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Compensation for Rotor Position Estimation Error due to Cross-Coupling Magnetic Saturation in Signal Injection Based Sensorless Control of PM Brushless AC Motors Z. Q. Zhu, Y. Li, D. Howe, and C. M. Bingham Department of Electronic and Electrical Engineering, University of Sheffield, Mappin St., Sheffield SI 3 JD, UK although the problem was not fully studied or solved. Recently, it was established in [7] [8] that the error was caused by the effect of cross-coupling magnetic saturation between the d- and Abstract- This paper proposes a simple method for reducing the rotor position estimation error caused by cross-coupling magnetic saturation between the d- and q-axes when signal injection based sensorless control is applied to a brushless AC (BLAC) motor. The error in the estimated rotor position, which results when conventional signal injection sensorless control is employed, is analyzed. Based on an improved model of a BLAC motor which accounts for the influence of dq-axis cross-coupling on the high-frequency components of the incremental winding inductances, as deduced by either finite element analysis or from (i.e., Ldqh#O), and was influenced by the machine design, but, as yet, no measures, from the control aspect, have been proposed to reduce the error. Nevertheless, it is well known that a mutual inductance exists between the d- and qaxes of a BLAC motor (Ldqh) as a result of cross-coupling due to magnetic saturation, as shown in [9], both experimentally and by finite element analysis. However, for simplicity, the . measurements, an improved signal injection based sensorless scheme is proposed. Its effectiveness is demonstrated by nfluence of . . ' cross-coupling magnetic . . saturaton1S usually neglected in sensorless rotor position control [1-6]. This paper improves the rotor position estimation accuracy in a signal injection based sensorless control scheme by accounting for the influence of cross-coupling due to magnetic saturation. Section II presents an improved model of a BLAC motor which accounts for the influence of the cross-coupling. Section III analyzes the position estimation error which arises when cross-coupling is neglected, and proposes a simple method of reducing the error, based on either predicted or measured incremental inductance characteristics. Measurements are presented in section IV which validate the effectiveness of the proposed method. measurements on a BLAC motor having an interior permanent magnet rotor. I. INTRODUCTION It is necessary to acquire rotor position information for a permanent magnet (PM) brushless AC (BLAC) motor in order to control the 3-phase stator currents, and, hence, the torque. Generally, either an encoder or a resolver is employed. However, such discrete position sensors not only increase the system complexity and cost, but may also compromise the reliability. It is desirable, therefore, to estimate the rotor position indirectly from the terminal voltages and currents. Sensorless methods based on high frequency signal injection exhibit excellent rotor position estimation performance at standstill and low speeds, when back-EMF based sensorless methods are problematic. The most common high frequency signal injection method uses a sinusoidal carrier voltage signal. It was originally developed for use with induction motors, but was subsequently applied to PM BLAC motors with geometric saliency [1-5]. The identification ofthe initial rotor position was obtained in [4] [5] by comparing sine and cosine terms in the 2nd_order harmonic component in the d-axis current. In [6], it was applied to a non-salient BLAC motor equipped with a surface-mounted