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# Control and Performance Analysis of an Indirect Matrix Converter Fed Triple Star Induction Motor

Ahmed Azib, Adel Oubelaid, Djamel Ziane, Mohamed Fouad Benkhoris

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Ahmed Azib, Adel Oubelaid, Djamel Ziane, Mohamed Fouad Benkhoris. Control and Performance Analysis of an Indirect Matrix Converter Fed Triple Star Induction Motor. Periodica Polytechnica Electrical Engineering and Computer Science, 2024, 10.3311/PPee.22832. http://04594037

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# Control and Performance Analysis of an Indirect Matrix Converter Fed Triple Star Induction Motor

Ahmed Azib<fnref n="1" />*, Adel Oubelaid<fnref n="1" />, Djamel Ziane<fnref n="1" />,<fnref n="2" />, Mohamed Fouad Benkhoris<fnref n="2" />

Received: 19 June 2023, Accepted: 25 March 2024, Published online: 27 May 2024

## Abstract

Efficient operation is a recommended factor in most of industrial applications. In this paper, an indirect matrix converter (IMC) topology is developed and proposed to substitute the classical (AC/DC/AC) three phase converter topologies; where the control strategy is frequently performed with a voltage source inverter (VSI). Large electrolytic capacitors are viewed as hazardous due to their short lifetime compared to ordinary capacitors and power switches, as well as their contribution to the total bulk and weight of the converter. Matrix converters may be the answer to getting rid of the capacitor, shrinking the converter, and improving its efficiency (MC). These advantages are offset by the fact that the (MC) calls for a tricky switching technique based on pulse width modulation, which is prone to commutation failure. The direct frequency conversion using the indirect matrix converter (IMC) with many advantages has appeared as an alternative to these conventional voltage inverters. This study investigates a field-oriented control (FOC) scheme for indirect matrix converter fed triple star induction motor (TSIM). After building the proposed IMC converter topology, an adequate field-oriented control strategy-based space vector modulation (SVM) is proposed and adapted for the control of the triple star induction motor. The FOC technique guarantees flux and torque decoupling while ensuring good operation performance as required in many industrial applications. The obtained numerical simulation results show and confirm the effectiveness of this control scheme.

## Keywords

indirect matrix converter, voltage source inverter, matrix converters, field-oriented control, triple star induction motor, space vector modulation

## 1 Introduction

Electric machines play a critical role in modern industry and are an essential part of many production processes. They are used in a wide range of applications, from powering industrial machinery to providing energy for lighting and heating systems. Their high efficiency and versatility make them ideal for use in many different industries, including manufacturing, mining, and construction. The use of electric machines has greatly increased productivity and reduced the cost of production, making them an essential component of modern industry. The increasing demand for renewable energy sources has also made electric machines even more important, as they are often used in the production of clean energy. The continued development and improvement of electric machines will continue to play a major role in

shaping the future of industry, making them an indispensable component of the modern manufacturing landscape.

The variable-speed drives are ubiquitous in manufacturing. AC motors are commonly used in industries. However, multiphase induction machines are increasingly employed as an alternative to conventional three-phase drives in industrial applications such as the automotive, aerospace, military, and nuclear sectors, where excellent reliability is essential. Among the many benefits of these machines are power segmentation, lower current stress on semiconductor devices, lower torque ripple, lower rotor harmonic currents and the ability to ensure safe operation. The ability to operate in a degraded mode with an acceptable torque ripple is another way a multiphase structure can boost dependability [1, 2].

<footnote n="1">¹ Laboratoire de Technologie Industrielle et de l'information (LTII), Faculté de Technologie, Université de Bejaia, 06000 Bejaia, Algeria</footnote>

<footnote n="2">² Nantes Université, Institut de Recherche en Énergie Électrique de Nantes, Atlantique, IREENA, UR 4642, 44600 Saint-Nazaire, France</footnote>

<footnote>* Corresponding author, e-mail: ahmed.azib@univ-bejaia.dz</footnote>
# Control and Performance Analysis of an Indirect Matrix Converter Fed Triple Star Induction Motor

Ahmed Azib<fnref n="1" />*, Adel Oubelaid<fnref n="1" />, Djamel Ziane<fnref n="1" />,<fnref n="2" />, Mohamed Fouad Benkhoris<fnref n="2" />

Received: 19 June 2023, Accepted: 25 March 2024, Published online: 27 May 2024

## Abstract

Efficient operation is a recommended factor in most of industrial applications. In this paper, an indirect matrix converter (IMC) topology is developed and proposed to substitute the classical (AC/DC/AC) three phase converter topologies; where the control strategy is frequently performed with a voltage source inverter (VSI). Large electrolytic capacitors are viewed as hazardous due to their short lifetime compared to ordinary capacitors and power switches, as well as their contribution to the total bulk and weight of the converter. Matrix converters may be the answer to getting rid of the capacitor, shrinking the converter, and improving its efficiency (MC). These advantages are offset by the fact that the (MC) calls for a tricky switching technique based on pulse width modulation, which is prone to commutation failure. The direct frequency conversion using the indirect matrix converter (IMC) with many advantages has appeared as an alternative to these conventional voltage inverters. This study investigates a field-oriented control (FOC) scheme for indirect matrix converter fed triple star induction motor (TSIM). After building the proposed IMC converter topology, an adequate field-oriented control strategy-based space vector modulation (SVM) is proposed and adapted for the control of the triple star induction motor. The FOC technique guarantees flux and torque decoupling while ensuring good operation performance as required in many industrial applications. The obtained numerical simulation results show and confirm the effectiveness of this control scheme.

## Keywords

indirect matrix converter, voltage source inverter, matrix converters, field-oriented control, triple star induction motor, space vector modulation

## 1 Introduction

Electric machines play a critical role in modern industry and are an essential part of many production processes. They are used in a wide range of applications, from powering industrial machinery to providing energy for lighting and heating systems. Their high efficiency and versatility make them ideal for use in many different industries, including manufacturing, mining, and construction. The use of electric machines has greatly increased productivity and reduced the cost of production, making them an essential component of modern industry. The increasing demand for renewable energy sources has also made electric machines even more important, as they are often used in the production of clean energy. The continued development and improvement of electric machines will continue to play a major role in

shaping the future of industry, making them an indispensable component of the modern manufacturing landscape.

The variable-speed drives are ubiquitous in manufacturing. AC motors are commonly used in industries. However, multiphase induction machines are increasingly employed as an alternative to conventional three-phase drives in industrial applications such as the automotive, aerospace, military, and nuclear sectors, where excellent reliability is essential. Among the many benefits of these machines are power segmentation, lower current stress on semiconductor devices, lower torque ripple, lower rotor harmonic currents and the ability to ensure safe operation. The ability to operate in a degraded mode with an acceptable torque ripple is another way a multiphase structure can boost dependability [1, 2].

<footnote n="1">¹ Laboratoire de Technologie Industrielle et de l'information (LTII), Faculté de Technologie, Université de Bejaia, 06000 Bejaia, Algeria</footnote>

<footnote n="2">² Nantes Université, Institut de Recherche en Énergie Électrique de Nantes, Atlantique, IREENA, UR 4642, 44600 Saint-Nazaire, France</footnote>

<footnote>* Corresponding author, e-mail: ahmed.azib@univ-bejaia.dz</footnote>
This study looks at the triple star induction machine, or TSIM for short, a multiphase machine with a squirrel cage-style rotor and three sets of three-phase windings that are spatially moved by 20 electrical degrees in the stator [3–5]. Yet, induction motors are difficult to manage due to their nonlinear architectures and the link between the flux and the generated electromagnetic torque. Many methods for distinguishing the two have therefore been proposed. These algorithms are based on many hypotheses and investigations. The first vector control strategy for induction motors was called field-oriented control (FOC) [6]. FOC can be divided into two main groups: indirect and direct. To do this, the motor equations are rewritten in a coordinate system that revolves around the flux vector of the rotor. You can be guaranteed that the flux and torque will be separated using the FOC method.

