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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 5, MAY 2012

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Current-Controlled Multiphase Slice Permanent Magnetic Bearingless Motors With Open-Circuited Phases: Fault-Tolerant Controllability and Its Verification Xiao-Lin Wang, Qing-Chang Zhong, Senior Member, IEEE, Zhi-Quan Deng, and Shen-Zhou Yue

Abstract—The fault-tolerant control of bearingless motors is vital for their safe and robust operation. In this paper, the operation of current-controlled multiphase slice permanent-magnet bearingless motors (PMBMs) with different open-circuited faulty phases is analyzed, and their fault-tolerant controllability in the general case is investigated. As an example, the feasibility of fault-tolerant control with arbitrary single, double, or triple open-circuited faulty phase(s) for a six-phase slice PMBM is discussed in detail. Simulation results from finite-element analysis are presented to demonstrate the operation of the motor under the proposed faulttolerant control strategy. Moreover, experimental results are also provided to further verify the theoretical development. Index Terms—Bearingless motor, controllability, current control, displacement control, failure analysis, fault tolerance, magnetic levitation, open circuited, permanent-magnet (PM) motors.

N OMENCLATURE APM D f1 , f2 , f3 Fx Fy h k ks kt

Amplitude of the permanent magnet (PM)’s MMF. Outside radius of the rotor. Coefficients for third-order minors. Levitation force in the x-direction. Levitation force in the y-direction. Axial length of the rotor. Sequence number of phases. Coefficient of the levitation force. Coefficient of the torque.

Manuscript received December 29, 2010; revised March 28, 2011 and May 29, 2011; accepted June 18, 2011. Date of publication June 30, 2011; date of current version February 3, 2012. This work was supported by the National Natural Science Foundation of China under Grant 50977043. An earlier version of this paper was presented at the XIX International Conference on Electrical Machines, Rome, Italy, September 2010. X.-L. Wang and Z.-Q. Deng are with the Department of Electrical Engineering, College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China (e-mail: [email protected]; [email protected]). Q.-C. Zhong is with the Department of Aeronautical and Automotive Engineering, Loughborough University, Leicestershire, LE11 3TU, U.K. (e-mail: [email protected]; [email protected]). S.-Z. Yue was with the Department of Electrical Engineering, College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China. He is now with Shanghai AeroSharp Electric Technologies Company, Ltd., Shanghai 201101, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2011.2161067

ik is leg N p Q R rk T W x, y α θ θk η λk Λ

Phase-k current. Current vector, = [i1 , . . . , ik , . . . , iN ]T . Length of the air gap. Number of stator teeth. Pair number of PM poles. = [Fx , Fy , T ]T . = diag{r1 , r2 , . . . , rN }. Resistance of phase-k stator windings. Torque. Turns of phase windings. x and y displacements of the rotor. Angle of the pole shoe. Mechanical angle of the rotor. Positional angle 2π(k − 1)/N of phase k. Lagrangian coefficient matrix. Status of phase-k windings. = diag{λ1 , λ2 , . . . , λN }. I. I NTRODUCTION

A

BEARINGLESS motor is a magnetically levitated electric machine that combines torque driving and selflevitation. Because it has no mechanical friction and noise caused by bearings, it could be applied to some high-speed areas, such as flywheel energy-storage systems, compact generators, turbomolecular pumps, etc. Bearingless motors can be PM ac motors, induction motors, switched reluctance motors, etc. The PM ones have better performance than other bearingless motors because the PM magnetic fields can be utilized as biased magnetic fields. PM bearingless motors also have the advantages of dense structure and low power loss. However, they normally have two sets of stator windings: one set for torque driving and one set for levitation. As a result, there are three kinds of magnetic fields in the air gap, which interact with each other, and the motor design and its control system become very complex. Moreover, when any phase is short circuited or open circuited, the motor does not work if its controller is not changed. It was shown in [1]–[3] that multiphase PM bearingless motors (PMBMs) with only one set of windings could work as magnetically levitated machines using a current control scheme to achieve normal operation with minimized

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resistive power loss. Both the torque and levitation forces can be controlled simultaneously by controlling the respective current components (torque and levitation) in the phase windings. This does not only lead to the simplification of the motor structure but also provide it with a certain redundancy and faulttolerant capability. Since then, a lot of researchers have made contributions to this area. A bridge configured winding was proposed for polyphase bearingless motors in [4] and [5] so that only one power supply is needed for torque production with the lateral forces produced by auxiliary power supplies of relatively low current and voltage ratings. It offers relatively low power loss and is possible to be extended to other polyphase machines. However, the system reliability is decreased because the torque currents are divided into two parallel paths in each phase and an isolated supply is needed to provide the levitation currents. An induction bearingless machine with multiphase windings powered by a multiphase inverter was proposed in [6], where the levitation forces were derived by using the virtual displacement method and a control algorithm based on the orientation of the rotor flux was presented to provide the exciting magnetic field of the levitation forces and torque. Obviously, the efficiency should be lower than that of PMBMs. Multiphase slice PMBMs were analyzed and designed in [7]–[9], where five different slice bearingless motors and converter topologies were comparatively evaluated and discussed. In [10], three different voltage-source inverter topologies that are suitable for driving two-phase slice bearingless motors were compared in terms of available power, voltage, number of required power electronic components, and level of compactness. In [11], a fault-tolerant controller was designed for five-phase PM motors with trapezoidal back electromotive forces under various open-circuited faults, using the fundamental and thirdharmonic current components for the excitation of the normal stator phases. The optimal solutions under single- and doublephase open-circuited fault conditions were presented. In [12] and [13], a back-electromotive-force observer was adopted to build fault-tolerant controllers for PM synchronous motors and flux-switching PM machines. Moreover, a redundant winding configuration and the corresponding suitable control strategy for the postfault operation were proposed, and the electromagnetic performance under normal and fault conditions was analyzed. In [14], fuzzy logic and sliding-mode controllers were compared for multiphase induction machines with one or more open-circuited phases, and experimental results were obtained on a setup based on a six-phase induction machine with a variable mechanical load. In [15], a multiphase two-level inverter was introduced for fault-tolerant multiphase brushless dc motors with a flexible and reliable field programmable gate array)/digital signal processor controller used for data acquisition, motor control, and fault monitoring. In [16], a faulttolerant control strategy and a prototype with an open-circuited phase for a six-phase PMBM were discussed. In order to be able to obtain other phase currents, a strategy to optimize the power loss was adopted for the fault-tolerant control in [1] and [16]. However, according to the proposed strategy, the decoupled control model depends on the phase that is open circuited. In other words, it needs to be derived again if the faulty phase is another one. Moreover, how to deal with the case with more