PM rotor, by utilizing the saliency effect which was caused by magnetic saturation. However, it was found experimentally in [2] that the error in the estimated rotor position increases with the load current, 1 -4244-0743-5/07/$20.OO ©2007 IEEE axes, q- I. ANALYSIS OF CROSS-COUPLINGEFFECT IN BRUSHLESS AC MOTOR For a 3-phase BLAC motor, the phase voltages, va, vb, and vc are given by: Fval b VC _ic_ /bdt ldt (1) where ia, ib, ic, I/a, VIb and qfc are the phase currents and flux-linkages, respectively, and Rs is the stator winding phase resistance. Since the 3-phase flux-linkages are functions of the phase currents and the rotor position, 0Sr, (1l) can be expressed as: 208 0ava 0aa aa dia V ai| dt Vb dib a1a 13iadt 0 fLah V/a dO, = [V a(ia+Ail,b, icAl) - VJa(ia, ib,h li, m)]/Ai 00a dt |Labh = IV a(ilaIb+A4iIc,). mr) Va(lalb ,Ici,m)]/Ai 0b aVb + ayl,b aob dObI Vb = Rsi5b + a_ dt where Ai is an incremental current, and the influence of the LVCJ j I permanent magnet flux, Dm, on the magnetic saturation is C dic ,C C a c dOc (2) alaei &3ib &3i jdt aoOc dt considered in the finite element calculation of ql,. aia O'b oic Since the incremental phase self- and mutual-inductances are |-ial -Es (Oa) La7h Labh Lach ia functions of the rotor position, 0r, the terms in the incremental =Rs ib + Lbah Lbh Lbch P ib + CWr -Es (0,b) inductance matrix are given by: LI~~ I!Lcah cbh ~~~~jJ)lP!bl + L-EjOj~~~~~~ Lcbh Lch FLah =L)ah(or) [Labh=La ab -;T where Lah, Lbh, and Lch, and Labh, Lbch, Lcah, Lbah, Lchh, and Lach are Lbh Lah (Or -227 / 3), Lbch =LCbh Labh (or -i) (8) the phase incremental self-inductances and incremental LLch =Lah (Or +2;T / 3) LLcah =Lach =Labh (Or+7iT/3) mutual-inductances, respectively. Oa Ob, and Oc are the position of the phase windings relative to the rotor, and defined as: Oa=6r, The d- and q-axis incremental self- and mutual-inductances Ob=Or-22t/3, and Oc=Or+2±lt3, and Es is the phase EMF. In a signal may be determined approximately from the phase inductances injection based sensorless control scheme, only the high using (6). By way of example, typical finite element calculated frequency terms in (2) are considered [6], i.e.: magnetic field distributions on both open-circuit and at rated load for the machine under consideration are shown in Fig. 1, ~Vahl rah 'abh ach 1 Flahl and corresponding phase incremental self- and (3) mutual-inductances, Lah and Labh, and d- and q-axis incremental Vbh L=|'bah Lbh Lbch IP ibh| Vch Lcah Lcbh Lch Jch self- and mutual-inductances, Ldh, Lqh, Ldqh, and Lqdh, are shown in Figs. 2 and 3, respectively. The direct calculation of d- and where Vah, Vbh, and Vch, and iah, ibh, and ich are the high frequency q-axis incremental self- and mutual- inductances will be components of the phase voltages and currents, respectively, discussed later. and p=d/dt. Equation (3) can be transformed into the dq-axis reference frame as: Fva 1 ?ia Fl 1 a alb a 0 I_C LLCah LL?dh LiC ah Lq ach abh bah Lda L cbh L*iL|td -LEs (O}C) I bch ch ]C hLl asF dh h. (4) P'~qh~ Lqdh Lqh where C is the 'Park' transformation matrix, and Ldh, Lqh, Ldqh and Lqdh are the d- and q-axis incremental self- and mutualinductances, i.e.: Fcosa0.