# Vector Weighting Approach and Vector Space Decoupling Transform in a Novel SVPWM Algorithm for Six-Phase Voltage Source Inverter

Authors: Peng Wu; Lei Yuan; Zhen Zuo; Junyu Wei
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

For six-phase permanent-magnet synchronous motor (PMSM) which has two sets of Y-connected three-phase windings spatially phase shifted by 30 electrical degrees, to increase the utilization ratio of the DC bus voltage, a novel space vector pulse width modulation (SVPWM) algorithm in full modulation range capability based on vector weighted method is proposed in this paper. The basic vector action time of SVPWM method is derived in detail, employing vector space decomposition transformation approach. Compared with the previous algorithm, this strategy is able to overcome the inherent shortcomings of the four-vector SVPWM, and it achieves smooth transitions from linear to overmodulation region. Simulation and experimental analyses demonstrate the effectiveness and feasibility of the proposed strategy.


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## Body

## 1. Introduction

Compared with the pulse width modulation (PWM) control algorithm for conventional three-phase motors, the PWM method for six-phase motors is more flexible [1–6]. At present, carrier-type PWM algorithm, permanent-magnet synchronous motor SVPWM algorithm, and current hysteresis PWM algorithm are three of the PWM algorithms that are commonly used for multiphase two-level voltage source inverter (VSI) [7, 8]. However, the current hysteresis PWM algorithm is unsuitable for high power algorithms. In recent years, carrier-type PWM algorithms and SVPWM algorithms that can be applied to multiphase motor systems have been extensively studied.The PWM algorithm for multiphase inverters aims to eliminate low-order harmonic components as much as possible, which is different from that for the three-phase motor system [9–11]. As these low-order harmonic components could result in the loss of the motor by leading to a large amount of stator harmonic current components in the x-y subspace. In order to increase the utilization ratio of the DC bus voltage, multiphase carrier-type PWM algorithm based on zero-sequence signal injection is applied for multiphase inverter system which is composed of several three-phase windings. In particular, the reference voltage vector can be decomposed into two three-phase inverters sharing a common DC bus voltage when used in practice. After that, the carrier-type PWM algorithm based on dual zero-sequence injection can be used [12, 13]. Although this method is easy for digital realization, it is not an optimal PWM algorithm for multiphase motors as there are fifth and seventh harmonic components in the stator current.As the multiphase AC motors have been widely used, the conventional inverter control algorithm is expected to be optimized for them. At present, many researches are devoted to the implementation of the SVPWM algorithm. One of the multiphase SVPWM algorithms is based on the maximum adjacent two vectors, in which the voltage of the subspace is sinus without considering the influence of the x-y subspace voltage [14]. In this connection, there are more harmonic voltages in the output and a larger fundamental voltage modulation factor can be gotten in this way [15]. While for multiphase AC motors, the SVPWM algorithm will become incredibly complex. And with the increase of the number of motor phases, the complexity of this direct calculation SVPWM algorithm will be greatly increased: sector judgment, vector duration calculation, vector action sequence arrangement, vector duration conversion into switching action time, and other complex issues and the difficulty will be greatly increased, and the performance requirements of the controller will undoubtedly be greatly improved.In order to further improve the DC bus utilization rate, a two-vector and improved four-vector algorithm can be used to increase the modulation factor by injecting the 5th and 7th harmonics of the vector in the x-y subspace. This method only increases the system loss and does not generate torque ripple. But it only reached the maximum linear modulation range. When it is necessary to continue to improve the bus voltage utilization rate, it can be said that the commonly used three-phase overmodulation technology [16, 17] is extended to the six-phase system, and the 11th and 13th harmonics are injected in the α-β subspace, so that the maximum bus voltage utilization can be obtained. However, the torque ripple will be poor.In order to increase the utilization ratio of the DC bus voltage, numerous studies have been done on improving the PWM algorithm for three-phase motor when overmodulation occurs. As for the PWM overmodulation strategies for multiphase motors, the fundamental amplitude of the output reference voltage will rise by using vector space decomposition (VSD) in [10] transform matrix decomposition to get fundamental component of the x-y subspace [18] while the harmonic content of the reference voltage in the linear modulation region also increases in that way.To solve above problems, and to increase the utilization ratio of the DC bus voltage, a novel SVPWM algorithm for six-phase VSI with wide modulation range capability based on vector weighted method is proposed in this paper. The concept of mid-vector presented in [18] is employed to obtain the expression for the time of algorithm of the active space voltage vectors, and using vector weighted method, the overmodulation region is divided into three sections, and the proposed SVPWM algorithm can also allow smooth transition from a linear modulation region to an overmodulation region. The control strategy is verified through simulation analysis.


