Permanent Magnet Synchronous Motors Based on. Z-Source Inverters. Muyang Li, Student Member, IEEE, Jiangbiao He, Student Member, IEEE, Nabeel A.O. ...
A Flux-Weakening Control Approach for Interior Permanent Magnet Synchronous Motors Based on Z-Source Inverters Muyang Li, Student Member, IEEE, Jiangbiao He, Student Member, IEEE, Nabeel A.O. Demerdash, Life-Fellow, IEEE Department of Electrical and Computer Engineering, Marquette University, WI, 53233 E-mail: muyang.li @mu.edu Abstract- This paper presents a flux-weakening solution by employing the voltage boost capability of Z-source inverters to overcome the voltage limitation in conventional drives for highspeed applications, e.g. hybrid/electric vehicles. In this control strategy, a flux-weakening algorithm has been developed to achieve constant output power of an interior permanent magnet synchronous motor (IPMSM). Correspondingly, a flux-weakening control strategy including a closed-loop speed control is presented. Compared to existing flux-weakening methods, this attractive control strategy allows an IPMSM to largely extend its constant power speed range and operate at higher speed with larger torque. Theoretical analysis is given in detail, and corresponding simulation results are presented to verify the presented control technique. Index Terms- flux-weakening, Z-source inverter, interior permanent magnet synchronous machine, constant power speed range (CPSR).
I.
CPSR
Z Z
max base
P
Te
P, T
Constant Power region (Flux-weakening region)
e
Constant torque region
Zbase
Zmax Speed
Fig. 1. Typical characteristic curves of torque/power vs. speed of IPMSMs.
INTRODUCTION
In recent decades, interior permanent magnet synchronous machines (IPMSMs) have attracted great attention in various industrial applications, particularly in propulsion systems for hybrid/electric vehicles (EVs). The attractiveness stems from the high power density, high efficiency, wide speed range, fast torque-speed response of IPMSMs, and the relatively decreasing price of permanent-magnet (PM) materials [1]. For traction applications in EVs, IPMSM-drive systems normally require a wide constant power speed range (CPSR) [2], in which the drive can maintain constant power at high speed, as shown in Fig. 1. However, CPSR is generally restricted by a limited dc bus voltage which is normally rated from 200 volts to 600 volts in EVs [3]. Such restriction of CPSR is due to fact that a specific power inverter cannot drive IPMSMs at high speed because of the fact that the backelectromotive force (back-EMF) is proportional to motor speed and air gap flux. Once the back-EMF becomes larger than the maximum output voltage of the drive, an IPMSM will be incapable of developing torque. Thus, a motor speed cannot be increased when the back-EMF reaches the maximum output voltage of the drive, unless the air gap flux can be weakened. Considering that the rotor magnetic field generated by PMs cannot be weakened directly, an extended speed range can be achieved by means of indirect flux-weakening (FW) control [4]. During the FW region, an opposing magneto motive force (MMF) is established by the stator current to counteract the “apparent” MMF established by PMs mounted on the rotor. As
978-1-4799-2262-8/14/$31.00 ©2014 IEEE
Fig. 2. Typical topology of a Z-source inverter.
a result, the air-gap flux is indirectly weakened and the motor speed is increased. Recently, some FW control strategies have been presented for the control of IPMSMs in the literature [4-6]. Closed-loop FW was first proposed in [4], which used a phase advance principle and d-axis current feedback. A voltage compensator was used in a current regulator for the FW control approach in [5]. The difference between dc link voltage and reference stator voltage was used to set up a FW regulator in [6]. Although these control strategies have good performance with wellregulated currents, they are constrained by the maximum output voltage of drives. The effects of terminal voltage limitation become critical in the deep FW region for a given drive system. Due to the increasing popularity of EVs and hybrid EVs in the past decade, a further extended CPSR becomes increasingly important for improving high speed performance of EVs and hybrid EVs. Aiming at solving the voltage constraint issue stated above, a novel FW solution was developed in this paper to extend the CPSR by utilizing the high voltage boost capability of Z-
Ellipses shrink while speed increases
Voltage limit ellipses
ܸ௦ ൌ ටݒௗଶ ݒଶ ܸ௫
MTPA trajectory
A
Region AB
Region OA
Current limit circle
B
O
Fig. 3. Current limiting circle, voltage limiting ellipse and maximum torque-per-ampere trajectory in d-q plane.
