https://doi.org/10.6113/JPE.2018.18.5.1424

ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718

Decoupled SVPWM for Five-Phase Permanent Magnet Machines with Trapezoidal Back-EMF

Zhipeng Lin^*, Guohai Liu^*, Wenxiang Zhao^*, and Qian Chen^†

^†,*School of Electrical and Information Engineering, Jiangsu University, Jiangsu, China

Abstract

This paper presents a novel space vector pulse-width modulation (SVPWM) to synthesize an arbitrary non-sinusoidal phase voltage. The key of the proposed method is that the switching vectors used to comprise the reference vectors in the α₁-β₁ frame and the α₃-β₃ frame are decoupled. In the α₁-β₁ frame, the reference vector is comprised by near two large vectors. The corresponding vector comprised by the two vectors in the α₃-β₃ frame is considered as a disturbance, which is restrained by close-loop control. In the α₃-β₃ frame, there are two methods to comprise the reference vector. Method I is a near two middle vectors method. Method II uses near four vectors (two middle and two little vectors). The proposed SVPWM using decoupled switching vectors can guarantee a maximum modulation index in the α₁-β₁ frame. The effectiveness of the proposed method is verified by simulated and experimental results under various operation conditions.

Key words: Decoupled concept, Five-phase permanent magnet machine, Non-sinusoidal voltage, Space vector pulse-width modulation (SVPWM), Trapezoidal back-EMF

Manuscript received Mar. 17, 2018; accepted Jul. 15, 2018

Recommended for publication by Associate Editor Zheng Wang.

^†Corresponding Author: chenqian0501@ujs.edu.cn Tel: +86-15252900600, Fax: +86-051188787773, Jiangsu University

*School of Electrical and Information Engineering, Jiangsu University, China

Ⅰ. INTRODUCTION

Multi-phase machines have been widely used in a variety of industrial applications for the last decade, due to their intrinsic features such as power splitting, better fault tolerance and lower torque ripple than three-phase machines [1]-[5]. During the design process, some five-phase permanent magnet (PM) machines are designed to have non-sinusoidal back-EMF in order to increase the maximum output torque [6]-[9]. In [6] and [7], harmonics are injected into the PM shape to maximize the output torque in five-phase PM machines. To realize the maximize output torque of five-phase machines with trapezoidal back-EMF, third harmonic current injection methods have been proposed in [10]-[14]. In order to improve the torque of five-phase unequal tooth surface-mounted PM machines, third harmonic currents are injected into the stator windings based on the ratio of the third harmonic back-EMF to the fundamental one [10]. A bifurcation analysis of a five- phase induction machine when a third harmonic is injected for torque-enhancement has been proposed in [11]. In [12], experimental results show that the output torque after an injection of the third harmonic current in a five-phase induction machine with trapezoidal back-EMF can increase by 11% when compared with only using the fundamental current.

In five-phase space vector pulse-width modulation (SVPWM) methods, the traditional near two large vectors method produces the harmonic vector and current in α₃-β₃ frame, due to the vector in α₃-β₃ frame without control [15]. In order to realize a sinusoidal phase voltage, near four vectors methods have been proposed in [16]-[18]. In [18], two SVPWM methods are proposed to reduce common-mode voltage. In [19], the method proposed in [18] was extended to the over-modulation zone. M. Jones [20] investigated the stator current ripple, caused by the switching harmonics, under two SVPWM techniques in which two large and two medium vectors or four large vectors are used. However, these methods only focus on how to produce a sinusoidal voltage. Based on a multiple d-q space concept, a novel analysis of a multiphase SVPWM for synthesizing an arbitrary non- sinusoidal phase voltage has been proposed in [21]. This method can comprise the reference voltages by near four vectors in the α₁-β₁ frame and the α₃-β₃ frame. However, the near four vectors used to comprise the reference vectors in the α₁-β₁ frame and the α₃-β₃ frame are coupled.