## 2. Dwell Time Calculation for Six-Phase VSI

The six-phase motor system fed by six-phase VSI is shown in Figure1. There are two sets of Y-coupled three-phase stator windings, which are spatially separated by 30° electrical degrees. Since there are two switch states for each bridge arm, the six-phase inverter has 26 switch states. According to the VSD transformation approach, all of the space vectors can be mapped to the α-β, x-y, and o1-o2 three mutually orthogonal subspaces. The fundamental component and harmonics with the order 12n±1 (n=1, 2, 3,…) are mapped into the α-β subspace. The harmonics with the order 6n±1 (n=1, 3, 5,…) and the harmonics with the order 6n±3 (n=1, 3, 5,…) are mapped into the x-y subspace and the o1-o2 subspace, respectively. The subspaces are divided into 12 sectors.Figure 1
Six-phase converter drive system.To suppress the harmonic current component of stator current, a good four-vector SVPWM method should satisfy that the amplitude of the reference voltage vector in theα-β subspace is maximum, and in the x-y subspace is minimum. In the α-β subspace, any three adjacent voltage vectors can be synthesized into a new vector, called mid-vector, and the mid-vector has a same direction as the vector in the middle position. As shown in Figure 2, three adjacent basic vectorsv1,v2, andv3 are employed to synthesize into a mid-vectorva. Supposing the dwell time of vectorv2 isaTs, the vectorsv1 andv3 will be 0.5(1-a)Ts. The amplitude ofva in α-β subspace and x-y subspace, respectively, are as follows:(1)vmax′=23+16Vdca+321-a(2)vmin′=23-16Vdca-321-awherea is a positive constant.Figure 2
SVPWM method based mid-vector.Similarly, the three adjacent basic vectorsv2, v3,andv4 are employed to synthesize into a mid-vectorvb. In this way, we can synthesize 12 voltage vectors which have the same distribution as the traditional two vector SVPWM algorithm, but the magnitude is different. So the reference voltage vectorvr in α-β subspace can be synthesized by using 12 mid-vectors, and the basic principle is the same as the traditional two vector SVPWM algorithm, the dwell time of mid-vectors Ta and Tb can be calculated according to (5), and vmax must be replaced by the variable vmax′, and the dwell time of vectorsv1, v2, v3,andv4 can be obtained by(3)t1=0.51-aTat2=aTa+0.51-aTbt3=0.51-aTa+aTbt4=0.51-aTbMoreover, the relationship betweenθ and δ is obtained by(4)θ=δ+π12-k-16πwherekis the sector, withk=1,2, 3…12.According to the traditional three-phase SVPWM method, the dwell time of mid-vector can be used by(5)Ta=vrvmaxsin⁡π/6sin2k-112π-δTsTb=vrvmaxsin⁡π/6sinδ-2k-312πTswhereTs is the switch period and vr is the amplitude of the reference voltage vector.In the linear modulation range, the control objective of four-vector SVPWM based on mid-vector method is to reduce the current harmonic component; i.e., the amplitude of reference is maximum in theα-β subspace and minimum in the x-y subspace. For this purpose, (1) is only to be satisfied with vmin′=0, i.e., a=23-3. Equation (2) will change into vmax′=(32-6)/3·Vdc. The dwell time of mid-vector can be obtained by replacing vmax with vmax′, and then the expression of dwell time can be achieved by (3).


## 3. A Novel SVPWM Technology with Wide Modulation Range

### 3.1. Overmodulation Region Division

To facilitate the analysis, the modulating index is defined by(6)m=πvr2Vdcwhen the reference voltage vector is more than this limiting value of vr=2Vdc/3 and the inverter moves into the overmodulation region for 2Vdc/3<vr<2Vdc/π. When the voltage vector is vr=2Vdc/π, the inverter operates in the twelve-step mode, as shown in Figure 3.Figure 3
Linear and overmodulation range.The overmodulation region is further divided into three submodes, the overmodulation region I (0.907<m≤0.977), the overmodulation region II (0.977<m≤0.988), and the overmodulation region III (0.988<m≤1).