-sinO 11 C cos Ob -sin Ob 1 Lcos Oc -sinOc L ~ L dhdqh 1 ~ ~ Ldqh j which-1 Lah and Labh are ahL abh Lah ~~Lcah can griven (5) 1ij L in L h ac Lcb,h0~~~~~~~~0 Lch whifle flu-irv1nkages be determined according to the (7), L6[=L P(Or 3) L L( dh Lih La and Lcah cbh Ldh Ldqh *1 dt thep definitions can beP R a_ (a) Open circuit, id-OA, iq=OA As Lah and Labh in Figs. 2(a) and 3(a) vary periodically with the rotor position, 0r, they can be expressed approximately as a Fourier series, i.e.: 60 E o0 cs 50 ------------------------------------- 30/\ 20 -a -J -k =i,2,3, Io a: Ea) -10 o: -30 -30 I 50 , 40 120 18 0R60t12 180 240 300 240 300 Rotor position (elec. Deg.) (a) Incremental phase self- and mutual-inductances 60 E 60 L- 20 - 1-0---a) 30 (a5kcos kOr+ b5ksin kOr) {Labh(O:)= amo+ (ankcosk+bmksinbk, (9 ) k1,2,3, where a5o, amo, a5k, b,k, amk, and bmk are the amplitudes ofthe DC terms and the kth-order terms ofthe Fourier series, respectively. Since the amplitude of the high order harmonics in Lah and Labh is relatively small, only terms up to k=8 are considered and transformed from 3-phase to dq- values using (6) and (7). The average values of Ldh, Lqh, Ldqh and Ldqh are calculated from: 360 36 L qh cu =a5+ E Sah(Or) 30 sx5 [Ldh =(a,0-a 0)+(a 12+a ) [-(LsOhmOd+ { Lqh = (as0 -amo) (as2/2+am2i) 2 Ldqh Lqdh m2 (10) (bs2/2+bm2) L dqh As well be evident from (9), the saliency which results in a BLAC motor equipped with an interior magnet rotor, Ldh#cLqh, is caused by 2nd order co-sine terms, a52, am2, in the Lah and Labh 0 60 120 180 240 300 360 waveforms, which are considered in [10]. The cross-coupling Rotor position (elec. Deg.) effect is caused by 2nd order sine terms, b52, bm2, in the Lah and Labh waveforms, which are usually neglected [10]. In (b) Incremental dq-axis inductances Fig. 2. Incremental inductances on no-load, iOA,iq=OA. subsequent equations, Lqdh is written as Ldqh. It can also be shown that the 5th and 7th order terms in Lah and Labh, i.e., as5, 60 E 50 bs5, aL7m5, as7, bs7, am7, and bm7, cause the variation of Ldh, Lqh, 40and Ldqh with rotor position every 60 elec. deg.. Other order w 40 -harmonics in Lah and Labh do not appear in Ldh, Lqh and Ldqh, as a an 30 Jr La o 20 result of (6). l0 Although the d- and q-axis incremental inductances can be Labhdetermined by transforming the waveforms of the incremental 5FU a) -10 t phase self- and mutual-inductances, Lah, and Labh, into the / 2 < a) dq-axis reference frame, this requires the flux-linkage, Vla, of __ _ _ _ _ _ _ phase a to be calculated for various d- and q- axis currents and rotor positions, which is time consuming. To reduce the finite 0 60 120 180 240 300 360 element calculation time, the incremental d- and q-axis Rotor position (elec. Deg.) inductances can be calculated directly from d- and q- axis nmflux-linkages by applying appropriate d- and q-axis currents in 60 the finite element analysis, equation (l1). Results calculated E 50 4 I Lqh a) 40 directly in this way are shown in Fig. 4. Ldh =[d(id+ Aid, i, qm =l au 30 iddiq, m)] | Aid dh = -, LqdLL,h Aiq, (D )-Vfqd(Id,ITm]/q (id,iq, D )]/Aiq [yfq (id I'q' ++ITm)VJ =[V~d('d ~~~~~~~~~~~~dqh 0 X E -10 o -20 -30 - bm5, - -1 ------------------------- ------------------- -_1 -_ E -10 L -- o -- 0 60 120 180 240 Rotor position (elec. Deg.) -Ld-tJ-(id- + Ai+d,Aiqn iq, O(mm )- q (dniq,iq, Dn/tm)/ AidAiq d41h -di -4( -Li 300 , -M c (/ O),, ) various d- and q-axis currents, id and 'qn the d- and q-axis incremental self- and mutual-inductances, Ldh, Lqh, Ldqh, vary with the d- and q- axis currents because of saturation. For the 360 (b) Incremental dq-axis inductances BLAC motor under consideration, Fig. 1, Ldh Lqh and ldqh are Inrmetlinutncsonfl-q Fig.3. 