In this paper, a novel SVPWM is presented to synthesize a non-sinusoidal phase voltage. In the α₁-β₁ frame, the reference vector is comprised by near two large vectors to guarantee the maximum voltage linear output area. The corresponding vector comprised by two vectors in the α₃-β₃ frame is considered as a disturbance, which is restrained by a close-loop control. In the α₃-β₃ frame, there are two methods for comprising the reference voltage. For Method I, near two middle vectors are used to maintain the maximum voltage linear output area in the α₃-β₃ frame. For Method II, near four vectors (two middle and two little vectors) are adopted to improve the robustness of the system on the basis of ensuring the maximum modulation index in the α₁-β₁ frame. In the proposed method, the switching vectors used to comprise the reference vectors in the α₁-β₁ frame and the α₃-β₃ frame are decoupled, which can guarantee the maximum modulation index in the α₁-β₁ frame.

Ⅱ. MOTOR TOPOLOGY AND MATHEMATICAL MODEL

Fig. 1 shows a 20-slot/18-pole five-phase PM machine [22]. It adopts an outer-rotor topology to achieve the prominent dynamic performance of electric vehicles, which can enhance its torque capability. Fig. 2 shows the measured back-EMF of a machine and fast Fourier transform results of measured back-EMF data. It is found that the back-EMF of a machine contains 22.2% third harmonic, 9.6% fifth harmonic and 1.2% seventh harmonic. The amplitude and phase of the fifth harmonic flux linkages are the same. Since the sum of the five-phase current is zero under a star connection, the fifth harmonic flux linkages do not produce torque and can be ignored. To simplify the mathematical model of a five-phase PM machine, only the fundamental and third harmonic components are considered. The parameters of a five-phase PM machine are listed in Table I.

Fig. 1. Cross section of a five-phase PM machine.

Fig. 2. Measured back-EMF and its THD: (a) Back-EMF, (b) Harmonic order.

(a)

(b)

TABLE I MOTOR PARAMETERS

Symbol	Description	Quantity
ψm1	Fundamental PM flux linkage	0.0411Wb
ψm3	3rd harmonic PM flux linkage	0.0033Wb
Ld1	Fundamental d-axis inductance	0.9323mH
Lq1	Fundamental q-axis inductance	1.2614mH
P	Pole pairs	9
Ns	Slot number	20

The back-EMFs are produced by the magnetic flux linkages, and the magnetic flux linkages of a five-phase PM machine are presented as follows:

           (1)

where ψ_m is the magnetic flux linkage vector, ψ_m1 and ψ_m3 represent the amplitudes of the fundamental and third harmonic of the PM flux linkages, θ_e is the rotor electrical position, and δ is the spatial shifting angle between adjacent phases, being δ=2π/5.

The fundamental and third harmonic components can be mapped into two orthogonal subspaces, which are decoupled from each other. The fundamental orthogonal subspace is referred as d₁-q₁ and the third harmonic orthogonal subspace is referred as d₃-q₃. The Clarke transformation matrix which transforms the components from the natural coordinate system into the rotating orthogonal coordinate system is shown as follows:

           (2)

The Park transformation matrix can be presented as follows:

          (3)

By applying transformation matrices (2) and (3), the d-q frame current components of a five-phase PM machine can be presented as:

        (4)

where i_d1, i_q1, i_d3 and i_q3 are the current components of the d₁-q₁ axis and the d₃-q₃ axis, and I_s represents the phase current vector.

The stator voltages in the d-q frame can be presented as:

      (5)

where R_s is the stator resistance, p is the pole pairs, ω is the rotor angular speed, and L_dq is the inductance in the d-q frame, which can be obtained as follows:

        (6)

where L_ls is the leakage inductance, and L_s is the phase inductance matrix.

By taking a derivate of the magnetic co-energy with respect to the rotor mechanical position (θ_m), the torque equation can be obtained as follows:

          (7)

where W_co represents the magnetic co-energy.

The electromagnetic torque can be obtained by substituting (1), (4) and (6) into (7):

         (8)

As expressed in (8), the injection of the third harmonic current improves the output torque.