than one open-circuited phase at the same time has not been discussed before. It is hence very important to find out solutions for these questions. Following on some preliminary results presented in [17], a current-controlled model of a multiphase slice PMBM is presented in this paper, and an attempt has been made to investigate the generic condition for its fault-tolerant controllability. It is found out that a PMBM is fault-tolerant controllable if and only if the rank of a matrix (to be defined later) is three. This is then applied to a six-phase slice PMBM as an example to analyze the feasibility of various open-circuited fault modes. It is feasible to implement fault-tolerant control when there is one phase that is open circuited or when there are two adjacent or opposite phases that are open circuited. It is not feasible if there are two intervened phases or three phases that are open circuited. Both simulation and experimental results will be presented. With comparison to [17], the major differences are as follows: 1) The theoretical analysis has been improved, e.g., the solution of the optimal power loss problem and the addition of the solution when a six-phase slice PMBM is under normal operation and when its phase 1 is open circuited; 2) the ripples of the levitation forces and the torque are analyzed; 3) simulations and experiments were redone, and the presentation of the results is improved; 4) simulation results are added to show the case when the slice PMBM is not under the faulttolerant control, which verifies the condition obtained; and 5) experimental results when the motor was accelerated from 800 to 2000 r/min are added to further demonstrate the analysis. II. C URRENT C ONTROL OF A M ULTIPHASE S LICE PMBM The structure of a multiphase slice PMBM is shown in Fig. 1(a), and its developed cross-sectional drawing is shown in Fig. 1(b). There are N teeth distributed equally in the stator and N phase coils wound around each tooth. 2p poles of PMs are attached to the surface of the rotor. The motor is designed to work in the linear magnetic region, and the PM shape can be optimized to create a sinusoidal PM magnetic field distribution. In the sequel, the following assumptions are made. 1) The leakage flux is negligible. 2) The fundamental component is considered only. The levitation force and torque models can be derived according to the principle of equivalent magnetic circuits and virtual energy [18], [19]. The current-controlled model of the multiphase slice PMBM can be expressed as Q = M(θ)is

(1)

where Q = [Fx , Fy , T ]T is the vector consisting of the torque T and the levitation forces Fx and Fy , is = [i1 , . . . , ik , . . . , iN ]T is the stator current vector of each phase, with k = 1, 2, . . . , N , and M(θ) is shown at the bottom of the 2 , and kt = next page. Here, ks = DhAPM W μ0 sin(α)/4leg DhAPM W μ0 sin(α/2)/leg , where D is the outside radius of the rotor, h is the axial length of the rotor, APM is the amplitude of the PM’s MMF, W denotes the turns of the phase windings, leg is the length of the air gap, and α is the angle of the pole shoe. θ is the mechanical angle of the rotor, and θk = 2π(k − 1)/N is the positional angle of phase k.

WANG et al.: SLICE PERMANENT MAGNETIC BEARINGLESS MOTORS WITH OPEN-CIRCUITED PHASES

Hence

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2i∗s + R−1 M(θ)T η = 0 M(θ)i∗s = Q∗ .

Multiplying the first equation with −(1/2)M(θ) from the left and then adding it to the second equation lead to 1 − M(θ)R−1 M(θ)T η = Q∗ 2 i.e., −1 ∗ η = −2 M(θ)R−1 M(θ)T Q. The desired currents can then be derived as 1 i∗s = − R−1 M(θ)T η = Z(θ)Q∗ 2

(5)

with −1 Z(θ) = R−1 M(θ)T M(θ)R−1 M(θ)T

Fig. 1. Structure of a multiphase slice PMBM. (a) Cross-sectional drawing (N = 6 and p = 1). (b) Developed cross-sectional drawing.

Based on the equation above, the levitation forces and torque are determined by the phase currents in the motor when the rotor angle is known. On the other hand, for the given levitation forces and torque Q∗ = [Fx∗ , Fy∗ , T ∗ ]T , the desired current vector i∗s needs to be calculated according to (1) for the purpose of control. However, the number of equations in (1) is three, which is normally less than the number of phase current variables (N here). In order to be able to obtain reasonable phase currents, certain additional restrictions should be considered. For example, the minimization of power loss is adopted in [1], [3], and [20] via solving min

∗T ∗ is Ris

(2)

where R = diag{r1 , r2 , . . . , rN } represents the resistance of the stator windings. In this paper, this strategy is followed with the assumption of r1 = r2 = · · · = R. In order to solve this optimization problem, the following Lagrangian function is constructed from (1) and (2): ∗ T ∗ ∗ L = i∗T s Ris + η (M(θ)is − Q )

(3)

where the Lagrangian coefficient matrix is η = [η1 , η2 , η3 ]T . The partial derivatives of L with respect to the currents i∗s and the coefficients η are ∂L T = 2i∗T s R + η M(θ) = 0 ∂i∗ s (4) ∂L ∗ ∗ ∂η = M(θ)is − Q = 0.

⎡

ks cos(2θ1 − θ) · · · M(θ) = ⎣ ks sin(2θ1 − θ) · · · kt sin(θ1 − θ) · · ·

which is an N × 3 matrix. Note that M(θ) is not square, so it is not invertible; however, M(θ)R−1 M(θ)T is square and invertible under normal operation conditions. In the rest of this section, N = 6 is taken as an example. In this case, Z(θ) can be found as Z(θ)

