IMPROVED TRAJECTORY CONTROL FOR AN INTERIOR PERMANENT MAGNET SYNCHRONOUS MOTOR DRIVE WITH EXTENDED OPERATING LIMIT M. E. Haque, L. Zhong and M. F. Rahman School of Electrical Engineering and Telecommunication The University of New South Wales Sydney NSW 2052 Australia E-mail: meh@student.unsw.edu.au Abstract This paper presents analysis of trajectory control for an interior permanent magnet (IPM) synchronous motor drive under PWM current control. An improved trajectory control with extended speed range has been introduced. The scheme allows the motor to be driven with maximum-torque-per-ampere characteristic (MTPA) below the base speed and it maintains the maximum voltage limit of the motor under wide flux-weakening and the motor current limit under all conditions of operation accurately. Following the analysis of the trajectory control, the results of extensive simulation and experiment are given. 1. INTRODUCTION The IPM synchronous motor possesses special features for adjustable-speed drives which distinguish it from other classes of ac machines, especially surface permanent magnet synchronous motor. Because of its special rotor configuration, pole saliency and relatively large armature inductance, the IPM synchronous motor is also more suitable for an extended speed range operation by the flux-weakening control. The maximum steady state torque of the IPM synchronous depends on the continuous armature current rating. The available output voltage of the inverter limits the maximum speed attainable at this torque. In traction and spindle drives, constant power operation and wide speed range are desirable. With dc motor drives, these are achieved by the appropriate reduction of the field current as the speed increases. In IPM synchronous motor, direct control of the magnet flux is not available. The air-gap flux, however, can be weakened by the demagnetizing current in the direct axis [1-5]. In this paper, the armature current control method expanding the operating limits is examined through analysis, modeling and experiment. 2. CONTROL OF IPM MOTOR DRIVE In a PWM current controlled drives, the currents are regulated by PI (proportional + integral) controllers. Traditionally, the current control executes in the stationary reference frame due to its simplicity. When the current controllers are viewed in the rotor reference frame, however, it is seen that there exist cross-coupling effects between d-axis and q-axis variables and back-emf disturbance. It is possible to compensate the back-emf disturbance by using backemf feed-forward compensation. Other advantages of the latter scheme have been reported in the literature, for instance, the cross-coupling effect can be easily compensated. Furthermore, field-weakening control becomes easier since the d-axis current is directly controlled. The current control in the rotor flux reference frame as a means of torque and flux control has been adopted in this paper. The block diagram of a PWM current controlled IPM motor drive is shown in Fig. 1. 3. CONTROL TRAJECTORIES The voltage and torque equations of an IPMSM may be expressed in the d-q reference frame as follows: vd r + pLd v = ω L d q −ω Lq id 0 + r + pLq iq ωλ f 3 T = P [λ f iq + ( Ld − Lq ) id iq ] 2 (1) (2) From equation (1), the steady-state phasor diagram of an IPMSM shown in Fig. 2 is obtained. In Fig. 2, β and γ are the leading angles of the stator current and voltage vectors from the q-axis respectively and δ is the torque angle. It is clear that the dq-axis components of the stator current are id = − I s sin β iq = I s cos β (3) (4) where Is is the amplitude of the stator current vector. Substituting (3) and (4) into torque equation (2), yields the expression for the torque in terms of the amplitude of the stator current as follows: * * ω* PI-control with limit ω Current control & dq-decoupling * iq * ω Calculation of id * id vdc vqc Voltage compensation PWM inverter dq-1 iq id abc Te ib θ Tr IPMSM θ Encoder Figure 1. PWM current controlled IPM motor drive. ωλf is iq γ β id Phase angle, β Figure 3. Effects of the current phase angle β. 3.1 Maximum Torque Per Ampere (MTPA) Trajectory q-axis vs ri s T ia dq • Speed observer Torque, N m * * vd vq * iq To obtain fast transient response and high torque, the current phase angle β must be controlled to develop the maximum torque. The relationship between the amplitude of the stator current and the phase angle β for the maximum torque can be derived by setting the derivative of (5) with respect to β to zero. λs δ λf d-axis 3 3 d T = − P λ f I s sin β + P( Lq − Ld ) I s2 cos2 β = 0 (6) 2 2 dβ Figure 2. Phasor diagram of an IPM motor. 