Research Article Development of Digital Control for High Power Permanent-Magnet Synchronous Motor Drives

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1 Mathematical Problems in Engineering Volume 214, Article ID , 1 pages Research Article Development of Digital Control for High Power Permanent-Magnet Synchronous Motor Drives Ming-Hung Chen an Hao-Ting Tseng Department of Electrical Engineering, Ming Chi University of Technology, 84 Gungjuan Roa, Taishan District, New Taipei City 2431, Taiwan Corresponence shoul be aresse to Ming-Hung Chen; mhchen@mail.mcut.eu.tw Receive 13 April 214; Accepte 1 May 214; Publishe 2 June 214 Acaemic Eitor: Her-Terng Yau Copyright 214 M.-H. Chen an H.-T. Tseng. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. This paper is concerne with the evelopment of igital control system for high power permanent-magnet synchronous motor (PMSM) to yiel goo spee regulation, low current harmonic, an stable output spee. The esign of controller is conucte by igitizing the mathematical moel of PMSM using impulse invariance technique. The preicte current estimator, which is insensitive to motor feeback currents, is propose to function uner stationary frame for harmonic current suppression. In the AC/DC power converter, mathematical moel an c-link voltage limit of the three-phase switch-moe rectifier are erive. In aition, a current controller uner synchronous frame is introuce to reuce the current harmonics an increase the power factor on the input sie. A igital control system for 75 kw PMSM is realize with igital signal processor (R5F563EDDFP). Experimental results inicate that the total harmonic istortion of current is reuce from 4.1% to 2.8% for 5 kw output power by the propose preicte current estimator technique. 1. Introuction Basically, the structure of PMSM, which consists of threephase winings on stator but permanent magnets on rotor, is similar to traitional synchronous motor. Because there are no excitation circuit, slip rings, an brushes, the PMSM features avantages of simple structure an high efficiency. In the vector-controlle PMSM rives, the output of spee controller provies the reference value for current controllers; therefore, it is suitable for proportional-integral (PI) controller in precision servo rive applications 1]. One of popular approaches to rive PMSM uses irect torque control (DTC) because of the merits of simple structure, quick ynamic response, an strong robustness against rotor parameters. However, it also presents some isavantages of large torque/flux ripples 2]. The DTC uses hysteresis comparators which only consier the signs of torque an flux errors so their amplitues are not ifferentiate 3]. In this paper, the high power PMSM (75 kw) is characterize by low equivalent resistance an inuctance. Thus it is ifficult to implement the motor rive. Currently, the PMSM rive system is mostly implemente by inverters with high switching frequency. Because of the low equivalent resistance an inuctance characteristic, currents feeback elay, an resolution limitation of igital controller, it introuces lots of current harmonics to motor so that it affects the performance of the electromagnetic torque output of the motor. Torque ripple is a critical concern in many high power PMSMs, an the existence of torque ripple egraes the control stability 4]. In orer to alleviate these rawbacks an reuce current harmonics in steay state, the motor mathematical moel is use to estimate the motor currents an thus to compensate for current harmonics cause by high switching frequency an to improve problems of current feeback elay an resolution limitation of igital control 5]. The preicte current estimator use in this paper obtains motor current feeback an accelerates the convergence with close-loop control. It can not only reuce the steay-state current harmonics, but also accelerate the spee an current transient response. Comparing with other papers, the propose system features the avantages of high power level, low current harmonic,anfastresponse.insummary,thecontrolstrategy