⎡

⎢ ⎢ ⎢ 1 =⎢ ⎢ 3ks ⎢ ⎣ 1 3ks

1 3ks

cos(θ) .. . cos(2θk −θ) .. . 4π cos 3 −θ

1 3ks

1 3ks

1 3ks

sin(−θ) .. . sin(2θk −θ) .. . 4π sin 3 −θ

⎤ sin(−θ) ⎥ .. ⎥ . ⎥ 1 ⎥ 3kt sin(θk −θ) ⎥ ⎥ .. ⎦ . 1 5π 3kt sin 3 −θ 1 3kt

and the partial derivatives of the Lagrangian function can be expressed as (6), shown at the bottom of the next page. Then, under the normal operation, the desired currents and the coefficients η can be obtained from the given levitation forces and torque as (7), shown at the bottom of the next page. When phase k is open circuited, ∂L/∂i∗k does not exist. In order to obtain the desired currents for the other normal phases, the equation ∂L/∂i∗k = 0 may be substituted with i∗k = 0 in (4). For example, if phase 1 is open circuited, then the first equation in (6) should be substituted with i∗1 = 0, which leads to the desired currents and coefficients in (8), shown at the bottom of the next page. Since the open-circuited fault could happen in any phase or in more than one phase at a time, each individual case has to be analyzed [11], [15], [16], [21]. Assume that there are two kinds of faults: one phase open circuited or two phases open

ks cos(2θk − θ) · · · ks sin(2θk − θ) · · · kt sin(θk − θ) · · ·

⎤ ks cos(2θN − θ) ks sin(2θN − θ) ⎦ kt sin(θN − θ)

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circuited. Then, the total number of fault cases is C61 + C62 = 6 + [(6 × 5)/(2 × 1)] = 21. The control system has to be very complicated in order to cope with these many fault conditions. Moreover, the issue whether it is feasible to find a fault-tolerant controller for each faulty case or not has not been reported in other literature. These questions will be answered later.

⎧ ∂L ∗ ⎪ ⎪ ⎪ ∂i1 ⎪ ∂L ⎪ ⎪ ⎪ ∂i∗2 ⎪ ⎪ ⎪ ∂L ⎪ ⎪ ⎪ ∂i∗ ⎪ 3 ⎪ ⎪ ∂L ⎪ ⎪ ⎪ ∂i∗ 4 ⎪ ⎪ ⎨ ∂L ∂i∗ 5

⎪ ∂L ⎪ ⎪ ∂i∗ ⎪ 6 ⎪ ⎪ ⎪ ⎪ ∂L ⎪ ⎪ ∂η1 ⎪ ⎪ ⎪ ⎪ ⎪ ∂L ⎪ ⎪ ∂η2 ⎪ ⎪ ⎪ ⎪ ⎩ ∂L ∂η3

III. FAULT-T OLERANT C ONTROLLABILITY A. Generic Condition for Fault-Tolerant Controllability Normally, each stator phase current is controlled independently. Hence, the normal phases can still be controlled when some phases are open circuited. Coefficient λk is defined to

= 2i∗1 r1 + ks cos(θ)η1 − ks sin(θ)η2 − kt sin(θ)η3 π 2π = 2i∗2 r2 + ks cos 2π 3 − θ η1 + ks sin 3 + θ η2 + kt sin 3 + θ η3 π π = 2i∗3 r3 + ks cos 2π 3 + θ η1 − ks sin 3 − θ η2 + kt sin 3 + θ η3 = 2i∗4 r4 + ks cos(θ)η1 − ks sin(θ)η2 + kt sin(θ)η3 π 2π = 2i∗5 r5 + ks cos 2π 3 − θ η1 + ks sin 3 + θ η2 − kt sin 3 + θ η3 π π = 2i∗6 r6 + ks cos 2π 3 + θ η1 − ks sin 3 − θ η2 − kt sin 3 + θ η3 √ = −Fx + ks 23 (i2 − i3 + i5 − i6 ) sin(θ) − 12 (−2i1 + i2 + i3 − 2i4 + i5 + i6 ) cos(θ) √ = −Fy + ks 23 (i2 − i3 + i5 − i6 ) cos(θ) + 12 (−2i1 + i2 + i3 − 2i4 + i5 + i6 ) sin(θ) √ = −T + ks 23 (i2 + i3 − i5 − i6 ) cos(θ) − 12 (2i1 + i2 − i3 + 2i4 − i5 + i6 ) sin(θ)

⎧ ∗ i1 = 3k1s cos(θ)Fx∗ − 3k1s sin(θ)Fy∗ − 3k1 t sin(θ)T ∗ ⎪ ⎪ ⎪ ∗ ∗ ∗ ⎪ 1 2π 1 4π ⎪ ⎪ i∗2 = 3k1s cos θ − 2π ⎪ 3 Fx − 3ks sin θ − 3 Fy + 3kt sin θ − 3 T ⎪ ∗ ∗ ∗ ⎪ ⎪ 1 4π 1 2π ⎪ i∗3 = 3k1s cos θ − 4π ⎪ 3 Fx − 3ks sin θ − 3 Fy − 3kt sin θ − 3 T ⎪ ⎪ ⎪ ⎪ i∗ = 3k1s cos(θ)Fx∗ − 3k1s sin(θ)Fy∗ + 3k1 t sin(θ)T ∗ ⎪ ⎪ ⎨ 4 ∗ ∗ ∗ 1 2π 1 4π i∗5 = 3k1s cos θ − 2π 3 Fx − 3ks sin θ − 3 Fy − 3kt sin θ − 3 T ∗ ∗ ⎪ ⎪ 1 4π 1 2π ∗ ⎪ i∗6 = 3k1s cos(θ − 4π ⎪ 3 )Fx − 3ks sin θ − 3 Fy + 3kt sin θ − 3 T ⎪ ⎪ ⎪ 2R ∗ ⎪ η1 = − 3k ⎪ 2 Fx ⎪ s ⎪ ⎪ ⎪ 2R ⎪ η2 = − 3k2 Fy∗ ⎪ ⎪ s ⎪ ⎪ ⎩ η3 = − 2R2 T ∗ 3k

(6)