During regenerative braking, energy from the machine is filtered and stored in a vast electrolytic capacitor connected to the DC link of an AC/DC/AC converter; this control approach is often implemented with a voltage source inverter (VSI). Large electrolytic capacitors are regarded as harmful because of their short lifetime compared to regular capacitors and power switches and their contribution to the converter's overall bulk and weight [7–10]. Matrix converters may be the answer to getting rid of the capacitor, shrinking the converter, and improving its efficiency (MC).

These are direct AC/AC power converters that can simultaneously change the amplitude and frequency of three-phase voltages. They have many advantages, including a smaller footprint and lower weight due to the absence of DC-link electrochemical capacitors, reversible power flow with an adjustable input power factor, and a nearly sinusoidal current flow at the input and output. These advantages are offset by the fact that the (MC) calls for a tricky switching technique based on pulse width modulation, which is prone to commutation failure [11–13].

Instead of using a traditional matrix converter, which has the drawbacks listed above, an indirect matrix converter topology (IMC) can be employed, which has all the benefits of a traditional matrix converter (MC). However, the primary use of IMC over a traditional matrix converter is the simplified commutation process. All load side switches commutate like a standard DC/AC converter [14–18], but all line side switches operate at zero current.

This paper introduces the implementation of the indirect field oriented control for (IMC) fed (TSIM). The novelty of the proposed work is given below.

* The nonlinear behavior of the indirect matrix converter, as well as the triple star induction motor

model, is critical for executing and improving the control performance of the given system. To obtain an accurate system model, a mathematical model is being developed.

* By simulation tests, the strength of the indirect field oriented technique is investigated in order to counteract the detaining effect caused by the link between the flux and the generated electromagnetic torque, reversible power flow with an adjustable input power factor, and a nearly sinusoidal current flow at the input and output.

This document is organized as follows: After the Introduction section, a description of IMC topology used to feed TSIM and its modelation are given in Section 2. The control startegy, which is indirect field oriented control is discussed in Section 3. In Section 4, the simulation results representing the analysis performances of IMCs-TSIM combination are discussed. Section 5 concludes the article.

## 2 Topology and Modulation for the indirect matrix converter

### 2.1 Topology of the IMC

Three voltage source inverters are typically used to feed a TSIM via a standard current source rectifier with dc link part. In this paper, an indirect matrix converter for TSIM is investigated, with a bidirectional current source converter on the mains side and three phase voltage inverters connected to a common imaginary dc-link, as shown in Figs. 1 and 2.

### 2.2 Modulation method

The indirect matrix converter's principal circuit is composed of an LC filter, bidirectional current source rectifier and three six-switch inverters. To show the quality of the waves obtained with this structure and the smooth nature of the switching, the presentation of SVM algorithm is more than necessary. This control strategy's goal is to synthesize output voltages from input voltages and input currents from output currents. The modulation is carried out by uniformly combining the switching states of the rectifier stage and the inverter stage by choosing the right vectors and establishing their duty cycles [17, 18].

This relationship describes the rectifier output voltage as a function of the input voltages. The parameters of all equations are included in (Appendix Table A.2):

$$ \begin{bmatrix} v_p \\ v_n \end{bmatrix} = \begin{bmatrix} m_{ap} & m_{bp} & m_{cp} \\ m_{an} & m_{bn} & m_{cn} \end{bmatrix} \begin{bmatrix} v_a \\ v_b \\ v_c \end{bmatrix} \quad (1) $$
This study looks at the triple star induction machine, or TSIM for short, a multiphase machine with a squirrel cage-style rotor and three sets of three-phase windings that are spatially moved by 20 electrical degrees in the stator [3–5]. Yet, induction motors are difficult to manage due to their nonlinear architectures and the link between the flux and the generated electromagnetic torque. Many methods for distinguishing the two have therefore been proposed. These algorithms are based on many hypotheses and investigations. The first vector control strategy for induction motors was called field-oriented control (FOC) [6]. FOC can be divided into two main groups: indirect and direct. To do this, the motor equations are rewritten in a coordinate system that revolves around the flux vector of the rotor. You can be guaranteed that the flux and torque will be separated using the FOC method.

During regenerative braking, energy from the machine is filtered and stored in a vast electrolytic capacitor connected to the DC link of an AC/DC/AC converter; this control approach is often implemented with a voltage source inverter (VSI). Large electrolytic capacitors are regarded as harmful because of their short lifetime compared to regular capacitors and power switches and their contribution to the converter's overall bulk and weight [7–10]. Matrix converters may be the answer to getting rid of the capacitor, shrinking the converter, and improving its efficiency (MC).

These are direct AC/AC power converters that can simultaneously change the amplitude and frequency of three-phase voltages. They have many advantages, including a smaller footprint and lower weight due to the absence of DC-link electrochemical capacitors, reversible power flow with an adjustable input power factor, and a nearly sinusoidal current flow at the input and output. These advantages are offset by the fact that the (MC) calls for a tricky switching technique based on pulse width modulation, which is prone to commutation failure [11–13].

Instead of using a traditional matrix converter, which has the drawbacks listed above, an indirect matrix converter topology (IMC) can be employed, which has all the benefits of a traditional matrix converter (MC). However, the primary use of IMC over a traditional matrix converter is the simplified commutation process. All load side switches commutate like a standard DC/AC converter [14–18], but all line side switches operate at zero current.

This paper introduces the implementation of the indirect field oriented control for (IMC) fed (TSIM). The novelty of the proposed work is given below.

* The nonlinear behavior of the indirect matrix converter, as well as the triple star induction motor

model, is critical for executing and improving the control performance of the given system. To obtain an accurate system model, a mathematical model is being developed.

* By simulation tests, the strength of the indirect field oriented technique is investigated in order to counteract the detaining effect caused by the link between the flux and the generated electromagnetic torque, reversible power flow with an adjustable input power factor, and a nearly sinusoidal current flow at the input and output.

This document is organized as follows: After the Introduction section, a description of IMC topology used to feed TSIM and its modelation are given in Section 2. The control startegy, which is indirect field oriented control is discussed in Section 3. In Section 4, the simulation results representing the analysis performances of IMCs-TSIM combination are discussed. Section 5 concludes the article.

## 2 Topology and Modulation for the indirect matrix converter

### 2.1 Topology of the IMC

Three voltage source inverters are typically used to feed a TSIM via a standard current source rectifier with dc link part. In this paper, an indirect matrix converter for TSIM is investigated, with a bidirectional current source converter on the mains side and three phase voltage inverters connected to a common imaginary dc-link, as shown in Figs. 1 and 2.

### 2.2 Modulation method

The indirect matrix converter's principal circuit is composed of an LC filter, bidirectional current source rectifier and three six-switch inverters. To show the quality of the waves obtained with this structure and the smooth nature of the switching, the presentation of SVM algorithm is more than necessary. This control strategy's goal is to synthesize output voltages from input voltages and input currents from output currents. The modulation is carried out by uniformly combining the switching states of the rectifier stage and the inverter stage by choosing the right vectors and establishing their duty cycles [17, 18].