2 2 Mathematical Problems in Engineering an pulse-with moulation (PWM) will be implemente byalowcostigitalsignalprocessor(dsp).theanalysisof preicte current estimator will be iscusse in etail. 2. System Configuration The configuration of the propose system shown in Figure 1 consists of two parts: power converter an igital signal control moule. As inicate above, the power converter inclues an AC/DC power converter, a high power inverter, gate rive, an current sensors. The propose system can thus transfer energy from electrical to mechanical forms with goo power quality. As to the igital signal control moule, it inclues a DSP to realize the controller of the propose system. A 12-bit analog-to-igital converter (ADC) anapwmmoulearebuiltinthedsptoreucethe harware complexity an improve the reliability as well as maintainability of the system. 3. Operation Principles In orer to reuce current harmonics to high power PMSM, a preicte current estimator for close-loop control is propose. By current sensors at the output of inverter, one can obtain three-phase currents an then transform them into stationary frame. The q- an -axis currents in stationary framethusyielthepeakvalueofoutputcurrent.thecurrents to PMSM can then be calculate by preicte current estimator. On the other han, a spee regulator ajusts the q-axis current from rotational encoer which is use to calculate rotor spee. Thus the motor spee can remain constant all the time. The etaile analysis is iscusse in the following part Mathematical Moel of PMSM. The equivalent circuit of PMSM is shown in Figure 2. Thevoltageequationcanbe written as 6] where V abcs = r s i abcs t λ abcs, (1) V abcs =V as V bs V cs ] T, i abcs =i as i bs i cs ] T, λ abcs =λ as λ bs λ cs ] T, r s r s = r s ]. r s ] V abcs an i abcs enote the phase voltages an line currents of motor, respectively; r s an λ abcs are the equivalent resistance an flux linkages of stator. Assume that the flux harmonics of rotor an stator are negligible. Uner this circumstance, the fluxes of rotor an stator are sinusoially istribute. Therefore, one has the flux linkage (2) λ abcs = L s i abcs λ m, (3) where L ls L ms 1 2 L ms 1 2 L ms L s = 1 2 L ms L ls L ms 1 2 L ms, 1 2 L ms 1 ] 2 L ms L ls L ms ] λ m =λ sin θ r m sin (θ r 12 )], sin (θ r 12 )] where θ r = ω r t; L ms =N 2 s /R m; θ r an ω r are rotor flux angle an spee, respectively; N s an L ls are equivalent turns an leakage inuctance of stator wining, respectively; R m is equivalent reluctance of stator; λ m is the equivalent flux linkage of rotor referre to stator. From (1) (4), one can obtain the following phase voltage of motor: (4) V abcs =r s i abcs L s t i abcs e abcs, (5) where e abcs enotes the internal voltage of motor an is proportional to ω r.thatis, e abcs =ω r λ cos θ r m cos (θ r 12 )]. (6) cos (θ r 12 )] For moelling an control esign, three-phase variables in stationary frame can be transforme into two-axis stationary frame. This yiels 7] f s qs f s s f s s ] = ] f as f bs ] = 1 f cs ] 1 ] ] 1 2 f 3 as f 2 bs ], 1 ] f cs ] 2 ] f s qs f s s f s s ], ] where subscripts q an areusetoenotethecomponents of q- an -axis in two-axis frame, respectively. The two-axis variables in stationary frame can be transferre to rotating frame as follows 7]: fr qs f r ]= cos θ r sin θ r ] fs qs sin θ s r cos θ r f s ], s fs qs f s ]= cos θ r sin θ r ] fr qs sin θ s r cos θ r f r ]. s (7) (8)

3 Mathematical Problems in Engineering 3 AC/DC power converter High power inverter 3φ, 22 V T r T T s t T a T b T c C c c PMSM Rotational encoer Loas Tr Ts Tt Ta Tb Tc Current sensors Gate rive Gate rive Current sensors Spee comman ADC DSP R5F563EDDFP ADC θ rm Figure 1: Block iagram of the propose system. bs as cs i bs i cs i as r s N s Figure 2: Equivalent circuit of the Y-connection PMSM. Transforming (5) an(6) intoq- stationary frame with (7), one obtains N s r s N s V s qs =r si s qs L s t is qs ω rλ m cos θ r, V s s =r si s s L s t is s ω rλ m sin θ r, where V s qs, Vs s, is qs,anis s are motor input voltages an currents reflecting in stationary frame, respectively. From (9), the state equation can be erive as follows: t is qs ] = r s L s r ] is qs s i s ] 1 1 s L s 1 ]us qs u s ], (1) s ] L s ] t is s r s (9) where u s qs = Vs qs ω rλ m cos θ r, (11) u s s = Vs s ω rλ m sin θ r. It is seen from (1) that inputs an outputs are linearity in a control perio if motor spee is fixe. Therefore, one can have igitalize state equation as follows 8]: is qs (k) i s s (k)] =Φ q is qs (k1) i s s (k1)]γ q us qs (k1) u s (12) s (k1)], where Φ q = e(r s/l s )T s e (r s/l s )T s ], Γ q = 1 r s 1e(r s/l s )T s 1e (r s/l s )T s ]. (13) T s enotes control cycle; k an (k 1) are current an last states, respectively. Equation (12) can be use in preicte current estimator. The electromagnetic torque equation in q- rotating frame can be written as T e = 3 N p 2 2 (L si r qs ir s λ m ir qs ), (14) where N p is numbers of poles. The motion equation is T e = 2 N P J t ω r 2 N P Bω r T L, (15) where J an B enote the inertia an friction of motor, respectively. ω r is rotor electrical spee an T L is the loa. This means that the spee of compressor can be controlle by i r qs an is use in this paper to implement the spee regulator of motor rive 9 11].