source inverters [7], which could output much higher voltage than a given dc bus voltage. Since all the conventional methods have to consider constraints from a stator winding terminal voltage, a new FW algorithm, which gives constant output power, was developed for generating current references with boosted voltage. In addition, a closed-loop speed control and open-loop FW control scheme were designed based on a general FW control scheme. The Z-source inverter, shown in Fig. 2, could be working at normal mode as a conventional inverter during the constant torque region and boost mode during the constant power region. The boost factor of Z-source inverter is adjusted according to the voltage references from current regulators. Simulation results for a 3.5 hp laboratory IPMSM are presented to verify the feasibility of proposed FW technique with a Z-source inverter-based ac drive. II.
FLUX-WEAKENING CONTROL OF IPMSMS BASED ON CONVENTIONAL INVERTERS
A. Modeling of IPMSMs In the synchronous rotating d-q reference frame, assuming the value of stator resistance is almost negligible, the IPMSM model in steady state can be approximated as follows [8]: ݒௗ ൌ െ߱ ܮ ݅
(1)
ݒ ൌ ߱ ൫ܮௗ ݅ௗ ߣ ൯
(2)
where, ݒǡ ݅and ܮ, are the voltage, current and inductance, while the subscripts d and q stand for the d and q-axes. Here, ߱ , is the electrical angular speed, and ߣ , is the per-phase flux linkage due to the permanent magnets. Accordingly, the developed torque equation can be expressed as follows [9]: ͵ (3) ܶ ൌ ൣߣ ݅ ൫ܮௗ െ ܮ ൯݅ௗ ݅ ൧ ʹʹ where, , is the number of poles. Here, the phase current and phase voltage constraints of motors are defined as: ܫመ௦ ൌ ට݅ௗଶ ݅ଶ ܫመ௫
(4)
(5)
where, ܫመ௫ , is the peak value of maximum inverter output current, and ܸ௫ , is the peak value of maximum inverter output voltage. After substituting (1) and (2) into (5), the voltage constraint is rendered current and speed dependent, yielding: ܮଶ ݅ଶ ሺܮௗ ݅ௗ ߣ ሻଶ ሺܸ௫ Ȁ߱ ሻଶ (6) Due to the different d- and q- axes inductances of IPMSMs, the voltage constraint forms an ellipse instead of a circle. The critical conditions of (4) and (6) are given by a “current limiting” circle and a “voltage limiting” ellipse as show in Fig. 3. B. Control in the Constant Torque Region For any current level, there is a particular d-axis current and q-axis current pair that maximizes the output torque of IPMSMs. Namely, this is the maximum torque per ampere (MTPA) algorithm which is implemented at constant torque region which is the region OA in Fig. 3, to fulfill the high output torque requirement for traction applications such as in EVs. When the motor speed is below base speed which is the maximum speed in the constant torque region as shown in Fig. 1, the voltage constraint will never be reached. Therefore, the torque capability in this region only depends on the current constraint which is (4). From (4), the q-axis current can be expressed as follows: ݅ ൌ ටܫመ௦ଶ െ ݅ௗଶ
(7)
Substituting (7) back into (3), for ݀ܶ Ȁ݀݅ௗ ൌ Ͳ, the d-axis current for MTPA operation is obtained as follows: ݅ௗǡெ் ൌ
ߣଶ ܫመ௦ଶ െඨ ଶ ʹ Ͷ൫ܮ െ ܮௗ ൯ ͳ൫ܮ െ ܮௗ ൯ ߣ
(8)