### 3.2. SVPWM Based on Vector Weighted Method

When the system is running in different overmodulation region as shown in Figure3, the basic idea of vector weighting method is to obtain a new integrated reference voltage vector by weighting the voltage vectors in different overmodulation regions, with using the concept of modulation coefficient, and use the reference voltage vector to calculate the duty cycle of PWM. In order to facilitate calculation, four voltage vectors are defined in this paper, such as Vsin1, Vsin2, Vdod, and Vtwe. Vsin1 is the voltage vector of the maximum linear modulation ratio when the four vector SVPWM algorithm is used, and the trajectory is a circle with a radius of Vdc/3. Vsin2 is the voltage vector corresponding to the two-vector SVPWM algorithm, and the trajectory is a dodecagon inscribed circle with a radius of 2(3+1)/6·Vdc·cosπ/12. Vdod is the voltage vector that rotates on the contour line of regular dodecagon. Vtwe is the voltage vector of twelve-step wave. The expression of the four voltage vectors will be shown as follows:(7)Vsin1=Vdc3ejδ≈0.577Vdc·ejδ(8)Vsin2=23+1Vdc6cos⁡π12ejδ≈0.622Vdc·ejδ(9)Vdod=23+1Vdc6cos⁡2k-1+1/12π-θcos⁡π12ejδ≈0.622Vdccos⁡2k-1+1/12π-θejδ(10)Vtwe=23+1Vdc6ejkπ/12≈0.644Vdc·ejkπ/12,k-16π≤δ≤k6πwhere δ is the angle between reference voltage vector vr and α axis, the relationship between Vsin1, Vsin2, Vdod, and vr is phase equality, and the amplitude of four-vector is 0.577Vdc, 0.622Vdc, 0.629Vdc, and 0.6366Vdc.


#### 3.2.1. Overmodulation Region I(0.907<m≤0.977)

It can be seen from (1) and (2) that the amplitude vmax′ of fundamental voltage vector will gradually become larger with an increase of a, but the amplitude of harmonic subspace voltage vector will also gradually increase. The modulation method especially will become two-vector SVPWM algorithm. To obtain the reference voltage vector vr, the amplitude need to be satisfied with vr=vmax′cos⁡π/12; it can be obtained as follows:(11)a=12v∗Vdc-23-3The unitary expression of dwell time can be achieved when (11) is substituted into (1), with considering function (3). When the reference voltage vector vr is in the modulation region I, the amplitude of vr will gradually increase to 0.622Vdc, and the value of a will also gradually increase from 23-3 to 1. Thus, the smooth transition from linear modulation region to overmodulation region I is realized.According to the principle of vector weighting method, the overmodulation coefficient of overmodulation I is defined as(12)k1=m-m1m2-m1,0≤k1≤1wherem1=0.907 andm2=0.977. The value k1 will be set to 0 when the vector vr is in linear modulation region, and the value k1 is set to 1 when vr rotates along the inner-circle of the regular dodecagon.Like overmodulation I region, to illustrate the process of adjusting the reference voltage vector, sector 2 shown in Figure4 will be taken as an example, and Vsin1 and Vsin2 will be used to reconstruct the reference voltage vector v∗, i.e.,(13)v∗=1-k1Vsin1+k1Vsin2Figure 4
The principle diagram reference voltage vector in overmodulation section.According to (7) and (8), the expression of v∗ in sector 2 can be obtained by(14)v∗=Vdc31-k1+23+12Vdc24a+321-a·k1The function (14) is substituted into (5), with considering (3), so the dell time of nonzero vector of each sector can be calculated and then get the PWM waveform of the overmodulation I.Figure5 shows the A phase voltage curve, in which Vdc=1V, and modulation ratiom increased from 0.907 to 0.977. As can be seen from the figure, with the gradual increase of modulation ratiom, the output voltage of the A phase voltage has a certain distortion. Whenm is close to 0.977, the reference voltage vector rotates along the inner-circle of the regular dodecagon.Figure 5
The relationship between fundamental amplitude of output phase voltage and m in mode I.