Ⅲ. PROPOSED SPACE VECTOR PWM

A. Switching Vectors in the α-β Frame

The common drive topology of a five-phase PM machine contains a five-phase half-bridge power inverter and a five-phase PM machine. S_A, S_B, S_C, S_D and S_E represent the switching state of phases A, B, C, D and E, respectively. If the lower switch of phase x is ON and the upper switch is OFF, then S_x=0. If the opposite case occurs, then S_x=1.

In a five-phase PWM inverter, there are two different types of basic equivalent circuit configurations, {C14} and {C23}, as shown in Fig. 3. Fig. 3(a) is where one upper switch is ON and four lower switches are OFF or in an opposite manner, like the switching patterns V₁₆(10000) and V₁₅(01111). Fig. 3 (b) is where two upper switches are ON and three lower switches are OFF or in an opposite manner, like the switching patterns V₂₄(11000) and V₇(00111).

Fig. 3. Basic equivalent circuit configuration: (a) C14, (b) C23.

(a)

(b)

By applying a Clarke transformation (2), the DC-link voltage U_dc and the switching state of a five-phase inverter, the switching vectors in the α₁-β₁ frame and the α₃-β₃ frame can be obtained as follows:

    (9)

     (10)

where V_s1 and V_s3 represent the switching vector in the α₁-β₁ frame and the α₃-β₃ frame, and is the rotating factor, being γ=e^jδ.

There is a total of 2⁵=32 switching states in a five-phase inverter. Therefore, there are 32 switching vectors in the α₁-β₁ frame and the α₃-β₃ frame, respectively. According to the magnitudes of the switching vectors, the 32 switching vectors in the α₁-β₁ frame and the α₃-β₃ frame can be classified into large, middle, little and zero switching vectors. The magnitudes of the large, middle and little switching vectors are 0.6472U_dc, 0.4U_dc and 0.2472U_dc, respectively. There are two zero switching vectors, V₀(00000) and V₃₁(11111). Table II lists the switching vectors in the α₁-β₁ frame and the α₃-β₃ frame. Fig. 4 shows the switching vectors in the α-β frame. As shown in Table II and Fig. 4, the middle vectors in the α₁-β₁ frame and the α₃-β₃ frame belong to C14. The large vectors and little vectors in the α₁-β₁ frame and the α₃-β₃ frame belong to C23. The switching patterns that belong to C23 can be classified into two categories. The first category is where the three phases with the same switching state are adjacent, which belong to the large vectors in the α₁-β₁ frame and the little vectors in the α₃-β₃ frame. The second category is where the three phases with the same switching state are not adjacent, which belong to the little vectors in the α₁-β₁ frame and the large vectors in the α₃-β₃ frame. In the second category, a phase with the opposite switching state is clamped between three phases with the same switching states. For example, the switching state of phase C is different from the switching states of phases B, D and E in the switching pattern V₁₁(01011). Hence, in the natural coordinate system, the direction of the voltage vector of phase B is opposite the voltage vector composed by phases D and E, which may cause the stator fluxes of the phases to counteract each other. The switching vectors which belong to the little vectors in the α₁-β₁ frame and the large vectors in the α₃-β₃ frame are not recommended.

Fig. 4. Switching vectors in: (a) α₁-β₁ frame, (b) α₃-β₃ frame.

(a)

(b)

TABLE II SWITCHING VECTORS GROUPING TABLE

Vectors grouping	Magnitude	α1-β1 frame	α3-β3 frame
Large vectors	0.6472Udc	V3,V6,V7,V12,V14,V17,V19,V24,V25,V28	V5,V9,V10,V11,V13,V18,V20,V21,V22,V26
Middle vectors	0.4Udc	V1,V2,V3,V8,V15,V16,V23,V27,V29,V30	V1,V2,V3,V8,V15,V16,V23,V27,V29,V30
Little vectors	0.2472Udc	V5,V9,V10,V11,V13,V18,V20,V21,V22,V26	V3,V6,V7,V12,V14,V17,V19,V24,V25,V28
Zero vectors	0	V0,V31	V0,V31

