Electromechanical Power Conversion

Chapter 2

Model

Kiran Singh

1. Introduction

17

Abstract

Analysis of Compensated

Six-Phase Self-Excited Induction

In this article, a mixed current-flux d-q modeling of a saturated compensated six-phase self-excited induction generator (SP-SEIG) is adopted during the analysis. Modeling equations include two independent variables namely stator current and magnetizing flux rather than single independent variables either current or flux. Mixed modeling with stator current and magnetizing flux is simple by having only four saturation elements and beneficial in study of both stator and rotor parameters. Performance equations for the given machine utilize the steady-state saturated magnetizing inductance (Lm) and dynamic inductance (L). Validation of the analytical approach was in good agreement along with three-phase resistive or resistive-inductive loading and also determined the relevant improvement in volt-

age regulation of machine using series capacitor compensation schemes.

Keywords: mixed double state space variables, self-excitation, six-phase,

Traditionally, synchronous generators have been used for power generation, but induction generators are increasingly being used these days because of their relative advantageous features over conventional synchronous generators. The need for external reactive power limits the application of an induction generator as isolated unit. The use of SEIG, due to its reduced unit cost, simplicity in operation and ease of maintenance are most suited in such system. These entire features facilitate the operation of induction generator in stand-alone mode to supply far-flung areas where extension of grid is economically not viable. The stand-alone SEIG can be used with conventional as well as non-conventional energy sources to feed remote single family, village community, etc. in order to expedite the electrification of rural and remote locations. A detailed dynamic performance of induction generator 'IG' operating in different modes i.e. isolated and grid-connected is necessary for the optimum utilization of its various favorable features. The investigations spread over last two decades also indicate the technical and economic vitality of using number

compensation, induction generator, non-conventional energy

Generator Using Double Mixed

State-Space Variable Dynamic

## Chapter 2

Analysis of Compensated Six-Phase Self-Excited Induction Generator Using Double Mixed State-Space Variable Dynamic Model

Kiran Singh

## Abstract

In this article, a mixed current-flux d-q modeling of a saturated compensated six-phase self-excited induction generator (SP-SEIG) is adopted during the analysis. Modeling equations include two independent variables namely stator current and magnetizing flux rather than single independent variables either current or flux. Mixed modeling with stator current and magnetizing flux is simple by having only four saturation elements and beneficial in study of both stator and rotor parameters. Performance equations for the given machine utilize the steady-state saturated magnetizing inductance (Lm) and dynamic inductance (L). Validation of the analytical approach was in good agreement along with three-phase resistive or resistive-inductive loading and also determined the relevant improvement in voltage regulation of machine using series capacitor compensation schemes.

Keywords: mixed double state space variables, self-excitation, six-phase, compensation, induction generator, non-conventional energy

## 1. Introduction

Traditionally, synchronous generators have been used for power generation, but induction generators are increasingly being used these days because of their relative advantageous features over conventional synchronous generators. The need for external reactive power limits the application of an induction generator as isolated unit. The use of SEIG, due to its reduced unit cost, simplicity in operation and ease of maintenance are most suited in such system. These entire features facilitate the operation of induction generator in stand-alone mode to supply far-flung areas where extension of grid is economically not viable. The stand-alone SEIG can be used with conventional as well as non-conventional energy sources to feed remote single family, village community, etc. in order to expedite the electrification of rural and remote locations. A detailed dynamic performance of induction generator 'IG' operating in different modes i.e. isolated and grid-connected is necessary for the optimum utilization of its various favorable features. The investigations spread over last two decades also indicate the technical and economic vitality of using number

of phases higher than three in AC machines for applications in marine ships, thermal power plant to drive induced draft fans, electric vehicles and circulation pumps in nuclear power plants etc. In this area, research is still in its early stage, yet some extremely great authority's findings have been reported in the previous literatures indicating the general expediency of multi-phase systems. The literature regarding multi-phase IG is nearly not available since it has only three findings before 2004. The first article on multi-phase induction generator is appeared in 2005, along with rest of theoretical and practical works on SP-SEIG using single state- space variables either stator and rotor d-q axis currents or stator and rotor d-q axis flux linkages so far as reported [1]. On the basis of previous article reviews, before companion paper of 2015 [1], there were no literatures on modeling and analysis of SP-SEIG, using d-q axis components of stator current and magnetizing flux as mixed state–space variables. In the view of novelty, such mathematical modeling and analysis were carried out in detail for three-phase SEIG only, in a very few available literatures and some of which are mentioned by [2]. The purpose of this article is also to accomplish a similar task for modeling and analysis of SP-SEIG with series compensation scheme using double state-space variables as were proposed by companion paper without compensation. The simulation is performed on series compensated SP-SEIG by using 4th order Runge-Kutta subroutine in Matlab software.