(7)

t

⎧ i∗ = 0 1 ⎪ √ √ √ ⎪ ⎪ ∗ cos(θ)−cos(3θ) ∗ ∗ ⎪ ⎪ = 2 3 sin(θ)−5 Fx + 2 3 cos(θ)+sin(θ)(2+cos(2θ)) Fy∗ + 2 3k3tcos(θ)−sin(θ) i 2 ⎪ 6k (3+cos(2θ)) 3k (3+cos(2θ)) (3+cos(2θ)) T s s ⎪ ⎪ √ √ √ √ √ ⎪ ⎪ cos(θ)− 3 sin(3θ) ∗ 3 cos(θ)+3 sin(θ) ∗ 3 cos(3θ)+6 sin(θ) ∗ ⎪ i∗3 = −3 3 sin(θ)−6 Fx + −3 3 cos(θ)− Fy + 2 3k T ⎪ 6ks (3+cos(2θ)) 3ks (3+cos(2θ)) ⎪ t (3+cos(2θ)) ⎪ ⎪ ⎪ sin(3θ) ∗ 2 sin(θ) ⎪ i∗4 = 56kcos(θ)+cos(3θ) Fx∗ + −33ksin(θ)−2 Fy + 3kt (3+cos(2θ)) T∗ ⎪ ⎪ s (3+cos(2θ)) s (3+cos(2θ)) ⎪ √ √ √ √ √ ⎨ cos(θ)+ 3 sin(3θ) ∗ 3 cos(3θ)+6 sin(θ) ∗ sin(θ) ∗ i∗5 = 3 3 sin(θ)−6 Fx + 3 3 cos(θ)+ Fy + −2 3k3tcos(θ)+3 T 6ks (3+cos(2θ)) 3ks (3+cos(2θ)) (3+cos(2θ)) ⎪ √ √ √ ⎪ ⎪ 3 sin(θ)+5 cos(θ)+cos(3θ) 3 cos(θ)+sin(θ)(2+cos(2θ)) 3 cos(θ)−sin(θ) 2 −2 −2 ⎪ i∗6 = Fx∗ + Fy∗ + 3kt (3+cos(2θ)) T ∗ ⎪ 6ks (3+cos(2θ)) 3ks (3+cos(2θ)) ⎪ ⎪ ⎪ ⎪ −8R−4 cos(2θ) ∗ sin(2θ) sin(2θ)) ∗ ⎪ η1 = 3k Fx + 3k22R Fy∗ + 3ks2R ⎪ 2 kt (3+cos(2θ)) T ⎪ s (3+cos(2θ)) s (3+cos(2θ)) ⎪ ⎪ ⎪ 2R sin(2θ) 2R cos(2θ)) ∗ −8R ⎪ Fy∗ + −2R+ ⎪ η2 = 3ks2 (3+cos(2θ)) Fx∗ + 3ks2 (3+cos(2θ)) ⎪ 3ks kt (3+cos(2θ)) T ⎪ ⎪ ⎩ sin(2θ) −2R+ 2R cos(2θ)) ∗ −8R ∗ ∗ η3 = 3k22R (3+cos(2θ)) Fx + 3k2 (3+cos(2θ)) Fy + 3ks kt (3+cos(2θ)) T s

s

(8)


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TABLE I T HIRD -O RDER M INORS OF M (θ) AND THE FAULT-T OLERANT C ONTROLLABILITY OF A S IX -P HASE S LICE PMBM

represent the status of phase-k windings: λk = 1 for normal operation and λk = 0 for open-circuited fault operation, with k ∈ {1, 2, . . . , N }. If phase k is open circuited, then i∗k = 0 as explained in the previous section. This is equivalent to that all the elements in the column k of matrix M(θ) are zero. In order to reflect this, the following matrix M (θ) is introduced: M (θ) = M(θ)Λ where Λ = diag{λ1 , λ2 , . . . , λN } is an N × N matrix. The equations corresponding to (4) with M (θ) can be rewritten as

2RT M (θ)T M (θ) 03×3

i∗s η

=

03×3 . Q∗

2RT M (θ)T MR (θ) = M (θ) 03×3

(10)

is invertible. In this case, the desired current is i∗s = Z (θ)Q∗

Since the elements of M (θ) involve trigonometric functions, it is difficult to calculate its row rank by using elementary transformations. The rank of M (θ) can be calculated via the largest order of all nonzero minors. For N = 6, all the minors with the largest order min(3, 6) = 3 are listed in the following: {f1 (λ1 λ3 λ6 , −λ1 λ5 λ6 , −λ2 λ3 λ6 , λ2 λ3 λ4 , λ3 λ4 λ6 , λ3 λ5 λ6 ), f2 (λ1 λ2 λ3 ,−λ1 λ2 λ5 ,−λ2 λ3 λ5 , λ2 λ4 λ5 , λ2 λ5 λ6 ,−λ4 λ5 λ6 ), f3 (λ1 λ2 λ4 ,−λ1 λ2 λ6 ,−λ1 λ3 λ4 , λ1 λ4 λ5 , λ1 λ4 λ6 , λ3 λ4 λ5 ), 0, 0}

(9)

There is a unique solution for i∗s if and only if the matrix

B. Detailed Analysis of the Fault-Tolerant Controllability When N = 6

(11)

where Z (θ) = R−1 M (θ)T (M (θ)R−1 M (θ)T )−1 is an N × 3 matrix that is dependent on the rotor angle θ. Since R is an invertible diagonal matrix, the nonsingularity of MR (θ) is equivalent to that of −M (θ)(2RT )−1 M (θ)T , which is equivalent to that M (θ) has a full row rank. Since Fx∗ , Fy∗ , and T ∗ are independent variables, it is not feasible to implement fault-tolerant control when MR (θ) is not invertible, i.e., when Rank(M (θ)) < 3. In conclusion, the fault-tolerant control of a slice PMBM is feasible if and only if the row rank of M (θ) is three, which is the number of degrees of freedom of a slice PMBM.

(12)

√ sin θ)/2), f2 = ks2 kt ((3 cos θ− where f1 = ks2 kt ((3 cos θ+ 3√ √ 3 sin θ)/2), and f3 = ks2 kt 3 sin θ. In total, there are C33 C63 = 20 minors. When a phase is open circuited, some minors are zero because the corresponding λ is zero. The coefficients f1 , f2 , and f3 may also be zero, depending on the value of θ. For θ ∈ [0, 2π) ⎧ 5π when θ = 2π ⎪ 3 , 3 ⎨ f1 = 0, (13) f2 = 0, when θ = π3 , 4π 3 ⎪ ⎩ f3 = 0, when θ = 0, π. Whether the rank of M (θ) is equal to three or not depends on whether all the third-order minors (12) are zero. Obviously, it is impossible that all of the minors are equal to zero at the same time (or at the same position θ) if there are no open-circuited phases, i.e., when λk = 1, with k ∈ {1, 2, . . . , 6}. Hence, under the normal operation mode, Rank(M (θ)) = 3, and (11) has a unique solution. The details of third-order nonzero minors and the faulttolerant controllability of the six-phase slice PMBM are analyzed in the following, with results summarized in Table I. 1) Single Open-Circuited Phase: When an arbitrary single phase k is open circuited, all elements in the column k of

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Fig. 2. Fault-tolerant control of a six-phase slice PMBM.