B. Synthesis of the Reference Vectors

The near four vectors modulation method has been widely used for the control of five-phase PM machines, which can realize a non-sinusoidal phase voltage. This method uses near four vectors to comprise the reference vectors in the α₁-β₁ frame and the α₃-β₃ frame. However, the maximum voltage linear output areas in the α₁-β₁ frame and the α₃-β₃ frame are coupled in this method. When the voltage vector in the α₁-β₁ frame realizes a maximum magnitude of 0.6112U_dc, the maximum magnitude of voltage vector in the α₃-β₃ frame is 0.1453U_dc. When the voltage vector in the α₃-β₃ frame realizes a maximum magnitude of 0.2351U_dc, the maximum magnitude of the voltage vector in the α₁-β₁ frame is 0.3804U_dc.

The novel SVPWM uses the decoupled concept to comprise the reference vectors in the α₁-β₁ frame and the α₃-β₃ frame separately. Then the phase duty cycles corresponding to the reference vectors in the α₁-β₁ frame and the α₃-β₃ frame are calculated. Finally, the calculated phase duty cycles in the α₁-β₁ frame and the α₃-β₃ frame are added to achieve a non-sinusoidal phase voltage.

1) Synthesis of the Reference Vector in the α₁-β₁ Frame

In order to realize the maximum voltage linear output area in the α₁-β₁ frame, ten large vectors and two zero vectors are chosen to comprise the reference voltage in the α₁-β₁ frame. For example, a α₁-β₁ frame reference vector located in sector I, as shown in Fig. 5(a), can be realized by:

     (11)

Fig. 5. Realization of a reference voltage vector located in sector I by near two vectors: (a) α₁-β₁ frame, (b) Corresponding vectors in the α₃-β₃ frame.

(a)

(b)

where T₁ and T₂ are the applying times of V₂₄ and V₂₅, respectively, and T_s is the sampling time.

T₁, T₂ and the applying time of the zero vectors T₀ can be obtained as follows:

    (12)

Then the duty cycles of phases A, B, C, D and E corresponding to the reference vector in the α₁-β₁ frame can be obtained as follows:

           (13)

The trajectory of the voltage vector in the α₁-β₁ frame with the maximum magnitude in the linear modulation region can be drawn as the inscribed circle of the decagon consisting of ten large vectors, as shown in Fig. 6. Then the maximum modulation index of this method can be calculated as:

      (14)

Fig. 6. Trajectory of the voltage vector with the maximum modulation index in the α₁-β₁ frame using the near two vectors method.

The near two vectors method can realize the maximum magnitude of a voltage vector in the linear modulation region. In the α₃-β₃ frame, the V_d3 shown in Fig. 5(b) is comprised by V₂₄ and V₂₅. In the traditional near two vectors method, due to the vector in the α₃-β₃ frame without control, V_d3 produces an uncontrolled third harmonic component. In the proposed SVPWM, V_d3 is considered as a disturbance in the α₃-β₃ frame. Since the vector in the α₃-β₃ frame is close-loop controlled, the disturbance V_d3 can be restrained.

2) Synthesis of the Reference Vector in the α₃-β₃ Frame

The ten large vectors in the α₃-β₃ frame will cause the stator fluxes of the phases to counteract each other, which has been explained in Section III part A. Hence, the ten large vectors in the α₃-β₃ frame are not recommended for use. The ten middle vectors, ten little vectors and two zero vectors can be used to comprise the reference vector in the α₃-β₃ frame.

a) Near Two Vectors Method in the α₃-β₃ Frame

To realize the maximum voltage linear output area in the α₃-β₃ frame, the ten middle vectors and two zero vectors are chosen to comprise the reference voltage in α₃-β₃ frame. The calculation method of applying time of near two middle vectors in the α₃-β₃ frame is similar to the method in the α₁-β₁ frame. For example, the reference vector located in sector I is comprised by V₁₆ and V₂₃ in the α₃-β₃ frame, as shown in Fig. 7.