Simply, the q-axis current for MTPA operation can be derived by substituting (8) back into (7), yielding: ଶ ݅ǡெ் ൌ ටܫመ௦ଶ െ ݅ௗǤெ்
(9)
C. Control in the Constant Power Region During the constant power region, FW operation is implemented by increasing the projection of the current vector on the negative d-axis direction. As a result, the air gap flux is reduced and the motor speed is increased. Since the air gap flux keeps decreasing as the speed is increased, the developed torque of such IPMSMs is reduced as shown in Fig. 1. Unlike the aforementioned MTPA control, the torque capability in constant power region is determined by both current and voltage limitations. In the constant power region, the maximum torque is produced at the crossing point of the current limiting circle and the voltage limiting ellipse [6]. If (7) is substituted into (6), the d-axis current in (6) becomes speed dependent. By satisfying the critical condition of the current limiting circle and voltage limiting ellipse, which are ܫመ௦ ൌ
ܫመ௫ andܸ௦ ൌ ܸ௫ respectively, the d-axis current trajectory for the voltage and current limited maximum torque (VCLMT) operation can be derived as in (10), as shown in the bottom of this page. Similar to MTPA control, the q-axis current can be obtained by substituting (10) back into (7), yielding: ଶ ݅ǡெ் ൌ ටܫመ௦ଶ െ ݅ௗǤெ்
(11)
When the d- and q-axes currents are controlled according to (10) and (11), the current vector in the d-q plane will rotate from OA to OB along the current limiting circle as shown in Fig. 3. Here, both the current and voltage are kept at maximum values to generate maximum torque during this VCLMT control. III.
FLUX-WEAKENING CONTROL OF IPMSMS BASED ON Z-SOURCE INVERTERS
A. Boosted Voltage Flux-Weakening Control It is noticed that the voltage limiting ellipses, shown in Fig. 3, have the same center point but progressively shrink while the speed increases, which implies a smaller operation region at deep flux-weakening operation of such IPMSMs. If such motor drives have voltage boost capabilities, the voltage limiting ellipse becomes controllable. From (6), the maximum speed can be obtained as follows: ܸ௫ ߱௫ ൌ (12) ଶ ଶ ඥܮ ݅ ሺܮௗ ݅ௗ ߣ ሻଶ If the voltage limit becomes controllable, from (12), it is obvious that the maximum speed can be further increased. Based on this idea, an extended CPSR can be achieved by boosting the output voltage of such motor drives. Therefore, special inverters with voltage boost capability such as Z-source inverters are needed. In general, the voltage boost operation can be applied in both the constant torque region and the FW region. When the voltage boost operation is implemented in the constant torque region with independent current limit, the maximum torque resulting from the MTPA control is not affected by the booted voltage since it is only current dependent. However, the base speed will increase because of the fact that a higher back-EMF can exist with the support of the boosted output voltage of such drives. As a result, the same speed range can be achieved by boosting the output voltage of drives even without using the aforementioned FW control. Since the speed increase has the same ratio with the voltage limiting increase which can be seen in (12), the application of voltage boost operation in the constant torque region has such a disadvantage that the voltage stress on drives could be quite high when the base speed is boosted to high speed levels. In addition, when operating in the