M (θ) are zero. However, because there is only one zero column, the rank of M (θ) is still three. Hence, the fault-tolerant control is feasible when an arbitrary single phase is open circuited. 2) Two Open-Circuited Phases: For simultaneous opencircuited two phases, the fault-tolerant controllability depends on the relative position of the open-circuited phases. There are three cases as follows.

TABLE II PARAMETERS OF THE S IX -P HASE S LICE PMBM U NDER S TUDY

Case 1) If two open-circuited phases are adjacent, such as phases 1 and 2 or phases 3 and 4, all third-order minors are not zero at the same position θ, and the rank of M (θ) is three. Thus, the fault-tolerant control is also feasible when two adjacent phases are open circuited. Case 2) If two open-circuited phases are opposite, such as phases 1 and 4 or phases 3 and 6, the conclusion is the same as that in Case 1. Case 3) If two open-circuited phases are intervened by a normal phase, such as phases 1 and 3 or phases 4 and 6, all third-order minors are equal to zero when θ = π/3 or 4π/3. Then, Rank(M (θ)) is not three for all θ ∈ [0, 2π). Hence, the condition of the fault-tolerant operation is not satisfied, and its faulttolerant control is not feasible.1 For example, when phases 1 and 3 are open circuited, (1) at θ = π/3 can be written as ⎡

0

⎢ ⎣0 0

ks √2 3ks 2

0

0

0

0

ks √2 s − 3k √ 2 3kt 2

ks √2 3ks 2

0

−ks 0

√ t − 3k 2

⎤ i∗1 ⎢ i∗2 ⎥ ⎡ ∗ ⎤ F ⎢ ⎥ ⎥ ⎢ i∗3 ⎥ ⎣ x∗ ⎦ ⎦ ⎢ ∗ ⎥ = Fy . ⎢ i4 ⎥ T∗ ⎣ ∗⎦ i5 i∗6 ⎤

⎡

Obviously, the rank of M (θ)|θ=π/3 is two. When Rank(M (θ)) = Rank([M (θ) Q∗ ]), there is no solution, and it is not fault-tolerant controllable. When Rank(M (θ)) =

1 Whenever

damage.

this happens, the system should be shut down to avoid further

Fig. 3.

Levitation forces and torque under the normal operation mode.

Rank([M (θ) Q∗ ]), there are infinitely many solutions. This leads to an extra constraint T∗ =

kt √ ∗ 3Fx − Fy∗ . 2ks


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Fig. 4. Levitation forces and torque with a single open-circuited phase. Without fault-tolerant control: 360◦ −720◦ . With fault-tolerant control: 720◦ −1080◦ . (a) Phase 1 open circuited. (b) Phase 2 open circuited. (c) Phase 3 open circuited. (d) Phase 4 open circuited. (e) Phase 5 open circuited. (f) Phase 6 open circuited.

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In general, this does not hold because these three variables must be independent from each other. Hence, it is not faulttolerant controllable either. 3) Three Open-Circuited Phases: When there are three open-circuited phases, Rank(M (θ)) < 3. Hence, the faulttolerant control is not feasible. All of the combinations of three open-circuited phases include the case with two intervened open-circuited phases. For example, three open-circuited phases 1, 2, and 3 include two intervened open-circuited phases, phase 1 and phase 3. Hence, it is not fault-tolerant controllable. Only three adjacent open-circuited phases are shown in Table I, and the other cases with three open-circuited phases are omitted. In summary, for a six-phase slice PMBM, there are 21 combinations of fault operation modes when the number of opencircuited phases is up to two. The six cases with two intervened open-circuited phases (namely, phases 1 and 3, phases 2 and 4, phases 3 and 5, phases 4 and 6, phases 1 and 5, and phases 2 and 6) are not controllable, and the other 15 cases are controllable to achieve the desired currents according to (11). If the number of open-circuited phases reaches three, then it is no longer controllable. IV. FAULT-T OLERANT C ONTROL Fig. 2 shows the block diagram of a fault-tolerant control system for a six-phase slice PMBM. For each phase, there is a power converter equipped with a current proportional–integral (PI) controller to track the reference current, and hence, each phase can be controlled independently. The power converters can be implemented by using H-bridge modules or power amplifiers [7], [15], [22]–[24]. Normally, four eddy-current sensors or Hall sensors can be installed on the stator to measure the rotor displacements in the x- and y-directions, and optoelectronic encoders or Hall sensors can be installed to measure the speed for the purpose of speed regulation. If Hall sensors are adopted to measure the PM magnetic field, the speed can be calculated via the number of revolutions of the PM’s MMF in one sampling period. The speed can also be calculated via the period of one revolution of the PM’s MMF, e.g., during start-up, to improve the accuracy. Note that the measurement accuracy is affected during acceleration or load variations because there are only two PM poles. These two methods can be combined to improve the accuracy of the speed measurement. The absolute angle of the rotor can then be obtained by integrating the speed in the digital controller. This is better than measuring the rotor angle directly and calculating the speed from the rotor angle because amplifying the measurement noise at high frequencies is avoided. In order to obtain a known initial angle, the phase currents should be set as some special values at the beginning. For example, if the current of phase 1 is forced to be a positive dc current and other phase currents are zero, the PM’s axis should be aligned with the axis of phase 1, and the initial angle is 0◦ . Once the initial angle is known, the absolute angular position can be obtained by the integrator in Fig. 2. When the output of a Hall sensor is zero, the integrator should be reset to avoid the drifting of the angle. There are two displacement PID controllers in the x- and y-directions, respectively, to generate

Fig. 5. Force and torque ripples with one open-circuited phase under the proposed control strategy.

references for levitation forces and one speed PI controller to generate the torque reference. The reference levitation forces and torque are interpreted to generate current references for the current controllers, according to (11), after taking into account the status of the phases detected by the open-circuit detector embedded in the controller. V. S IMULATION R ESULTS The characteristics of the fault-tolerant control system are analyzed by using the finite-element analysis package ANSYS. The major parameters of the six-phase slice PMBM under simulations are given in Table II. The PM shape was optimized to create a sinusoidal PM magnetic field distribution. The process of simulation is given as follows. 1) Set θ = 0, Fx∗ = 5 N, Fy∗ = 5 N, and T ∗ = 0.1 N · m. 2) Update M (θ) according to the phase status. 3) Substitute M (θ) and the given forces and torque into (11) to calculate the desired currents for each phase. 4) Load the calculated results into the ANSYS model to simulate the actual levitation forces and torque. 5) Repeat steps 2–4 with θ increased by 20◦ . Stop when θ = 1080◦ . There are three stages during the simulations, each for one revolution. The first one (from 0◦ to 360◦ ) is the normal operation mode without any open-circuited phases. During the second stage, one or two phases became open circuited at the point of θ = 360◦ , but the controller was not changed. The desired currents were still calculated according to the


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Fig. 6. Levitation forces and torque with two adjacent open-circuited phases. (a) Phases 1 and 2 open circuited. (b) Phases 2 and 3 open circuited. (c) Phases 3 and 4 open circuited. (d) Phases 4 and 5 open circuited. (e) Phases 5 and 6 open circuited. (f) Phases 6 and 1 open circuited.

