Space Vector Modulation Methods with Modified Zero Vector Distribution for Electrical Vehicle Drives with Six-Phase Induction Motor Operating Under Direct Field-Oriented Control

Abstract

This paper presents a Space Vector Modulation (SVM) method with a novel zero vector distribution system for electrical vehicle drives with a six-phase induction motor working under the Direct Field-Oriented Control (DFOC) method. Different SVM methods are described and compared, and a new approach with long vectors only and a special zero vector distribution, that compensates for the third harmonic component is proposed. The DFOC method is described and the influence of the applied modulation method on six-phase motor currents is shown. Results of our experimental studies on the DFOC method are presented and discussed. The proposed modulation method for a six-phase Voltage Source Inverter can be applied in fault-tolerant electrical vehicles.

1. Introduction

Conventional three-phase squirrel-cage induction motors are commonly used in electrical vehicles [1,2] and industrial drive systems [3,4]. However, three-phase motors are not characterized by high operating reliability, since an interruption even in only one phase of the stator winding eliminates the possibility of the further operation of the motor. In recent years there has been a great interest in using squirrel-cage induction motors with more than three stator phases. Motors of this design are called multi-phase induction motors [5,6,7].

These multi-phase motors ensure the greater reliability of drive system operation [7,8]. Greater reliability is connected with the possibility of operating the motor conditionally even if one of the inverter or motor phases is damaged. Moreover, the motor phase currents are smaller compared to a three-phase motor when the same motor power and voltage are considered. Additionally, the higher harmonic components of motor currents and DC-link current are reduced.

Due to these properties, multi-phase machines are used in electric vehicles since they require high reliability [5,9]. Moreover, they can be applied in electrical traction systems [10,11], wind power systems [12,13] and battery chargers [14,15].

Multi-phase motors can have different stator winding designs. In the scientific literature and in practical applications there are examples of multi-phase motors with symmetrical distributions of their phase axes, for example, five-phase [6,16,17], six-phase [6,18,19], seven-phase [20], nine-phase [14] and even fifteen-phase induction motors [21]. Motors with double stator winding [22,23,24] and motors with open-end stator winding [25] have also been analyzed. In this article a six-phase induction motor with symmetrical phase-axis distribution will be analyzed. In six-phase motors it is possible to use a single-star stator winding connection or a double-star stator winding connection, which have been analyzed in various articles [6,8]. In this paper both solutions are considered, but with an emphasis on the single-star connection due to its greater reliability and resistance to stator phase failures. However, it is a more difficult solution for stator voltage modulation methods.

In drive systems with six-phase motors, similarly to three-phase motors, the DFOC [6,8,22], DTC-ST [8,26] and DTC-SVM [20] vector control methods are used. Both of the most commonly used control methods—DFOC and DTC-SVM—require voltage modulation. There are many modulation methods that can be adapted from three-phase modulation principles to multi-phase control schemes, for example, sinusoidal and current tracking modulation [17,22], but vector modulation seems to be the natural choice, because DFOC and DTC-SVM control structures define the refence voltage vector with two components directly. There are only a few articles on the various Space Vector Modulation methods for symmetrical six-phase motors with a common neutral point. They present the known solution for three-phase systems when using only long voltage vectors [18,19,27] and a novel approach using medium vectors only [18,19]. All of them show that when using only long vectors, additional voltage components appear and make the voltage and current significantly distorted. Therefore, the usage of medium vectors only is proposed. However, this makes the voltage that can be obtained limited. Moreover, the mentioned papers lack the analysis of higher harmonics, there are no fundamental frequency amplitude characteristics presented, and they include only simulation test results. Additionally, analysis of the conversion of voltage vectors into duty cycles is missing.

In this paper, particular attention is paid to modulation methods with long vectors, which provide the highest possible voltage and longest linear modulation range. The novelty of this paper is in its Space Vector Modulation algorithm with long voltage vectors only and modified zero vector distribution, considering zero vectors other than the two basic ones. Moreover, the paper focusses on the comparison of modulation methods with vectors of different lengths: long, medium and short. Theoretical analysis, experimental studies and a comparison of four vector modulation methods for a six-phase voltage inverter are conducted. The tests were carried out within a DFOC structure. Detailed characteristics of the fundamental component and low-order higher harmonics are presented as well.

The paper is organized as follows. First, in Section 2 the mathematical model of the six-phase squirrel-cage induction motor is presented. Then, in Section 3 the mathematical model of the six-phase VSI is described. The experimental setup is presented in Section 4. Section 5 deals with the description of the DFOC method. The SVM methods are discussed in Section 6. Section 7 is dedicated to the description of the duty cycle approach. The experimental results are presented and discussed in Section 8 and modulation characteristics are presented and discussed in Section 9. Section 10 is dedicated to the concluding remarks.

2. Mathematical Model of Six-Phase Squirrel-Cage Induction Motor

The mathematical model of a six-phase squirrel-cage induction motor is formulated based on commonly used simplifying assumptions presented in detail in [8,20,22]. The mathematical model is described by a system of differential equations in which the values of the mutual induction coefficients are variable as functions of the rotor rotation angle. In order to eliminate this inconvenience, the transformation of the motor phase variables is carried out using a transformation matrix.