210 FVe 1 FLdh Ldqh- -1 FVdh1 Fl T I e =T(AO) P .e =T(AO) L (AO) Z dqh qh Vqh j -1qhj LVqh 12 Lavg -Ldjcos(2AO+Om) Ldjesin(2A0+Om (1Fi 25mH, 32mH and -7mH, respectively, when iL=OA and iq=4A. Clearly, since the magnitude of Ldqh iS comparable with that of Ldh and Lqh, its influence cannot be neglected in the high frequency voltage equations. L 60 £ 50 -- (usss 40 0 -a -C X ^ * * * * cos(AO) sin(AO) j $ "== 8_ -_ T(AO)= Id=1A -sin(AO) cos(AO) i 30 @I d=OA * * Lavg (Lqh+Ldh)/2, Ld =(Lqh-Ldh)2 hL2( -4- Id=-IA >~ - + + E 2() -i Lno2 I d=-2A -4 -3 1 -1 0 2 3 q -axis current (A)(a) d-axis incremental inductance, Ldh. -2 (13) i Ldif + dif+Ldqh 4 q-aisc = )Vsig avg-Ld cos(2A0+Om) Ldif sin(2AO+Om) 0 1iLh1 P p Ldif sin(2AO+Om) Lavg+Ldif cos(2A0+Om) IqhL The resulting d- and q-axis high frequency currents in the E-US a)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(4 Id=3A c40 estimated reference frame are obtained by solving (14): r 30 | _ _q | 20 3 l() evSi d=O~ IA )[Lag +Ld cos(2AO+O,m)] sig2~ ~ ~IL2 +~~~~~~ --Id:-1A | dh 1 ___ -4 15 ___ -,3 o l'h ___ -2 ___ -t ___ 0 ___ 1 q--axis current (A) ___ 2 ___ 3 4 E' 0 E 0t ------------4_ dif) Ldj sin(2A0+Om,) the estimated rotor position as a result ofthe incremental mutual inductance, Ldqh, which exists between d- and q-axes. The error -Id=2A -~~~~~~~~~~~~~~~ L signal injection based sensorless schemes [6], an error exists in Id=3A 5 ,avg Thus, ifthe high frequency component in the q-axis current is controlled to be zero, i.e. iqhe=O, as was in the conventional isgivenby: +Id=1A ()-------- o -i P(' L (b) q-axis incremental inductance, Lqh. 10 I_, -4a) hd) Since the high frequency sinusoidal voltage, vsig, is applied to the estimated d-axis, i.e., Vdhe=Vsig, VqhOe=0 (12) becomes: 60 £ ( Om =arctan(Ldqh/Ldf), Id0-3A C: 0 Iqh j where AO=Ore_Or is the error in the estimated rotor position, and Id=3A -_ Id=2A I_, Lavg+Ldif cos(2AO+Om)j Ldif sin(2AO+Om) ~~~d-A2 -4Id30A Id=-2A I d=O 0 1 60-m ( 2Lh (16) Ldh -Lqh 2 -rca |- Idd=-3A 6T Id= 3 0~~~~~~~~~~~~~~~ a)d -xlm -1 mcrmeta 0 -4 -3 -2 ta q-axis currentt (A) a) 3 4 21 ------ o ------ 0 60 a) o( 0 3 500 andq-axiscurrents.~ ~ incremental ~ ~ ~~~~~~~~~~ C inductance, Ldqh=Lqdh. (c) dq-axis mutual o a) .p 4. Variation of incremental selfand mutual-inductances with dFig. dq-axis -_F and q-axis currents. O01 0_ III. ANALYSIS OF ROTOR POSITIoN ERROR CAUSED BY CROSS-COUPLING transformed into the estimated rotor position, Ore, i.e.: 211 _ _ 0 --------0 0element ed=OA0 _+ d O +I d=2A _ 60 45 .0 co 30 l 215 E .o a) oo 0 sampling, the speed control loop, and the PWM are all set to 5kHz, and the injected signal is 35V, 330Hz. The actual rotor position, 0r, is obtained from a 1024 pulse-per-revolution encoder, which is used as a reference for the estimated rotor position Ore. The rotor position error compensation factor, Kr, was Kr=8.0 '/A, as obtained from the experimental results given in Fig. 5(b). i- Id 3A - -4---I-----I d= 2A Id= - lA Id=O A 0~~~~~~~~~~~~~~d1 lA -Id --d2 15 O ow ----- o ° 30 Id=3A - ,Lv cl 45 1) 60 4 3 2 1 0 1 q-axis current (A) 2 3 4 (b) Directly measured error, Kr8.00/A. P11) -- Fig. 5. Comparison of predicted and measured rotor position estimation for various d- and q-axis currents. A1AC id error to I Clearly, the error in the estimated rotor position will only be when Ldqh=O. The stronger the cross-coupling between the d- and q-axes, the larger will be the error in the estimated rotor position. For the BLAC motor under consideration, the variation of the predicted error with the d- and q-axis currents is zero AC ..sed shown in Fig.5(a), the predictions being obtained by employing iie element lmn cluaediceena Fg theth finite calculated incremental inductances, Fig. 4,,iin (16). The error in the estimated rotor position has also been measured by driving the BLAC motor with the actual rotor position obtained from a precision encoder, so that id=id, and 6. Signal injection sensorless control with compensation for rotor position Fig. error due to cross-coupling. nucacs Iqe=iq. The high frequency voltage was 60 °6- 45 -- E)& 315 injected into the estimated d-axis, and the estimated high frequency q-axis O 2 °a current, iqhe, was forced