high speed region, most EVs and hybrid EVs do not require a torque as high as in the constant torque region where the vehicles start. Therefore, the application of voltage boost operation is not recommended in the constant torque region. It is necessary to develop a new control algorithm for the voltage boost operation in the FW region since the VCLMT control is limited by a voltage constraint from such drives. The principle of the boosted voltage flux-weakening (BVFW) control is that the output power of IPMSMs remains constant during this operation and the current limit is still a constraint. As the speed increased, the current space vector rotates counterclockwise in the d-q plane which produces s stronger demagnetizing current component to reduce the output torque of such IPMSMs. The voltage level, which could be much higher than the dc bus voltage of such drives, depends on the speed and the d, q-axes currents which follows from (1) and (2). The details of this algorithm are given next: The general output power equation of such motors can be written as follows: ܲ௧ௗ ൌ ܶ ߱
where, ܲ௧ௗ , is the output power at the end of the MTPA operation, which is defined as rated power here, and ߱ , is the motor’s mechanical speed. Correspondingly, the current and voltage at the rated power are defined as rated current and rated voltage. By substituting the developed torque ܶ expression from (3), (13) can be rewritten as follows: ͵ (14) ܲ௧ௗ ൌ ൣߣ ݅ ൫ܮௗ െ ܮ ൯݅ௗ ݅ ൧߱ ʹ Since the current limit is still a constraint here, by substituting ଶ ݅ ൌ ටܫመ௧ௗ െ ݅ௗଶ into (14), the d-axis current becomes speed
dependent only, which is given as follows: ͵ ଶ ଶ ܲ௧ௗ ൌ ቈߣ ටܫመ௧ௗ െ ݅ௗଶ ൫ܮௗ െ ܮ ൯݅ௗ ටܫመ௧ௗ െ ݅ௗଶ ߱ ʹ
(15)
Once the rated power, ܲ௧ௗ , is determined, Equation (15) can be further simplified as a quartic equation in ݅ௗ : ܽ݅ௗସ ܾ݅ௗଷ ܿ݅ௗଶ ݀݅ௗ ݁ ൌ Ͳ
where,
ଶ
(16)
ܽ ൌ െሺܮௗ െ ܮ ሻଶ , ܾ ൌ െʹߣ ሺܮௗ െ ܮ ሻ, ଶ െ ߣଶ , ܿ ൌ ሺܮௗ െ ܮ ሻଶ ܫመ௧ௗ ଶ , ݀ ൌ ʹߣ ሺܮௗ െ ܮ ሻଶ ܫመ௧ௗ ଶ ଶ െ Ͷܲ௧ௗ Ȁͻ߱ଶ . ݁ ൌ ߣଶ ܫመ௧ௗ
ଶ
ଶ െߣ ܮௗ ට൫ߣ ܮௗ ൯ െ ൫ܮଶௗ െ ܮଶ ൯ ቀߣଶ ܮଶ ܫመ௫ െ ൫ܸ௫ Ȁ߱ ൯ ቁ
݅ௗǡெ் ൌ
(13)
൫ܮଶௗ െ ܮଶ ൯
(10)
It was found that only one of the four roots of Equation (16) is a valid solution which should be a negative real root. This was verified through the simulation results given below. Once Equation (16) is solved, the q-axis current can be obtained by substituting the solution of (16) back into (7), yielding: ଶ ݅ǡிௐ ൌ ටܫመ௦ଶ െ ݅ௗǡிௐ
Shoot-though
Carrier Wave
Amplit ude
(17) Reference Wave
Here, both the ݅ௗǡிௐ and the ݅ǡிௐ are speed-dependent variables. Since there is no voltage constraint, such IPMSMs can run at any speed as long as the voltage components in the d-q reference frame fulfill (18) and (19) as follows: ݒௗǡிௐ ൌ െ߱ ܮ ݅Ǥிௐ
(18)
ݒǡிௐ ൌ ߱ ൫ܮௗ ݅ௗǡிௐ ߣ ൯
(19)
Shoot-though
Fig. 4. Simple boost control for Z-source inverter.