This relationship describes the rectifier output voltage as a function of the input voltages. The parameters of all equations are included in (Appendix Table A.2):

$$ \begin{bmatrix} v_p \\ v_n \end{bmatrix} = \begin{bmatrix} m_{ap} & m_{bp} & m_{cp} \\ m_{an} & m_{bn} & m_{cn} \end{bmatrix} \begin{bmatrix} v_a \\ v_b \\ v_c \end{bmatrix} \quad (1) $$
<figure><img src="image_1.png" /><figcaption>Fig. 1 Main circuit of indirect matrix converter fed TSIM</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Fig. 2 Representation of the windings of TSIM</figcaption></figure>

The voltage relationship for the inverters stages is provided by:

$$ \begin{bmatrix} v_{Ak} \\ v_{Bk} \\ v_{Ck} \end{bmatrix} = \begin{bmatrix} m_{pAk} & m_{nAk} \\ m_{pBk} & m_{nBk} \\ m_{pCk} & m_{nCk} \end{bmatrix} \begin{bmatrix} v_p \\ v_n \end{bmatrix} \quad (2) $$

Where: $k = 1, 2, 3$

As a result, the relationship between IMC's output and input voltages is given by

$$ \begin{bmatrix} v_{Ak} \\ v_{Bk} \\ v_{Ck} \end{bmatrix} = \begin{bmatrix} m_{pAk} & m_{nAk} \\ m_{pBk} & m_{nBk} \\ m_{pCk} & m_{nCk} \end{bmatrix} \begin{bmatrix} m_{ap} & m_{bp} & m_{cp} \\ m_{an} & m_{bn} & m_{cn} \end{bmatrix} \begin{bmatrix} v_a \\ v_b \\ v_c \end{bmatrix} \quad (3) $$

Where: $k = 1, 2, 3$

### 2.2.1 Rectifier stage control

The rectifier's bidirectional switches are regulated to prevent input line shorts, deliver DC voltage to the inverter
<figure><img src="image_3.png" /><figcaption>Fig. 1 Main circuit of indirect matrix converter fed TSIM</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2 Representation of the windings of TSIM</figcaption></figure>

The voltage relationship for the inverters stages is provided by:

$$ \begin{bmatrix} v_{Ak} \\ v_{Bk} \\ v_{Ck} \end{bmatrix} = \begin{bmatrix} m_{pAk} & m_{nAk} \\ m_{pBk} & m_{nBk} \\ m_{pCk} & m_{nCk} \end{bmatrix} \begin{bmatrix} v_p \\ v_n \end{bmatrix} \qquad (2) $$

Where: $k = 1, 2, 3$

As a result, the relationship between IMC's output and input voltages is given by

$$ \begin{bmatrix} v_{Ak} \\ v_{Bk} \\ v_{Ck} \end{bmatrix} = \begin{bmatrix} m_{pAk} & m_{nAk} \\ m_{pBk} & m_{nBk} \\ m_{pCk} & m_{nCk} \end{bmatrix} \begin{bmatrix} m_{ap} & m_{bp} & m_{cp} \\ m_{an} & m_{bn} & m_{cn} \end{bmatrix} \begin{bmatrix} v_a \\ v_b \\ v_c \end{bmatrix} \qquad (3) $$

Where: $k = 1, 2, 3$

### 2.2.1 Rectifier stage control

The rectifier's bidirectional switches are regulated to prevent input line shorts, deliver DC voltage to the inverter
stage, and maintain sinusoidal input currents with the potential to control power factor. To ensure all this, the rectifier switches can only be configured in six different ways, which are then presented by six active inputs current $I_1$ to $I_6$ vectors (Fig. 3) [19, 20].

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3 Switching state vector of current source rectifier (a) and its SVM in sector 1 (b)</figcaption></figure>

During a switching phase, $I_{in}$ is formed from the neighboring active vectors $I_γ$ and $I_δ$, which have duty cycles of $d_γ$ and $d_δ$, respectively. If the input currents are assumed to be constant over a switching period $T_{sw}$, the reference current vector is as follows.

$$I_{in} = d_{\gamma} I_{\gamma} + d_{\delta} I_{\delta} + d_{0} I_{0} \quad (4)$$

The calculation of the duty cycles $d_γ$ and $d_δ$ are relative to the currents $I_γ$ and $I_δ$, are given by the following equations:

$$d_{\gamma} = \frac{t_{r1}}{T_{sw}} = m_{R} \sin\left(\frac{\pi}{3} - \theta_{i}\right) \quad (5)$$

$$d_{\delta} = \frac{t_{r2}}{T_{sw}} = m_{R} \sin(\theta_{i}) \quad (6)$$

$$d_{0} = 1 - d_{\gamma} - d_{\delta} \quad (7)$$

Where, $m_R$ is modulation factor; $0 \le m_R \le 1$

In the case where the null vector is eliminated from the switching sequence and the duty cycles are scaled back to the unit so as to satisfy the following expression:

$$I_{in} = d_{\gamma}^{R} I_{\gamma} + d_{\delta}^{R} I_{\delta} \quad (8)$$

The adjusted duty cycles of the rectifier stage are determined by the relationship (9), (10), where the modulation index $m_R = 1$ for maximum voltage at the DC bus [21]:

$$d'_{\gamma} = \frac{d_{\gamma}}{d_{\gamma} + d_{\delta}} \quad (9)$$

$$d'_{\delta} = \frac{d_{\delta}}{d_{\gamma} + d_{\delta}} \quad (10)$$

### 2.2.2 Inverter stage control

The three-phase output voltage inverter can, like any three-phase quantity ($v_{Ak}$, $v_{Bk}$, $v_{Ck}$) be decomposed into three other quantities: direct $V_α$, inverse $V_β$ and homopo-lar $V_0$ using the Concordia or Park matrix [P(0)]. Since the voltage of each arm can only take two values, then there are eight possible combinations between these voltages for each inverter Figs. 4–6.

The principle of the space vector modulation control is to determine the output state of the inverter such that the three-phase system it generates, when brought back to two-phase, is approximately equal to the reference vector. Generally, the duty cycles $d_α$, $d_β$ and $d_0$ for each inverter can be determined from Figs. 4–6.
stage, and maintain sinusoidal input currents with the potential to control power factor. To ensure all this, the rectifier switches can only be configured in six different ways, which are then presented by six active inputs current $I_1$ to $I_6$ vectors (Fig. 3) [19, 20].

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3 Switching state vector of current source rectifier (a) and its SVM in sector 1 (b)</figcaption></figure>

During a switching phase, $I_{in}$ is formed from the neighboring active vectors $I_γ$ and $I_δ$, which have duty cycles of $d_γ$ and $d_δ$, respectively. If the input currents are assumed to be constant over a switching period $T_{sw}$, the reference current vector is as follows.

$$I_{in} = d_{\gamma} I_{\gamma} + d_{\delta} I_{\delta} + d_{0} I_{0} \quad (4)$$

The calculation of the duty cycles $d_γ$ and $d_δ$ are relative to the currents $I_γ$ and $I_δ$, are given by the following equations:

$$d_{\gamma} = \frac{t_{r1}}{T_{sw}} = m_{R} \sin\left(\frac{\pi}{3} - \theta_{i}\right) \quad (5)$$

$$d_{\delta} = \frac{t_{r2}}{T_{sw}} = m_{R} \sin(\theta_{i}) \quad (6)$$

$$d_{0} = 1 - d_{\gamma} - d_{\delta} \quad (7)$$

Where, $m_R$ is modulation factor; $0 \le m_R \le 1$

In the case where the null vector is eliminated from the switching sequence and the duty cycles are scaled back to the unit so as to satisfy the following expression:

$$I_{in} = d_{\gamma}^{R} I_{\gamma} + d_{\delta}^{R} I_{\delta} \quad (8)$$

The adjusted duty cycles of the rectifier stage are determined by the relationship (9), (10), where the modulation index $m_R = 1$ for maximum voltage at the DC bus [21]:

$$d'_{\gamma} = \frac{d_{\gamma}}{d_{\gamma} + d_{\delta}} \quad (9)$$

$$d'_{\delta} = \frac{d_{\delta}}{d_{\gamma} + d_{\delta}} \quad (10)$$

### 2.2.2 Inverter stage control

The three-phase output voltage inverter can, like any three-phase quantity ($v_{Ak}$, $v_{Bk}$, $v_{Ck}$) be decomposed into three other quantities: direct $V_α$, inverse $V_β$ and homopo-lar $V_0$ using the Concordia or Park matrix [P(0)]. Since the voltage of each arm can only take two values, then there are eight possible combinations between these voltages for each inverter Figs. 4–6.