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Fig. 7. Levitation forces and torque with two opposite open-circuited phases. (a) Phases 1 and 4 open circuited. (b) Phases 2 and 5 open circuited. (c) Phases 3 and 6 open circuited.

equations under the normal condition. The proposed faulttolerant controller was activated at θ = 720◦ while the single or double phases were kept open circuited. The desired currents were calculated based on the fault-tolerant control, i.e., M (θ) was updated according to the open-circuited phases. The results during the normal operation mode, i.e., for θ = 0◦ −360◦ , are shown in Fig. 3. The levitation forces and torque were kept almost constant as 5 N and 0.1 N · m, respectively. At the same time, the desired currents were sinusoidal but with

different initial angles and amplitudes because different phases have different positions. The simulation results with one open-circuited phase are shown in Fig. 4. When a phase was open circuited, the levitation forces and torque went out of control. In particular, Fx rose to almost 8.0 N at 520◦ and 700◦ in Fig. 4(b), and Fy dropped below zero at 460◦ and 640◦ in Fig. 4(a). When the fault-tolerant control strategy was activated at 720◦ , the open-circuited phase currents became zero, and the other phase


currents were recalculated. The motor resumed working properly again, although the maximal range of the currents increased to ±0.6 A from ±0.35 A. The ripples of the levitation forces were less than 3%, and the ripple of the torque was less than 0.8%, as shown in Fig. 5. This means that the operation of the motor was quite smooth. The ranges of the levitation forces are different for open-circuited phases at different positions. However, the torque ripples do not depend on the position of the faulty phase. The results have demonstrated that the levitation forces and torque could be established if the currents of the multiphase PMBM are controlled by the fault-tolerant strategy when one phase becomes open circuited. The simulation results for two open-circuited phases are shown in Fig. 6 when the two phases were adjacent and in Fig. 7 when the two phases were opposite. In both cases, the proposed strategy was able to bring the system back to proper operation. The ripples of the levitation forces and torque are shown in Fig. 8. As expected, the ranges of the levitation forces were larger than those in the cases with a single open-circuited phase shown in Fig. 4. The maximum ripple of the levitation forces was increased to about 5%, and the torque ripple was increased to about 1.3%. The levitation force ripples depend not only on the number of open-circuited phases but also on their positions. However, the torque ripples do not depend on the position of the open-circuited phases. It can also be seen that the fluctuation of the levitation forces with two adjacent open-circuited phases is larger than the fluctuation with two opposite ones if there is a common open-circuited phase. For example, the ripples in Fig. 6(a) are larger than the ones in Fig. 7(a), both having open-circuited phase 1. This is because, if two adjacent phases are open circuited, the distribution of the remaining phases is asymmetrical in space and the harmonics of the levitation forces are increased. As explained before, when there are two intervened or three adjacent open-circuited phases, the fault-tolerant control is not feasible. These cases are shown in Fig. 9, only with the phase currents. The levitation forces and torque cannot be simulated because the currents are not continuous. Take open-circuited phases 1 and 3 as an example; the current is infinite at 780◦ (i.e., 60◦ ) or 960◦ (i.e., 240◦ ). This is in line with the theoretical analysis that all third-order minors are equal to zero when θ = π/3 or 4π/3 if phases 1 and 3 are open circuited. Moreover, all the break points in Fig. 9(a), (b), and (d) appear in Fig. 9(f) because open-circuited phases 1, 3, and 5 are the combination of phases 1 and 3, phases 3 and 5, and phases 5 and 1. VI. E XPERIMENTAL R ESULTS The fault-tolerant strategy proposed earlier has been verified with a six-phase slice PMBM prototype. The main parameters of the structure are the same as those given in Table II. Moreover, the material of PM slices is Nd–Fe–B. Six Hall sensors are installed to measure the radial displacement and the rotor speed. Six H-bridge modules are used to drive the phase windings. The open-circuited fault can be applied by six relays which are connected between the phase windings and the power modules. All of the experiments were carried out at the rated

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Fig. 8. Force and torque ripples with two adjacent or opposite open-circuited phases under the proposed control strategy.

speed of 2000 r/min except for the acceleration from 800 to 2000 r/min.

A. With One Phase Open Circuited Fig. 10 shows the dynamic response, including the displacements in the x- and y-directions and the six phase currents, when there was a single open-circuited phase. In Fig. 10(a) and (c), the experiment was started with a normal operation mode at the rated speed of 2000 r/min and then entered into the fault-tolerant mode after an open-circuited fault was detected in phases 1 and 2, respectively. Before the current of the open-circuited phase became zero, the motor worked under the normal mode. During the period of the normal mode, the range of the x displacement fluctuated between about ±65 μm, and the range of the y displacement fluctuated between nearly ±90 μm, which are less than the air gap of 2 mm. When the relay was turned off, the current of the open-circuited phase dropped to zero rapidly, and once this was detected, the controller entered into the fault-tolerant mode. It is obvious that the displacement rose to the maximum during this transition. The transient response took about 200 ms before the rotor was levitated stably again. Fig. 10(b) shows a close look of the curves when the motor worked under the fault-tolerant mode with open-circuited phase 1. The experiment was repeated when the motor was accelerated from 800 to 2000 r/min, and the results are shown in Fig. 10(d) and (e). It is obvious that the speed and displacements were affected by the open-circuited faults, and the currents of other phases were bigger than before. It is worth noting that the torque was generated by the speed PI

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Fig. 9. Cases not feasible: Two intervened or three open-circuited phases. (a) Phases 1 and 3 open circuited. (b) Phases 3 and 5 open circuited. (c) Phases 2 and 4 open circuited. (d) Phases 5 and 1 open circuited. (e) Phases 2, 3, and 4 open circuited. (f) Phases 1, 3, and 5 open circuited.

controller, and hence, the amplitude of the phase currents varied when the operation condition was changed.

