Direct torque and indirect flux control of brushless DC motor with non sinusoidal back EMF without position sensor

Direct Torque and Indirect Flux Control of Brushless DC Motor with Non-sinusoidal Back-EMF without position sensor Abstract: In this paper, the position sensorless direct torque and indirect flux control (DTIFC) of BLDC motor with nonsinusoidal (non-ideal trapezoidal) Back-EMF has been extensively investigated using three-phase conduction scheme with six-switch inverter. In the literature, several methods have been proposed to eliminate the low-frequency torque pulsations for BLDC motor drives such as Fourier series analysis of current waveforms and either iterative or least-mean-square minimization techniques. Most methods do not consider the stator flux linkage control, therefore possible high-speed operations are not feasible. In this work,by using sliding mode observer the mechanical rotor position is removed. The proposed sensorless DTC method controls the torque directly and stator flux amplitude indirectly using d–axis current. Since stator flux is controllable, flux-weakening operation is possible. Moreover, this method also permits to regulate the varying signals. Simple voltage vector selection look-up table is designed to obtain fast torque and flux control. Furthermore, to eliminate the low-frequency torque oscillations, the new method has been used to estimate the electromagnetic torque. Simulation results confirm that the proposed three-phase conduction DTC of BLDC motor drive with six-switch inverter scheme. Keywords: Brushless dc (BLDC) motor, sliding mode observer, direct torque control (DTC), stator flux control, fast torque response. 1. Introduction Permanent magnet brushless machines are widely used for servo drives, ship propulsion systems and traction drives. Brushless AC (BLAC) drive back-EMF waveform is sinusoidal and is supplied with SPWM voltage source inverter, while the Brushless DC (BLDC) stator phase currents are rectangular and its back-EMF is trapezoidal due to concentrated windings. BLDC motor has better efficiency, higher torque density, lower cost and simpler structure, compared to BLAC motors. Further, BLAC drive requires an accurate encoder sensor, while BLDC drive needs to discrete position sensor such as Hall Effect device. Cost reduction of variable speed drives is growing interest over these years. Elimination of the position sensor are important subjects in low cost drives. Up to now, many researches have been reported to control of BLDC motor using six-switch three-phase inverter (SSTPI). Direct torque control (DTC) scheme was first proposed by Takahashi [1] and Depenbrock [2] for induction motor drives in the mid 1980s. More than a decade later, in the late 1990s, DTC techniques for both interior and surface-mounted synchronous motors (PMSM) were analyzed [3]. More recently, application of DTC scheme is extended to BLDC motor drives to minimize the low-frequency torque ripples and torque response time as compared to conventional PWM current controlled BLDC motor drives [4], [5] and [6]. In [4], [5] and [6], the voltage space vectors in a two-phase conduction mode are defined and a stationary reference frame electromagnetic torque equation is derived for surface-mounted permanent magnet synchronous machines with non-sinusoidal back-EMF (BLDC, and etc.). It is shown in [5] that only electromagnetic torque in the DTC of BLDC motor drive under two-phase conduction mode can be controlled. Flux control is not trivial due to the sharp changes whose amplitudes are unpredictable depending on several factors such as load torque, dc-link voltage, winding inductance, etc. In this work, the torque control method with three- phase conduction BLDC motor presented in [7] is used. Also, unlike [7] to estimate the