The principle of the space vector modulation control is to determine the output state of the inverter such that the three-phase system it generates, when brought back to two-phase, is approximately equal to the reference vector. Generally, the duty cycles $d_α$, $d_β$ and $d_0$ for each inverter can be determined from Figs. 4–6.
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4 Switching state vector of first voltage inverter (a) and its SVM in sector 1(b)</figcaption></figure>

$$d_{\alpha} = \frac{T_{\alpha}}{T_{sw}} = m_{v} \cdot \sin\left(\frac{\pi}{3} - \theta_{v}\right) \qquad (11)$$

$$d_{\beta} = \frac{T_{\beta}}{T_{sw}} = m_{v} \cdot \sin(\theta_{v}) \qquad (12)$$

$$d_{0} = 1 - d_{\alpha} - d_{\beta} \qquad (13)$$

Where, $m_v$ is modulation factor, $0 \le m_v \le 1$.

Finally, coordination between the control switches of the rectifier and the switches of the inverter is required to manage the indirect matrix converter under the restrictions

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5 Switching state vector of the second voltage inverter (a) and its SVM in sector 1(b)</figcaption></figure>

of a variable input power factor, the two-way transit of power, and zero-current switching commutation. When the current is zero, the rectifier level is activated. Fig. 7 shows a control cycle during the sector 2 for the input current and sector 1 for the output voltage to demonstrate this coordination. The active vectors of the first inverter may
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4 Switching state vector of first voltage inverter (a) and its SVM in sector 1(b)</figcaption></figure>

$$d_{\alpha} = \frac{T_{\alpha}}{T_{sw}} = m_{v} \cdot \sin\left(\frac{\pi}{3} - \theta_{v}\right) \qquad (11)$$

$$d_{\beta} = \frac{T_{\beta}}{T_{sw}} = m_{v} \cdot \sin(\theta_{v}) \qquad (12)$$

$$d_{0} = 1 - d_{\alpha} - d_{\beta} \qquad (13)$$

Where, $m_v$ is modulation factor, $0 \le m_v \le 1$.

Finally, coordination between the control switches of the rectifier and the switches of the inverter is required to manage the indirect matrix converter under the restrictions

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 5 Switching state vector of the second voltage inverter (a) and its SVM in sector 1(b)</figcaption></figure>

of a variable input power factor, the two-way transit of power, and zero-current switching commutation. When the current is zero, the rectifier level is activated. Fig. 7 shows a control cycle during the sector 2 for the input current and sector 1 for the output voltage to demonstrate this coordination. The active vectors of the first inverter may
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 6 Switching state vector of the third voltage inverter (a) and its SVM in sector 1 (b)</figcaption></figure>

obviously be applied to the second one while taking into account the shifted angle of π/9 between them. Therefore, the following relationships can be used to obtain the duty ratios that correspond to the inverters stage [22]:

<figure><img src="image_11.png" /><figcaption>Fig. 7 Switching sequences of IMC (input current sector 2 and output voltage sector 1)</figcaption></figure>

$$
\left\{
\begin{aligned}
t_{r1} &= d_{\gamma}^{R} \cdot T_{sw} \\
t_{i1} &= \frac{1}{2} d_{\gamma}^{R} \cdot d_{0} \cdot T_{sw} \\
t_{i2} &= d_{\gamma}^{R} \cdot d_{\alpha} \cdot T_{sw} \\
t_{i3} &= d_{\gamma}^{R} \cdot d_{\beta} \cdot T_{sw} \\
t_{i4} &= \frac{1}{2} d_{\delta}^{R} \cdot d_{0} \cdot T_{sw} \\
t_{i5} &= \frac{1}{2} d_{\delta}^{R} \cdot d_{\beta} \cdot T_{sw} \\
t_{i6} &= \frac{1}{2} d_{\delta}^{R} \cdot d_{\alpha} \cdot T_{sw}
\end{aligned}
\right.
\qquad (14)
$$

3 Field-orientd control (FOC) for TSIM

The field-oriented control works in the same way that an
independently excited DC motor does. Both the current
and the torque can be adjusted individually in this motor.
PI regulators are all that is needed to implement the con-
trol algorithm. If the induction motor's coordinate system
is tied to the rotor's flux vector, flux and torque can indeed
be controlled independently.

To summarize, Fig. 8 shows an illustration of the IFOC
in a block form for TSIM. The main components of this
type of control are the speed control loop, the θ_s calcula-
tion block and the direct PARK transformations. Speed
is regulated through the block's external loop. The out-
put of its regulator is the reference electromagnetic torque
T em* or the reference currents i sq*. Parallel to this inter-
nal loop is a control loop for i sd*. The reference i sd* cur-
rents are calculated from the flux to be imposed. This
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 6 Switching state vector of the third voltage inverter (a) and its SVM in sector 1 (b)</figcaption></figure>

obviously be applied to the second one while taking into account the shifted angle of $\pi/9$ between them. Therefore, the following relationships can be used to obtain the duty ratios that correspond to the inverters stage [22]:

<figure><img src="image_13.png" /><figcaption>Fig. 7 Switching sequences of IMC (input current sector 2 and output voltage sector 1)</figcaption></figure>

$$ \left\{ \begin{aligned} t_{r1} &= d_{\gamma}^{R} \cdot T_{sw} \\ t_{i1} &= \frac{1}{2} d_{\gamma}^{R} \cdot d_{0} \cdot T_{sw} \\ t_{i2} &= d_{\gamma}^{R} \cdot d_{\alpha} \cdot T_{sw} \\ t_{i3} &= d_{\gamma}^{R} \cdot d_{\beta} \cdot T_{sw} \\ t_{i4} &= \frac{1}{2} d_{\delta}^{R} \cdot d_{0} \cdot T_{sw} \\ t_{i5} &= \frac{1}{2} d_{\delta}^{R} \cdot d_{\beta} \cdot T_{sw} \\ t_{i6} &= \frac{1}{2} d_{\delta}^{R} \cdot d_{\alpha} \cdot T_{sw} \end{aligned} \right. \qquad (14) $$

### 3 Field-oriented control (FOC) for TSIM

The field-oriented control works in the same way that an independently excited DC motor does. Both the current and the torque can be adjusted individually in this motor. PI regulators are all that is needed to implement the control algorithm. If the induction motor's coordinate system is tied to the rotor's flux vector, flux and torque can indeed be controlled independently.

To summarize, Fig. 8 shows an illustration of the IFOC in a block form for TSIM. The main components of this type of control are the speed control loop, the $\theta_s$ calculation block and the direct PARK transformations. Speed is regulated through the block's external loop. The output of its regulator is the reference electromagnetic torque $T_{em}^*$ or the reference currents $i_{sq}^*$. Parallel to this internal loop is a control loop for $i_{sd}^*$. The reference $i_{sd}^*$ currents are calculated from the flux to be imposed. This
<figure><img src="image_12.png" /><figcaption>Fig. 8 An illustration of the IFOC in a block form for TSIM</figcaption></figure>

flux corresponds to its nominal value for the speed range below the base speed and finally, the inverse PARK.