## 2. Modeling description

Concerns about mathematical modeling of SP-SEIG, short-shunt series compensation capacitors and static resistive 'R' and balanced three-phase reactive 'R-L'

Figure 1. Distribution of 6-phases in 36 slots of 6 pole induction machine.

Analysis of Compensated Six-Phase Self-Excited Induction Generator Using Double Mixed… DOI: http://dx.doi.org/10.5772/intechopen.82323

loads as previously discussed in [1, 3] are not described in this section, only briefly summarized along with newly addition of long-shunt series compensation capacitors.

#### 2.1 SP-SEIG model

A basic two-pole, six-phase induction machine is schematically described by its stator and rotor axis [1]. In which, six stator phases of both sets, a, b, c and x, y, z (set I and II, respectively) are arranged to form two sets of uniformly distributed star configuration, displaced by an arbitrary angle of 30 electrical degree gravitate asymmetrical winding structure. The distribution of 6-phases in 36 slots of 6 pole induction machine is also shown in Figure 1. Previously, voltage equations using single state-space variable namely flux linkage were used in the expanded form for six-phase induction machine.

After simplification of voltage equations using double mixed state-space variables namely stator currents and magnetizing fluxes in the machine model following form occurs from Eqs. (1)–(39) of [1]:

$$\begin{bmatrix} \mathbf{V\_{dq}} \end{bmatrix} = \begin{bmatrix} \mathbf{H} \end{bmatrix} \begin{bmatrix} \mathbf{dX\_{dq}} \end{bmatrix} \Big/ \mathbf{dt} + \begin{bmatrix} \mathbf{J} \end{bmatrix} \begin{bmatrix} \mathbf{X\_{dq}} \end{bmatrix} \tag{1}$$

where [Vdq] = [Vd1 Vq1 Vd2 Vq2 0 0]t , [Xdq] = [id1 iq1 id2 iq2 ψdm ψqm] <sup>t</sup> and matrices [H] and [J] are given by Eqs. (2) and (3), respectively.

$$
\begin{bmatrix}
\mathbf{0} & -\mathcal{L}\_{\sigma 1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \\
\mathbf{0} & \mathbf{0} & -\mathcal{L}\_{\sigma 2} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & -\mathcal{L}\_{\sigma 2} & \mathbf{0} & \mathbf{1} \\
\mathbf{L}\_{\sigma r}' & \mathbf{0} & \mathbf{L}\_{\sigma r}' & \mathbf{0} & \left(\mathbf{1} + \frac{\mathbf{L}\_{\sigma r}'}{\mathbf{L}\_{\mathbf{dd}}'}\right) & \frac{\mathbf{L}\_{\sigma r}'}{\mathbf{L}\_{\mathbf{dd}}} \\
\mathbf{0} & \mathbf{L}\_{\sigma r}' & \mathbf{0} & \mathbf{L}\_{\sigma r}' & \frac{\mathbf{L}\_{\sigma r}'}{\mathbf{L}\_{\mathbf{dd}}} & \left(\mathbf{1} + \frac{\mathbf{L}\_{\sigma r}'}{\mathbf{L}\_{\mathbf{eq}}}\right)
\end{bmatrix}
\tag{2}
$$