electromagnetic torque is no need to use proposed 22 × matrix in [7], because the new method has been used to estimate the electromagnetic torque. As well as, in order to elimination of mechanical position sensor a sliding mode observer is used to estimate the rotor position. As opposed to the prior two-phase conduction methods, this DTC technique can control both torque and stator flux of the BLDC motor simultaneously, therefore field-weakening operation is possible. The proposed sensorless DTC method controls the torque directly and Reza Heidari*, GholamReza Arab Markadeh**, and Saeed Abazari*** *Department of Electrical Engineering, Shahrekord University, Shahrekord, Iran, E-mail: Heidari_reza@Hotmail.com **Department of Electrical Engineering, Shahrekord University, Shahrekord, Iran, E-mail: arab- gh@eng.sku.ac.ir ***Department of Electrical Engineering, Shahrekord University, Shahrekord, Iran, E-mail: Saeedabazari@yahoo.com stator flux amplitude indirectly using d–axis current. Unlike those for motor with sinusoidal back-EMFs, optimal current references for a nonsinusoidal back-EMF motor (BLDC) in the synchronous reference frame are not constant, therefore current wave shapes require very fast controllers in particular at high speed. Classical bandwidth of the controller (such as proportional– integral) does not allow tracking all of the reference current harmonics [8]. Since the hysteresis controllers used in the proposed DTC scheme are not fast controllers like PI, they can easily regulate not only constant but also the varying references (torque and flux). Simulation results are presented to illustrate the validity and effectiveness of the sensorless three-phase conduction DTC of a BLDC motor drive with six-switch inverter. 2. The Proposed Sensorless DTC of Four-Switch BLDC Motor Drive Using Three-Phase Conduction 2.1 Principles of the Proposed Method Title Block Indirect torque control method of BLDC motor explained in [9] was extended to a direct torque and indirect flux control technique for six-switch inverter BLDC motor in [7] which is suitable for field-weakening operations. In this work, unlike [7] to estimate the electromagnetic torque, fluxs and currents in dq reference frame have been used. The stator flux-linkage vector can be obtained from the measured stator voltages α s V and β s V and currents α s i and β s i as ∫ ∫ −= −= dtRiV dtRiV sss sss )( )( βββ ααα ψ ψ (1) Where R is the stator winding resistance. As was expressed earlier, in this paper, to estimate the electromagnetic torque, fluxs and currents in dq reference frame have been used. In order to establish this transformation between any two frames of reference, let x denote the reference frame to witch the variables are being transformed, then x qdos yx y qdos fkf = (2) Where f is a variable such as, voltage or current and so on.           −− −−− = 100 0)cos()sin( 0)sin()cos( xyxy xyxy yx k θθθθ θθθθ (3) Where x θ and y θ are electrical rotor position of first and second reference farme, respectively. In order to transformation between stationary reference frame and rotor reference frame, relations will be as follows:             − =       α β ψ ψ θθ θθ ψ ψ )sin()cos( )cos()sin( rr rr d q (4)       − = )sin()cos( )cos()sin( rr rr qd k θθ θθ βα (5) Where r θ is the electrical rotor position. 2.2 Electromagnetic Torque Estimation in dq Reference Frame Because of the rotor position dependant terms in the dq frame stator flux, conventional torque estimation in stator reference frame used for DTC of sinusoidal ac motors is no longer valid for BLDC motor, therefore a new torque estimation algorithm is derived in dq frame consisting of dq–axes fluxs and currents. The torque estimation is the key factor in the proposed DTC scheme that with the following relationship can be obtained. )( 4 3 dsqsqsdse ii P T ψψ −= (6) Where P is the number of poles, ds ψ , qs ψ , ds i and qs i are the dq-axes fluxs and currents, respectively. As it can be noticed that the equation (6) eliminates the speed term in the denominator which causes problem at zero and near zero speeds. 