to zero by adjusting the estimated rotor The measured rotor estimation position, OrU. directly position error is shown in Fig. 5(b), which compares well with the o predicted results shown in Fig. 5(a). ry IV. COMPENSATION OF ROTOR POSITION ERROR DUE TO CROSS-COUPLING AND COMPARISON WITH CONVENTIONAL METHOD The error in the estimated rotor position can be compensated for by either employing (16) or using the results shown in Fig. 5. However, from both the predicted and measured results shown in Fig. 5, it can be seen that the error in estimated rotor position is approximately proportional to the q-axis current, i.e. AIO.Kriq. Thus, the error can simply be compensated for according to the q-axis current, i', by applying a compensation factor, Kr This significantly simplifies the implementation of an error compensation scheme. The proposed sensorless control scheme is shown in Fig. 6, and the parameters of the interior permanent magnet motor, whose incremental inductance characteristics were shown in Figs. 4 and 5, are given in Table I. The control strategy is implemented on a TMS32OC3 1 DSP, together with a PIC 1 8F443 1 MCU, which serves as the PWM generator and the encoder interface. The frequencies of the AD 212 - 0 Id=3A lA --Id=O -4Id=2A ' 15 - 30 1~~~~~~~~~--d=3A * -6- -45 - -60 -4 60 0 -1 1 2 3 4 q-axis current (A) (a) Conventional method, RMS(Ore-Or)=21. 10 -3 -2 4 . c E& 30 ag - iId=-lA 15 o ° °Id=OA 15 o - Q- 1 -- Id=lA -Id=2A 3 --Id=3A 0 t Fig. 7. -45 -60 Id=-2A - -4 -3 -2 -1 0 1 2 3 q-axis current (A) (b) Proposed method, RMS(Ore~Or)=3.2° Measured rotor position estimation error with proposed signal injection based sensorless methods. 4 conventional and the rotor speed command is changed from -1OHz to +1OHz, i.e., -200rpm to +200rpm. As will be seen in Fig. 8(a), when the conventional signal injection based sensorless scheme is employed, the estimated rotor position error increases significantly with the load current, and is 250 when iq=4A. By applying the proposed error correction method this reduces to 50, TABLE I SPECIFICATION OF BLAC MOTOR Number of pole-pairs 3 Rated speed Rated torque Rated phase voltage (peak) 1000rpm 4.0Nm 158V Rated phase current (peak) 4.0A Phase resistance R, 6.0Q Fig. 8(b). V. CONCLUSIONS Finite element analysis and measurements have shown that the neglect of cross-coupling between the d- and q-axes of a BLAC motor which results due to magnetic saturation, may lead to significant errors in the rotor position estimation when signal injection based sensorless control is employed. However, by applying a simple error correction method, a significant improvement in the accuracy of the rotor position estimation can be achieved. Fig. 7 compares the measured steady-state error in the estimated rotor position for various d- and q-axis currents, ie when the estimated rotor position is used for position and feedback. With conventional signal injection based sensorlessi control, Fig. 7(a), the rotor position estimation error increases with both the magnitude of iq and increasing positive values of id, since the magnetic circuit then becomes more heavily saturated and the influence of dq-axis cross-coupling becomes more significant. For example, the error increases to 450 when ird3A, iq=4A, while the root mean square (RMS) error in Fig. 7(a) is 21.1'. However, when the proposed error compensation method is applied, the RMS rotor position estimation error is reduced to only 3.20, Fig. 7(b). jle, REFERENCES Ogasawara, and H. Akagi, "Implementation and position control performance of a position-sensorless IPM motor drive system based on magnetic saliency," IEEE Trans. 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