#### 3.2.2. Overmodulation Region II(0.977<m≤0.988)

When the vectorvr is in overmodulation II, the outside of the regular dodecagon cannot be synthesized by the switch vector, the amplitude or phase of vr must be adjusted so that the average value of output voltage for the entire period is equal to the reference voltage. The overmodulation coefficient of over modulation II is defined as(15)k2=m-m2m3-m2wherem3=0.988. In addition, the value ofk2 is set to 0 when voltage vector is located in overmodulation I, and the value of value ofk2 is set to 1 when voltage vector rotates along the regular dodecagon.According to (8) and (9), the expression of v∗ in sector 2 can be obtained by(16)v∗=23+12Vdc241-k2+23+12Vdc24cos⁡π/6-φk2In addition, when the reference voltage vector is located outside the regular dodecagon, an unreasonable situation that the dwell time of the zero vector is less than 0 will happen. So (3) is modified by (17)ti=tit1+t2+t3+t4,i=1,2,3,4t0=0Figure6 shows the A phase voltage curve, in which Vdc=1V, and modulation ratiom increased from 0.974 to 0.988. As can be seen from the diagram, with the gradual increase of the modulation ratiom, the distortion of the output phase voltage is more and more obvious; whenm is close to 0.988, the output voltage vector will rotate along the track of the regular dodecagon.Figure 6
The relationship between fundamental amplitude of output phase voltage and m in mode II.


#### 3.2.3. Overmodulation Region III(0.988<m≤1)

The overmodulation coefficient of overmodulation III is defined as(18)k3=m-m31-m3where the value ofk3 is set to 0 when voltage vector is located in overmodulation II, and the value ofk3 is set to 1 when voltage vector rotates along the twelve-step wave.Like overmodulation II region, to illustrate the process of adjusting the reference voltage vector, sector 2 shown in Figure7 will be taken as an example, and Vdod and Vtwe will be used to reconstruct the reference voltage vector v∗, i.e.,(19)v∗=Vdod1-k3+Vtwek3Figure 7
The principle diagram reference voltage vector in overmodulation section III.Although the introducedVtwe vector makes the phase of reconstructed voltage vector different from the reference voltage vector, it increases the output amplitude of voltage vector. According to (9) and (10), the magnitude of the reconstructed voltage vector on the α-β axis can be obtained as(20)v∗α=23+12cos⁡φ·Vdc24cos⁡π/6-φ1-k3+23+1cos⁡π/4·Vdc6k3(21)v∗β=23+12sin⁡φ·Vdc24cos⁡π/6-φ1-k3+23+1sin⁡π/4·Vdc6k3The amplitude and phase of the reconstructed voltage vector can be obtained by(22)v∗=v∗α2+v∗β2γ=arctan⁡v∗βv∗αFigure8 shows the A phase voltage curve, in which Vdc=1V, and modulation ratiom increased from 0.987 to 1. As can be seen from the diagram, with the gradual increase of the modulation ratiom, the distortion of the output phase voltage is more and more obvious; whenm is close to 1, the output voltage vector will rotate along the track of the regular dodecagon vertex.Figure 8
The relationship between fundamental amplitude of output phase voltage and m in mode III.


## 3.1. Overmodulation Region Division

To facilitate the analysis, the modulating index is defined by(6)m=πvr2Vdcwhen the reference voltage vector is more than this limiting value of vr=2Vdc/3 and the inverter moves into the overmodulation region for 2Vdc/3<vr<2Vdc/π. When the voltage vector is vr=2Vdc/π, the inverter operates in the twelve-step mode, as shown in Figure 3.Figure 3
Linear and overmodulation range.The overmodulation region is further divided into three submodes, the overmodulation region I (0.907<m≤0.977), the overmodulation region II (0.977<m≤0.988), and the overmodulation region III (0.988<m≤1).


## 3.2. SVPWM Based on Vector Weighted Method

When the system is running in different overmodulation region as shown in Figure3, the basic idea of vector weighting method is to obtain a new integrated reference voltage vector by weighting the voltage vectors in different overmodulation regions, with using the concept of modulation coefficient, and use the reference voltage vector to calculate the duty cycle of PWM. In order to facilitate calculation, four voltage vectors are defined in this paper, such as Vsin1, Vsin2, Vdod, and Vtwe. Vsin1 is the voltage vector of the maximum linear modulation ratio when the four vector SVPWM algorithm is used, and the trajectory is a circle with a radius of Vdc/3. Vsin2 is the voltage vector corresponding to the two-vector SVPWM algorithm, and the trajectory is a dodecagon inscribed circle with a radius of 2(3+1)/6·Vdc·cosπ/12. Vdod is the voltage vector that rotates on the contour line of regular dodecagon. Vtwe is the voltage vector of twelve-step wave. The expression of the four voltage vectors will be shown as follows:(7)Vsin1=Vdc3ejδ≈0.577Vdc·ejδ(8)Vsin2=23+1Vdc6cos⁡π12ejδ≈0.622Vdc·ejδ(9)Vdod=23+1Vdc6cos⁡2k-1+1/12π-θcos⁡π12ejδ≈0.622Vdccos⁡2k-1+1/12π-θejδ(10)Vtwe=23+1Vdc6ejkπ/12≈0.644Vdc·ejkπ/12,k-16π≤δ≤k6πwhere δ is the angle between reference voltage vector vr and α axis, the relationship between Vsin1, Vsin2, Vdod, and vr is phase equality, and the amplitude of four-vector is 0.577Vdc, 0.622Vdc, 0.629Vdc, and 0.6366Vdc.