$$\begin{bmatrix} -\mathbf{r}\_1 & \mathbf{w}\mathbf{L}\_{\mathrm{el}1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{-w} \\ -\mathbf{w}\mathbf{L}\_{\mathrm{el}1} & -\mathbf{r}\_1 & \mathbf{0} & \mathbf{0} & \mathbf{w} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & -\mathbf{r}\_2 & \mathbf{w}\mathbf{L}\_{\mathrm{el}2} & \mathbf{0} & -\mathbf{w} \\ \mathbf{0} & \mathbf{0} & -\mathbf{w}\mathbf{L}\_{\mathrm{el}2} & -\mathbf{r}\_2 & \mathbf{w} & \mathbf{0} \\ \mathbf{r}\_t' & -(\mathbf{w} - \mathbf{w}\_t)\mathbf{L}\_{\mathrm{er}}' & \mathbf{r}\_t' & -(\mathbf{w} - \mathbf{w}\_t)\mathbf{L}\_{\mathrm{er}}' & \frac{\mathbf{r}\_t'}{\mathbf{L}\_{\mathrm{en}}} & -(\mathbf{w} - \mathbf{w}\_t)\left(\mathbf{1} + \frac{\mathbf{L}\_{\mathrm{er}}'}{\mathbf{L}\_{\mathrm{en}}}\right) \\ (\mathbf{w} - \mathbf{w}\_t)\mathbf{L}\_{\mathrm{er}}' & \mathbf{r}\_t' & (\mathbf{w} - \mathbf{w}\_t)\mathbf{L}\_{\mathrm{er}}' & \mathbf{r}\_t' & -(\mathbf{w} - \mathbf{w}\_t)\left(\mathbf{1} + \frac{\mathbf{L}\_{\mathrm{er}}'}{\mathbf{L}\_{\mathrm{en}}}\right) & \frac{\mathbf{r}\_t'}{\mathbf{L}\_{\mathrm{er}}} \\ \end{bmatrix} \tag{3}$$

The nonlinear equations of voltage and current across the shunt excitation capacitor and series compensation capacitors (short-shunt and long-shunt) can be transformed into d-q axis by using reference frame theory, i.e. Park's (dq0) transformation [4], are given by Section 2.2 of [1] and (Section 2.3 of [1] and by following Section 2.2), respectively. Modeling of static loads is also given in Section 3 of [1].

#### 2.2 Modeling of long-shunt capacitors

Current through series capacitors Cls1 and Cls2 (in case of long shunt), connected in series with winding set I and II, respectively, is same as the machine


Table 1. Machine model symbols.

current. The machine current along with series capacitance determine the voltage across series long-shunt capacitor and when transformed in to d-q axis by using Park's transformation is given in Eqs. (4) and (5) [5–7].

$$\begin{aligned} \rho V\_{\text{q1s}} &= \mathbf{i}\_{\text{q1}} / \mathbf{C}\_{\text{ls1}} \\ \rho V\_{\text{d1s}} &= \mathbf{i}\_{\text{d1}} / \mathbf{C}\_{\text{ls1}} \\ \rho V\_{\text{q2s}} &= \mathbf{i}\_{\text{q2}} / \mathbf{C}\_{\text{ls2}} \\ \rho V\_{\text{d2s}} &= \mathbf{i}\_{\text{d2}} / \mathbf{C}\_{\text{ls2}} \end{aligned} \tag{4}$$

and the load terminal voltage is expressed as

$$\begin{aligned} \mathbf{V\_{Lq1}} &= \mathbf{V\_{q1}} + \mathbf{V\_{q1s}} \\ \mathbf{V\_{Ld1}} &= \mathbf{V\_{d1}} + \mathbf{V\_{d1s}} \\ \mathbf{V\_{Lq2}} &= \mathbf{V\_{q2}} + \mathbf{V\_{q2s}} \\ \mathbf{V\_{Ld2}} &= \mathbf{V\_{d2}} + \mathbf{V\_{d2s}} \end{aligned} \tag{5}$$

The remaining symbols of machine model have their usual meanings from Ref. [1] and Table 1.

#### 3. Methodology

In this section, a numerical method is introduced to the solution of Eqs. (1)–(3); where double mixed current flux state space model is discussed by [1]. The ordinary linear differential equations can be solved by the analytical technique rather than approximation method. Eq. (1) is non-linear differential and cannot be solved exactly with high expectations, only approximations are estimated numerically by computer technique using 4th order Runge-Kutta method or classical Runge-Kutta method or often referred as "RK4" as so commonly used [8]. The analytical response of compensated SP-SEIG in only single operating mode is carried out under significant configuration using RK4 subroutine implemented in Matlab Mfile. The dynamic performances were determined under no load, R load and R-L loading condition in only the single mode of excitation capacitor bank, and in both modes of compensating series capacitor bank. The following analytical dynamic responses of series compensated SP-SEIG is considered for the validity of proposed approaches in this chapter.


Analysis of Compensated Six-Phase Self-Excited Induction Generator Using Double Mixed… DOI: http://dx.doi.org/10.5772/intechopen.82323

Figure 2.