4 4 Mathematical Problems in Engineering s L e rs x e ss L x e ts L x T r r x i rs r x i ss r x i ts Tr rs T s T s ss T t T t c ts i c C c High power inverter It is note that (21) characterizes the balance of power between the utility an c-link when the utility output is sinusoial with unity power factor. Base on (21), a controller is esigne to yiel the current comman for rectifier input. For the sake of simplicity in analysis an control, the three-phase voltages, currents, an switching functions of (16) (18) can be transforme into synchronous frame. This gives Figure 3: Equivalent circuit of the AC/DC power converter. e qs]= ω el x e s L x pr x L x pr x ] i qs] V qs], (22) ω e L x i s V s 3.2. Analysis of AC/DC Power Converter. The AC/DC power converter shown in Figure 1 is implemente by three-phase switche-moe evices. The ynamic equations with inuctor currents an capacitor voltage can be erive as L x t i rs =e rs r x i rs V rs, (16) L x t i ss =e ss r x i ss V ss, (17) L x t i ts =e ts r x i ts V ts, (18) C c t V c =i rs S r i ss S s i ts S t i c, (19) where e rs, e ss,ane ts an i rs, i ss,ani ts enote the phase voltages an line currents of utility, respectively; V rs, V ss, an V ts aretheinputvoltagesofconverter;s r, S s,ans t are switching functions; V c an i c are the c-link voltage an output current, respectively; r x an L x are resistance an inuctance of utility; C c is the capacitance of c-link capacitor. Assume that the switching frequency of the propose system is high enough to yiel negligible harmonic current from the system. Uner this circumstance, one can erive the following inequality for proper operation of the converter: V c >2V m, (2) where V m = (V m i m r x ) 2 (ω e L x i m ) 2 ; V m an i m are the peak values of voltage an current from utility, respectively, an ω e is the frequency of the utility output. The c-link voltage comman of the propose system is V c = 31 V. From(2) an the relation between V m an ω e, one fins that the utility frequency must lie between 5 Hz an 8 Hz. This will result in a stable an fixe c-link voltage for the inverter in the next stage to prouce a balance threephase 6 Hz voltage with a peak value of 22 2V.Ifc-link voltagecannotsatisfytheinequalityconstraintof(2), the systemwillbeoutofcontrol.similarly,from(19) onecan obtain the voltage-current equation for the c-link as C c t V c = p x V c i c, (21) where p x (3/2)V m i m is the average power from utility. where p is the operator /t; e qs, e s, i qs,ani s are q- axis voltage an current, respectively, while V qs an V s are the transforme q- an-axis voltage of the rectifier input, respectively. Equation (22)canberewrittenas L x t i qs =e qs V qs r x i qs ω e L x i s, L x t i s =e s V s r x i s ω e L x i qs. (23) From (23),onecanobtainthefollowingvoltagecomman for three-phase rectifier: where V qs =e qs u qs ω el x i s, V s =e s u s ω el x i qs, u qs G iqs (i qs i qs), u s G is (i s i s), (24) (25) an G iqs an G is are the gains of the q- an -axis proportional-integral current controller, respectively. The symbol enotes proportional-integral operation. Applying the inverse transformation to the voltage commans obtaine from (24) (25)givesthethreephaseoutputvoltage commans V rs, V ss,anv ts. After comparing with the counter values in DSP, which correspon to an 8 khz high frequency triangular carrier of sinusoial PWM, the uty cycles are etermine to switch the power transistors an thus forcing the actual current to follow the commaning current within one carrier cycle. This results in fast transient response in the rectifier. Further investigation of the rectifier circuit gives the following equations for the average real an reactive powers: p s = 3 2 V mi qs, (26) q s = 3 2 V mi s. (27) Equation (26) inicates that p s epens only on i qs, while (27)showsthatsettingi s equal to zero implies zero reactive power an thereby results in unity input power factor an raises the efficiency of the rectifier. For the sake of keeping c voltage at output of the AC/DC power converter constant,