B. Z-Source Inverters The Z-source inverter, shown in Fig. 2, is a buck-boost converter which employs a unique impedance network to couple the inverter main circuit to the dc link. It was mentioned in [7] that a Z-source inverter-based ac drive has several attractive merits including the availability of larger desired output voltages, ride-through ability during voltage sags, and reduced harmonics. In addition, this topology avoids the outright shoot-through risk which could cause catastrophic damage in conventional PWM voltage source drives. A Z-source inverter has two operation modes: the normal mode and the boost mode [7]. The operation principle of the normal mode has no difference from that of traditional PWM inverters. However, during the boost mode, the PWM pattern allows one to turn-on both the upper switch and the lower switch in the same phase leg to charge the inductors in the impedance network, which would not be allowed in conventional voltage source inverter topologies. Similar to a dc-dc boost converter, the Z-source inverter achieves voltage boost ability through the charged inductors. It is noticed that a dc-dc boost converter can be implemented to boost the dc bus voltage in a drive system. As a result, such drive system can achieve the voltage boost ability as a Z-source inverter. However, compared to the Z-impedance network, a dc-dc boost converter normally requires extra switching devices, larger capacitor banks, and larger volume. Therefore, the Z-source inverter is a better choice. There are several improved Z-source topologies, e.g. quasiZ-source inverters [10], improved Z-source inverters [11], switched inductor Z-source inverters [12], etc. Each of them has its pros and cons. Since such Z-source inverters have unique shoot-though states, the control of Z-source inverters for the whole duty cycle is special. Generally, there are three commonly used control methods, namely, simple boost control [7], maximum boost control [13], and maximum constant boost control [14]. The principle for all the control methods listed above is that a special command, which turns on both the two switches in one phase leg, are implemented in the PWM
control to determine the shoot-though time during one duty cycle. The higher the shoot-though duty ratio, the higher the output voltage. The major difference between these methods is the different voltage stress on switching devices at the same voltage gain in the boost mode. Here, for simplicity, only the original Z-source inverter, shown in Fig. 2, is simulated with simple boost control. The voltage gain of such a Z-source inverter can be expressed as follows [7]: ܸௗ (20) ܸ ൌ ܯή ܤή ʹ where ܸ is the peak voltage value of phase a, ܯis the modulation index, ܤis the boost factor, and ܸௗ is the dc bus voltage. During the simple boost control which is described in [7], the Z-source inverters can operate in the boost mode once the value of the triangular carrier waveform becomes higher than the amplitude of the sinusoidal reference waveform as shown in Fig. 4. The boost factor for the simple boost control can be expressed in terms of M as follows [13]: ͳ (21) ܤൌ ʹ ܯെ ͳ By substituting (21) into (20), the modulation index can be calculated depending on the demanded output voltage of such Z-source inverters, yielding: ͳ ܯൌ (22) ͳ ʹ െ ܸௗ Ȁܸ ʹ C. BVFW Control Scheme Here, Fig. 5 illustrates a block diagram of an IPMSM-drive system with a closed-loop speed control and the presented BVFW control. This control scheme was developed based on the control strategy presented in [8]. Such IPMSM, shown in Fig.5, operates in the constant torque region with MTPA control and in the constant power region with BVFW control.
Position counter
, a,b,cd,q
,
,
Current sensor
Part A
ωm,ref + Σ _ ωm
d,qa,b, c
Current regulator Speed regulator
MTPA
+
Σ
IPMS M
Z-source inverter
BVFW
+
Speed sensor
Part B
ωe
ωm
Times the number of pole pairs
Fig. 5. Block diagram of an IPMSM-drive system with closed-loop speed control and open-loop flux weakening control.
IV.
SIMULATION RESULTS
To verify the feasibility of the proposed BVFW control with Z-source inverters, a simulation model of a motor-drive system including a mechanical load was set-up in an ANSYS Simplorer environment. In addition, this IPMSM was simulated in a MATLAB environment for both the constant torque region and the FW region. The parameters of the modeled 3.5 hp laboratory IPMSM are ܮௗ ൌ ͷǤͶ݉ܪ, ܮ ൌ ͳͲǤͷ݉ܪ, ߣ ൌ
8 MTPA control VCLMP control BVFW control
7
Developed Torque, N/m
6
5
4
3
2
1
0
0
500
1000
1500 2000 Electrical Speed, rad/s
2500
3000
3500
Fig. 6. Comparison of output torque between BVFW and VCLMT. 15
Amplitude of phase current d-axis current q-axis current 10
Currents, Amperes