2.3 Control of Stator Flux Linkage Amplitude Since BLDC motor does not have sinusoidal back- EMF, the stator flux trajectory is not pure circle as in PMSM. It is more like a decagon shape as shown in Fig. 1. Thus, direct stator flux amplitude control in a BLDC motor is not trivial as in PMSM such that rotor position varying flux command should be considered. However, this is a complicated way to control the stator flux linkage amplitude [5]. Therefore, in this work similar [7] instead of s ϕ itself its amplitude is indirectly controlled by d– axis current. In [7] for six-switch BLDC motor drive, in the constant torque region * r ds i was controlled as zero and in the flux-weakening region it was decreased for a certain amount depending on the operational speed to achieve maximum torque.The switching table for controlling both the amplitude and rotating direction of the stator flux linkage is given in Table I. Fig. 1: Decagon trajectory of stator flux linkage in the stationary αβ – plane. TABLE 1. Switching Table For DTC of Four-Switch BLDC Motor Using Three-Phase Conduction Electrical rotor position T ϕ 6 θ 5 θ 4 θ 3 θ 2 θ 1 θ 1 v 6 v 5 v 4 v 3 v 2 v 1 1 5 v 4 v 3 v 2 v 1 v 6 v -1 2 v 1 v 6 v 5 v 4 v 3 v 1 -1 4 v 3 v 2 v 1 v 6 v 5 v -1 2.4 Sliding Mode Observer Sliding mode observer can be used to estimate the non-sinusoidal back-EMF values of BLDC motor. The estimated stator currents and Back-EMF based on sliding mode observer is obtained as:            = = ++− − = ++− − = ) ~ ( ˆ ) ~ ( ˆ ) ~ ( ˆ ˆˆ ) ~ ( ˆ ˆˆ 2 2 1 1 ββ αα β ββ ββ α αα αα ss ss ss ss s s s s ss ss s s s s iSgnKe iSgnKe iSgnK L V L e i L R i iSgnK L V L e i L R i & & & & (7) Where βαβα eeii ss ˆ , ˆ , ˆ , ˆ are the estimation of βα − axes stator currents and Back-EMFs respectively, 1s K and 2s K are constant coefficients, Sgn(.) is a sign function and sliding surfaces are defined as:      −== −== ββββ αααα sss sss iiiS iiiS ˆ ~ ˆ ~ (8) If sampling period is significantly less than electrical and mechanical time constant then back-EMF value can be assumed to remain constant. During each sampling period and so: 0/,0/ == dtdedtde βα . By subtracting (4) from (1), the estimation error of stator currents and Back-EMF can be expressed as            −= −= −− − = −− − = ) ~ ( ~ ) ~ ( ~ ) ~ ( ~ ~~ ) ~ ( ~ ~~ 2 2 1 1 ββ αα β β χβ α α αα ss ss ss s s s s s ss s s s s s iSgnKe iSgnKe iSgnK L e i L R i iSgnK L e i L R i & & & & (9) Defining positive definite Lyapunov function V(x) as: ) ~~ ( 2 1 22 βα ss iiV += (10) Derivation of V(x) with respect to time concludes 0 ~~ ) ~ ~ ~ ~ ( 1 ) ~~ ( 2 1 2 1 22 <−− +−+−= βα ββααβα ssss ss s ss s s iKiK ieie L ii L R V & (11) The lower bound of 1s K can be selected such that V & became negative definite and then the estimated stator current components converge asymptotically to their measured values. 2 1 ~ ~ ~ ~ ~ 1 s ss ss s s i ieie LL R K ββαα + −−> (12) It is sufficient that: ) ~~ ( 1 1 βα ee L K s s +> (13) To decrease the chattering effects of large discontinuous control law, near the sliding surfaces the sgn(.) function is replaced with saturation function as:        −≤− < ≥ =         qi qii qi q i Sat s ss s s ~ 1 ~~ ~ 1 ~ (14) Where q is the sliding surfaces band. In order to decrease the pure integrator influences in Back-EMF estimation in (7), the integral operator is replaced by a low pass filter. Finally, the back-EMF sliding mode observer will become: ) ~ ( ~~ ) ~ ( ~~ 2 2 βββ ααα sst sst iSgnKeKe iSgnKeKe +−= +−= & & (15) Where t K is the higher cut-off frequency of low pass filter. 