The transformation matrix C transforms the phase variables into the variables denoted by subscripts α and β, expressed in a coordinate frame stationary with respect to the stator winding; subscripts z₁ and z₂, expressed in the additional coordinate system; and zero variables, denoted by subscripts 0₁ and 0₂. The transformation matrix C must be a square matrix whose number of rows and columns is equal to the number of motor phases. It can be formulated in various ways. In this paper the following matrix C is assumed [28]:

$$\left[\begin{array}{c}{u}_{s\alpha }\\ u_{s\beta }\\ u_{sz1}\\ u_{sz2}\\ u_{s01}\\ u_{s02}\end{array}\right]=C\left[\begin{array}{c}{u}_{sA}\\ u_{sB}\\ u_{sC}\\ u_{sD}\\ u_{sE}\\ u_{sF}\end{array}\right]=\sqrt{{\displaystyle \frac{2}{n}}}\left[\begin{array}{cccccc}1& \cos \alpha & \cos 2\alpha & \cos 3\alpha & \cos 4\alpha & \cos 5\alpha \\ 0& \sin \alpha & \sin 2\alpha & \sin 3\alpha & \sin 4\alpha & \sin 5\alpha \\ 1& \cos 2\alpha & \cos 4\alpha & \cos 6\alpha & \cos 8\alpha & \cos 10\alpha \\ 0& \sin 2\alpha & \sin 4\alpha & \sin 6\alpha & \sin 8\alpha & \sin 10\alpha \\ 1/\sqrt{2}& 1/\sqrt{2}& 1/\sqrt{2}& 1/\sqrt{2}& 1/\sqrt{2}& 1/\sqrt{2}\\ 1/\sqrt{2}& -1/\sqrt{2}& 1/\sqrt{2}& -1/\sqrt{2}& 1/\sqrt{2}& -1/\sqrt{2}\end{array}\right]\left[\begin{array}{c}{u}_{sA}\\ u_{sB}\\ u_{sC}\\ u_{sD}\\ u_{sE}\\ u_{sF}\end{array}\right]$$

(1)

where n is the number of motor phases, n = 6; α = 2π/n; u_sA, …, u_sF are motor phase voltages; u_sα and u_sβ are components of the stator voltage vector in the α–β system; u_sz1 and u_sz2 are components of the stator voltage vector in the z₁–z₂ frame; and u_s01 and u_s02 are components of the stator voltage vector in the 0₁–0₂ frame.

The general equations of the six-phase induction motor, expressed in the transformed coordinate systems are presented below [8,23]:

-

The stator and rotor voltage equations in the α–β coordinate system:

$$u_{s\alpha }=R_s{i}_{s\alpha }+p{\psi }_{s\alpha }$$

(2)

$$u_{s\beta }=R_s{i}_{s\beta }+p{\psi }_{s\beta }$$

(3)

$$0=R_r{i}_{r\alpha }+p{\psi }_{r\alpha }+{\omega }_e{\psi }_{r\beta }$$

(4)

$$0=R_r{i}_{r\beta }+p{\psi }_{r\beta }-{\omega }_e{\psi }_{r\alpha }$$

(5)

-

The stator voltage equations in the additional coordinate system z₁–z₂:

$$u_{sz1}=R_s{i}_{sz1}+p{\psi }_{sz1}=R_s{i}_{sz1}+L_{ls}p{i}_{sz1}$$

(6)

$$u_{sz2}=R_s{i}_{sz2}+p{\psi }_{sz2}=R_s{i}_{sz2}+L_{ls}p{i}_{sz2}$$

(7)

-

The stator voltage equations in the 0₁–0₂ coordinate system:

$$u_{s01}=R_s{i}_{s01}+p{\psi }_{s01}=R_s{i}_{s01}+L_{ls}p{i}_{s01}$$

(8)

$$u_{s02}=R_s{i}_{s02}+p{\psi }_{s02}=R_s{i}_{s02}+L_{ls}p{i}_{s02}$$

(9)

-

The equation of the electromagnetic torque:

$$T_e={\displaystyle \frac{n}{2}}{p}_p{\displaystyle \frac{L_m}{L_r}}\left({\psi }_{r\alpha }{i}_{s\beta }-{\psi }_{r\beta }{i}_{s\alpha }\right)$$

(10)

where p is the Laplace operator (derivative); i_sα and i_sβ are components of the stator current vector in the α–β system; i_rα and i_rβ are components of the rotor current vector in the α–β system; ψ_sα, ψ_sβ, ψ_rα and ψ_rβ are components of the stator and rotor flux vector in the α–β coordinate system; i_sz1 and i_sz2 are components of the stator current vector in the z₁–z₂ frame; i_s01 are i_s02 components of the stator current in the 0₁–0₂ frame; ψ_sz1 and ψ_sz2 are components of the stator flux vector in the z₁–z₂ frame; ψ_s01 are ψ_s02 are components of the stator flux vector in the 0₁–0₂ frame; ω_e is the electrical angular speed of the rotor; T_e is the electromagnetic torque; R_s and R_r are the stator and rotor phase resistance; L_m is the magnetization inductance; L_r = L_m + L_rl gives the total inductance of the rotor winding; L_s = L_m + L_sl gives the total inductance of the stator winding; L_sl and L_rl are the stator and rotor leakage inductance; and p_p is the number of pole pairs.