### 3.2.1. Overmodulation Region I(0.907<m≤0.977)

It can be seen from (1) and (2) that the amplitude vmax′ of fundamental voltage vector will gradually become larger with an increase of a, but the amplitude of harmonic subspace voltage vector will also gradually increase. The modulation method especially will become two-vector SVPWM algorithm. To obtain the reference voltage vector vr, the amplitude need to be satisfied with vr=vmax′cos⁡π/12; it can be obtained as follows:(11)a=12v∗Vdc-23-3The unitary expression of dwell time can be achieved when (11) is substituted into (1), with considering function (3). When the reference voltage vector vr is in the modulation region I, the amplitude of vr will gradually increase to 0.622Vdc, and the value of a will also gradually increase from 23-3 to 1. Thus, the smooth transition from linear modulation region to overmodulation region I is realized.According to the principle of vector weighting method, the overmodulation coefficient of overmodulation I is defined as(12)k1=m-m1m2-m1,0≤k1≤1wherem1=0.907 andm2=0.977. The value k1 will be set to 0 when the vector vr is in linear modulation region, and the value k1 is set to 1 when vr rotates along the inner-circle of the regular dodecagon.Like overmodulation I region, to illustrate the process of adjusting the reference voltage vector, sector 2 shown in Figure4 will be taken as an example, and Vsin1 and Vsin2 will be used to reconstruct the reference voltage vector v∗, i.e.,(13)v∗=1-k1Vsin1+k1Vsin2Figure 4
The principle diagram reference voltage vector in overmodulation section.According to (7) and (8), the expression of v∗ in sector 2 can be obtained by(14)v∗=Vdc31-k1+23+12Vdc24a+321-a·k1The function (14) is substituted into (5), with considering (3), so the dell time of nonzero vector of each sector can be calculated and then get the PWM waveform of the overmodulation I.Figure5 shows the A phase voltage curve, in which Vdc=1V, and modulation ratiom increased from 0.907 to 0.977. As can be seen from the figure, with the gradual increase of modulation ratiom, the output voltage of the A phase voltage has a certain distortion. Whenm is close to 0.977, the reference voltage vector rotates along the inner-circle of the regular dodecagon.Figure 5
The relationship between fundamental amplitude of output phase voltage and m in mode I.


### 3.2.2. Overmodulation Region II(0.977<m≤0.988)

When the vectorvr is in overmodulation II, the outside of the regular dodecagon cannot be synthesized by the switch vector, the amplitude or phase of vr must be adjusted so that the average value of output voltage for the entire period is equal to the reference voltage. The overmodulation coefficient of over modulation II is defined as(15)k2=m-m2m3-m2wherem3=0.988. In addition, the value ofk2 is set to 0 when voltage vector is located in overmodulation I, and the value of value ofk2 is set to 1 when voltage vector rotates along the regular dodecagon.According to (8) and (9), the expression of v∗ in sector 2 can be obtained by(16)v∗=23+12Vdc241-k2+23+12Vdc24cos⁡π/6-φk2In addition, when the reference voltage vector is located outside the regular dodecagon, an unreasonable situation that the dwell time of the zero vector is less than 0 will happen. So (3) is modified by (17)ti=tit1+t2+t3+t4,i=1,2,3,4t0=0Figure6 shows the A phase voltage curve, in which Vdc=1V, and modulation ratiom increased from 0.974 to 0.988. As can be seen from the diagram, with the gradual increase of the modulation ratiom, the distortion of the output phase voltage is more and more obvious; whenm is close to 0.988, the output voltage vector will rotate along the track of the regular dodecagon.Figure 6
The relationship between fundamental amplitude of output phase voltage and m in mode II.