B. With Two Phases Open Circuited Fig. 11(a) shows the experimental results when phases 1 and 2, which are adjacent, were open circuited at the same time, and Fig. 11(b) shows the experimental results when phases 1 and 4, which are opposite, were open circuited at the same time, when the motor was operated at the steady state. The experiments were repeated when the motor was accelerated from 800 to 2000 r/min, and the results are shown in Fig. 11(c) and (d), respectively. The peak–peak currents of other phases were larger than those in Fig. 10, and the ranges of the displacements were increased to about ±150 μm. In the normal mode, the harmonics of the levitation forces are less than those in the open-circuited cases because the phase windings are distributed symmetrically. As a result, the range of

the displacements was smaller. During the period of transition, there was obvious fluctuation in the transition because it took time to detect the fault and it also took time for the fault-tolerant controller to respond to the fault. For the other cases with two open-circuited phases adjacent or opposite, the results are similar to those shown in Fig. 11 and are hence omitted.

VII. C ONCLUSION The current-controlled model of a multiphase slice PMBM has been analyzed and applied to its fault-tolerant control. The general condition of fault-tolerant controllability for a multiphase PMBM has been analyzed when there are opencircuited phases. It has been found out that the fault-tolerant control of a multiphase slice PMBM is feasible if and only if the row rank of M (θ) is three. Based on this condition, whether the fault-tolerant control with individual open-circuited fault


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Fig. 11. Experimental results with two adjacent or opposite phases open circuited at the rated speed and during the acceleration from 800 to 2000 r/min. (a) Adjacent phases 1 and 2 became open circuited at the rated speed. (b) Opposite phases 1 and 4 became open circuited at the rated speed. (c) Phases 1 and 2 became open circuited when accelerating. (d) Phases 1 and 4 became open circuited when accelerating.

ACKNOWLEDGMENT Fig. 10. Experimental results with a single phase open circuited at the rated speed and during the acceleration from 800 to 2000 r/min. (a) Phase 1 became open circuited at the rated speed. (b) Phase 1 was open circuited at the rated speed. (c) Phase 2 became open circuited at the rated speed. (d) Phase 1 became open circuited when accelerating. (e) Phase 2 became open circuited when accelerating.

case is feasible or not can be analyzed. For a six-phase slice PMBM, it is feasible if there is a single open-circuited phase or there are two adjacent or opposite open-circuited phases. It is not feasible if there are two intervened open-circuited phases or three (or more) open-circuited phases. Finite-element analysis of the operations of a six-phase slice PMBM under different fault-tolerant control modes has verified the generic conditions. Moreover, experimental results have shown that the slice PMBM worked well for the cases with a single open-circuited phase and with two adjacent or opposite opencircuited phases at the rated speed or during the process of acceleration from 800 to 2000 r/min, which has further verified the analysis.

The authors would like to thank the Associate Editor and the Reviewers for their constructive comments, which have considerably improved the quality of this paper. R EFERENCES [1] S. Silber and W. Amrhein, “Power optimal current control scheme for bearingless PM motors,” in Proc. 7th Int. Symp. Magn. Bearings, Aug. 2000, pp. 401–406. [2] S. Silber, W. Amrhein, K. Nenninger, G. Trauner, and M. Reisinger, “Performance optimization of a bearingless pump using feedback linearization techniques,” in Proc. 8th Int. Symp. Magn. Bearings, Mito, Japan, Aug. 2002, pp. 379–382. [3] S. Silber, W. Amrhein, P. Bosch, R. Schob, and N. Barletta, “Design aspects of bearingless slice motors,” IEEE/ASME Trans. Mechatronics, vol. 10, no. 6, pp. 611–617, Dec. 2005. [4] W. Khoo, “Bridge configured winding for polyphase self-bearing machines,” IEEE Trans. Magn., vol. 41, no. 4, pp. 1289–1295, Apr. 2005. [5] W. Khoo, R. L. Fittro, and S. D. Garvey, “AC polyphase self-bearing motors with a bridge configured winding,” in Proc. 7th Int. Symp. Magn. Bearings, Mito, Japan, Aug. 2002, pp. 47–52. [6] M. Kang, J. Huang, J.-Q. Yang, and H.-B. Jiang, “Analysis and experiment of a 6-phase bearingless induction motor,” in Proc. 11th ICEMS, Wuhan, China, 2008, pp. 990–994.