Algorithm for Runge-Kutta method implemented for SP-SEIG under constant rated speed at 1000 RPM.

Both analytical responses are well detailed in Section 4. The analytical study of compensated SP-SEIG is given in Section 4 by using an explicit MATLAB program incorporates the RK4 method. An Algorithm of RK4 method for the analysis of compensated SP-SEIG is also shown in Figure 2. The parameters of studied machine and saturation dependent coefficients of system matrix [H] are also reported by [1] for further dynamic analysis of saturated compensated SP-SEIG using double mixed state space variable model under constant rotor speed along with appropriate initial estimated variables values which are also responsible in the development of rated machine terminal voltage and it depends on other machine variables.

#### 4. Analytical response

Analytical performances of compensated SP-SEIG are illustrated in Figures 3–6 along with sudden switching of R load of 200 Ω and a balanced three-phase R-L load (200 Ω in series with 500 mH) at t = 2 s when short-shunt and long-shunt compensation along both three-phase winding sets. These waveforms are

correspond to d-q axis stator voltage, d-q axis stator current, d-q axis load current, d-q axis magnetizing flux, d-q axis series capacitor voltage and magnetizing current. In the case of R-L load, generated voltage and current amplitude has been dropped few more volts and amperes, respectively, compared to R load. Combined Analysis of Compensated Six-Phase Self-Excited Induction Generator Using Double Mixed… DOI: http://dx.doi.org/10.5772/intechopen.82323

#### Figure 4.

Analytical waveforms during sudden switching of RL load at t = 2 s.

amplitude of magnetizing flux, steady-state saturated magnetizing inductance and dynamic (tangent slope) inductance along with the combined amplitude waveform of d-axis stator current and load current during no-load and sudden switching of R load of 200 Ω at t = 2 s are also shown in Figures 3–6. As per discussion of [1], generated RMS value of steady state terminal voltage and current is about 225 V line to line and 4.46 A at rated speed of 1000 RPM with the value of excitation capacitance of 38.5 μF per phase in simple-shunt configuration for the selected machine in this article. In proposing stator current and magnetizing flux mixed variable model,

Figure 5. Analytical waveforms during sudden switching of R load of 200 Ω at t = 2 s.

it has been observed that SP-SEIG terminal voltage and current build-up from their initial values to final steady state values entirely depends upon its initial few Weber values of d-and q-axis magnetizing flux at constant rated speed of 1000 RPM.

## 4.1 Short-shunt series compensated SP-SEIG

Performance of short-shunt SP-SEIG in only its single mode of operation using the values of excitation and series capacitor banks of 38.5 and 108 μF per phase, respectively, has been predicted from the built explicit MATLAB program using RK4 subroutine. Computed waveforms are shown in Figures 3 and 4. Application

Analysis of Compensated Six-Phase Self-Excited Induction Generator Using Double Mixed… DOI: http://dx.doi.org/10.5772/intechopen.82323

Figure 6. Analytical waveforms during sudden switching of R-L load at t = 2 s.

of short-shunt scheme results in overvoltage across the generator terminals as shown in Figure 3a, the per phase voltage level is more than the voltage level of Figure 5a and it is illustrated in Figure 3c.

## 4.1.1 When both three-phase winding sets are connected in short-shunt configuration with independent R loading

The analytical d-q waveform of voltage, current, magnetizing flux and magnetizing current during no-load and sudden switching of R load of 200 Ω at t = 2 s are shown in corresponding Figure 3a, b, d and g. In addition, combined amplitude

waveforms of magnetizing flux, steady-state saturated magnetizing inductance and dynamic (tangent slope) inductance is shown in Figure 3f. The angular displacements of the magnetizing current space vector with respect to the d-axis of the common reference frame are also shown in Figures 3j and k. The combined d-q axis voltage drop across the series short-shunt capacitor is given in Figure 3c and combined amplitude waveform of d-axis stator current and load current during noload and sudden switching of R load of 200 Ω at t = 2 s is given in Figure 3e. The d- and q-axis load currents are also depicted in Figure 3h and i, respectively, during sudden switching of R load of 200 Ω at t = 2 s.

