## Abstract

The yet growing demand for higher demanding industrial applications and the global concern about harmful emissions in the atmosphere have increased the interest for new developments in electric machines and power converters. To meet these new requirements, multiphase machines have become a very attractive solution, offering potential advantages over three-phase classical solutions. Multiphase machine’s power demand can be split over more than three phases, thus reducing the electric field stress on each winding (protecting the insulation system) and the requirements on maximum power ratings, for semiconductor devices. Moreover, only two degrees of freedom (i.e. two independently controllable currents) are required for independent flux and torque control. Due to the previous facts, the use of multiphase drives has become very attractive for applications and developments in areas such as electric ship propulsion, more-electric aircraft, electric and hybrid electric road vehicles, electric locomotive traction and in renewable electric energy generation. As a consequence of this multiphase drive tendency, the development of power converter topologies, capable of dealing with high power ratings and handling multiphase winding distributions, has encourage the development of new converter topologies, control strategies and mathematical tools, to face this new challenge.

### Keywords

- multiphase AC drive
- neutral-point clamped
- cascaded H-bridge
- nine-switch converter
- 11-switch converter
- field-oriented control (FOC)
- direct torque control (DTC)

## 1. Introduction

Multiphase variable-speed drives, based on multiphase AC machines, are nowadays the most natural solution for high-demanding industrial, traction and power generation and distribution applications [1].

The types of multiphase machines for variable-speed applications are in principle the same as their three-phase counterparts. These are synchronous machines, which depending on the excitation can be subclassified into wound rotor, permanent magnet or reluctance machines and induction machines.

In a multiphase winding, the stator winding distribution becomes more concentrated, rather than distributed, as in the case of three-phase windings. This fact and the particularity of using quasi-sinusoidal, rather than sinusoidal voltages because of the inverting process in the power conversion stage, have several advantages that can be summarized as lower field harmonics content and better fault tolerance, because of extra degrees of freedom and less susceptibility to torque pulsations, due to an excitation field with less harmonic content [2].

As a consequence, the use of a multiphase winding configuration improves the MMF spatial distribution, by reducing its harmonic content and losses, due to flux leakage, and increasing the machine’s efficiency. These facts have increased the interest for research and development for transportation applications, such as cargo ships, aircraft and road vehicles, thus also contributing to the reduction of greenhouse effect emissions.

## 2. Mathematical modeling of multiphase machines

All electrical machines can be found to be variations of a common set of fundamental principles, which apply alike the number of phases of which the machine is constructed.

In this context, multiphase machines can be treated as belonging to an n-dimensional space, corresponding to the respective state variables. The machine model, on its original phase-variable form, can be transformed into

Each new two-dimensional subspace is orthogonal to each other, so there is no mutual coupling and they are represented in a stationary reference frame, respecting all other state variables. These new two-dimensional subspaces are could be denoted as

Let us consider an arbitrary state variable

In the case of

The last row of

Using the above-presented mathematical decomposition, any machine can be represented by an equivalent two-axis idealized machine, called the Kron’s primitive machine. The Kron’s primitive machine corresponds to the representation of the n-dimensional space, in the

The mathematical model describing a generalized n-dimensional electrical machine is given in Eqs. (3) and (4), for dynamics, respectively:

Considering symmetrical windings for both stator and rotor, then

where due to the machine’s symmetry

It has to be noticed that for both stator and rotor windings,

The resulting generalized machine representation is obtained by applying the linear transformation operator

The corresponding stator and rotor flux linkages given in Eqs. (7) and (8) are transformed into the

The presented methodology considers a n-phase winding with uniform distribution that is

The developed electromechanical torque can be expressed in terms of the electromechanical energy conversion as in Eqs. (16) and (17):

Following the electromechanical torque can be expressed as in Eq. (20):

The generalized machine representation given by Eqs. (12), (13) and (20) can be expressed in an arbitrary orthogonal synchronous reference frame