3. Mathematical Model of Six-Phase Voltage Source Inverter

A block diagram of a six-phase Voltage Source Inverter (VSI) is shown in Figure 1. Figure 1a shows the converter connected to a six-phase induction motor stator winding using the single-star topology when the neutral point is common for all phases. Figure 1b shows the double-star topology with two neutral points, N₁ and N₂.

Depending on the winding connection type, shown in Figure 1, the phase voltages vary as shown in (11)—the matrix’s left part indicates the single-star topology, while the right part indicates the double-star connection [11]. This difference will be visible in the voltage vectors generated.

$$\left[\begin{array}{c}{u}_{sA}\\ u_{sB}\\ u_{sC}\\ u_{sD}\\ u_{sE}\\ u_{sF}\end{array}\right]=\left[\begin{array}{c}{u}_{AN}\\ u_{BN}\\ u_{CN}\\ u_{DN}\\ u_{EN}\\ u_{FN}\end{array}\right]={\displaystyle \frac{u_d}{6}}\left[\begin{array}{cccccc}5& -1& -1& -1& -1& -1\\ -1& 5& -1& -1& -1& -1\\ -1& -1& 5& -1& -1& -1\\ -1& -1& -1& 5& -1& -1\\ -1& -1& -1& -1& 5& -1\\ -1& -1& -1& -1& -1& 5\end{array}\right]\left[\begin{array}{c}{S}_A\\ S_B\\ S_C\\ S_D\\ S_E\\ S_F\end{array}\right]\left[\begin{array}{c}{u}_{sA}\\ u_{sB}\\ u_{sC}\\ u_{sD}\\ u_{sE}\\ u_{sF}\end{array}\right]=\left[\begin{array}{c}{u}_{A{N}_1}\\ u_{B{N}_2}\\ u_{C{N}_1}\\ u_{D{N}_2}\\ u_{E{N}_1}\\ u_{F{N}_2}\end{array}\right]={\displaystyle \frac{u_d}{3}}\left[\begin{array}{cccccc}2& 0& -1& 0& -1& 0\\ 0& 2& 0& -1& 0& -1\\ -1& 0& 2& 0& -1& 0\\ 0& -1& 0& 2& 0& -1\\ -1& 0& -1& 0& 2& 0\\ 0& -1& 0& -1& 0& 2\end{array}\right]\left[\begin{array}{c}{S}_A\\ S_B\\ S_C\\ S_D\\ S_E\\ S_F\end{array}\right]$$

(11)

where S_A…F are the switching signals of the inverter legs; S = 1 when the upper switch of the inverter leg is on and S = 0 when the lower switch is on; and u_d is the voltage of the DC bus of the inverter.

The six-phase VSI generates a system of six-phase voltages on the AC side, which can be represented by the corresponding stator voltage vectors in selected coordinate systems, calculated with (1). Figure 2 shows the sets of stator voltage vectors generated by the six-phase VSI for different combinations of conduction states of the inverters power electronics switches [18,19,28]. The number of possible states is equal to 2ⁿ = 64. Figure 2a shows voltage vectors in the α–β coordinate system, Figure 2b shows the voltage vectors in the additional z₁–z₂ coordinate system and Figure 2c shows the 0₁–0₂ coordinate system. In the figures, the numbering of the individual voltage vectors by consecutive numbers in the decimal system is adopted. Each decimal number specifying the voltage vector number can be converted into a six-position number in the binary system. The consecutive bits of the binary number determine the logical values corresponding to the states of the individual branch switches of the VSI.

As can be seen in Figure 2a, the generated vectors in the α–β frame amplitudes can be long (marked in red), medium (marked in blue), short (green) or zero (black). All the lengths of the vectors are gathered in

Table 1. Amplitudes of voltage vectors in different coordinate frames.

		Long Vectors	Medium Vectors	Short Vectors	Zero Vectors
$\alpha -\beta$		${\displaystyle \frac{2\sqrt{3}}{3}}{u}_d$	$u_d$	${\displaystyle \frac{\sqrt{3}}{3}}{u}_d$	0
$z_1-z_2$		0	${\displaystyle \frac{\sqrt{3}}{3}}{u}_d$	${\displaystyle \frac{\sqrt{3}}{3}}{u}_d$$, u_d$	$0, {\displaystyle \frac{2\sqrt{3}}{3}}{u}_d$
$0_1-0_2$	single star	${\displaystyle \frac{\sqrt{3}}{3\sqrt{2}}}{u}_d$	0	${\displaystyle \frac{\sqrt{3}}{3\sqrt{2}}}{u}_d$$,{\displaystyle \frac{2\sqrt{3}}{3\sqrt{2}}}{u}_d$	$0, {\displaystyle \frac{\sqrt{3}}{\sqrt{2}}}{u}_d$
	double star	0	0	0	0

.

