ω] p The elf an mutual inuctance wa etimate by applying a tranient excitation [4] to the machine. The excitation euence (t1-t5) i illutrate in Figure 9. The time interval between t1/t an t4/t5 wa 1.5 u an t/t3 an t3/t4 wa 5 u.

5 NORPIE 4, Tronheim, Norway Control ignal to inverter leg A,B,C A B C inverter V I a A U c V V U a U b a n b c IPMSM Figure 1 Meaurement of elf an mutual inuctance t1 t t3 t4 t5 Figure 9 Tranient excitation in the zero vector Figure 1 how the connection between the inverter an the IPMSM. The pule were applie acro the a-phae an neutral point while the b an c phae wa open. The current in the a-phae, the applie voltage an the inuce voltage in the b an c-phae wa meaure. Current an voltage meaurement where ample at time intance t, t3 an t4. From thee meaurement the elf an mutual inuctance can be erive a: t4 t aa = Ua I ) I 3) + I 4) (3) t4 t ab = Ub I ) I 3) + I 4) (4) t4 t ac = Uc I ( t ) I ( t ) + I ( t ) (5) a a 3 a 4 The rotor wa turne by han an the inuctance a a function of the poition wa etimate by current an voltage meaurement. In Figure 11 there i a large ifference in the poition epenent part of the elf an mutual inuctance. In claical machine moeling [9] the elf an mutual inuctance are moele a: a( θ) = a + a gco( θ) (6) a ab( θ) = g co(θ 1) (7) Where: a - leakage inuctance phae a -average inuctance (minu leakage) phae a a -poition epenent inuctance phae a g t In the aliency moeling in the previou ection variation in the leakage inuctance where aume. If thee variation are inclue in the moel the elf inuctance can be expree a: a ( θ) = a ( a t a y)co( θ) (8) ( θ) = + + ( )co( θ) (9) a a a at ay g Where: -average leakage inuctance phae a a -tooth aturation epenent term phae a a t -yoke aturation epenent term phae a a y The b an c-phae i not conuct any current an the leakage inuctance in thee phae where not meaure. The expreion in (7) i therefore aume to be correct for the mutual inuctance term. The claical moeling ue the average inuctance a the bai for the moel. In thi paper a maximum inuctance wa ue. In orer to relate the two moel (9) wa ue in the tationary inuctance matrix an tranforme to the repreentation. By comparing the -inuctance the relation between the moel where foun (Table). lab lac.3..1 la rotor poition[el.eg] Figure 11 Self an mutual inuctance meaure with tranient excitation Table Relation between Theoretical moel an meaurement

6 NORPIE 4, Tronheim, Norway Theoretical moel Moifie tanar moel(meaurement) 3 + a + + g r t y t y a t a y + a From the meaurement one in thi work it i ifficult to etimate all the component in the aliency moel. Table 3 how the term that can be erive from the meaurement. Table 3 Meaure inuctance term a a a t a yt g V. REFERENCES [1] M.W. Degner; Flux, Poition, an Velocity Etimation in AC Machine Uing carrier ignal injection, Ph.D. Diertation, Dep. Of Mechanical Engineering, Univerity of Wiconin - Maion 1998 [] M.J. Corley, R.D. orenz; Rotor poition an velocity etimation for a alient-pole permanent magnet ynchronou machine at tantill an high pee Inutry Application, IEEE Tranaction on, Volume: 34 Iue:4,July-Aug.1998 Page(): [3] M. Schroel; Senorle Control of A.C. Machine, Ph.D. thei, Wien, 199 [4] S. Ovrebo, R. Nilen; New elf ening cheme bae on INFORM, heteroyning an uenberger oberver, IEMDC'3. IEEE International, Volume: 3, 1-4 June 3,Page: vol.3 [5] J. Ha,K. Ie, T. Sawa, S. Sul, Senorle poition control an initial poition etimation of an interior permanent magnet motor Thirty-Sixth IAS Annual Meeting. Conference Recor of the 1 IEEE [6] M.. Aime, M.W. Degner, R.D. orenz; Saturation meaurement in AC machine uing carrier ignal injection, IAS, The 1998 IEEE, Volume: 1, 1-15 Oct Page: vol.1 [7] T Wolbank, J Machl Aniotropy in Inuction Machine amination an it Influence on Mechanical Senorle Control an Conitioning Monitoring EPE Touloue 3-1- [8] M. ekell,. Harnefor, H.P. Nee; Machine Deign Conieration for Senorle Control of PM Motor, Proceeing International Conference on Electrical Machine, Itanbul, September 1998 [9] A.E. Fitzgeral, C. Kigley, S.D. Uman; Electrichal Machinery, McGraw-Hill Inc The magnitue of the poition epenent term in the mutual an elf inuctance inicate that there i a poition epenency in the leakage inuctance. Thee variation can be incorporate in the machine moel by uing aliency moeling a preente in thi paper. Table 3 how the etimate inuctance term for the tet machine ue in thi work obtaine by tranient excitation (Figure 9). The term ( a t a yt) ecribe the poition epenent term in the leakage inuctance. In claical machine moeling thi term will eual zero. IV. CONCUSION A new aliency moel i preente in thi paper. The moel incorporate the combine aturation effect in the tator teeth/yoke, inuce current in the magnet an loa epenency. The moel ugget that the leakage inuctance houl be moele with a poition epenent part. Meaurement on an IPMSM how that there can be a ifference in the poition epenent term in the elf an mutual inuctance when tranient excitation i ue to etimate the inuctance. Thi ifference can be incorporate in the machine moel by aing poition epenent part to the leakage inuctance. The new aliency moel ecribe: the ifferent ource of aliency, the combine effect of aliency in main an leakage inuctance, aliency ue to inuce current in magnet an the loa epenency in the machine.