Fig. 9 depicts the d-q rotational coordinate system.

In our case, a d-q frame that rotates with the rotor flux vector $\Phi_r$ is chosen. This yields Eq. (15) shown below:

$$ \begin{cases} \Phi_{rd} = \Phi_r \\ \Phi_{rq} = 0 \end{cases} \qquad (15) $$

By imposing, $\Phi_{rd} = \Phi_r$ and $\Phi_{rq} = 0$, the TSIM equations in a reference frame related to the rotating field become:

$$ \left\{ \begin{aligned} V_{sd1} &= R_{s1} i_{sd1} + \frac{d\Phi_{sd1}}{dt} - w_s \Phi_{sq1} \\ V_{sq1} &= R_{s1} i_{sq1} + \frac{d\Phi_{sq1}}{dt} - w_s \Phi_{sd1} \\ V_{sd2} &= R_{s1} i_{sd2} + \frac{d\Phi_{sd2}}{dt} - w_s \Phi_{sq2} \\ V_{sq2} &= R_{s1} i_{sq2} + \frac{d\Phi_{sq2}}{dt} - w_s \Phi_{sd2} \\ V_{sd3} &= R_{s1} i_{sd3} + \frac{d\Phi_{sd3}}{dt} - w_s \Phi_{sq3} \\ V_{sq3} &= R_{s1} i_{sq3} + \frac{d\Phi_{sq3}}{dt} - w_s \Phi_{sd3} \\ 0 &= R_r i_{rd} + \frac{d\Phi_r}{dt} \\ 0 &= R_r i_{rd} + (w_s - w_r) \Phi_r \end{aligned} \right. \qquad (16) $$

with

$$ \left\{ \begin{aligned} \Phi_{sd1} &= L_{sd1} i_{sd1} + L_m (i_{sd1} + i_{sd2} + i_{sd3} + i_{rd}) \\ \Phi_{sq1} &= L_{sq1} i_{sq1} + L_m (i_{sq1} + i_{sq2} + i_{sq3} + i_{rq}) \\ \Phi_{sd2} &= L_{sd2} i_{sd2} + L_m (i_{sd1} + i_{sd2} + i_{sd3} + i_{rd}) \\ \Phi_{sq2} &= L_{sq2} i_{sq2} + L_m (i_{sq1} + i_{sq2} + i_{sq3} + i_{rq}) \\ \Phi_{sd3} &= L_{sd3} i_{sd3} + L_m (i_{sd1} + i_{sd2} + i_{sd3} + i_{rd}) \\ \Phi_{sq3} &= L_{sq3} i_{sq3} + L_m (i_{sq1} + i_{sq2} + i_{sq3} + i_{rq}) \\ \Phi_r &= L_r i_{rd} + L_m (i_{sd1} + i_{sd2} + i_{sd3} + i_{rd}) \\ 0 &= L_r i_{rq} + L_m (i_{sq1} + i_{sq2} + i_{sq3} + i_{rq}) \end{aligned} \right. \qquad (17) $$

The expression of the electromagnetic torque will in this case have the following form:

$$ T_{em} = \frac{L_m}{L_r + L_m} P(\Phi_r (i_{sq1} + i_{sq2} + i_{sq3})) \qquad (18) $$

From the Eq. (7) of the two systems (16) and (17), the expression of the rotor flux becomes

$$ \frac{d\Phi_r}{dt} = \frac{r_r L_m}{(L_m + L_r)} (i_{sd1} + i_{sd2} + i_{sd3}) - \frac{r_r}{(L_m + L_r)} \Phi_r \qquad (19) $$

From Eqs. (18) and (19) one can see that if the rotor flux is kept constant, the torque depends solely on the quadra-ture component ($i_{sq1} + i_{sq2} + i_{sq3}$), while only the direct com-ponent ($i_{sd1} + i_{sd2} + i_{sd3}$) influences the amplitude of the rotor flux. As a result, a structure that is comparable to that of a separately excited MCC is produced.
<figure><img src="image_14.png" /><figcaption>Fig. 8 An illustration of the IFOC in a block form for TSIM</figcaption></figure>

flux corresponds to its nominal value for the speed range below the base speed and finally, the inverse PARK.

Fig. 9 depicts the d-q rotational coordinate system.

In our case, a d-q frame that rotates with the rotor flux vector $\Phi_r$ is chosen. This yields Eq. (15) shown below:

$$ \begin{cases} \Phi_{rd} = \Phi_r \\ \Phi_{rq} = 0 \end{cases} \qquad (15) $$

By imposing, $\Phi_{rd} = \Phi_r$ and $\Phi_{rq} = 0$, the TSIM equations in a reference frame related to the rotating field become:

$$ \left\{ \begin{aligned} V_{sd1} &= R_{s1} i_{sd1} + \frac{d\Phi_{sd1}}{dt} - w_s \Phi_{sq1} \\ V_{sq1} &= R_{s1} i_{sq1} + \frac{d\Phi_{sq1}}{dt} - w_s \Phi_{sd1} \\ V_{sd2} &= R_{s1} i_{sd2} + \frac{d\Phi_{sd2}}{dt} - w_s \Phi_{sq2} \\ V_{sq2} &= R_{s1} i_{sq2} + \frac{d\Phi_{sq2}}{dt} - w_s \Phi_{sd2} \\ V_{sd3} &= R_{s1} i_{sd3} + \frac{d\Phi_{sd3}}{dt} - w_s \Phi_{sq3} \\ V_{sq3} &= R_{s1} i_{sq3} + \frac{d\Phi_{sq3}}{dt} - w_s \Phi_{sd3} \\ 0 &= R_r i_{rd} + \frac{d\Phi_r}{dt} \\ 0 &= R_r i_{rd} + (w_s - w_r) \Phi_r \end{aligned} \right. \qquad (16) $$

with

$$ \left\{ \begin{aligned} \Phi_{sd1} &= L_{sd1} i_{sd1} + L_m (i_{sd1} + i_{sd2} + i_{sd3} + i_{rd}) \\ \Phi_{sq1} &= L_{sq1} i_{sq1} + L_m (i_{sq1} + i_{sq2} + i_{sq3} + i_{rq}) \\ \Phi_{sd2} &= L_{sd2} i_{sd2} + L_m (i_{sd1} + i_{sd2} + i_{sd3} + i_{rd}) \\ \Phi_{sq2} &= L_{sq2} i_{sq2} + L_m (i_{sq1} + i_{sq2} + i_{sq3} + i_{rq}) \\ \Phi_{sd3} &= L_{sd3} i_{sd3} + L_m (i_{sd1} + i_{sd2} + i_{sd3} + i_{rd}) \\ \Phi_{sq3} &= L_{sq3} i_{sq3} + L_m (i_{sq1} + i_{sq2} + i_{sq3} + i_{rq}) \\ \Phi_r &= L_r i_{rd} + L_m (i_{sd1} + i_{sd2} + i_{sd3} + i_{rd}) \\ 0 &= L_r i_{rq} + L_m (i_{sq1} + i_{sq2} + i_{sq3} + i_{rq}) \end{aligned} \right. \qquad (17) $$

The expression of the electromagnetic torque will in this case have the following form:

$$ T_{em} = \frac{L_m}{L_r + L_m} P(\Phi_r (i_{sq1} + i_{sq2} + i_{sq3})) \qquad (18) $$

From the Eq. (7) of the two systems (16) and (17), the expression of the rotor flux becomes

$$ \frac{d\Phi_r}{dt} = \frac{r_r L_m}{(L_m + L_r)} (i_{sd1} + i_{sd2} + i_{sd3}) - \frac{r_r}{(L_m + L_r)} \Phi_r \qquad (19) $$

From Eqs. (18) and (19) one can see that if the rotor flux is kept constant, the torque depends solely on the quadra-ture component ($i_{sq1} + i_{sq2} + i_{sq3}$), while only the direct com-ponen ($i_{sd1} + i_{sd2} + i_{sd3}$) influences the amplitude of the rotor flux. As a result, a structure that is comparable to that of a separately excited MCC is produced.
<figure><img src="image_13.png" /><figcaption>Fig. 9 Position of the frame to the rotor flux vector</figcaption></figure>

The FOC is essentially based on the use of a coordinate transformation system. The latter requires the knowledge of the PARK angle $\theta_s$. Depending on how this angle is determined. Two types of FOC techniques can be distinguished: the direct and indirect field oriented control (DFOC, IFOC) [6–9].