All the long vectors have zero z₁–z₂ components but non-zero short amplitudes in the 0₁–0₂ frame. On the other hand, the medium vectors have short amplitudes in the z₁–z₂ frame, but zero 0₁–0₂ components. The short vectors have short amplitudes in the z₁–z₂ frame but can have short or medium amplitudes in the 0₁–0₂ frame. Zero vectors in the α–β frame can have zero, medium or long amplitudes in the z₁–z₂ frame. Only two vectors are zero in all frames: 0 and 63. Only two more zero vectors also have zero amplitudes in the z₁–z₂ frame: 21 and 42. However, they are the only vectors with long amplitudes in the 0₁–0₂ frame.

The difference between the single-star and double-star topology is simple—in the case of the double-star connection, all 0₁–0₂ frame vectors disappear, while the rest of the vectors is invariant. Thus, the voltage modulation is more difficult in the case of the single-star topology, therefore this case is analyzed further in the paper.

According to (10) the electromagnetic torque in multi-phase motors is generated only with the u_sαβ vector. However, the action of the stator voltage vectors u_sz1z2 and u_s0102 should be considered in the analysis, since these voltage vectors force the flow of additional phase current components in the stator winding. These components can cause an increase in the amplitudes of the stator phase currents and thus increase the electrical power losses.

4. Experimental Setup

The experimental setup was equipped with a six-phase induction motor of 1.5 kW, specially rewound from a 3-phase Besel Sh 90-4L industrial motor (Brzeg, Poland). Its parameters are given in Appendix A.

The motor was supplied from a specially designed 6-phase IGBT-based VSI. Each switch of the inverter was controlled with fiber optics so that galvanic isolation was ensured. The hardware dead-time was ensured by two IXYS modules. In order to generate the load torque, another, wound-type induction motor (Indukta, Indukta SZUe 24b, Cieszyn, Poland) was used. The rapid prototyping unit NI PXIe-1071 with an NI PXIe-8840 Quad Core RT processor and an FPGA NI PXI-7851R R Series card were used. NI VeriStand 2021 R3 and MATLAB-Simulink 2020b software were used to prototype and verify the control structure and modulation methods, as well as to acquire the experimental data.

The speed was measured using an incremental encoder from Kübler 8.5020.8844.3600 with 3600 imp/rev. The DC voltage and 6-phase currents were measured by LEM transducers. A block diagram of the experimental setup is presented in Figure 3.

5. Direct Field-Oriented Control System

The Direct Field-Oriented Control (DFOC) method for six-phase squirrel-cage induction motors is considered in [6,8,22]. A diagram of the control system is shown in Figure 4.

Two outer control loops with PI controllers are applied in this control structure: the control loop of the motor speed and the control loop for the magnitude of the rotor flux vector. The controller of the angular speed defines the torque-producing current (the y-axis component of the stator current vector). The rotor flux controller defines the flux-producing current (the x-axis component of the stator current vector). Two inner current control loops are responsible for ensuring that the reference values of the x and y current components are equal to their measured equivalents. Reference values of the stator voltage vector components are transformed to the α–β coordinate system using the inverse Park transformation and further used by the Space Vector Modulation (SVM) block. The SVM determines the switching states of the six-phase VSI. An estimator of the rotor flux vector, based on the stator phase currents and motor speed, is applied in this control system.

The experimental studies of the DFOC of the six-phase induction motor included rapid changes in the reference motor speed (Figure 5). It can be stated that the real speed follows the reference one with a small overregulation and without any steady-state errors. Thus, the analyzed method provides fast and accurate control of the motor speed (Figure 5a).

The magnitude of the rotor flux vector reaches the set value in a very short time, and during the speed changes the rotor flux vector maintains its value at a constant level. This is shown in Figure 5b. The rotor flux amplitude is controlled by the x-axis stator current component. It is constant during the control process as well (Figure 5d).

In order to change speed rapidly, the y-axis stator current component reaches high values during speed reversals, as shown in Figure 5e. Its changes correspond to changes in motor phase currents (Figure 5c).

6. Space Vector Modulation Methods

6.1. Classical Space Vector Modulation Method (Only Long Vectors and Two Different Zero Vectors)

Since the long vectors (7, 14, 28, 35, 49, 56) are identical to those in the case of a three-phase system, the same SVM strategy could be used in the case of a six-phase system, assuming traditional zero vector (0, 63) usage [18,19,27]. However, this can be justified only when the double-star winding topology is used. Otherwise, 0₁–0₂ vector components appear and low-order higher harmonics are generated in both the voltage and phase currents (as will be shown in the following section). It should be noted that when using only long vectors and these two zero vectors the z₁–z₂ components are always eliminated.

In this case the reference stator voltage vector ${\underset{¯}{u}}_s^{ref}$ is synthesized using only two long voltage vectors, located in the corresponding sector, and two zero voltage vectors. In the modulation algorithm, six sectors are defined, with the angles between neighboring vectors equal to π/3. An example of SVM operation is shown in Figure 6. Sector number 1 is presented in Figure 6a; hence, the long vectors 49 and 56 are used. The reference vector is obtained by applying vector 49 for time τ₄₉ and vector 56 for time τ₅₆. The remaining time must be divided into zero vectors. In most cases the symmetrical approach is applied, and the vectors are distributed to half of the switching time T_s each.