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[7] M. T. Bartholet, T. Nussbaumer, D. Krahenbuhl, F. Zurcher, and J. W. Kolar, “Modulation concepts for the control of a two-phase bearingless slice motor utilizing three-phase power modules,” IEEE Trans. Ind. Appl., vol. 46, no. 2, pp. 831–840, Mar. 2010. [8] M. Bartholet, S. Silber, T. Nussbaumer, and J. Kolar, “Performance investigation of two-, three- and four-phase bearingless slice motor configurations,” in Proc. Int. Conf. PEDS, Bangkok, Thailand, 2007, pp. 9–16. [9] M. T. Bartholet, T. Nussbaumer, S. Silber, and J. W. Kolar, “Comparative evaluation of polyphase bearingless slice motors for fluid-handling applications,” IEEE Trans. Ind. Appl., vol. 45, no. 5, pp. 1821–1830, Sep./Oct. 2009. [10] M. T. Bartholet, T. Nussbaumer, and J. W. Kolar, “Comparison of voltagesource inverter topologies for two-phase bearingless slice motors,” IEEE Trans. Ind. Electron., vol. 58, no. 5, pp. 1921–1925, May 2011. [11] S. Dwari and L. Parsa, “Fault-tolerant control of five-phase permanentmagnet motors with trapezoidal back EMF,” IEEE Trans. Ind. Electron., vol. 58, no. 2, pp. 476–485, Feb. 2011. [12] W. Zhao, M. Cheng, W. Hua, H. Jia, and R. Cao, “Back-EMF harmonic analysis and fault-tolerant control of flux-switching permanent-magnet machine with redundancy,” IEEE Trans. Ind. Electron., vol. 58, no. 5, pp. 1926–1935, May 2011. [13] A. Akrad, M. Hilairet, and D. Diallo, “Design of a fault-tolerant controller based on observers for a PMSM drive,” IEEE Trans. Ind. Electron., vol. 58, no. 4, pp. 1416–1427, Apr. 2011. [14] M. Fnaiech, F. Betin, G.-A. Capolino, and F. Fnaiech, “Fuzzy logic and sliding-mode controls applied to six-phase induction machine with open phases,” IEEE Trans. Ind. Electron., vol. 57, no. 1, pp. 354–364, Jan. 2010. [15] L. De Lillo, L. Empringham, P. W. Wheeler, S. Khwan-On, C. Gerada, M. N. Othman, and X. Huang, “Multiphase power converter drive for fault-tolerant machine development in aerospace applications,” IEEE Trans. Ind. Electron., vol. 57, no. 2, pp. 575–583, Feb. 2010. [16] S. Yue, X. Wang, Q. Liao, and Z. Deng, “Auto diagnosis error tolerant operation of a bearingless slice motor,” Trans. China Electrotech. Soc., vol. 25, no. 5, pp. 76–81, May 2010. [17] X.-L. Wang, Q.-C. Zhong, Z.-Q. Deng, and S.-Z. Yue, “Fault-tolerant control of multi-phase permanent magnetic bearingless motors,” in Proc. 19th ICEM, Rome, Italy, Sep. 2010, pp. 1–6. [18] M. Ooshima, A. Chiba, T. Fukao, and M. Rahman, “Design and analysis of permanent magnet-type bearingless motors,” IEEE Trans. Ind. Electron., vol. 43, no. 2, pp. 292–299, Apr. 1996. [19] S. Zhang and F. L. Luo, “Direct control of radial displacement for bearingless permanent-magnet-type synchronous motors,” IEEE Trans. Ind. Electron., vol. 56, no. 2, pp. 542–552, Feb. 2009. [20] P. Karutz, T. Nussbaumer, W. Gruber, and J. Kolar, “Accelerationperformance optimization for motors with large air gaps,” IEEE Trans. Ind. Electron., vol. 57, no. 1, pp. 52–60, Jan. 2010. [21] Z. Sun, J. Wang, G. Jewell, and D. Howe, “Enhanced optimal torque control of fault-tolerant PM machine under flux-weakening operation,” IEEE Trans. Ind. Electron., vol. 57, no. 1, pp. 344–353, Jan. 2010. [22] P. Barriuso, J. Dixon, P. Flores, and L. Moran, “Fault-tolerant reconfiguration system for asymmetric multilevel converters using bidirectional power switches,” IEEE Trans. Ind. Electron., vol. 56, no. 4, pp. 1300– 1306, Apr. 2009. [23] W. Song and A. Q. Huang, “Fault-tolerant design and control strategy for cascaded H-bridge multilevel converter-based STATCOM,” IEEE Trans. Ind. Electron., vol. 57, no. 8, pp. 2700–2708, Aug. 2010. [24] M. A. Parker, C. Ng, and L. Ran, “Fault-tolerant control for a modular generator-converter scheme for direct-drive wind turbines,” IEEE Trans. Ind. Electron., vol. 58, no. 1, pp. 305–315, Jan. 2011.

Xiao-Lin Wang was born in Sichuan, China, in 1976. He received the B.Sc. and Ph.D. degrees in power electronics and motor drives from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 1999 and 2004, respectively. Since December 2004, he has been with Nanjing University of Aeronautics and Astronautics as a Lecturer in the Department of Electrical Engineering, College of Automation Engineering. In 2007, he was promoted to an Associate Professor. From September 2009 to September 2010, he was also a Postdoctoral Researcher with the University of Liverpool, Liverpool, U.K. His current research focuses on magnetically levitated bearingless motors, solid-state power converters, renewable energy, and electric storage systems.

Qing-Chang Zhong (M’04–SM’04) received the Diploma in electrical engineering from Hunan Institute of Engineering, Xiangtan, China, in 1990, the M.Sc. degree in electrical engineering from Hunan University, Changsha, China, in 1997, the Ph.D. degree in control theory and engineering from Shanghai Jiao Tong University, Shanghai, China, in 1999, and the Ph.D. degree in control and power engineering (awarded the Best Doctoral Thesis Prize) from Imperial College London, London, U.K., in 2004. He was with Technion—Israel Institute of Technology, Haifa, Israel; Imperial College London; the University of Glamorgan, Cardiff, U.K.; and the University of Liverpool, Liverpool, U.K. In summer 2010, he has been with Loughborough University, Loughborough, U.K., as a Professor of control engineering. He is the author or coauthor of Robust Control of Time-Delay Systems (Springer-Verlag, 2006), Control of Integral Processes with Dead Time (Springer-Verlag, 2010), and Control of Power Inverters for Distributed Generation and Renewable Energy (Wiley-IEEE Press, 2011). Dr. Zhong is a Fellow of the Institution of Engineering and Technology and was a senior research fellow of the Royal Academy of Engineering/Leverhulme Trust, U.K. (2009–2010). He serves as an Associate Editor for the IEEE T RANSACTIONS ON P OWER E LECTRONICS and the Conference Editorial Board of the IEEE Control Systems Society.

Zhi-Quan Deng was born in Hubei, China, in 1969. He received the B.S. degree in mechanical engineering from Xi’an Institute of Metallurgy and Construction Engineering, Xi’an, China (which has been known as Xi’an University of Architecture and Technology since 1994), in 1990 and the M.S. and Ph.D. degrees in engineering machinery from Northeastern University, Shenyang, China, in 1993 and 1996, respectively. In 1996, he was a Postdoctoral Fellow with Nanjing University of Aeronautics and Astronautics, Nanjing, China, where he joined the Department of Electrical Engineering in 1998 and is currently a Professor with the College of Automation Engineering. His current research interests include bearingless motor drive systems, magnetic bearings, and superhigh-speed electrical machines.

Shen-Zhou Yue was born in Zhejiang, China, in 1983. He received the B.S. degree in automation engineering from Wenzhou University, Wenzhou, China, in 2007 and the M.S. degree in electric machines and electric apparatus from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2010. Since 2010, he has been with Shanghai AeroSharp Electric Technologies Company, Ltd., Shanghai, China, where he is currently working on grid-connected inverters for solar and wind applications.