### 3.2.3. Overmodulation Region III(0.988<m≤1)

The overmodulation coefficient of overmodulation III is defined as(18)k3=m-m31-m3where the value ofk3 is set to 0 when voltage vector is located in overmodulation II, and the value ofk3 is set to 1 when voltage vector rotates along the twelve-step wave.Like overmodulation II region, to illustrate the process of adjusting the reference voltage vector, sector 2 shown in Figure7 will be taken as an example, and Vdod and Vtwe will be used to reconstruct the reference voltage vector v∗, i.e.,(19)v∗=Vdod1-k3+Vtwek3Figure 7
The principle diagram reference voltage vector in overmodulation section III.Although the introducedVtwe vector makes the phase of reconstructed voltage vector different from the reference voltage vector, it increases the output amplitude of voltage vector. According to (9) and (10), the magnitude of the reconstructed voltage vector on the α-β axis can be obtained as(20)v∗α=23+12cos⁡φ·Vdc24cos⁡π/6-φ1-k3+23+1cos⁡π/4·Vdc6k3(21)v∗β=23+12sin⁡φ·Vdc24cos⁡π/6-φ1-k3+23+1sin⁡π/4·Vdc6k3The amplitude and phase of the reconstructed voltage vector can be obtained by(22)v∗=v∗α2+v∗β2γ=arctan⁡v∗βv∗αFigure8 shows the A phase voltage curve, in which Vdc=1V, and modulation ratiom increased from 0.987 to 1. As can be seen from the diagram, with the gradual increase of the modulation ratiom, the distortion of the output phase voltage is more and more obvious; whenm is close to 1, the output voltage vector will rotate along the track of the regular dodecagon vertex.Figure 8
The relationship between fundamental amplitude of output phase voltage and m in mode III.


## 3.2.1. Overmodulation Region I(0.907<m≤0.977)

It can be seen from (1) and (2) that the amplitude vmax′ of fundamental voltage vector will gradually become larger with an increase of a, but the amplitude of harmonic subspace voltage vector will also gradually increase. The modulation method especially will become two-vector SVPWM algorithm. To obtain the reference voltage vector vr, the amplitude need to be satisfied with vr=vmax′cos⁡π/12; it can be obtained as follows:(11)a=12v∗Vdc-23-3The unitary expression of dwell time can be achieved when (11) is substituted into (1), with considering function (3). When the reference voltage vector vr is in the modulation region I, the amplitude of vr will gradually increase to 0.622Vdc, and the value of a will also gradually increase from 23-3 to 1. Thus, the smooth transition from linear modulation region to overmodulation region I is realized.According to the principle of vector weighting method, the overmodulation coefficient of overmodulation I is defined as(12)k1=m-m1m2-m1,0≤k1≤1wherem1=0.907 andm2=0.977. The value k1 will be set to 0 when the vector vr is in linear modulation region, and the value k1 is set to 1 when vr rotates along the inner-circle of the regular dodecagon.Like overmodulation I region, to illustrate the process of adjusting the reference voltage vector, sector 2 shown in Figure4 will be taken as an example, and Vsin1 and Vsin2 will be used to reconstruct the reference voltage vector v∗, i.e.,(13)v∗=1-k1Vsin1+k1Vsin2Figure 4
The principle diagram reference voltage vector in overmodulation section.According to (7) and (8), the expression of v∗ in sector 2 can be obtained by(14)v∗=Vdc31-k1+23+12Vdc24a+321-a·k1The function (14) is substituted into (5), with considering (3), so the dell time of nonzero vector of each sector can be calculated and then get the PWM waveform of the overmodulation I.Figure5 shows the A phase voltage curve, in which Vdc=1V, and modulation ratiom increased from 0.907 to 0.977. As can be seen from the figure, with the gradual increase of modulation ratiom, the output voltage of the A phase voltage has a certain distortion. Whenm is close to 0.977, the reference voltage vector rotates along the inner-circle of the regular dodecagon.Figure 5
The relationship between fundamental amplitude of output phase voltage and m in mode I.