## 4.1.2 When both three-phase winding sets are connected in short-shunt configuration with independent R-L loading

In the same order of figure numbers, all computed waveforms are shown in Figure 4 with sudden switching of R-L load (200 Ω resistance in series with 500 mH inductor) at t = 2 s. Analytical generated RMS steady state voltage and corresponding current at rated speed of 1000 RPM are shown in Figure 4a and b. Drops in d-q-axis generating and lagging load currents are shown in Figure 4b, h and i.

## 4.2 Long-shunt series compensated SP-SEIG

It is also seen that like short-shunt compensation, long-shunt compensation is also self- regulating in nature. In the same manner, analysis of long-shunt SP-SEIG along with a single mode of excitation and the series capacitor banks of 38.5 and 350 F respectively, have also been computed and predicted by using the RK4 subroutine and illustrated in Figures 5 and 6. Here, the value of series capacitor is more than the twice of short-shunt series capacitor.

## 4.2.1 When both three-phase winding sets are connected in long-shunt configuration with independent R loading

The analytical waveform of voltage, current, magnetizing flux and magnetizing current during no-load and sudden switching of R load of 200 Ω at t = 2 s with longshunt compensation along both three-phase winding sets are respectively shown in Figure 5a, b, d and g. Combined amplitude waveforms of magnetizing flux, steadystate saturated magnetizing inductance and dynamic (tangent, slope) inductance is shown in Figure 5f. The angular displacements of the magnetizing current space vector with respect to the d-axis of the common reference frame are also shown in Figure 5j and k. The application of the long-shunt scheme results in less overvoltage or reduced terminal voltage across the generator terminals. As it is shown in Figure 5a, the per phase voltage level is less than the voltage level of Figure 3a. It gives evidence that long-shunt SP-SEIG is able to deliver output power at reduced terminal voltages, as shown in Figure 5c. The combined d-q axis voltage drop across the series short-shunt capacitor is given in Figure 5c and combined amplitude waveform of d-axis stator current and load current during no-load and sudden switching of resistive load of 200 Ω at t = 2 s is given in Figure 5e. The d- and q-axis load currents are also depicted in Figure 5h and i respectively, during sudden switching of resistive load of 200 Ω at t = 2 s. The steady state no-load voltage is generated at rated speed of 1000 RPM.

Analysis of Compensated Six-Phase Self-Excited Induction Generator Using Double Mixed… DOI: http://dx.doi.org/10.5772/intechopen.82323

4.2.2 When both three-phase winding sets are connected in long-shunt configuration with independent R-L loading

In the same order of figure numbers, all computed waveforms are also shown in Figure 6 with sudden switching of R-L load (200 Ω resistance in series with 500 mH inductor) at t = 2 s. The RMS analytical value of steady state voltage is generated at rated speed of 1000 RPM. Same like as Section 4.1, the d-q axis drops in generating and lagging load currents are also given in Figure 6b, h and i.

## 5. Discussion

In the machine model, two mixed state space variables have been chosen in place of single state space variable (general case). Double mixed stator current and airgap flux state space model belongs to one of the more complex model types compared to remaining mixed variable model of stator flux linkages-stator current, rotor flux linkages-rotor current, rotor flux linkages-stator current, stator flux linkages-rotor current, air-gap flux-rotor current, magnetizing current-stator flux linkages and rotor flux linkages-magnetizing current. On the other perspective, considerably simpler than d-q axis winding current model and has simple matrix model. Short-shunt compensation results in overvoltage across the generator terminals during no-load. Whereas, long-shunt compensation gives reduced terminal voltage as compared to short-shunt compensation. In long-shunt configuration, deep saturation provides higher level of magnetizing (or stator) current against large load current at sudden switching of R (or RL) load. While, sometimes, in long-shunt compensation scheme, even reduced generated no load terminal voltage as compared to short-shunt scheme, can be capable of almost same generated total output power. In this fashion, when one moves from simple-shunt to short-shunt (or long-shunt) compensated SP-SEIG, voltage regulation has to be improved by maintaining almost similar magnitude of voltage response after load, as was in simple-shunt SP-SEIG at no-load in Figure 5a of Ref. [1]. Proposed saturated machine model will be applied in air-gap flux field orientation vector control strategy.

## 6. Conclusion

Mixed stator current and air-gap flux as a double state-space variables model preserves information about both stator and rotor parameters. A careful value selection of the combination, i.e. shunt and series capacitors may avoid the excessive voltage across the terminals of SP-SEIG during sudden switching of machine load. It is noticed that involvement of extra supplied reactive power as per selfregulating nature of short-shunt as compared to long-shunt series capacitors in each lines, retains the similar output profile of simple-shunt scheme, when machine load is suddenly switched on after few seconds. In both cases (R and R-L loading), the little bit of marked voltage drops were occurred during R-L loading compared to the R loading when variation from no load to full load.