The duty cycles of phases A, B, C, D and E corresponding to the reference vector located in sector I under the α₃-β₃ frame can be obtained as follows:

     (15)

where T₀, T₁ and T₂ are the applying times of the zero vectors, V₁₆ and V₂₃, respectively.

Fig. 7. Realization of a reference voltage vector located in sector I by near two middle vectors: (a) In the α₃-β₃ frame, (b) The corresponding vector in the α₁-β₁ frame.

(a)

(b)

The trajectory of the voltage vector in the α₃-β₃ frame with the maximum magnitude in the linear modulation region can be drawn as the inscribed circle of the decagon consisting of ten middle vectors shown in Fig. 8. The maximum modulation index of this method can be calculated as:

     (16)

Fig. 8. Trajectory of the voltage with the maximum modulation index in the α₃-β₃ frame using the near two vectors method.

In the α₁-β₁ frame, the V_d1 shown in Fig. 7(b) is comprised by V₁₆ and V₂₃. In the proposed SVPWM, V_d1 is also considered as a disturbance in the α₁-β₁ frame. The disturbance V_d1 can be restrained, due to the closed-loop control of the vector in the α₁-β₁ frame.

b) Near Four Vectors Method in the α₃-β₃ Frame

The near two vectors method mentioned above can realize the maximum magnitude of a voltage vector in the linear modulation region in the α₃-β₃ frame. However, it produces a disturbance vector in the α₁-β₁ frame. In order to improve the robustness of the system, the near four vectors method can be adopted.

This method chooses the near four vectors (two middle and two little vectors) to comprise the reference vector in the α₃-β₃ frame. For example, the reference vector located in sector I is comprised by V₆, V₁₆, V₂₃ and V₂₈ in the α₃-β₃ frame, as shown in Fig. 9. In addition, T₁, T₂, T₃ and T₄ are applying times of V₆, V₁₆, V₂₃ and V₂₈, respectively. In order for the value of the vector comprised in the α₁-β₁ frame to be zero, the applying time of the middle vectors and little vectors should satisfy the following equation:

           (17)

Fig. 9. Realization of a reference voltage vector located in sector I by near four vectors: (a) In the α₃-β₃ frame, (b) The corresponding vector in the α₁-β₁ frame.

(a)

(b)

In the α₃-β₃ frame, V₆ and V₂₈ are used to comprise λV_ref3, and V₁₆ and V₂₃ are used to comprise (1-λ)V_ref3. In addition, λ can be calculated by the following equation:

     (18)

where λ=0.2764.

T_1, T₂, T₃, T₄ and the applying time of the zero vectors T₀ can be obtained as follows:

       (19)

The duty cycles of phases A, B, C, D and E corresponding to the reference vector in the α₃-β₃ frame in this method can be obtained as follows:

        (20)

The maximum voltage vector of this method which uses the near four vectors to comprise the vector in the α₃-β₃ frame can be obtained as follows:

          (21)

The maximum modulation index in the α₃-β₃ frame of this method is presented as follows:

   (22)

3) Generation of the Switching Signals

The total duty cycles of the phases can be obtained as follows:

           (23)

where X is A, B, C, D and E.

Then the switching signal of every phase can be modulated by the corresponding duty cycle and triangular carrier. The maximum value of the triangle carrier is 1, and the minimum value is 0. The frequency of the triangle carrier can be selected according to the maximum frequency of the inverter.

In the methods mentioned above (section III part B (1) and (2)), the switching vectors used to comprise the reference vectors in the α₁-β₁ frame and the α₃-β₃ frame are decoupled. The method using the near two vectors in the α₁-β₁ frame and the α₃-β₃ frame is denoted as Method I, the method using the near two vectors in the α₁-β₁ frame and the near four vectors in the α₃-β₃ frame is denoted as Method II. Method I can realize the maximum magnitude of the voltage vector in the linear modulation region under the α₁-β₁ frame and the α₃-β₃ frame, respectively. Method II can improve the robustness of the system on the basis of ensuring the maximum modulation index in the α₁-β₁ frame. A diagram of the proposed SVPWM is shown in Fig. 10.