## 3.2.2. Overmodulation Region II(0.977<m≤0.988)

When the vectorvr is in overmodulation II, the outside of the regular dodecagon cannot be synthesized by the switch vector, the amplitude or phase of vr must be adjusted so that the average value of output voltage for the entire period is equal to the reference voltage. The overmodulation coefficient of over modulation II is defined as(15)k2=m-m2m3-m2wherem3=0.988. In addition, the value ofk2 is set to 0 when voltage vector is located in overmodulation I, and the value of value ofk2 is set to 1 when voltage vector rotates along the regular dodecagon.According to (8) and (9), the expression of v∗ in sector 2 can be obtained by(16)v∗=23+12Vdc241-k2+23+12Vdc24cos⁡π/6-φk2In addition, when the reference voltage vector is located outside the regular dodecagon, an unreasonable situation that the dwell time of the zero vector is less than 0 will happen. So (3) is modified by (17)ti=tit1+t2+t3+t4,i=1,2,3,4t0=0Figure6 shows the A phase voltage curve, in which Vdc=1V, and modulation ratiom increased from 0.974 to 0.988. As can be seen from the diagram, with the gradual increase of the modulation ratiom, the distortion of the output phase voltage is more and more obvious; whenm is close to 0.988, the output voltage vector will rotate along the track of the regular dodecagon.Figure 6
The relationship between fundamental amplitude of output phase voltage and m in mode II.


## 3.2.3. Overmodulation Region III(0.988<m≤1)

The overmodulation coefficient of overmodulation III is defined as(18)k3=m-m31-m3where the value ofk3 is set to 0 when voltage vector is located in overmodulation II, and the value ofk3 is set to 1 when voltage vector rotates along the twelve-step wave.Like overmodulation II region, to illustrate the process of adjusting the reference voltage vector, sector 2 shown in Figure7 will be taken as an example, and Vdod and Vtwe will be used to reconstruct the reference voltage vector v∗, i.e.,(19)v∗=Vdod1-k3+Vtwek3Figure 7
The principle diagram reference voltage vector in overmodulation section III.Although the introducedVtwe vector makes the phase of reconstructed voltage vector different from the reference voltage vector, it increases the output amplitude of voltage vector. According to (9) and (10), the magnitude of the reconstructed voltage vector on the α-β axis can be obtained as(20)v∗α=23+12cos⁡φ·Vdc24cos⁡π/6-φ1-k3+23+1cos⁡π/4·Vdc6k3(21)v∗β=23+12sin⁡φ·Vdc24cos⁡π/6-φ1-k3+23+1sin⁡π/4·Vdc6k3The amplitude and phase of the reconstructed voltage vector can be obtained by(22)v∗=v∗α2+v∗β2γ=arctan⁡v∗βv∗αFigure8 shows the A phase voltage curve, in which Vdc=1V, and modulation ratiom increased from 0.987 to 1. As can be seen from the diagram, with the gradual increase of the modulation ratiom, the distortion of the output phase voltage is more and more obvious; whenm is close to 1, the output voltage vector will rotate along the track of the regular dodecagon vertex.Figure 8
The relationship between fundamental amplitude of output phase voltage and m in mode III.


## 4. Simulation Results Analysis

To verify the validity of the proposed SVPWM algorithm of six-phase VSI with wide modulation range, the simulation model is built in Matlab/Simulink environment. Figure9 shows the phase voltage and phase voltage harmonic spectra in different modulation regions. The voltage is per unit value. When m=0.906, the harmonic voltages vx and vy are zero in x-y subspaces, and the phase voltage is sine wave, THD=0. As the modulation ratio increases, the THD of phase voltage gradually increases, and the low-order harmonic components are mainly from α-β subspace and x-y subspace.Figure 9
Phase voltage and phase voltage harmonic spectra in different modulation regions.
(a)
m=0.906 (b)
m=0.95 (c)
m=0.98 (d)
m=1Figure10 shows the voltage vector of the different modulation ratios in the α-β and x-y subspaces. Among them, the left side of Figures 9(a), 9(b), and 9(c) is α-β subspace, and the right side is x-y subspace. It can be seen from Figure 9, with the increase of the modulation ratiom, the fundamental amplitude of voltage vector on α-β subspace increases gradually, and the modulation ratio ism=0.988, the output voltage vector tracks along the regular dodecagon, and the output voltage vector trajectory rotates along the track of the regular dodecagon vertex with the modulation ratio ofm=1, which verify the correctness of the above theory analysis.Figure 10
The voltage vector ofα-β and x-y subspaces in different modulation regions.
(a)
Overmodulation region I (m=0.974) (b)
Overmodulation region II(m=0.988) (c)
Overmodulation region III(m=1)In addition, although the harmonic component in the x-y subspace is gradually increasing, the utilization ratio of DC voltage is increased to a certain extent, and the distortion rate and harmonic content of phase voltage increase gradually, with the increase of modulation ratio. According to the above analysis, when the modulation ratio is higher than the linear modulation region (m>0.907), modulation strategy completely degenerates into two-vector SVPWM algorithm, the distortion rate of harmonic distribution is applicable to all the two-largest-vector synthesis, and only the harmonic amplitude is different.Figure11 shows the modulation curve of SVPWM method with different modulation ratio; it can be seen that when the modulation ratio ism=1, the modulation wave becomes a square wave signal.Figure 11
The modulated wave with different modulation ratio.To verify the validity of the proposed SVPWM algorithm achieves the smooth transitions from linear to overmodulation region, Figure12 shows the curve of phase voltage, when the modulation ratio ism increased from 0.907 to 1. It can be seen that as the modulation ratio increases stepwise, the phase voltage waveform achieves a smooth transition from linear modulation region to overmodulation region.Figure 12
The modulated wave in different modulation regions.In order to illustrate the superiority of the control algorithm proposed in this paper, Figure13 shows the comparison with the traditional two-vector SVPWM algorithm and the simulation results. As can be seen from Figure 13, the THD content of the proposed control strategy is relatively small. Thus, it is verified that the proposed algorithm has better control performance.Figure 13
The comparison with the traditional two-vector SVPWM algorithm.