Electric Power Conversion

## Author details

Kiran Singh Department of Electrical Engineering, Indian Institute of Technology, Roorkee, Uttarakhand, India

\*Address all correspondence to: kiransinghiitr@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapteris distributed underthe terms oftheCreative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Analysis of Compensated Six-Phase Self-Excited Induction Generator Using Double Mixed… DOI: http://dx.doi.org/10.5772/intechopen.82323

## References

[1] Singh K, Singh GK. Modelling and analysis of six-phase self-excited induction generator using mixed stator current and magnetizing flux as statespace variables. Electric Power Components and Systems. 2015;43: 2288-2296

[2] Liao YW, Levi E. Modelling and simulation of a stand-alone induction generator with rotor flux oriented control. Electric Power Systems Research. 1998;46:141-152

[3] Singh K, Singh GK. Stability assessment of isolated six-phase induction generator feeding static loads. Turkish Journal of Electrical Engineering & Computer Sciences. 2016;24:4218-4230

[4] Krause PC. Analysis of Electric Machinery. New York: McGraw Hill Book Company; 1986

[5] Khan MF, Khan MR. Generalized model for investigating the attributes of a six-phase self-excited induction generator over a three-phase variant. International Transactions on Electrical Energy Systems. 2018;28:e2600

[6] Khan MF, Khan MR. Performance analysis of a three phase self-excited induction generator operating with short shunt and long shunt connections. IEEE Biennial International Conference on Power and Energy Systems: Towards Sustainable Energy (PESTSE). 2016:1-6

[7] Chermiti D, Abid N, Khedher A. Voltage regulation approach to a selfexcited induction generator: Theoretical study and experimental validation. International Transactions on Electrical Energy Systems. 2017;27(5):e2311

[8] Kohler Werner E, Johnson Lee W. Elementary Differential Equations with Boundary Value Problems. 2nd ed. Addison-Wesley; 2006

Chapter 3

Abstract

Carlos A. Reusser

torque control (DTC)

generation and distribution applications [1].

nent magnet or reluctance machines and induction machines.

to an excitation field with less harmonic content [2].

1. Introduction

31

Power Converter Topologies for

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

Multiphase variable-speed drives, based on multiphase AC machines, are nowadays the most natural solution for high-demanding industrial, traction and power

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, perma-

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

Multiphase Drive Applications

## Chapter 3

## Power Converter Topologies for Multiphase Drive Applications

Carlos A. Reusser

## 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 ndimensional space, corresponding to the respective state variables. The machine model, on its original phase-variable form, can be transformed into n=2 twodimensional subspaces, for a machine with an even number of phases. If the number of phases is odd, then the original n-dimensional subspace can be decomposed into a ðn � 1Þ=2 two-dimensional subspaces and a single-dimensional quantity, which corresponds to a common mode subspace.

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 uk vk, where k stands for the respective new orthogonal subspace.

<sup>T</sup> Let us consider an arbitrary state variable <sup>λ</sup>, defined as <sup>λ</sup> ¼ ½λ<sup>1</sup> <sup>λ</sup><sup>2</sup> … <sup>λ</sup><sup>n</sup> � , and let Th i be a linear operator, which transforms the n-dimensional space into n=2 twodimensional subspaces, for n even, as in Eq. (1):

$$T(\cdot) = \sqrt{\frac{2}{n}} \begin{bmatrix} 1 & \cos\left(a\right) & \cos\left(2a\right) & \dots & \cos\left(2a\right) & \cos\left(a\right) \\ 0 & \sin\left(a\right) & \sin\left(2a\right) & \dots & -\sin\left(2a\right) & -\sin\left(a\right) \\ 1 & \cos\left(2a\right) & \cos\left(4a\right) & \dots & \cos\left(4a\right) & \cos\left(2a\right) \\ 0 & \sin\left(2a\right) & \sin\left(4a\right) & \dots & -\sin\left(4a\right) & -\sin\left(2a\right) \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \cos\left(\frac{n}{2}\right)a & \cos\left2\left(\frac{n}{2}\right)a & \dots & \cos 2\left(\frac{n}{2}\right)a & \cos\left(\frac{n}{2}a\right) \\ 0 & \sin\left(\frac{n}{2}\right)a & \sin 2\left(\frac{n}{2}\right)a & \dots & -\sin 2\left(\frac{n}{2}\right)a & -\sin\left(\frac{n}{2}\right)a \\ & & & & \end{bmatrix} \tag{1}$$