In the presented example for sector 1, the synthesis of the reference vector can be described by the following equation:

$${\underset{¯}{u}}_s^{ref}={\underset{¯}{u}}_0{\tau }_0+{\underset{¯}{u}}_{49}{\tau }_{49}+{\underset{¯}{u}}_{56}{\tau }_{56}+{\underset{¯}{u}}_{63}{\tau }_{63}$$

(12)

Thus, the times are given by

$$\begin{array}{l}{\tau }_{49}=\sqrt{3}{\displaystyle \frac{\left|{\underset{¯}{u}}_s^{ref}\right|}{u_d}}\sin \left(\pi /3-\vartheta \right)\\ {\tau }_{56}=\sqrt{3}{\displaystyle \frac{\left|{\underset{¯}{u}}_s^{ref}\right|}{u_d}}\sin \left(\vartheta \right)\\ {\tau }_0={\tau }_{63}=\left(1-{\tau }_{49}-{\tau }_{56}\right)/2\end{array}$$

(13)

where τ = t/T_s is the normalized time and ϑ is the angle of the reference stator voltage vector towards vector ${\underset{¯}{u}}_{49}$. The example (13) is presented for sector 1. However, the remaining voltage sectors are calculated in a similar manner.

As was mentioned before, the z₁–z₂ components are zero, since only long vectors are used. Figure 6b shows the 0₁–0₂ components of the generated stator voltage when the single-star topology is used. Although vectors 49 and 56 have the same amplitudes and opposite directions, the components are not balanced over the switching period since their times are different. In the case of the double-star topology, the components are zero.

6.2. Vector Modulation Method for Six-Phase VSI Using Medium Vectors

The drawback of the SVM method adapted from three-phase systems, described in the previous section, is the generation of low-order higher harmonics due to the non-balanced 0₁–0₂ components. To minimize their content in the output voltage, the usage of only medium vectors (48, 57, 24, 60, 12, 30, 6, 15, 3, 39, 33, 51) is proposed in [18,19]. The synthesis of the reference voltage in the α–β frame is like the traditional SVM described in Section 6.1 and the number of the voltage sectors is similar, equal to six. The angle of each sector is the same, but the sectors are distributed in a different way. For example, sector 1 is shown in Figure 7a, and it extends from an angle of 30 degrees to 90 degrees. The zero vectors used are also the same, 0 and 63.

Using medium vectors only, the z₁–z₂ components are present in the voltage, as shown in Figure 2b. Therefore, a special sequence must be defined; i.e., the two vectors with the same α–β coordinates must be applied for the same time to balance the z₁–z₂ components, as shown in Figure 7b. In this example vector ${\tau }_{48}{\underset{¯}{u}}_{48}$ is compensated by ${\tau }_{57}{\underset{¯}{u}}_{57}$ and ${\tau }_{24}{\underset{¯}{u}}_{24}$ is compensated by ${\tau }_{60}{\underset{¯}{u}}_{60}$.

Thus, in this modulation method the required stator voltage vector is synthesized using four medium voltage vectors located in the corresponding sector and two zero voltage vectors. The selected voltage vectors used to implement the SVM algorithm in sector 1 are shown in Figure 7.

The modulation algorithm is described by the following equation:

$${\underset{¯}{u}}_s^{ref}={\underset{¯}{u}}_0{\tau }_0+{\underset{¯}{u}}_{24}{\tau }_{24}+{\underset{¯}{u}}_{60}{\tau }_{60}+{\underset{¯}{u}}_{48}{\tau }_{48}+{\underset{¯}{u}}_{57}{\tau }_{57}+{\underset{¯}{u}}_{63}{\tau }_{63}$$

(14)

where times are calculated as

$$\begin{array}{l}{\tau }_{24}={\tau }_{60}={\displaystyle \frac{\sqrt{3}}{3}}{\displaystyle \frac{\left|{\underset{¯}{u}}_s^{ref}\right|}{u_d}}\sin \left(\pi /3-\vartheta \right)\\ {\tau }_{48}={\tau }_{57}={\displaystyle \frac{\sqrt{3}}{3}}{\displaystyle \frac{\left|{\underset{¯}{u}}_s^{ref}\right|}{u_d}}\sin \left(\vartheta \right)\\ {\tau }_0={\tau }_{63}=\left(1-{\tau }_{24}-{\tau }_{60}-{\tau }_{48}-{\tau }_{57}\right)/2\end{array}$$

(15)

The 0₁–0₂ components are eliminated if only medium voltage vectors are used, as is shown in Figure 2c.

6.3. Modified Space Vector Modulation Method Using Long and Four Zero Voltage Vectors

In this paper we propose modifying classical SVM with only long vectors by using a different distribution of the zero vectors. In a traditional approach only two vectors (0, 63) are applied, while in this new approach two more zero vectors are applied (21, 42). Only these four zero vectors guarantee that the α–β and z₁–z₂ components are zero.