3 3 T = P λ f I s cos β + P( Lq − Ld ) I s2 sin 2 β 2 4 (5) where id, iq – d-and q-axes armature current, vd,vq – d- and q-axis armature voltage, Ld, Lq – d- and q- axes axis inductance, λf – magnet flux, r – armature resistance, Is – amplitude of the armature current P – number of pole pairs, p – differential operator d/dt, T – Torque and ω - rotor speed. The first term of (5) is the excitation torque Te and the second term is the reluctance torque Tr. Figure 3 shows T, Te and Tr as functions of the current phase angle β for the motor in Table 1 when the amplitude of the stator current is kept at the rated value. As is seen from the fig. 3, Te is maximum at β = 0 and Tr is maximum at β = 45o. Therefore, the total torque, T, is maximum within the range of 0o < β < 45o. For the IPM motor, Lq-Ld may be large, so the reluctance torque is no longer negligible. From the above analysis, to operate an IPM synchronous motor at high torque and efficiency, id should be determined by (4) with β corresponding to the maximum torque for a given stator current. By substituting equation (3) and (4) into equation (6), one obtains 3 3 − P λ f I s sin β + P( Lq − Ld ) I s2 (cos2 β − sin2 β ) = 0 2 2 (7) From equation (7), one can obtain id = λf 2( Lq − Ld ) − λ 2f 4( Lq − Ld ) 2 + iq2 (8) Equation (8) implies that the maximum torque-perampere (MTPA) is obtained if id is determined by equation (8) for any iq. . It should be noted here that the torque is not directly proportional to iq. This is why the torque control via current control is called indirect torque control. 3.2 Current and Voltage Limit Trajectories When an IPM synchronous motor is fed from an inverter, the maximum stator current and voltage are limited by the inverter/motor current and dc-link voltage ratings respectively. These constraints can be expressed as I s = id2 + iq2 ≤ I sm (9) Vs = vd2 + vq2 ≤ Vsm (10) where Ism and Vsm are the available maximum current and voltage of the inverter/motor. Substituting equation (1) at steady state into equation (10), one can obtain Vs = (rid − ω Lq iq )2 + ( riq + ω Ld id + ωλ f )2 ≤ Vsm (11) Equation (11) can be simplified as equation (12) if the stator resistance is neglected. ( Lqiq )2 + ( Ld id + λ f ) 2 ≤ ( Vsm 2 ) ω (12) id can be calculated from equation (12) as (13) if resistance is neglected. id = − λf Ld + 1 Ld 2 Vsm ω2 − ( Lq iq ) 2 = − λf Ld + 1 Ld id 1 (13) where id 1 = 2 Vsm − ( Lqiq ) 2 ω2 (14) 3.3 Voltage Limited Maximum Output Trajectory The armature current vector i3(id3,,iq3) producing maximum output power under the voltage limit condition is derived as follow [6]: id = − λf Ld − ∆id (15) 2 Vsm − ( Ld ∆id ) ω iq = ρ Ld (16) V − ρλ f + ( ρλ f ) 2 + 8( ρ − 1) 2 sm ω ∆id = 4( ρ − 1) Ld where, ρ = 2 (17) Lq Ld The current vector trajectory of the voltage limited maximum output is shown in Fig. 4. The rotor speed iq, A Maximum torque per ampere trajectory Voltage limited maximum output id= -λf/Ld trajectory G B D A2 ω = ωp A3 C E 4. CONTROL MODE SELECTION The MTPA and current limit trajectories are independent of the rotor speed and are only determined by the motor parameters and inverter current rating. However, the voltage limit trajectory varies with the change in rotor speed. It is seen that the voltage limit trajectory is an ellipse, which contracts when the rotor speed increases. When the rotor speed increases infinitely, the voltage limit ellipse becomes a point on the d axis. If this point is inside the current limit circle, theoretically, the motor has an infinite maximum speed, otherwise, has a finite maximum speed. The intersection of MTPA, current limit trajectories is the operation point A1 at which the motor has the rated current and voltage and will produce the rated torque at the base speed. The control modes, i.e., maximum torque-per-ampere and flux weakening controls, are selected according to the analysis of fig. 4. The q-axis commanded current i q* is determined by the outer control loop and the d-axis commanded current i *d is decided by equation (8) in maximum torque-per-ampere control mode, or by (12) in flux weakening control mode. Whether MTPA mode or the field weakening mode should be selected is determined by the rotor speed and the load. According to the rotor speed, the motor operation is divided into three sections, that is, below the base speed ωb, above the crossover speed ωc and between the base and crossover speeds. The crossover speed ωc is the speed at which the back-emf voltage of the unloaded motor equals to the maximum voltage. As a result, the control mode is selected as follows. 