In the case of n odd, then the corresponding transformation is given as in Eq. (2):

$$T(\cdot) = \sqrt{\frac{2}{n}} \begin{bmatrix} 1 & \cos(a) & \cos(2a) & \cdots & \cos(2a) & \cos(a) \\ 0 & \sin(a) & \sin(2a) & \cdots & -\sin(2a) & -\sin(a) \\ 1 & \cos(2a) & \cos(4a) & \cdots & \cos(4a) & \cos(2a) \\ 0 & \sin(2a) & \sin(4a) & \cdots & -\sin(4a) & -\sin(2a) \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \cos\left(\frac{(n-1)}{2}\right)a & \cos 2\left(\frac{(n-1)}{2}\right)a & \cdots & \cos 2\left(\frac{(n-1)}{2}\right)a & \cos 2\left(\frac{(n-1)}{2}\right)a \\ 0 & \sin\left(\frac{(n-1)}{2}\right)a & \sin 2\left(\frac{(n-1)}{2}\right)a & \cdots & \sin 2\left(\frac{(n-1)}{2}\right)a & -\sin 2\left(\frac{(n-1)}{2}\right)a \\ \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} & \cdots & \sqrt{\frac{1}{2}} & \sqrt{\frac{1}{2}} \end{bmatrix} a \tag{2}$$

Power Converter Topologies for Multiphase Drive Applications DOI: http://dx.doi.org/10.5772/intechopen.81977

The last row of Th i in Eq. (2) corresponds to the projection of the state variable onto the common mode subspace.

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 u<sup>1</sup> v<sup>1</sup> or αβ subspace, using the linear transformation Th in!αβ; this definition can be found as the generalized theory of electrical machines.

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

$$\left[\boldsymbol{\nu}\_{s} = \left[\boldsymbol{R}\_{s}\right]\dot{\mathbf{i}}\_{s} + \frac{d}{dt}\boldsymbol{\Psi}\_{s} \right. \tag{3}$$

$$\boldsymbol{\nu}\_r = \left[ \boldsymbol{R}\_r \right] \dot{\mathbf{i}}\_r + \frac{d}{dt} \boldsymbol{\Psi}\_r \tag{4}$$

$$\boldsymbol{\sigma}\_{s} = \begin{bmatrix} \boldsymbol{v}\_{s1} \boldsymbol{v}\_{s2} \dots \boldsymbol{v}\_{\boldsymbol{m}} \end{bmatrix}^{\mathrm{T}} \qquad \dot{\mathbf{t}}\_{s} = \begin{bmatrix} \dot{\boldsymbol{i}}\_{s1} \dot{\boldsymbol{i}}\_{s2} \dots \dot{\boldsymbol{i}}\_{\boldsymbol{m}} \end{bmatrix}^{\mathrm{T}} \qquad \boldsymbol{\Psi}\_{s} = \begin{bmatrix} \boldsymbol{\Psi}\_{s1} \boldsymbol{\Psi}\_{s2} \dots \boldsymbol{\Psi}\_{\boldsymbol{m}} \end{bmatrix}^{\mathrm{T}} \tag{5}$$

$$\boldsymbol{\mathfrak{v}}\_{r} = \begin{bmatrix} \boldsymbol{v}\_{r1} \boldsymbol{v}\_{r2} \dots \boldsymbol{v}\_{m} \end{bmatrix}^{T} \qquad \dot{\mathbf{t}}\_{r} = \begin{bmatrix} \dot{\boldsymbol{i}}\_{r1} \dot{\boldsymbol{i}}\_{r2} \dots \dot{\boldsymbol{i}}\_{m} \end{bmatrix}^{T} \qquad \boldsymbol{\mathfrak{w}}\_{r} = \begin{bmatrix} \boldsymbol{\psi}\_{r1} \boldsymbol{\psi}\_{r2} \dots \boldsymbol{\psi}\_{rm} \end{bmatrix}^{T} \tag{6}$$