The reference voltage in α–β is synthetized with the same long vectors as described in Section 6.1 and Figure 6a. The voltage components in z₁–z₂ frame are always zero, as shown in Figure 2b. However, the 0₁–0₂ components appear as mentioned in Section 6.1. In order to compensate for the 0₁–0₂ components, two additional zero vectors are applied. The application of this idea to sector 1 is shown in Figure 8—vector ${\tau }_{49}{\underset{¯}{u}}_{49}$ is geometrically opposite to vector ${\tau }_{21}{\underset{¯}{u}}_{21}$ (similarly, vector ${\tau }_{56}{\underset{¯}{u}}_{56}$ should be compensated by ${\tau }_{42}{\underset{¯}{u}}_{42}$). Since the amplitude of the ${\underset{¯}{u}}_{21}$ vector is three times longer than the amplitude of the ${\underset{¯}{u}}_{49}$ vector in the 0₁–0₂ frame, the following zero voltage vector distribution is defined:

$$\begin{array}{l}{\tau }_{21}={\tau }_{49}/3\\ {\tau }_{42}={\tau }_{56}/3\\ {\tau }_0={\tau }_{63}=\left(1-{\tau }_{21}-{\tau }_{42}\right)/2\end{array}$$

(16)

The proposed modulation algorithm is described by the following formula:

$${\underset{¯}{u}}_s^{ref}={\underset{¯}{u}}_0{\tau }_0+{\underset{¯}{u}}_{21}{\tau }_{21}+{\underset{¯}{u}}_{49}{\tau }_{49}+{\underset{¯}{u}}_{56}{\tau }_{56}+{\underset{¯}{u}}_{42}{\tau }_{42}+{\underset{¯}{u}}_{63}{\tau }_{63}$$

(17)

When the reference voltage value increases, the required times of zero vector (16) can be longer than the remaining time in the T_s period reserved for zero vector ${\tau }_{zero}$. Thus, the new zero vectors can no longer be large enough to compensate for the long vectors in 0₁–0₂ frame. In this case, i.e., when ${\tau }_{21}+{\tau }_{42}>{\tau }_{zero}$, it is proposed to distribute the times in a symmetrical way:

$$\begin{array}{l}{\tau }_{21}={\tau }_{21}/\left({\tau }_{21}+{\tau }_{42}\right)\cdot {\tau }_{zero}\\ {\tau }_{42}={\tau }_{42}/\left({\tau }_{21}+{\tau }_{42}\right)\cdot {\tau }_{zero}\\ {\tau }_0={\tau }_{63}=0\end{array}$$

(18)

6.4. Space Vector Modulation Method Using Short and Zero Voltage Vectors

If the reference voltage vector becomes short, intuitively, it can be assumed that the usage of short vectors can provide better-quality motor voltage and currents in terms of the generated higher harmonics. Therefore, this opportunity is evaluated in this section.

There are a lot of short vectors to be selected for each of the six possible sectors. However, to eliminate the z₁–z₂ components in voltage and current, the vectors should be selected in pairs. This means, for example, that in sector 1, vectors 17 and 32 could be selected to create the α component of the reference vector, while vectors 16 and 40 could create the β component. This approach, however, favors vectors containing zeros in a binary convention, and lower inverter switches would be mainly used. Therefore, in this paper we elected to use not four but eight short vectors, as shown in Figure 9a. In such an algorithm the switches are utilized in a balanced manner.

The eight vectors are used symmetrically, i.e., the normalized time of each vector is the same:

$$\begin{array}{l}{\tau }_{16}={\tau }_{40}={\tau }_{58}={\tau }_{61}={\displaystyle \frac{\sqrt{3}}{2}}{\displaystyle \frac{\left|{\underset{¯}{u}}_s^{ref}\right|}{u_d}}\sin \left(\pi /3-\vartheta \right)\\ {\tau }_{17}={\tau }_{32}={\tau }_{53}={\tau }_{59}={\displaystyle \frac{\sqrt{3}}{2}}{\displaystyle \frac{\left|{\underset{¯}{u}}_s^{ref}\right|}{u_d}}\sin \left(\vartheta \right)\\ {\tau }_0={\tau }_{63}=\left(1-{\tau }_{16}-{\tau }_{40}-{\tau }_{58}-{\tau }_{61}-{\tau }_{17}-{\tau }_{32}-{\tau }_{53}-{\tau }_{59}\right)/2\end{array}$$

(19)

The modulation can be described using the following formula:

$${\underset{¯}{u}}_s^{ref}={\underset{¯}{u}}_0{\tau }_0+{\underset{¯}{u}}_{17}{\tau }_{17}+{\underset{¯}{u}}_{32}{\tau }_{32}+{\underset{¯}{u}}_{53}{\tau }_{53}+{\underset{¯}{u}}_{59}{\tau }_{59}+{\underset{¯}{u}}_{16}{\tau }_{16}+{\underset{¯}{u}}_{40}{\tau }_{40}+{\underset{¯}{u}}_{58}{\tau }_{58}+{\underset{¯}{u}}_{61}{\tau }_{61}+{\underset{¯}{u}}_{63}{\tau }_{63}$$

(20)