Fig. 10. Diagram of the proposed SVPWM.

Ⅳ. VERIFICATION

A. Simulated Results

To evaluate the performance of the proposed SVPWM strategy, a simulation model of a five-phase PM machine drive is developed. Fig. 11 shows a schematic of the five-phase PM machine drive control system including the proposed SVPWM method, which can realize the control of the objective in the α₃-β₃ frame. The SVPWM block in Fig. 11 is detailed in Fig. 10. Fig. 12 shows the reference current and sampled current of phase A using Method I and Method II under same operation condition. The simulated results show that the two proposed SVPWM methods can regulate non-sinusoidal phase currents including the third harmonic component without steady-state error.

Fig. 11. Control scheme with the proposed SVPWM method.

Fig. 12. The waveforms of the phase A reference current and sampled current using: (a) Method I, (b) Method II.

(a)

(b)

B. Experimental Results

In order to verify the suitability of the proposed SVPWM, a test platform has been set up as shown in Fig. 13. The experimental set is composed of a five-phase PM machine, a five-phase half-bridge inverter, and a DC generator as the load. A dSPACE1005 controller is utilized for the implementation of the overall control algorithm. The machine torque is measured by a high precision torque transducer (HBM T20WN/20NM). The frequency of the IGBTs is fixed at 10 kHz, and the DC-link voltage is 60 V.

Fig. 14 shows the torque and phase current waveforms using Method I. The load torque is 6.2 Nm. Fig. 15 shows the sampled and reference current components in the d₁-q₁ axis and the d₃-q₃ axis. In addition, i_d1, i_q1, i_d3 and i_q3 represent the sampled current components of the d₁-q₁ axis and the d₃-q₃ axis, respectively. Meanwhile, i_q1r and i_q3r are the reference currents of the q₁-axis and the q₃-axis, respectively. In this experiment, the reference currents of the d₁-axis and the d₃-axis are 0 and the reference current in the q₃-axis is 2 A. In this experiment, the q₁-axis current i_q1 is 8.7 A. The q₃-axis reference current can be calculated by multiplying i_q1 by the ratio of the third harmonic 22.2% [10]. The result is about equal to 2 A. The reference current of the q₃-axis is set as 2 A, which can improve the output torque and is also convenient for checking the value of the feedback current i_q3. As shown in Fig. 15, the sampled currents can track the reference currents accurately in the d₁-q₁ axis and the d₃-q₃ axis. Fig. 16 shows trajectories of the sampled current vectors in the α₁-β₁ axis and α₃-β₃ axis. The experimental results (Figs. 14-16) reveal that non-sinusoidal phase currents, including the third harmonic component, can be regulated without steady-state error by employing Method I.

Fig. 13. Experimental test platform.

Fig. 14. Torque and phase current waveforms using Method I; T_e is scaled to 2 Nm/div, the phase current is scaled to 5A/div.

Fig. 15. Sampled and reference current components in the d₁-q₁ axis and the d₃-q₃ axis using Method I.

Fig. 16. Trajectories of the sampled current vectors using Method I in the: (a) α₁-β₁ axis, (b) α₃-β₃ axis.

(a)

(b)

Fig. 17 exhibits torque and phase current waveforms using Method II. Fig. 18 shows the sampled and reference current components of the d₁-q₁ axis and the d₃-q₃ axis. The reference current components of the d₁-axis and the d₃-axis are 0, and the reference current of the q₃-axis is 2 A. As shown in Fig. 18, the sampled current can track the reference currents accurately in the d₁-q₁ axis and the d₃-q₃ axis. Fig. 19 shows the trajectories of the sampled current vectors in the α₁-β₁ axis and the α₃-β₃ axis. Experimental results (Figs. 17-19) show that Method II can realize control of the third harmonic current without steady-state error.

Fig. 17. Torque and phase current waveforms using Method II; T_e is scaled to 2 Nm/div, the phase current is scaled to 5A/div.

Fig. 18. Sampled and reference current components of the d₁-q₁ axis and the d₃-q₃ axis using Method II.