4.1 Operation below the base speed Voltage limit trajectory ω = 1500 rpm = ωbase ω = 2200 rpm ω = 2400 rpm= ωc ω = 2800 rpm= ω1 ω > 2800 rpm A1 ωp is the minimum speed for the voltage-limited maximum-output operation. Below this speed, the voltage-limited maximum-output operating point can not be reached, because the voltage-limited maximumoutput trajectory intersects the voltage limit trajectory outside the current limit circle. If (λ f Ld ) > I sm , the voltage-limited maximum output trajectory is outside the current-limit trajectory. Therefore, voltage-limited maximum-output trajectory needs not to be considered. O id, A Figure 4. Control trajectories in id-iq plane. It is seen in fig.4 that the voltage limit ellipse corresponding to the operation below the base speed is larger than the one for base speed. Therefore, if the stator current vector is controlled according to the maximum torque-per-ampere trajectory and satisfying the current limit, which is the trajectory A1O, it must satisfy the voltage limit since the current vector is inside the voltage limit ellipse. Therefore, maximum torque-per-ampere mode is selected for constant torque operation. The d- and q-axis currents, idA1 and iqA1, with which the maximum torque is produced, are determined by (8) and (9) when Is = Ism. idA1 = λf 2( Lq − Ld ) − λ 2f 4( L − L)2 2 2 + I sm − idA 1 (18) Solving (12), one gets idA1 = λf 4( Lq − Ld ) − λ 2f 16( Lq − Ld )2 + 2 I sm 2 2 2 iqA1 = I sm - idA1 (19) (20) This is the operating point at which the motor produces the maximum torque and the limit value of the outer control loop for constant torque operation is then determined by (20). 4.2. Operation above the crossover speed To run the motor above the crossover speed, the flux λf, which is the flux linkage along the rotor d axis, has to be reduced. Although the flux is already reduced with the MTPA control, the voltage limitation is no longer satisfied when the rotor speed is above the crossover speed. The stator current vector is therefore controlled according to the voltage limit trajectory instead of MTPA trajectory. Thus, id and iq are determined according to the voltage and current limit equation (12) and (9). The limit value, idv and iqv, of the outer control loop for such a field weakening operation is therefore determined by these two equations with Is = Ism, which is inversely proportional to the rotor speed. idv = − λ f Ld a + 1 λ 2f L2d − a b a (21) where a = L2d − L2q 2 2 b = I sm Lq + λ 2f − (22) 2 Vsm ω2 (23) current vector moves from A3 to E (fig. 4) along the voltage-limited maximum output trajectory. 4.3 Operation between the base and crossover speeds (3-24) When the rotor speed is in the range from ωb to ωc, the control mode is determined by the load. If the motor is unloaded, it may operate near the crossover speed with MTPA control. When the motor is fully loaded, it has to be controlled according to the voltage limit trajectory right above the base speed. For instance, if the motor runs at 2200 rpm, the corresponding voltage limit trajectory is BCO as shown in fig. 4. Thus, if the motor is heavily loaded and the current vector is along the trajectory BC, it has to be controlled according to the voltage limit trajectory. Otherwise, the motor can be still controlled under MTPA control along the trajectory CO. The determination of the control mode is based on the calculated id from both equation (8) and (13). If the calculated id from equation (9) is smaller than the one calculated from (13), the current vector is controlled to the MTPA trajectory for constant torque operation. Otherwise, voltage limit trajectory is used to control the current vector for field weakening operation. The limit value for the outer control loop is still determined by equation (24). 5. MODELING AND EXPERIMENTAL RESULTS Modeling and experiment were performed on an IPM synchronous motor. The specification of the motor used is shown in Table 1. The rated and crossover speeds for this motor are 1500 and 2400 rpm, respectively. The sampling time of the inner control loop is 150 µs and that of the outer control loop is 750 µs. The drive system was implemented on a TMS320C31 digital signal processor with a clock speed of 33 MHz. The rotor position and speed were obtained from an incremental encoder with 5000 pulses per revolution. A three-phase insulated gate bipolar transistor (IGBT) intelligent power-module is used for an inverter, which is supplied at a dc link voltage of 570 V. than ω1. The trajectory for id = −λ f / Ld has shown in Figure 5 shows the modeling results of speed, iq, and id with respect to a