Rotation into the

The generalized machine dynamic representation in the

Equations (23) and (24) can be written in their matrix form as in Eq. (26), becoming a generalized impedance model, as described in Eq. (27):

### 2.1 Multiphase synchronous machines

The use of multiphase synchronous machines has been focused on its application for medium- and high-power generation systems, being their primary use, on wind energy conversion systems (WECS). The previous statement is based on the fact that most of wind energy conversion systems operate in the low-voltage range, principally due to restrictions of winding insulation. This fact has stimulated the development of multiphase generator topologies, thus gaining increasing interest in the research for new converter topologies [3, 4]. Some of their main advantages are the following:

The total power can be divided into lower power converters.

Due to that each converter is insulated from each other, there is no circulating current between converters, which leads to no power derating for the converters.

The increase of the number of phases in the generator voltages is phase-shifted so that low-order harmonics are reduced and consequently smaller filters can be used.

Reduced torque pulsations.

Fault-tolerant redundancy under winding fault conditions.

Additional degrees of freedom which can be used to improve the machine performance.

Arrangement of multiple three-phase windings has become a very popular construction technique, for multiphase machines. In this field, the most common configuration is the six-phase machine, based on two independent three-phase windings, which are spatially shifted in 30 degrees, as shown in Figure 1.

The use of multiple three-phase windings has the advantage of guaranteeing full decoupling under faulty conditions, thus preventing the circulation of common mode currents and pulsating torque.

The linear transformation operator

The equivalent model in the

Equations (31) and (32) correspond to the damping winding effect:

The corresponding flux linkages, considering the general case of an anisotropic machine, are given in the following equations:

and the electromechanical torque is given in Eq. (37):

### 2.2 Multiphase induction machines

Modern industrial high-demanding processes are commonly based on induction machines. They are very attractive for these kind of applications because of their simplicity and capability to work under extreme torque demanding conditions [5].

Within the previous context, multiphase induction machines have become very popular for applications where high redundancy and power density are required. In particular, the use of multiple three-phase windings (six and nine phases) in naval propulsion systems has aroused much interest and encouraged the research of new multiphase converter topologies and control schemes [6, 7].

Based on the Kron’s primitive machine model developed in Eq. (26), the dynamical model of the multiphase induction machine is given in Eqs. (38)–(41):

Due to the isotropy of the induction machine and the absence of an MMF source in the rotor (permanent magnets or field winding), the corresponding flux linkages are given in Eqs. (42)–(45):

The electromechanical torque expression can be derived from Eq. (37) by considering

## 3. Multiphase power converters

As stated previously, multiphase machines have many advantages over traditional three-phase-based machine drives, by reducing the impact of low-frequency torque pulsations and the dc-link current harmonic content. They also, due to the nature of their winding configuration, improve the system reliability, by introducing redundant operation conditions. As a consequence the use of multiphase converter topologies with multiphase drive arrangements has been proved as a viable approach for its application in high-demanding industrial applications.

In this field, the development of power converters capable to deal with the multiphase machine structure has capture much attention in the recent years; thus several topologies have been introduced in the last decades. This topologies can consist of arrangements of conventional two-level three-phase voltage source converters (2LVSC), multilevel converters (MLVSC) or in more specialized and dedicated topologies such as multiphase matrix converters.

### 3.1 Classical topologies

Classical topologies for multiphase converters are commonly based in arrangements of parallel-connected fundamental cells (multiple legs), or in multiple channel configuration, of voltage source converter topologies.

Commonly used topologies for multiphase applications are H-bridge converter (HBC), neutral-point clamped converter (NPC) and the cascaded H-bridge (CHB) topology, which are shown in Figure 2.