5 Mathematical Problems in Engineering 5 θ e T r c c Δ c I m i qs G c Voltage regulator i s = Limiter i s i qs G is G iqs Current regulator us uqs s qs q to abc SPWM Tr T s Ts T t Tt Gate rive ω e L x ω e L x θ e i qs i s abc to q ADC ADC Current sensor i rs i ts e s = e qs Figure 4: Control block iagram of the AC/DC power converter. ADC Voltage sensor c s qs (z) uqs s (z) 1 γ q z φ q î s qs (z) e s qs (z) G qe i s qs (z) (a) s s (z) u s s (z) 1 γ z φ î s s (z) e s s (z) G e i s s (z) (b) Figure 5: Control block iagram of the preicte current estimator: (a) q-axis current estimator; (b) -axis current estimator. ω rm T L Δ iqs r Gs K T z J(z 1) controller in orer to ajust c-link voltage at output of the AC/DC power converter. The current comman can be written as I m =(G Vc ΔV c ), (28) Figure 6: Control block iagram of the spee regulator. c-link voltage is monitore to control the input currents to ensure the balance of power between utility an c-link (see Figure 3). This is conucte by voltage regulator which can generate the current comman for preicte current where ΔV c V c V c; G Vc isthegainoftheproportionalintegral voltage controller. For reucing the orer of the close-loop transfer function of voltage control loop, the pole-zero cancellation metho was use to etermine the controller parameters which yiel the proportional-integral controller parameters of voltage regulator, k Vp =.62 an k Vi = The control block iagram of the AC/DC power

6 6 Mathematical Problems in Engineering 5 A/iv 5 A/iv i rs i ss 5 A/iv (a) 1 ms/iv 15 V/iv (b) 1 ms/iv i ts c (c) 1 ms/iv () 1 ms/iv Figure 7: Experimental results of the propose AD/DC power converter in steay state: (a) phase current i rs ;(b)phasecurrenti ss ;(c)phase current i ts ;()c-link voltage V c. converter is shown in Figure 4, where preicte current controller uner synchronous frame is use Analysis of Preicte Current Estimator. The propose high power inverter is shown in Figure 1. Asmentione above, constant c voltage is supplie from the c-link of the last stage. Proper control of the power transistors will transfer power from c to three-phase ac. Generally, the high frequency switching of inverters, resolution limitation of analog-to-igital converters, an sampling elays from current feebacks introuce a large number of current harmonics. Therefore, it is ifficult to esign igital controller for high power rives. In orer to suppress the current harmonics an improve the performance of the current regulator, the preicte current estimator is use to obtain real currents. The feeback currents from motor estimate by q- anaxis in stationary frame reuce time elay problems. After z-transform to (12), the control block iagram of preicte current estimator is shown in Figure 5. The preicte current estimator suppresses ripples of voltage commans to reuce current harmonics. From (12) (13), the estimate currents ı s s an ı s s are ı s qs (z) =Vs qs (z) es qs (z)] γ q zφ q ı s s (z) =Vs s (z) es s (z)] γ, zφ (29) where e s qs =ω r λ m cos θ r an e s s =ω r λ m sin θ r.thecurrent regulators of preicte current estimator can be written as z G qe =k pqe k iqe z1, (3) z G e =k pe k ie z1, where k pqe, k iqe, k pe,ank ie are gains of the proportionalintegral current regulators, respectively. If u s qs =an us s =, the transfer functions of preicte current estimator can be erive as i s qs (z) ı qs s (z) = (k pqe k iqe )zk iqe, z 2 (k pqe k iqe φ q )zk iqe i s s (z) ı s s (z) = (k pe k ie )zk ie. z 2 (k pe k ie φ )zk ie (31) The preicte current estimator improves the accuracy of current feeback an reuces current harmonics to motor. Itnotonlycanimprovethecurrentsamplingelay,butalso hasthecharacteristicsoflow-passfilterstoenhancesystem stability Analysis of Spee Regulator. It is seen from (14) that the electromagnetic torque of motor is proportional to q- axis current. This means that the output spee of motor canbecontrollebyq-axis current i r qs anisuseinthis paper to implement the spee regulator. Therefore, the q- axis current comman i r qs canbecalculatebyspeeerror to spee regulator G s. It is important to note that the friction of this motor is ignore ue to high inertia in the high power PMSM. This yiels z G s =k ps k is z1, (32)