Fig. 19. Trajectories of the sampled current vectors using Method II in the: (a) α₁-β₁ axis, (b) α₃-β₃ axis.

(a)

(b)

Comparing the experimental results of Fig. 15 and Fig. 18, the current ripples of Method II are less than those of Method I in the d₁-q₁ axis and the d₃-q₃ axis. Fig. 20 shows fast Fourier transformation results of the phase current using Method I and Method II. The fundamental currents of Method I and Method II are 8.7 A and 8.5 A, respectively. The third harmonic currents of both methods are 2 A, which are equal to the reference values. As shown in Fig. 20, the amplitudes of the seventh, ninth and eleventh harmonic currents in Method II are less than those in Method I. By comparing experimental results of Method I and Method II, it can be condluded that the experimental effectiveness of Method II is better than that of Method I. This conclusion is consistent with the theoretical analysis mentioned above (Section III part B (3)).

Fig. 20. Amplitude of the harmonic currents.

Ⅴ. CONCLUSIONS

In this paper, a novel decoupled SVPWM has been proposed to realize a non-sinusoidal phase voltage. In the α₁-β₁ frame, the reference vector has been comprised by near two large vectors to guarantee the maximum voltage linear output area. The corresponding vector comprised by two vectors in the α₃-β₃ frame has been considered as a disturbance, which has been restrained by close-loop control. In the α₃-β₃ frame, there are two methods of comprising the reference vector. For Method I, near two middle vectors have been used to maintain the maximum voltage linear output area in the α₃-β₃ frame. For Method II, near four vectors (two middle and two little vectors) have been adopted to improve the robustness of the system on the basis of ensuring the maximum modulation index in the α₁-β₁ frame. In the proposed method, the switching vectors used to comprise the reference vectors in the α₁-β₁ frame and the α₃-β₃ frame are decoupled, which can guarantee the maximum modulation index in the α₁-β₁ frame. The proposed SVPWM has been experimentally verified. Both of these results demonstrate that the proposed SVPWM can regulate non-sinusoidal phase current including the third harmonic component without steady-state error.

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Zhipeng Lin received his B.S. degree in Control Engineering from Jiangsu University, Zhenjiang, China, in 2016, where, he is presently working towards his M.S. degree in Control Science and Engineering. His current research interests include the power-electric control of electric machines and fault tolerant control.

Guohai Liu received his B.S. degree in Electrical Engineering from Jiangsu University, Zhenjiang, China, in 1985; and his M.S. and Ph.D. degrees in Control Engineering from Southeast University, Nanjing, China, in 1988 and 2002, respectively. Since 1988, he has been with Jiangsu University, where he is presently working as a Professor and as the Dean of the School of Electrical Information Engineering. From 2003 to 2004, he was a Visiting Professor in the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, ENG, UK. His current teaching and research interests include electrical machines, motor drives for electric vehicles and intelligent control. He has authored or coauthored over 200 technical papers and 4 textbooks in these areas. He also holds 30 chinese patents. Dr. Liu is a Fellow of the IET

Wenxiang Zhao received his B.S. and M.S. degrees in Electrical Engineering from Jiangsu University, Zhenjiang, China, in 1999 and 2003, respectively; and his Ph.D. degree in Electrical Engineering from Southeast University, Nanjing, China, in 2010. Since 2003, he has been with Jiangsu University, where he is presently working as a Professor in the School of Electrical Information Engineering. From 2008 to 2009, he was a Research Assistant in the Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong, China. From 2013 to 2014, he was a Visiting Professor in the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, ENG, UK. His current research interests include electric machine design, modeling, fault analysis, and intelligent control. He has authored and co-authored over 130 technical papers in these areas.

Qian Chen received his B.S. degree in Electrical Engineering and his Ph.D. degree in Control Engineering from Jiangsu University, Zhenjiang, China, in 2009 and 2015, respectively. Since 2015, he has been with Jiangsu University, where he is presently working as an Associate Professor in the School of Electrical Information Engineering. His current research interests include electric machine design, modeling, fault analysis, and intelligent control.