7. Duty Cycles

To minimize the commutation losses, only one switching on and one switching off transition should take place in each phase during one switching period of the SVM. Moreover, this approach is natural for modern microcontrollers applied in the industry—they require the definition of their duty cycle in the range of 0 to 100%. The following formula can be used to calculate the duty cycle in the case of the classical SVM method with long vectors only and two zero vectors (Section 6.1):

$$\left[\begin{array}{c}{d}_A\\ d_B\\ d_C\\ d_D\\ d_E\\ d_F\end{array}\right]=\left[\begin{array}{cccc}{U}_0\left(1\right)& U_{49}(1)& U_{56}(1)& U_{63}(1)\\ U_0\left(2\right)& U_{49}(2)& U_{56}(2)& U_{63}(2)\\ U_0\left(3\right)& U_{49}(3)& U_{56}(3)& U_{63}(3)\\ U_0\left(4\right)& U_{49}(4)& U_{56}(4)& U_{63}(4)\\ U_0\left(5\right)& U_{49}(5)& U_{56}(5)& U_{63}(5)\\ U_0\left(6\right)& U_{49}(6)& U_{56}(6)& U_{63}(6)\end{array}\right]\left[\begin{array}{c}{\tau }_0\\ {\tau }_{49}\\ {\tau }_{56}\\ {\tau }_{63}\end{array}\right]$$

(21)

where U_Y(k) is the binary component of the vector which can be equal to 0 or 1. The duty cycle of the remaining methods can be calculated using the following generalized formula:

$$d_X={\displaystyle \sum _{k=1}^{k_Y}{U}_Y(X){\tau }_Y}$$

(22)

Thus, the duty cycle of phase X is equal to the sum of the vector binary components multiplied by their corresponding normalized times.

It must be noted here that this approach does not modify the average phase voltage over the switching period. However, it can rearrange and modify the voltage vectors obtained. This is shown in Figure 10, where the SVM with medium vectors only is selected as an example. It can be seen that in the case of the default approach (Figure 10a), only medium vectors are used—48, 24, 57 and 60—alongside two zero vectors, 0 and 63. If the duty cycles are recalculated (21), two medium vectors remain unchanged—24 and 60—but two others appear: 16 and 61. Additionally, long vector 56 appears (Figure 10b). Thus, if long vectors appear, the modulation characteristics must be different in the overmodulation region between the two approaches.

The approach shown in Figure 10b, as it is a microcontroller-based solution, will be used further in this paper.

8. Experimental Test Results

8.1. Duty Cycles in Experimental Tests

The comparison between the duty cycles obtained for all four analyzed methods is shown in Figure 11. It can be seen that in the cases of the traditional SVM method adapted from three-phase systems (Figure 11a) and the SVM method with short vectors only (Figure 11d), a similar characteristic shape is visible. This reduces the amplitude of the duty cycle compared to a sinusoidal signal but also adds a third harmonic into the voltage and motor currents inherently, as will be shown in the following section. However, in the case of the proposed SVM (Figure 11b) and the SVM with medium vectors only (Figure 11d), the duty cycles are circular, similar to sinusoidal modulation.

Table 2. 0₁–0₂ component compensation level in function of modulation index for proposed modulation method.

Region	Modulation Index		01–02 Component Compensation	Total Time for Zero Vectors	${\mathit{\tau }}_{21} + {\mathit{\tau }}_{42} > {\mathit{\tau }}_{\mathit{z}\mathit{e}\mathit{r}\mathit{o}}$
	from	to
I	0	$\sqrt{3}/2$	always full compensation	never zero	never
II	$\sqrt{3}/2$	1	sometimes full compensation/sometimes partial compensation	never zero	sometimes
III	1	$2\sqrt{3}/3$	always partial compensation	never zero	always
IV	$2\sqrt{3}/3$	$4/3$	sometimes partial compensation/sometimes no compensation	sometimes zero	always
V	$4/3$	∞	no compensation	always zero	always

and Figure 12 summarize the operation of the proposed modulation method for different modulation indices. It can be seen that if the index is smaller than $\sqrt{3}/2$, the shape of the duty cycle is sinusoidal (Figure 12a) and the sum of τ₂₁ and τ₄₂ is always lower than τ_zero. This indicates that the 0₁–0₂ frame components are fully compensated.

If the modulation index is between $\sqrt{3}/2$ and 1, full compensation is not possible and the required time for vectors 21 and 42 is sometimes longer than the available time for all zero vectors. The shape of the duty cycles is no longer sinusoidal (Figure 12b). However, the proposed SVM method tends to limit the duty cycle between 0% and 100%. This preserves the linear dependence between the reference and the obtained voltage. If the modulation index is between 1 and $2\sqrt{3}/3$, the compensation is always partial, but τ_zero is never zero (Figure 12c). Between $2\sqrt{3}/3$ and 4/3, the time for the zero vectors sometimes becomes zero (Figure 12e), and above 4/3 it is always zero (Figure 12f), so the compensation of the 0₁–0₂ components is not possible at all.

8.2. Effect of Modulation Type on Stator Currents of Induction Motor Operating Under DFOC

The phase currents of a six-phase induction motor operating under DFOC are shown in Figure 13. The waveforms were obtained in the same conditions shown in Figure 5. It can be seen that the amplitude of the currents is significantly increased when classical SVM with long vectors (Figure 13a) or SVM with short vectors (Figure 13d) only are used, compared to the remaining two methods presented (Figure 13b,c). These increases in amplitude are connected with an additional third harmonic component, especially visible in Figure 14. The figure presents a zoom in on the phase currents—the very poor quality of the currents when no compensation is applied should be easily noted.