Classical multiphase VSC topologies consist of an arrangement of

Modulation of each individual half-cell circuit is implemented in such a way, to obtain

SPWM methods can be implemented for two-level or multilevel half-cells. In the first case, only one high-frequency carrier is required for the respective switching signals per phase

where

For multilevel parallel arrangements (NPC or CHB), the use of multiple carries, as an extension of the two-level PWM methods, has been proven as a suitable solution. Level-shifted PWM (LSPWM) has become a very popular modulation technique, because it fits for any multilevel converter topology and ensures low harmonic distortion. The corresponding switching states are given in Eq. (49):

where

The main advantages of this topology are its simplicity by using half-bridges for each leg (phase), and the requirement of a single dc-link the feed the n-phase inverting stage. The use of a parallel arrangement of fundamental half-cells generates

Multicell topology also enables the possibility to have common or split connected loads. In the first case, each phase in the load side is connected to a common neutral point

Split connected load arrangement is possible, if

Figure 6a shows a split-phase single-channel arrangement where the corresponding phase for each half-cell is derived as

For both previously described topologies, SPWM is achieved by implementing the modulation as given in Eqs. (47) and (48) considering the asymmetry (if required)

In the case of space vector modulation for multiphase converters, it is necessary to extend the space vector representation into its corresponding subspaces. So as presented previously in this chapter, the output voltage space vector can be decomposed into

For each orthogonal subspace, there exits two active vectors,

In the case of

Eqs (53) and (54) can be interpreted as the partitioning of the

### 3.2 Matrix converter

The matrix converter is a direct AC-AC converter, which uses an arrangement of bidirectional switches, to connect each input phase, with a single corresponding output phase, thus generating an arrangement of

Due to the absence of a dc-link stage, the output voltages should be synthesized by selecting segments of the input voltages, by generating the adequate switching states. However, some switching state restrictions have to be taken into account, because of the particular topology of the matrix converter. Let’s consider a generalized switching state

For every instant

Sinusoidal PWM scheme is implemented via the *Venturini* method [8], which is based on the solution of the relational input-output equations of the matrix converter, given in Eqs. (59)–(61):

where

On the other hand, space vector PWM formulation has no difference, as it is implemented in VSC. However, its complexity lies in the fact in that the absence of a nearly constant dc-link voltage, from which the reference voltage space vectors are synthesized. So both space vectors are to be composed using the input and output, voltage and current space vectors simultaneously. Moreover, the total number of possible switching states is

The matrix converter has several advantages over multiphase voltage source converters, for multiphase drive applications. It is capable to synthesize nearly sinusoidal output voltage and currents, with low-order harmonics, thus improving the MMF distribution in machine air gap, eliminating torque ripple and preventing mechanical stresses on the output shaft. It provides bidirectional energy flow and provides full power factor control. Moreover, due to the absence of dc-link, it presents more power density, because of the lack of large capacitors, becoming an alternative for integrated drive converter configurations [10, 11]. However the complexity in the implementation of SPWM and SVPWM schemes, and the complex commutation strategy for the bidirectional switches, makes the matrix converter less attractive than the voltage source-based multiphase solutions.

There exist several variations of the direct matrix converter topology presented previously, such as the indirect and sparse matrix converter topologies, which are extensively described in literature [12]. These topological variations relay on the same basis but differ in the number and type of power switches used.

### 3.3 Nine-switch converter

The topology is derived from two three-phase voltage source converters that share a positive and a negative busbar, respectively, as shown in Figure 7.

The nine-switch converter has the ability to operate in back-to-back mode, as rectifier (A stage input) and inverter (B stage output), as two channel rectifier (A and B stages input) and as two stage inverter (A and B stages output), enabling the converter to handle a six-phase systems with just one channel, in spite of commonly 12-switch back-to-back topologies.