3. Simulation Results The drive system shown in Fig. 2 has been simulated in order to demonstrate the validity of the proposed three- phase conduction DTC of a six-switch BLDC motor drive scheme with parameters has been shown in Table 2. The controller sampling period is selected as 32 µs. The magnitudes of the torque and flux hysteresis bands are 0.001N·m, and 0.001 Wb, respectively. Implementations of torque, q– and d–axis rotor reference frame stator currents and three-phase currents and Back-EMFs responses of the proposed DTC of a BLDC motor drive scheme are demonstrated in Fig. 3(a) through (e), respectively under 0.5 N.m load torque condition. The torque reference is changed suddenly from 0.5 N·m to 0.25 N·m at 1 second. As seen in Fig. 3(a) that fast torque response is obtained and the estimated torque tracks the reference torque closely. The high frequency ripples observed in the torque and current can be minimized by properly selecting the dc-link voltage and torque hysteresis band size. Fig. 2: Block diagram of the direct torque and indirect flux control of four-switch BLDC motor drive using three-phase conduction mode without position sensor. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Fig. 3. Simulation results of direct torque and indirect Flux Control of Six-Switch Brushless DC Motor without position sensor when refe T − is changed from 0.5 N.m to 0.25 N.m at t=1 sec under Ai r ds 0 * = , (a) generated torque, (b) q-axis stator current, (c) d- axis stator current, (d) actual electrical rotor position, (e) estimated electrical rotor position, (f) error of actual and estimated electrical rotor position, (g) Three-phase currents, (h) Three-phase Back- EMFs, (i) stator flux locus, (j) rotor flux locus. βα SS VV , BLDC * em T * r ds i e T + − − + dq/ αβ )( 4 3 r dsq r qsde ii P T ϕϕ −= ∫ ∫ −= −= dtiRV dtiRV ssss ssss )( )( βββ ααα ϕ ϕ ( ) 3/2 baS aS iii ii += = β α dq/ αβ ObserverModeSliding ee qisatkeke qisatkeke bemf SSt SSt )/(tan )/( )/( 1 2 2 ∧∧ − ∧∧∧ ∧∧∧ = −−= −−= • • αβ βββ ααα θ 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time ( sec ) Torque ( N.m ) 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time ( sec ) i q s ( In the rotor reference frame ) 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time ( sec ) i d s ( In the rotor reference frame ) 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 0 1 2 3 4 5 6 7 Time ( sec ) Actual electrical position 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 0 1 2 3 4 5 6 7 Time ( sec ) Estimated electrical position 1.018 1.019 1.02 1.021 1.022 1.023 1.024 1.025 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 Time ( sec ) θ actual - θ estimated 1.44 1.45 1.46 1.47 1.48 1.49 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 Time ( sec ) Three - phase currents ( A ) 1.44 1.45 1.46 1.47 1.48 1.49 1.5 -80 -60 -40 -20 0 20 40 60 80 Time ( sec ) Three - phase Back - EMFs -0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1 Alfa - axis Stator Flux Linkage ( Wb ) Beta - axis Stator Flux Linkage ( Wb ) -0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1 Alfa -axis Rotor Flux Linkage ( Wb ) Beta -axis Rotor Flux Linkage ( Wb ) In Fig. 4, the possibility of the flux-weakening region operation is simulated when * r ds i is changed from 0 A to -1 A. As it can be seen in Fig. 4 that the shape of stator flux linkage trajectory is kept same, however its amplitude is smaller compared to the initial case which means that the flux in the machine is weakened to obtain maximum possible torque above the base speed. It is concluded that in the proposed control scheme flux- weakening operation is viable by properly selecting the d–axis current reference as in PMSM drives. As a result, there is no need to use position-varying stator flux linkage amplitude )( res θϕ as a reference which is complicated to obtain especially in the field-weakening region. (a) (b) Fig. 4. Simulated indirectly controlled flux linkage trajectory under the sensorless three-phase conduction DTC of a BLDC motor drive when * r ds i is changed from 0 A to -1 A under 0.5 N.m load torque, (a) d-axis stator current, (b) stator flux locus. Table 2. Parameters of BLDC motor )(Ω s R 0.56 )(mHL s 0.85 Flux induced by magnets (Wb) 0.07627 ).