step change in speed reference from 0 to 1500 rpm. It is seen from these figures that the current vector is at the intersection of MTPA and current limit trajectories, i.e., point A1 during acceleration and moves along the MTPA trajectory when the speed approaches its reference. Since the load torque is taken to be zero, the current vector finally settles down at origin, O. figure 4. If the rotor speed ω ≥ ω p , id and iq has to be determined from equation (15) and (16) and the Figure 6 shows the dynamic responses with respect to a step change of speed reference from 0 to 2800 2 2 iqv = I sm - idv (24) When the rotor speed is in the range from ωc to ω1, id is determined from equation (13). However, as the rotor speed exceeds ω1 for which id1 is –ve and id becomes a complex number. So id should be calculated as id = −λ f / Ld for a rotor speed greater Speed, rpm Speed, rpm Time, sec. Time, sec. (a) Speed response. (a) Speed response. iq, A iq, A 0.4 Time, sec. Time,sec (c) iq response (b) iq response. id, A id, A Time, sec. (c) id respnse. Time, sec. (c) id respnse. iq, A MTPA iq, A MTPA Current limit Current lim it A O id, A (d) Locus of current vector. id , A (d) Locus of current vector. Figure 6. Modeling results for flux-weakening operation under trajectory control ωr >ωb. Figure 5. Modeling results for constant torque operation under trajectory control ωr ≤ ωb. rpm. It is clear that the smooth transition between constant torque and field weakening controls occurs when the rotor speed exceeds the base speed (1500 rpm). It is also clear that the stator current vector moves along the maximum-torque-per armature current trajectory when the speed is below the base speed and shifts to the voltage limit ellipse when the speed increases above base speed. Figure 7 shows the torque and power versus speed characteristics of the motor. It is seen that field weakening occurs once the speed goes above the base speed (1500 rpm). It is seen that constant power operation is possible with the trajectory control discussed in this paper. T o r q u e, N m P o w e r, W 2 800 400 0 0 1000 2000 3000 4000 0 1000 2000 3000 Speed, rpm 4000 Speed, rpm Figure 7. Torque and power versus speed. Figures 8 and 9 show the experimental results for constant torque operation and flux-weakening operation, respectively. The experimental results conform to the modeling results quite well. iq, A Speed, rpm 2 1600 0.1 id, A id, A 1.5 0 1200 1 -0.1 1 800 -0.2 0.5 400 -0.3 0 0 0 0.1 0.2 0.3 0.1 0.4 0.2 0.3 -0.4 0.4 0.1 T i m e, s e c. T i m e, s e c. (a) speed response. 0.2 0.3 0.4 0 -1 0 1 T i m e, s e c. (b) iq response. (c) id response. iq, A (d) Locus of current vector. Figure 8. Experimental results for constant torque operation under trajectory control. i q, A Speed, rpm iq, A id, A 3000 2 0 1.5 -0.5 1 -1 0.5 1.5 2000 1 0.5 1000 0 0 0 -0.5 0.2 0.4 0.6 Time, sec (a) speed response. -1.5 0.2 0.4 Time, sec (b) iq response. 0 0.2 0.4 0.6 Time, sec. (c) id response. -1.5 -1 -0.5 0 0.5 1 1.5 id, A (d) Locus of current vector. Figure 9. Experimental results for flux-weakening operation under trajectory control. REFERENCES 6. CONCLUSION This paper analyses the control trajectories for an IPM synchronous motor. The control trajectories, namely the maximum torque-per-ampere, current and voltage limit trajectories, have been derived and expressed in terms of id and iq, and then drawn in the id -iq plane. Based on the analysis of these trajectories, the selection of constant torque and flux-weakening modes has been discussed. It has been established that the maximum torque-per-ampere trajectory should be selected for constant torque operation and the voltage limit trajectory should be selected for flux-weakening operation. It has been shown by the modeling and experimental results that flux-weakening is very well incorporated in this drive. The transition between constant torque and flux-weakening is very smooth. It is also shown that constant power operation is possible with the improved trajectory control discussed in this literature. TABLE 1. Parameters of the IPM Motor used. 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Zhong and K. W. Lim, “A DSP based instantaneous torque control strategy for interior permanent magnet synchronous motor drive with wide speed range and reduced torque ripples”, IEEE Industry applications society Annual Meeting, vol. 1, pp. 518-524, 1996. [6] S. Morimoto, M. Sanada, Y. Takeda, “Expansion of operating limits for permanent magnet motor by current vector control considering inverter capacity”, IEEE Trans. on Ind. Appl., vol., no. 5, 26, pp. 866-871, 1990.