In the speed control loop, the reference phase current, ܫመ௦ , is controlled by the difference between the reference speed and the measured speed. The current control, which yields d- and q-axes current references, can be divided into two parts: Part A and Part B. Through Part A, the current trajectories of the MTPA control are obtained by decomposing the reference phase current into d- and q-axes components, according to (8) and (9). Part B is the presented BVFW regulator which generates extra demagnetizing current references during FW operation based on the solution of (16). In the constant torque region, Part B is not triggered due to the low speed, which gives ο݅ௗ ൌ Ͳ. Therefore, only Part A is activated according to MTPA control in the constant torque region. However, this control strategy requires both Part A and B to be activated in the constant power region. During the FW operation, an extra demagnetizing current command, ο݅ௗ , is added to the d-axis current of the MTPA control, which is expressed by ݅ௗǡெ் in Fig. 5, to synthesize the d-axis current of BVFW control. It is noticed that the q-axis current reference, ݅ , can always be generated according to the current limit and d-axis current. Also, it can be indicated that this open-loop BVFW regulator can be implemented by using look-up tables to minimize computational burdens. The current regulator is designed according to the decoupling current controllers presented in [5]. The crosscoupling effects can be cancelled by the feed-forward compensation. Finally, the voltage references generated from the current regulator are used to control an original Z-source inverter as shown in Fig. 2. The modulation index of simple boost control can be obtained by substituting the phase voltage, ଶ ଶ ݒ , back into (22). which is ܸ ൌ ටݒௗ
5
0
-5
-10
0
500
1000
1500
2000
2500
3000
3500
Electrical Speed, rad/s
Fig. 7. Current trajectories of the MTPA control and the BVFW control.
ͲǤͳͶͺܾݓ, ൌ , ܫ ൌ ͳͲܣ. The load inertia was set to ܬൌ ͲǤͲʹ݉݃ܭଶ and the dc bus voltage was 400 volts. Here, Fig. 6, 7, and 8 give the results based on the mathematical model of the aforementioned IPMSM without the activation of any control loop. Fig. 6 shows the output torque versus speed of such IPMSM in both the constant torque region and the constant power region. It is obvious that the BVFW control has wider speed range and can produce higher torque than the VCLMT at high speeds. Fig. 7 shows the current
V.
400
Amplitude of phase voltage d-axis voltgae q-axis voltage
300
Voltages, Volts
200
100
0
-100
-200
0
500
1000
1500
2000
2500
3000
3500
Electrical Speed, rad/s
Fig. 8. Voltages trajectories of the MTPA control and the BVFW control. Motor Mechanical Speed
CONCLUSIONS
In this paper, a novel FW control algorithm for IPMSMs has been presented. The principal feature of the proposed algorithm is that it eliminates the dc bus voltage constraint to FW operation by implementing a boost converter which is the Z-source inverter in this paper. Theoretical analysis and simulation were conducted to verify the feasibility and advantages of such control strategy. Compared with the conventional FW strategies, the presented BVFW control allows an IPMSM to largely extend its CPSR and operate at higher speed with higher torque. It should be noticed that such merits have significant importance to traction applications, e.g. EVs. The major drawback of the presented strategy is that the open-loop BVFW control is still sensitive to motor parameters. Closed-loop BVFW control and experiments are going to be presented in future work. ACKNOWLEDGMENT
MTPA and BVFW Control
Curve Info
600
The authors would like to acknowledge the financial support received from the University of Minnesota-DOE project. Also, we would wish to acknowledge ANSYS Corporation (Dr. Marius Rosu and Mr. Mark Solveson) for making Simplorer 11.0 software available for use by these authors.
VM_ROT1.OMEGA TR
Mechanical Speed [rad_per_sec]
500
400
300
REFERENCES
200
[1]
100
0
[2] 0.00
0.50
1.00
1.50
2.00
2.50
Time [s]
Fig. 9. Mechanical speed curve of the IPMSM. d,q-axes voltage references
[3]
MTPA and BVFW Control
300 Curve Info Vd_ins_ref.VAL[0]
[4]
TR Vq_ins_ref.VAL[0] TR
200
Voltages, volts
[5] 100
0
[6]
-100
[7] -200 0.00
0.50
1.00
1.50
2.00
2.50
[8]
Time [s]
Fig. 10. D, q-voltage references of the current regulator. [9]
trajectories of the MTPA control in the constant torque region and the BVFW control in the constant power region, while Fig. 8 illustrates the voltage trajectories that sustain such currents. The mechanical speed of the IPMSM, controlled by the scheme of Fig. 5, is shown in Fig. 9. The transition from constant torque region to constant power region has very small distortion. Fig. 10 shows the d, q-reference voltages from the current regulator. It should be noticed that the q-axis voltage reference is boosted higher than the maximum amplitude of the phase voltage in conventional PWM inverters, which is 200 volts here. It indicates that the output voltage of the Z-source inverter is boosted higher than the dc bus voltage during the BVFW operation.