## 5. Experimental Results Analysis

To demonstrate the effectiveness of the proposed a novel SVPWM algorithm in full modulation region for six-phase voltage source inverter, the drive operation is examined at experiment platform shown in [19]. Figures 14 and 15 illustrate a 2MW IPMSM driving system and detailed experiment setup; a high-power back-to-back converter system is used to feed the IPMSM system and the parameters of IPMSM are present in Table 1. All three-phase currents and voltages are measured using magnetic current and voltage transducers.Table 1
Parameters of IPMSM prototype.
Rate power [kW] 2180  Rate speed [r/min] 17  Rate frequency [Hz] 8.5  Rotor inertia [kg⋅m2] 16000  Rate phase to phase voltage [V(rms)] 690  Rate current [A] 1960  Number of pole pairs 30  Stator resistance per phase(R) [Ω] 0.0192  d-axis inductance (Ld) [mH] 4  q-axis inductance (Lq) [mH] 5  Permanent magnet flux (Ψf) [Wb] 10.228Figure 14
The overall experiment setup.Figure 15
The detailed experiment setup.The system control algorithms are developed in Matlab/Simulink, followed by implementation on an OPAL RT-Lab (Real-time Digital Simulator) controller board. In this paper, the method of changing the amplitude of a given signal is used to change the modulation ratio m value. The motor is controlled by V/F and the dead time is 10us. The phase voltage waveforms under different modulation degrees are observed by an oscilloscope as shown in Figure16. It can be seen that with different modulation ratios along with the increasing modulation, the modulated wave in the PWM also gradually changes from a sine wave to a square wave signal. This is the same as the simulation result shown in Figure 9, and the phase voltage also transitions from the linear modulation region to the overmodulation region.Figure 16
Phase A voltage experimental waveforms in different modulation index.
(a)
m=0.9 (b)
m=0.98 (c)
m=1U A and UU phase voltage reconstruction waveforms are shown in Figure 17. It can be seen that the experimental waveform is consistent with the simulation waveform shown in Figure 10. Phase voltage UU lags UA 30 electrical angles. As the modulation index m increases, the voltage utilization rate increases gradually, and the phase voltage gradually transitions from a sine wave to a twelve-step wave.Figure 17
Phases A and U voltage experimental waveforms in different modulation index.
(a)
m=0.9 (b)
m=0.98 (c)
m=1


## 6. Conclusion

To improve the utilization ratio of DC bus voltage and restrain the stator current harmonics and torque ripple, a novel SVPWM modulation strategy based on vector space decoupling transform approach and vector weighting method is proposed in this paper. The overmodulation of the six-phase inverter is divided into 3 regions, such as overmodulation I, overmodulation II, and overmodulation III, and the dwell time of the SVPWM algorithm based on vector weighting method is deduced. Simulation and experimental analyses demonstrate the effectiveness and feasibility of the proposed strategy.



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*Source: 8729125-2019-06-13.xml*