( 2 mkgJ 0.0003617 ) ( smNB 0.9444*e-4 Rotor pole numbers 16 DC link voltage (V) 36 Rated speed (rad/sec) 30 3. Conclusion This study has successfully demonstrated application of the proposed three-phase conduction direct torque control (DTC) scheme for BLDC motor drives. It is shown that the BLDC motor could also operate in the field-weakening region by properly selecting the d–axis current reference in the proposed DTC scheme. Also, a Sliding mode observer is used to estimate the electrical rotor position. Then, stator flux-linkage vector are obtained from the measured stator voltages α s V and β s V and currents α s i and β s i , then by using (1) and (4) q- and d-axis flux linkage are drived. Also, by usig (4) q- and d-axis current are calculated. Finally, by using (6) electromagnetic torque is estimated. A look-up table for the three-phase voltage vector selection is designed similar to a DTC of PMSM drive to provide fast torque and flux control. The simulation results show the effectiveness of the proposed method. References [1] I. Takahashi and T. Noguchi, “A new quick-response and high efficiency control strategies of an induction motor,” IEEE Trans. Ind. Appl., vol. 22, no. 5, pp. 820–827, Sep./Oct. 1986. [2] M. Depenbrock, “Direct self-control of inverter-fed induction machine,” IEEE Trans. Power Electron., vol. 3, no. 4, pp. 420– 429, Oct. 1988. [3] L. Zhong, M. F. Rahman, W. Y. Hu, and K. W. Lim, “Analysis of direct torque control in permanent magnet synchronous motor drives,” IEEE Trans. Power Electron., vol. 12, no. 3, pp. 528 536, May 1997. [4] Y. Liu, Z. Q. Zhu, and D. Howe, “Direct torque control of brushless dc drives with reduced torque ripple,” IEEE Trans. Ind. Appl., vol. 41, no. 2, pp. 599–608, Mar./Apr. 2005. [5] S. B. Ozturk and H. A. Toliyat, “Direct torque control of brushless dc motor with non-sinusoidal back-EMF,” in Proc. IEEE-IEMDC Biennial Meeting, Antalya, Turkey, May 3-5, 2007. [6] G.R. Arab Markadeh and S.I. Mousavi, ‘Position sensorless direct torque control of brushless DC motor based on back-EMF vector’ , ELECTROMOTION. pp. 128-138, 2009. [7] S. B. Ozturk and H. A. Toliyat, “Sensorless Direct Torque and Indirect Flux Control of Brushless DC Motor with Non-sinusoidal Back-EMF,” IEEE, pp. 1373-1378, 2008. [8] F. Bodin, S. Siala, “New reference frame for brushless dc motor drive,” in Proc. IEE-PEVD Annu. Meeting, London, UK, Sep. 21- 23, 1998, pp. 554-559. [9] P. J. Sung,W. P. Han, L. H. Man, and F. Harashima, “A new approach for minimum-torque-ripple maximum-efficiency control of BLDC motor,” IEEE Trans. Ind. Electron., vol. 47, no. 1, pp. 109–114, Feb. 2000. [10] J. Hu, B. Wu, “New integration algorithms for estimating motor flux over a wide speed range,” IEEE Trans. Power Electrons., vol. 13, pp. 969–977, Sep. 1998. 0.9 0.92 0.94 0.96 0.9 8 1 1.02 1.04 1.06 1.08 1.1 -2 -1.5 -1 -0.5 0 0.5 Time ( sec ) i d s ( In the rotor reference frame ) -0.1 -0.05 0 0.05 0.1 -0.1 -0.05 0 0.05 0.1 Alfa - axis stator flux linkage ( Wb ) Beta - axis stator flux linkage ( Wb ) . Direct Torque and Indirect Flux Control of Brushless DC Motor with Non-sinusoidal Back-EMF without position sensor Abstract: In this paper, the position. 2009. [7] S. B. Ozturk and H. A. Toliyat, “Sensorless Direct Torque and Indirect Flux Control of Brushless DC Motor with Non-sinusoidal Back-EMF, ” IEEE, pp.