Unlike DFOC, IFOC does not use the magnitude of the rotor flux but only its position [8, 9]. It eliminates the need for a flux sensor or estimator or observer, but requires the use of a speed sensor. The phase of the flux is then determined using a mathematical model that can be incorporated into the control structure. The PARK angle $\theta_s$ is calculated from the stator pulsation, which is itself reconstructed using the machine speed and the rotor pulsation $w_r$.

$$w_s = w_{gl} + P w_r \qquad (20)$$

Where, $w_{gl}$ denotes the slip angular speed. It is available as follows:

$$w_{gl} = \frac{1}{(i_{sd1} + i_{sd2} + i_{sd3})} \frac{R_r}{(L_r + L_m)} (i_{sq1} + i_{sq2} + i_{sq3}) \qquad (21)$$

## 4 Simulation results

To demonstrate the effectiveness of the proposed method, a numerical simulation of the dynamic behaviour of TSIM (Appendix Table A.1) fed by indirect matrix converters and controlled by IFOC is performed. The simulation parameters are: The reference flux $\Phi_{rref} = 1$ web, $T_L = 14$ N m for $2 \le t(s) \le 3$; $w_{rref} = 157.07$ rps for $0 \le t(s) \le 3.5$ and $w_{rref} = -157.07$ rps for $3.5 \le t(s) \le 5$.

The electrical rotor speed is depicted in Fig. 10. It can be noticed that it tracks precisely its reference. The rotor speed keeps tracking its reference even in case of load torque variation ($T_L = 14$ N m for $2 \le t(s) \le 3$) and speed reference modification from $w_{rref} = 157.07$ rps to $w_{rref} = -157.07$ rps at $t = 3.5$ s.

The rotor flux waveforms in d-q axis components are shown in Figs. 11 and 12, it is clear that the both $\Phi_q$ and $\Phi_q$ reach their references and remain almost constant which it guarantees the decoupling between the rotor speed, electromagnetic torque and the rotor flux.

Figs. 13–16 show simulation results for the torque dynamic, the stator currents d-q axis components $i_{sq1}$, $i_{sq2}$, and $i_{sq3}$, respectively, one can see that the Electromagnetic torque form is the same as the $i_{sq1,2,3}$ currents and follows the Electromagnetic torque reference.

Figs. 17 and 18 show the waveforms of the stator currents and their zoom.

<figure><img src="image_14.png" /><figcaption>Fig. 10 Rotor speed</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 11 Allure of direct rotor flux $\Phi_{rd}$</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Fig. 9 Position of the frame to the rotor flux vector</figcaption></figure>

The FOC is essentially based on the use of a coordinate transformation system. The latter requires the knowledge of the PARK angle $\theta_s$. Depending on how this angle is determined. Two types of FOC techniques can be distinguished: the direct and indirect field oriented control (DFOC, IFOC) [6–9].

Unlike DFOC, IFOC does not use the magnitude of the rotor flux but only its position [8, 9]. It eliminates the need for a flux sensor or estimator or observer, but requires the use of a speed sensor. The phase of the flux is then determined using a mathematical model that can be incorporated into the control structure. The PARK angle $\theta_s$ is calculated from the stator pulsation, which is itself reconstructed using the machine speed and the rotor pulsation $w_r$.

$$w_s = w_{gl} + P w_r \qquad (20)$$

Where, $w_{gl}$ denotes the slip angular speed. It is available as follows:

$$w_{gl} = \frac{1}{(i_{sd1} + i_{sd2} + i_{sd3})} \frac{R_r}{(L_r + L_m)} (i_{sq1} + i_{sq2} + i_{sq3}) \qquad (21)$$

## 4 Simulation results

To demonstrate the effectiveness of the proposed method, a numerical simulation of the dynamic behaviour of TSIM (Appendix Table A.1) fed by indirect matrix converters and controlled by IFOC is performed. The simulation parameters are: The reference flux $\Phi_{rref} = 1$ web, $T_L = 14$ N m for $2 \le t(s) \le 3$; $w_{rref} = 157.07$ rps for $0 \le t(s) \le 3.5$ and $w_{rref} = -157.07$ rps for $3.5 \le t(s) \le 5$.

The electrical rotor speed is depicted in Fig. 10. It can be noticed that it tracks precisely its reference. The rotor speed keeps tracking its reference even in case of load torque variation ($T_L = 14$ N m for $2 \le t(s) \le 3$) and speed reference modification from $w_{rref} = 157.07$ rps to $w_{rref} = -157.07$ rps at $t = 3.5$ s.

The rotor flux waveforms in d-q axis components are shown in Figs. 11 and 12, it is clear that the both $\Phi_q$ and $\Phi_q$ reach their references and remain almost constant which it guarantees the decoupling between the rotor speed, electromagnetic torque and the rotor flux.

Figs. 13–16 show simulation results for the torque dynamic, the stator currents d-q axis components $i_{sq1}$, $i_{sq2}$, and $i_{sq3}$, respectively, one can see that the Electromagnetic torque form is the same as the $i_{sq1,3}$ currents and follows the Electromagnetic torque reference.

Figs. 17 and 18 show the waveforms of the stator currents and their zoom.

<figure><img src="image_16.png" /><figcaption>Fig. 10 Rotor speed</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 11 Allure of direct rotor flux $\Phi_{rd}$</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Fig. 12 Allure of quadrature rotor flux $\Phi_{rq}$</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 15 Stator current $i_{sq2}$ waveform</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 13 Electromagnetic torque trajectory</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 16 Stator current $i_{sq3}$ waveform</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 14 Stator current $i_{sq1}$ waveform</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 17 Stator currents waveforms</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Fig. 12 Allure of quadrature rotor flux $\Phi_{rq}$</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 15 Stator current $i_{sq2}$ waveform</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 13 Electromagnetic torque trajectory</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 16 Stator current $i_{sq3}$ waveform</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 14 Stator current $i_{sq1}$ waveform</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 17 Stator currents waveforms</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Fig. 18 Zoom of stator currents waveforms</figcaption></figure>

Figs. 19 and 20 illustrate the DC link voltage waveform and the indirect matrix output voltage waveform.

Finally, Figs. 21 and 22 depict the input voltage, current, and filtered input current, respectively. As it can be seen, the control has a unity input power factor.

## 5 Conclusions

The efficiency of an indirect field oriented control controlled triple star induction motor (TSIM) with an indirect matrix converter is investigated in this paper. The converter is composed of three inverters linked by a fictitious DC link to a standard bidirectional current source rectifier. For the converter, a space vector modulation was envisaged; in this method, three conventional SVM of indirect matrix converters with particular switching vector location patterns are used. In the rectifier stage, the space vector modulation strategy results in an adjustable input displacement power factor and near sinusoidal input current. The simulation findings showed the decoupling of motor speed

and rotor flux while maintaining excellent drive performance and a power factor at unity. Add to this the benefits of the IFOC, which ensures precise decoupling between rotor flux and motor speed in dynamic and steady states.