Figure 15 presents the currents in the stationary α–β reference frame. All the waveforms are almost identical as they are defined by the control structure in the same conditions of motor speed and load torque. Figure 16 proves that all the presented methods compensate for the z₁–z₂ components of the motor current in the analyzed motor operating conditions. Finally, Figure 17 shows that the 0₁–0₂ components are not compensated in the cases of the traditional SVM method and the SVM method with short vectors only. However, they are significantly smaller if the SVM method with medium vectors only or the proposed modulation method are considered.

9. Modulation Characteristics

Figure 18 shows the odd harmonics of the phase voltage of the inverter, starting from the fundamental component (Figure 18a) up to the ninth component (Figure 18e). All components are normalized to the maximum value of the first harmonic. The characteristics are shown for four different analyzed SVM methods with short, medium, long and modified long vectors, each one in two different versions: the default one and the one based on duty cycle recalculation. The results were prepared with the MATLAB-Simulink models with the Euler solver (ode1) and a fixed-step size equal to 1 × 10⁻⁷ s. The switching frequency was set to 5 kHz, while the fundamental frequency of the reference voltage was equal to 60 Hz.

Figure 18a shows that in a linear range (region I and region II), e.g., when the modulation index is lower than one, the first harmonic of the voltage increases in a linear way. This means that the reference voltage is the same as the output voltage. In this range, all the characteristics, except the one for default short vectors, are identical. If the classical SVM or the proposed SVM with long vectors and modified zero vectors are applied, the linear range is extended up to the end of region III. However, in the case of the SMV with medium vectors only, the characteristics become strongly nonlinear and enter the range of overmodulation (region III and higher). Remarkably, the default SVM with medium vectors only is saturated and never reaches the maximum possible VSI voltage. Its duty cycle version is not limited, though. This occurs because the long vectors are produced then, as well. It is clear that the linear range of the default SVM with short vectors only is limited to 50% of the linear range of the SVM with long vectors. However, the duty-cycle-based version of the SVM with short vectors is similar to other SVM methods regarding the fundamental component of the voltage.

Figure 18b presents the third harmonic component, which is visible here only because the motor is connected in the single-star topology. Otherwise, it would be zero. The classical SVM method with long vectors as well as SVM with short vectors both produce the largest third harmonic contents, starting as early as in region I and up to region V. When the proposed modification of the SVM is applied, this harmonic component is eliminated up to the beginning of region II. Then, it increases and reaches a similar level as in traditional SVM methods around the transition between region III and region IV. The limited voltage value of the default SVM with medium voltage is rewarded with a completely zero value of the third harmonic component (green line in Figure 18b). Recalculating the duty cycles in the case of this method introduces this harmonic component starting from region III, i.e., when the modulation index is larger than 1.

The default SVM with medium vectors only, that is characterized with the lowest third harmonic component, introduces a large fifth harmonic component starting from when the modulation index is equal to one (Figure 18c). In this case recalculating the duty cycle helps to reduce the fifth harmonic. There are no differences between the classical and the proposed SVM methods when this component is considered. Moreover, SVM with short vectors in both versions, especially in the default version, produces the largest fifth harmonic component of all the analyzed methods.

The SVM method with short vectors only produces the largest seventh harmonic component. There are no differences between the SVM methods with long vectors when the seventh harmonic component is considered (Figure 18d). However, the default versions produce much larger contents of this component in the output voltage than the duty-cycle-based versions. Similarly to the third harmonic, the classical SVM method with long vectors produces the highest content of the ninth harmonic, starting from region I, while default SVM with medium vectors only produces no harmonic content, as shown in Figure 18e.

10. Conclusions

This paper presents a comprehensive comparison of Space Vector Modulation methods applied to control a six-phase Voltage Source Inverter supplying a six-phase induction motor. Four different possibilities are considered: SVM with short vectors only, SVM with medium vectors, classical SVM with long vectors only and the one proposed in this study—an SVM method with long vectors only and a modified distribution of zero vectors. Additionally, the differences between the default approach and duty-cycle-based modulation are presented. The influence of the analyzed methods on Direct Field-Oriented Control is presented in terms of the motor phase currents and all six components in stationary reference frames. Simulation and experimental test results are included as well. A detailed low-order higher harmonics analysis is shown and compared for each of the SVM methods. The paper describes the difference between the single- and double-star topologies of motor windings, but focuses on the former, as it is more reliable, more difficult in terms of modulation and produces higher harmonics.

It is proved that the classical SVM method with long vectors only, adapted from three-phase systems, cannot be applied in six-phase systems with single neutral point, as it produces third harmonic components in both voltage and current over the entire range of voltage variation. The same is true for SVM with short vectors only. Additionally, the voltage that can be achieved in the case of the second method is constrained. The only method that does not generate any third (and its multiple) harmonic components is the default version of the SVM method with medium vectors. However, its amplitude is inherently limited in the overmodulation region. On the other hand, it produces the largest fifth harmonic among all the analyzed methods. A reasonable solution is the proposed modulation method—SVM with long vectors and a modified distribution of zero voltage vectors, applied to compensate the components responsible for third harmonic generation. This method is characterized by a longer linear operating range compared to the modulation with medium vectors only, no third harmonic being produced in the linear range and relatively low harmonic content.