7 Mathematical Problems in Engineering 7 35 A/iv 35 A/iv i r qs i r qs 5 ms/iv 5 ms/iv (a) (b) 1 A/iv 1 rpm/iv i r s 5 ms/iv 5 ms/iv (c) () 1 V/iv 1 V/iv r qs r s 5 ms/iv 5 ms/iv (e) (f) 5 A/iv 5 A/iv i as i as THD i = 2.8% (g) 5 ms/iv N 6 (Hz) (h) Figure 8: Experimental results of the propose system in steay state: (a) q-axis current comman i r qs ;(b)q-axis current ir qs ;(c)-axis current i r s ; () motor spee ;(e)q-axis voltage comman V r qs ;(f)-axis voltage comman Vr s ;(g)outputcurrenti as; (h) total harmonic istortion of output current i as. where k ps an k is are gains of the proportional-integral current regulators, respectively. The transfer functions of spee regulator can be erive as ωrm T L = = K T (k ps k is )z 2 k p K T z (k ps K T k is K T J)z 2 (k ps K T 2J) z J, z 2 z = T L ω rm = (k ps K T k is K T J)z 2 (k ps K T 2J) z J. (33) The control block iagram of spee regulator is shown in Figure 6. Experimental evaluation of the propose system will be given in the following section Parameters Selection. For fixe switching frequency, the switching an conucting losses are almost proportional to the amplitue of the correlative currents. Therefore, the total loss of switching evices can be compute from the circuit parameters an the q- an -axis currents of the AC/DC power converter an the inverter. The machine an electric

8 8 Mathematical Problems in Engineering 1 rpm/iv 1 rpm/iv ω rm 5 nt-m/iv (a) 1 ms/iv 5 A/iv (b) 1 ms/iv T e i as i r qs 5 A/iv (c) 1 ms/iv i r qs 5 A/iv () 1 ms/iv i r s 1 A/iv (e) 1 ms/iv i r s 1 A/iv (f) 1 ms/iv (g) 1 ms/iv (h) 1 ms/iv Figure 9: Experimental results of the propose system uner 22 rpm spee variations: (a) motor spee comman ω rm ; (b) motor spee ; (c) electromagnetic torque T e ; () phase current i as ;(e)q-axis current comman i r qs ;(f)q-axis current ir qs ;(g)-axis current comman i r s ;(h)-axis current ir s. parameters of PMSM are given in Table 1. Forreucingthe orer of the close-loop transfer function of voltage an current loops in AC/DC power converter an spee loop in high power inverter, the pole-zero cancellation metho is use to etermine the control parameters of voltage regulator, k Vp =.52, k Vi = 81.36; current regulator, k iqp = 3.2, k iqi = ; an spee regulator, k sp = , k si = 13.52,respectively. 4. Experimental Results An experimental prototype of a 75 kw motor rive for PMSM was built with machine an electric parameters shown in Table 1. The DSP R5F563EDDFP mentione above was use to implement the control scheme. The power controller was forme by insulate-gate bipolar transistors (IGBT), which operates at 8kHz. Figure7 shows measure waveforms Table 1: Machine an electric parameters of PMSM. Machine parameters Rate spee: 36 rpm Rate torque: 2 nt-m Rate power: 75 kw Line voltage: 22 Vrms Line current: 2 Arms Pole number: 12 Electric parameters λ m =.55 V-s/ra r s = 6.2 mω L s =.34 mh of the input currents an c-link voltage of the AC/DC power converter uner loa of 5 kw in the steay state. It illustrates that the input currents were almost sinusoial with low harmonic istortion, whereas the c-link voltage V c remaine constant. Figure 8 shows experimental results of