<figure><img src="image_23.png" /><figcaption>Fig. 20 Output IMC voltage waveform</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 21 Input voltage and input current</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Fig. 19 DC link voltage</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 22 Input voltage and filtered input current</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Fig. 18 Zoom of stator currents waveforms</figcaption></figure>

Figs. 19 and 20 illustrate the DC link voltage waveform and the indirect matrix output voltage waveform.

Finally, Figs. 21 and 22 depict the input voltage, current, and filtered input current, respectively. As it can be seen, the control has a unity input power factor.

## 5 Conclusions

The efficiency of an indirect field oriented control controlled triple star induction motor (TSIM) with an indirect matrix converter is investigated in this paper. The converter is composed of three inverters linked by a fictitious DC link to a standard bidirectional current source rectifier. For the converter, a space vector modulation was envisaged; in this method, three conventional SVM of indirect matrix converters with particular switching vector location patterns are used. In the rectifier stage, the space vector modulation strategy results in an adjustable input displacement power factor and near sinusoidal input current. The simulation findings showed the decoupling of motor speed

and rotor flux while maintaining excellent drive performance and a power factor at unity. Add to this the benefits of the IFOC, which ensures precise decoupling between rotor flux and motor speed in dynamic and steady states.

<figure><img src="image_25.png" /><figcaption>Fig. 20 Output IMC voltage waveform</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Fig. 21 Input voltage and input current</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Fig. 19 DC link voltage</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Fig. 22 Input voltage and filtered input current</figcaption></figure>
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[1] Levi, E., Bojoi, R., Profumo F., Toliyat, H. A., Williamson, S. "Multiphase induction motor drives - A technology status review", IET Electric Power Applications, 1(4), pp. 489–516, 2007.
https://doi.org/10.1049/iet-epa:20060342

[2] Levi, E. "Multiphase electric machines for variable-speed applications", IEEE Transations on Industrial Electronics, 55(5), pp. 1893–1909, 2008.
https://doi.org/10.1109/TIE.2008.918488

[3] Rubino, S., Mandrile, F., Armando, E., Bojoi, I. R., Zarri, L. "Fault-Tolerant Torque Controller Based on Adaptive Decoupled Multi-Stator Modeling for Multi-Three-Phase Induction Motor Drives", IEEE Transactions on Industry Applications, 58(6), pp. 7318–7335, 2022.
https://doi.org/10.1109/TIA.2022.3197547

[4] Rubino, S., Bojoi, R., Cittanti, D., Zarri, L. "Decoupled and Modular Torque Control of Multi-Three-Phase Induction Motor Drives", IEEE Transactions on Industry Applications, 56(4), pp. 3831–3845, 2020.
https://doi.org/10.1109/TIA.2020.2991122

[5] Rubino, S., Bojoi, R., Cittanti, D., Zarri, L. "Decoupled torque control of multiple three-phase induction motor drives", In: IEEE Energy Conversion Congress and Exposition (ECCE), Baltimore, MD, USA, 2019, pp. 4903–4910. ISBN 9781728103969
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# Appendix

<table><caption>Table A.1 Parameters of 4.5 Kw three-star induction motor</caption><thead><tr><th>Power</th><th>P<sub>n</sub> = 4.5 Kw</th></tr></thead><tbody><tr><td>Based speed</td><td>N<sub>r</sub> = 2922 rpm</td></tr><tr><td>Stator winding resistance</td><td>R<sub>s</sub> = 3.75 Ω</td></tr><tr><td>Stator winding resistance</td><td>R<sub>r</sub> = 2.12 Ω</td></tr><tr><td>Stator leakage inductance</td><td>L<sub>ls</sub> = 0.022 H</td></tr><tr><td>Rotor leakage inductance</td><td>L<sub>lr</sub> = 0.006 H</td></tr><tr><td>Mutual inductance</td><td>L<sub>m</sub> = 0.3672 H</td></tr><tr><td>Number of pole pairs</td><td>P = 1</td></tr><tr><td>Moment of inertia</td><td>J = 0.0625 Kg m<sup>2</sup></td></tr></tbody></table>

<table><caption>Table A.2 Nomenclature, Greek letters and abbreviations</caption><thead><tr><th>Nomenclature</th><th></th></tr></thead><tbody><tr><td>A<sub>s1</sub>, B<sub>s1</sub>, C<sub>s1</sub></td><td>Three TSIM stator winding systems.</td></tr><tr><td>A<sub>s2</sub>, B<sub>s2</sub>, C<sub>s2</sub></td><td></td></tr><tr><td>A<sub>s3</sub>, B<sub>s3</sub>, C<sub>s3</sub></td><td></td></tr><tr><td>A<sub>r</sub>, B<sub>r</sub>, C<sub>r</sub></td><td>Rotor winding system</td></tr><tr><td>v<sub>j</sub> (j = a, b, c)</td><td>Single voltages relative to grid neutral</td></tr><tr><td>v<sub>p</sub>, v<sub>n</sub></td><td>Positive and negative potential of the fictitious IMC DC bus</td></tr><tr><td>v<sub>j</sub> (j = Ak, Bk, Ck)</td><td>Single voltages with respect to TSIM neutral</td></tr><tr><td>t<sub>1</sub>, t<sub>2</sub>, t<sub>3</sub>, t<sub>0</sub></td><td>The application times of the vectors bordering a sector and the zero vector</td></tr><tr><td>T<sub>sw</sub></td><td>Switching period</td></tr><tr><td>V<sub>sd1</sub>, V<sub>sq1</sub></td><td>Star voltages 1, 2 and 3 in PARK reference frame (d, q)</td></tr><tr><td>V<sub>sd2</sub>, V<sub>sq2</sub></td><td></td></tr><tr><td>V<sub>sd3</sub>, V<sub>sq3</sub></td><td></td></tr><tr><td>V<sub>rd</sub>, V<sub>rq</sub></td><td>Rotor voltages in PARK reference frame (d, q)</td></tr><tr><td>i<sub>sd1</sub>, i<sub>sq1</sub></td><td>Star currents 1, 2 and 3 in PARK reference frame (d, q)</td></tr><tr><td>i<sub>sd2</sub>, i<sub>sq2</sub></td><td></td></tr><tr><td>i<sub>sd3</sub>, i<sub>sq3</sub></td><td></td></tr><tr><td>i<sub>rd</sub>, i<sub>rq</sub></td><td>Rotor currents in PARK reference frame (d, q)</td></tr><tr><td>i<sup>*</sup><sub>sd1</sub>, i<sup>*</sup><sub>sq1</sub></td><td>Reference currents of star 1, 2 and 3 in PARK reference frame</td></tr><tr><td>i<sup>*</sup><sub>sd2</sub>, i<sup>*</sup><sub>sq2</sub></td><td></td></tr><tr><td>i<sup>*</sup><sub>sd3</sub>, i<sup>*</sup><sub>sq3</sub></td><td></td></tr><tr><td>i<sup>*</sup><sub>rd</sub>, i<sup>*</sup><sub>rq</sub></td><td>Fundamental electrical pulse of rotor magnitudes and its reference</td></tr><tr><td>T<sub>em</sub>, T<sub>em</sub><sup>*</sup></td><td>Electromagnetic torque and its reference</td></tr><tr><td>Greek letters</td><td></td></tr><tr><td>α, β</td><td>Stator reference frame</td></tr><tr><td>d<sub>r</sub>, d<sub>δ</sub></td><td>Input current vector duty cycle I<sub>r</sub>, I<sub>δ</sub></td></tr><tr><td>d<sub>r</sub><sup>R</sup>, d<sub>δ</sub><sup>R</sup></td><td>Adjusted input current vector duty cycle I<sub>r</sub>, I<sub>δ</sub></td></tr><tr><td>d<sub>a</sub>, d<sub>β</sub></td><td>Output voltage vector duty cycle V<sub>a</sub>, V<sub>β</sub></td></tr><tr><td>Φ<sub>sd1</sub>, Φ<sub>sq1</sub></td><td>Star flux 1, 2 and 3 in PARK's reference frame (d, q)</td></tr><tr><td>Φ<sub>sd2</sub>, Φ<sub>sq2</sub></td><td></td></tr><tr><td>Φ<sub>sd3</sub>, Φ<sub>sq3</sub></td><td></td></tr><tr><td>Φ<sub>rd</sub>, Φ<sub>rq</sub></td><td>Rotor flux in the PARK frame of reference (d, q)</td></tr><tr><td>θ<sub>s</sub></td><td>PARK angle</td></tr><tr><td>Φ<sub>r</sub>, Φ<sub>rref</sub></td><td>TSIM rotor flux vector and its reference</td></tr><tr><td>Abbreviations</td><td></td></tr><tr><td>IMC</td><td>Indirect matrix converter</td></tr><tr><td>VSI</td><td>Voltage source inverter</td></tr><tr><td>CM</td><td>Matrix converter</td></tr><tr><td>FOC</td><td>Field oriented control</td></tr><tr><td>TSIM</td><td>Triple star induction motor</td></tr><tr><td>SVM</td><td>Space vector modulation</td></tr></tbody></table>
# Appendix