[10] [11] [12] [13] [14]
T.M. Jahns, G.B. Kliman, and T.W. Neumann, “Interior permanent magnet synchronous motors for adjustable-speed drives,” IEEE Trans. on Ind. Appl., vol. IA-22, no. 4, pp. 738-747, Jul./Aug. 1986. W.L. Soong and N. Ertugrul, “Field-weakening performance of interior permanent magnet motors”, IEEE Trans. on Ind. Appl., vol. 38, no. 5, pp. 1251-1258, Sep./Oct. 2002. H. Wen, W. Xiao, H. Li, and X. Wen, “Analysis and minimisation of dc bus voltage for electric vehicle applications”, IET Electrical Systems in Transportation, vol. 2, pp. 68-76, Jun. 2012. T.M. Jahns, “Flux-weakening regime operation of an interior permanent magnet synchronous motor drive”, IEEE Trans. on Ind. Appl., vol. IA23, no. 4, pp. 681-689, Jul./Aug. 1987. S. Morimoto, M. Sanada and Y. Takeda, “Wide-speed operation of interior permanent magnet synchronous motors with high-performance current regulator”, IEEE Trans. on Ind. Appl., vol. 30, no. 4, pp. 920926, Jul./Aug. 1994. J. Kim and S. Sul, “Speed control of interior permanent magnet synchronous motor drive for the flux weakening operation”, IEEE Trans. on Ind. Appl., vol. 33, no. 1, pp. 43-48, Jan./Feb. 1997. F.Z. Peng, “Z-source inverter”, IEEE Trans. on Ind. Appl., vol. 39, no. 2, pp. 504-510, Mar./Apr. 2003. V.R. Jevremovic and D.P. Marcetic, “Closed-loop flux-weakening for permanent magnet synchronous motors”, in the 4th IET Conference on Power Electronics, Machines and Drives, pp. 717-721, Apr. 2008. P. Zhang, G.Y. Sizov, M. Li, D.M. Ionel, N.A.O. Demerdash, S.J. Stretz, and A.W. Yeadon, “Multi-objective tradeoffs in the design optimization of a brushless permanent magnet machine with fractional-slot concentrated windings”, in IEEE Energy Conversion Congress and Exposition (ECCE), pp. 2842-2849, Sep. 2013, accepted for IEEE Trans. on Ind. Appl., 2014. J. Anderson and F.Z. Peng, “Four quasi-Z-source inverters”, in IEEE Power Electronics Specialists Conference, Jun. 2008, pp. 2743-2749. Y. Tang, S. Xie, C. Zhang, and Z. Xu, “Improved Z-source inverter with reduced Z-source capacitor voltage stress and soft-start capability”, IEEE Trans. on Power Electron., vol. 24, no. 2, pp. 409-415, Feb. 2009. M. Zhu, K. Yu, and F.L. Luo, “Switched inductor Z-source inverter”, IEEE Trans. on Power Electron., vol. 25, no. 8, pp. 2150-2158, Aug. 2010. F.Z. Peng, M. Shen, and Z. Qian, “Maximum boost control of the Zsource inverter”, IEEE Trans. on Power Electron., vol. 20, no. 4, pp. 833838, Jul. 2005. M. Shen, J. Wang, A. Joseph, F.Z. Peng, L.M. Tolbert, and D.J. Adams, “Maximum constant boost control of the Z-source inverter”, IEEE Trans. Ind. Appl., vol. 42, no. 3, pp. 770-777, May/Jun. 2006