However, this topological advantage introduces some drawbacks to the nine-switch topology, because of the fact that

Carrier-based PWM modulation schemes such as sinusoidal PWM (SPWM), space vector modulation (SVM) and min-max (third harmonic injection) PWM [14] can be applied to the nine-switch converter, using two independent SPWM modulation schemes, one for each converter stage, as in the 12-switch back-to-back converter. However, because of the restrictions introduced to the modulation, pattern by the middle switches

The implementation of SPWM considers the use of two voltage reference space vectors

For the constant frequency mode operation (CFM), both reference space vectors

By introducing a phase shifting of

Recently, the NSC has been gaining much attention in various applications like isolated wind-hydro hybrid power system, power quality enhancement and hybrid electric vehicles because of its ability to interconnect multiphase power systems in common or split configurations and also independent three-phase-based systems, in constant or variable frequency modes. Figure 11 shows simulation results for a wind energy conversion system (WECS) based on a nine-switch back-to-back converter topology [15].

Another important feature of the NSC is the capability to rearrange its switching states under single- or multiple-switch faulty conditions [16], as shown in Figure 12.

### 3.4 11-switch converter

The 11-switch converter, presented in [17], consists of a modified topology of the 9-switch converter topology, previously discussed. As shown in Figure 13, this topology introduces two additional switches

## 4. Control of multiphase electric drives

As discussed previously in this chapter, the electromechanical torque developed by the multiphase machine depends only on the state variables in

An additional control loop in the

### 4.1 Field-oriented control

The characteristics of field-oriented control (FOC) have made this control strategy the most widely used for high-demanding industrial applications. Field-oriented control is based in the decoupling of the current space vector, into a flux-producing component and a torque-producing component. This is achieved by rotating the current space vector from

In this way the magnetizing flux can be controlled, so that the machine operates with nominal flux under any condition, and also torque can be controlled only by

Figure 16 shows simulation results for a dual three-phase induction machine drive using FOC scheme, under load impact.

The main drawbacks of FOC are the requirement in the estimation of the rotor flux linkage space vector and the rotation of the state space variables into this synchronous reference frame, becoming a very complex process. Also, as the dynamic response, this control strategy is limited by maximum bandwidth achievable for the PI controllers, which represents one of the fundamental limitations of linear controllers.

### 4.2 Direct torque control

Direct torque control (DTC) is based on the estimation of torque and flux directly from the state variables of the AC machine. The torque and flux can be controlled by applying the suitable voltage vector, synthesized by the available switching states of the converter.

The required voltage vector is chosen via a switching table, as function of the actuation of the torque and flux loop hysteresis controllers (in terms of increasing or decreasing flux or torque for a certain operational point). Implementation of the DTC scheme for a multiphase AC drive is shown in Figure 17.

The main characteristics of DTC are its simple implementation and a fast dynamic response achieved by using hysteresis controllers. Furthermore, the required switching states are directly assigned from the switching table algorithm, so no modulator is needed.

## 5. Conclusions

In this chapter, the main advantages of multiphase machine drives for its application in high-demanding industrial processes, traction and renewable energy grid interfacing were presented, with their main focus on multiphase power converter topologies. Various technical issues related to classical multiphase converters, based on multicell arrangements, were discussed, as well as some new converter topologies, such as the nine-switch converter (NSC) and 11-switch converter (ESC), were introduced.

The main advantages of classical multiphase converter topologies, based on voltage source converters (VSC), are mostly referred to their topological simplicity and their capability to implement conventional sinusoidal-PWM-based techniques. On the other hand, the increasing number of semi-conductors and dc-link capacitors (in the case of multichannel arrangements) and the need for common mode current compensation are their major drawbacks. In this field of application, matrix converters arise as suitable alternative, which enables the possibility to handle multiple output phases. However, maximum voltage gain limitations and the complexity of the modulation and commutation strategies are their main disadvantages, when compared to classical topologies.

Nine-switch and eleven-switch converters appear as a middle-point alternative between multicell and matrix converter topologies. These topologies allow the use of sinusoidal-PWM-based modulation techniques, without the need of complex modulation and commutation strategies (as in the case of the matrix converter), using a single dc-link stage.

Implementation of control strategies for multiphase drive, such as field-oriented control (FOC) and direct torque control (DTC), has been easily achieved by using multi-space decomposition of the each state space vectors (representing a state space variable).