<table><caption>**Table A.1 Parameters of 4.5 Kw three-star induction motor**</caption><tr><td>Power</td><td>P<sub>n</sub> = 4.5 Kw</td></tr><tr><td>Based speed</td><td>N<sub>r</sub> = 2922 rpm</td></tr><tr><td>Stator winding resistance</td><td>R<sub>s</sub> = 3.75 Ω</td></tr><tr><td>Stator winding resistance</td><td>R<sub>r</sub> = 2.12 Ω</td></tr><tr><td>Stator leakage inductance</td><td>L<sub>ls</sub> = 0.022 H</td></tr><tr><td>Rotor leakage inductance</td><td>L<sub>lr</sub> = 0.006 H</td></tr><tr><td>Mutual inductance</td><td>L<sub>m</sub> = 0.3672 H</td></tr><tr><td>Number of pole pairs</td><td>P = 1</td></tr><tr><td>Moment of inertia</td><td>J = 0.0625 Kg m<sup>2</sup></td></tr></table>

<table><caption>**Table A.2 Nomenclature, Greek letters and abbreviations**</caption><thead><tr><th>Nomenclature</th><th></th></tr></thead><tbody><tr><td>A<sub>s1</sub>, B<sub>s1</sub>, C<sub>s1</sub></td><td>Three TSIM stator winding systems.</td></tr><tr><td>A<sub>s2</sub>, B<sub>s2</sub>, C<sub>s2</sub></td><td></td></tr><tr><td>A<sub>s3</sub>, B<sub>s3</sub>, C<sub>s3</sub></td><td></td></tr><tr><td>A<sub>r</sub>, B<sub>r</sub>, C<sub>r</sub></td><td>Rotor winding system</td></tr><tr><td>v<sub>j</sub> (j = a, b, c)</td><td>Single voltages relative to grid neutral</td></tr><tr><td>v<sub>p</sub>, v<sub>n</sub></td><td>Positive and negative potential of the fictitious IMC DC bus</td></tr><tr><td>v<sub>j</sub> (j = Ak, Bk, Ck)</td><td>Single voltages with respect to TSIM neutral</td></tr><tr><td>t<sub>1</sub>, t<sub>2</sub>, t<sub>3</sub>, t<sub>0</sub></td><td>The application times of the vectors bordering a sector and the zero vector</td></tr><tr><td>T<sub>sw</sub></td><td>Switching period</td></tr><tr><td>V<sub>sd1</sub>, V<sub>sq1</sub></td><td>Star voltages 1, 2 and 3 in PARK reference frame (d, q)</td></tr><tr><td>V<sub>sd2</sub>, V<sub>sq2</sub></td><td></td></tr><tr><td>V<sub>sd3</sub>, V<sub>sq3</sub></td><td></td></tr><tr><td>V<sub>rd</sub>, V<sub>rq</sub></td><td>Rotor voltages in PARK reference frame (d, q)</td></tr><tr><td>i<sub>sd1</sub>, i<sub>sq1</sub></td><td>Star currents 1, 2 and 3 in PARK reference frame (d, q)</td></tr><tr><td>i<sub>sd2</sub>, i<sub>sq2</sub></td><td></td></tr><tr><td>i<sub>sd3</sub>, i<sub>sq3</sub></td><td></td></tr><tr><td>i<sub>rd</sub>, i<sub>rq</sub></td><td>Rotor currents in PARK reference frame (d, q)</td></tr><tr><td>i<sup>*</sup><sub>sd1</sub>, i<sup>*</sup><sub>sq1</sub></td><td>Reference currents of star 1, 2 and 3 in PARK reference frame</td></tr><tr><td>i<sup>*</sup><sub>sd2</sub>, i<sup>*</sup><sub>sq2</sub></td><td></td></tr><tr><td>i<sup>*</sup><sub>sd3</sub>, i<sup>*</sup><sub>sq3</sub></td><td></td></tr><tr><td>W<sub>r</sub>, W<sub>rref</sub></td><td>Fundamental electrical pulse of rotor magnitudes and its reference</td></tr><tr><td>T<sub>em</sub>, T<sub>em</sub><sup>*</sup></td><td>Electromagnetic torque and its reference</td></tr><tr><td>Greek letters</td><td></td></tr><tr><td>α, β</td><td>Stator reference frame</td></tr><tr><td>d<sub>r</sub>, d<sub>δ</sub></td><td>Input current vector duty cycle I<sub>r</sub>, I<sub>δ</sub></td></tr><tr><td>d<sub>r</sub><sup>R</sup>, d<sub>δ</sub><sup>R</sup></td><td>Adjusted input current vector duty cycle I<sub>r</sub>, I<sub>δ</sub></td></tr><tr><td>d<sub>a</sub>, d<sub>β</sub></td><td>Output voltage vector duty cycle V<sub>a</sub>, V<sub>β</sub></td></tr><tr><td>Φ<sub>sd1</sub>, Φ<sub>sq1</sub></td><td>Star flux 1, 2 and 3 in PARK's reference frame (d, q)</td></tr><tr><td>Φ<sub>sd2</sub>, Φ<sub>sq2</sub></td><td></td></tr><tr><td>Φ<sub>sd3</sub>, Φ<sub>sq3</sub></td><td></td></tr><tr><td>Φ<sub>rd</sub>, Φ<sub>rq</sub></td><td>Rotor flux in the PARK frame of reference (d, q)</td></tr><tr><td>θ<sub>s</sub></td><td>PARK angle</td></tr><tr><td>Φ<sub>r</sub>, Φ<sub>rref</sub></td><td>TSIM rotor flux vector and its reference</td></tr><tr><td>Abbreviations</td><td></td></tr><tr><td>IMC</td><td>Indirect matrix converter</td></tr><tr><td>VSI</td><td>Voltage source inverter</td></tr><tr><td>CM</td><td>Matrix converter</td></tr><tr><td>FOC</td><td>Field oriented control</td></tr><tr><td>TSIM</td><td>Triple star induction motor</td></tr><tr><td>SVM</td><td>Space vector modulation</td></tr></tbody></table>