9 Mathematical Problems in Engineering 9 THD i (%) (2) (1) 1 4.1% 2.8% P out (kw) of rive system. In short, it coul be consierable potentially to implement the high power servo rives. Conflict of Interests The authors eclare that there is no conflict of interests regaring the publication of this paper. Acknowlegments The authors wish to express their sincere appreciation to UCHI OPTOELECTRONIC (M) SDN. BHD. for supporting thisresearch.theauthorswoulliketothanktheanonymous reviewers for their valuable comments an suggestions to improve the quality of the paper. (1) Without preicte current estimator (2) With preicte current estimator Figure 1: Total harmonic istortion of currents versus output power. the propose system uner the rotor spee 3 rpm with 5kWpoweroutput.Itcanbeseenthatthecurrentripples of q- an-axis are small when -axis current comment sets to zero. Figure 9 illustrates motor spee comman ω rm, motor spee, electromagnetic torque T e,phasecurrent i as, q-axis current comman i r qs, q-axis current ir qs, -axis current comman i r s,an-axis current ir s of the propose system uner the motor spee of 22 rpm. It shows that the spee response of PMSM was fast an the torque ripple was small ue to preicte current estimator. It is also seen from Figure 1 that the measure total harmonic istortion of currents reuce with preicte current estimator, an the total harmonic istortion of currents graually ecline when the output power increases. Besies, the total harmonic istortion of currents ecreases to 2.8% in 5 kw power output. It is obvious to say that the electromagnetic torque ripple is very small uner this circumstance. Therefore, the propose system satisfies the esign requirement. 5. Conclusions This paper is focuse on the esign of igital controller for high power PMSM rives. The parameters of igital controller are erive by propose preicte current estimator to reuce total harmonic istortion of motor currents. With theoretical erivation of esign parameters, the igital controller maintains rive operating stably by setting -axis current to zero with motor spee range from to 3 rpm. Comparing without preicte current estimator in 5 kw power output, experimental results show measure total harmonic istortion of currents are 1.3% ecrease. In this paper, a DSP performs as the core controller to reuce the complexity of harware circuit. The entire igital control algorithms are conucte by software to make high stability References 1] A.V.Sant,K.R.Rajagopal,anN.K.Sheth, Permanentmagnet synchronous motor rive using hybri PI spee controller with inherent an noninherent switching functions, IEEE Transactions on Magnetics,vol.47,no.1,pp ,211. 2]Y.ZhanganJ.Zhu, Anovelutycyclecontrolstrategyto reuce both torque an flux ripples for DTC of permanent magnet synchronous motor rives with switching frequency reuction, IEEE Transactions on Power Electronics,vol.26,no. 1, pp , ] Y.Zhang,J.Zhu,W.Xu,anY.Guo, Asimplemethotoreuce torque ripple in irect torque-controlle permanent-magnet synchronous motor by using vectors with variable amplitue an angle, IEEE Transactions on Inustrial Electronics, vol.58, no.7,pp ,211. 4] H. Zhu, X. Xiao, an Y. Li, Torque ripple reuction of the torque preictive control scheme for permanent-magnet synchronous motors, IEEE Transactions on Inustrial Electronics, vol.59,no.2,pp ,212. 5] F. Zhao, T. A. Lipo, an B.-I. Kwon, A novel two-phase permanent magnet synchronous motor moeling for torque ripple minimization, IEEE Transactions on Magnetics, vol. 49, no.5,pp ,213. 6]Y.ZhanganJ.Zhu, Directtorquecontrolofpermanent magnet synchronous motor with reuce torque ripple an commutation frequency, IEEE Transactions on Power Electronics,vol.26,no.1,pp ,211. 7] T.-C. Jeong, W.-H. Kim, M.-J. Kim et al., Current harmonics lossanalysisof15-kwtractioninteriorpermanentmagnet synchronous motor through co-analysis of -q axis current control an finite element metho, IEEE Transactions on Magnetics,vol.49,no.5,pp ,213. 8] D.Q.Wei,B.Zhang,D.Y.Qiu,anX.S.Luo, Effectsofcurrent time-elaye feeback on the ynamics of a permanent-magnet synchronous motor, IEEE Transactions on Circuits an Systems II: Express Briefs,vol.57,no.6,pp ,21. 9] R. Ortega, L. Praly, A. Astolfi, J. Lee, an K. Nam, Estimation of rotor position an spee of permanent magnet synchronous motors with guarantee stability, IEEE Transactions on Control Systems Technology,vol.19,no.3,pp ,211. 1]Y.Ge,C.Wang,X.Zhou,anH.Wang, Researchonrotor position sensing of a permanent magnet synchronous motor

10 1 Mathematical Problems in Engineering base on high-frequency voltage injection an Kalman filter, in Proceeings of the International Conference on Electrical an Control Engineering (ICECE 1),pp ,June21. 11] Y. Lei, X. Fei, S. Jian-qing, C. Ming-liang, S. Qiao-ming, an Q.-F. Li, Sensorless control of high-power interior permanentmagnet synchronous motor rives at very low spee, IET Electric Power Applications,vol.7,no.3,pp ,213.

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