Considering symmetrical windings for both stator and rotor, then ½ � R and ½ � <sup>s</sup> Rr ð Þ are diagonal <sup>n</sup> � <sup>n</sup> matrices, thus <sup>½</sup>Rs ð Þ <sup>n</sup> <sup>½</sup>Rr ð Þ <sup>n</sup> � ¼ diag Rs ð Þ and � ¼ diag Rr . The stator and rotor flux linkages can be found as in Eqs. (7) and (8):

$$\mathbf{w}\_s = \begin{bmatrix} L\_s \end{bmatrix} \dot{\mathbf{z}}\_s + \begin{bmatrix} L\_{sr} \end{bmatrix} \dot{\mathbf{z}}\_r \tag{7}$$

$$\mathbf{w}\_r = \begin{bmatrix} L\_r \end{bmatrix} \dot{\mathbf{r}}\_r + \begin{bmatrix} L\_{rs} \end{bmatrix} \dot{\mathbf{r}}\_s \tag{8}$$

where due to the machine's symmetry ½Lrs � ¼ ½Lsr � T ; the corresponding stator and rotor inductance matrices are described as follows:

$$\begin{aligned} \begin{bmatrix} L\_{\iota} \end{bmatrix} &= \begin{bmatrix} L\_{\iota 11} & L\_{\iota 12} & \dots & L\_{\iota 1(n-1)} & L\_{\iota 1n} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ L\_{\iota n1} & L\_{\iota n2} & \dots & L\_{\iota n(n-1)} & L\_{\iota mn} \end{bmatrix} \\ \begin{bmatrix} L\_{r} \end{bmatrix} &= \begin{bmatrix} L\_{r11} & L\_{r12} & \dots & L\_{r1(n-1)} & L\_{r1n} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ L\_{rn1} & L\_{rn2} & \dots & L\_{rn(n-1)} & L\_{rnn} \end{bmatrix} \end{aligned} \tag{10}$$

It has to be noticed that for both stator and rotor windings, L<sup>11</sup> ¼ L<sup>22</sup> ¼ … ¼ Lnn which correspond to the winding self-inductance L; for the corresponding stator and rotor self-inductances within each one, due to the machine's symmetry, it fulfills that Ljk ¼ Lkj ¼ Lm ∀j, k j ¼6 k. Then for the stator and rotor winding selfinductances, it can be stated that Ljj ¼ L<sup>σ</sup> þ Lm where L<sup>σ</sup> corresponds to the stator or rotor winding leakage inductance. On the other hand, the stator-to-rotor mutual inductance matrix ½Lsr � can be found as in Eq. (11):

$$[L\_{\mathcal{V}}] = \mathcal{M} \begin{bmatrix} \cos\left(\theta\_{r}\right) & \cos\left(\theta\_{r} - (n-1)a\right) & \cos\left(\theta\_{r} - (n-2)a\right) & \dots & \cos\left(\theta\_{r} - a\right) \\ \cos\left(\theta\_{r} - a\right) & \cos\left(\theta\_{r}\right) & \cos\left(\theta\_{r} - (n-1)a\right) & \dots & \cos\left(\theta\_{r} - 2a\right) \\ \cos\left(\theta\_{r} - 2a\right) & \cos\left(\theta\_{r} - a\right) & \cos\left(\theta\_{r}\right) & \dots & \cos\left(\theta\_{r} - 3a\right) \\ \vdots & \vdots & \vdots & \dots & \vdots \\ \cos\left(\theta\_{r} - (n-1)a\right) & \cos\left(\theta\_{r} - (n-2)a\right) & \cos\left(\theta\_{r} - (n-3)a\right) & \dots & \cos\left(\theta\_{r}\right) \end{bmatrix} \tag{11}$$

The resulting generalized machine representation is obtained by applying the linear transformation operator Th i αβ to Eqs. (3) and (4), resulting in the follow- <sup>n</sup>! ing dynamic representation, in the αβ subspace, in Eqs. (12) and (13):

$$\boldsymbol{\sigma}\_{s}^{(a\boldsymbol{\beta})} = \boldsymbol{R}\_{s}\mathbf{i}\_{s}^{(a\boldsymbol{\beta})} + \frac{d}{dt}\boldsymbol{\Psi}\_{s}^{(a\boldsymbol{\beta})} \tag{12}$$

$$\boldsymbol{\sigma}\_{r}^{(a\boldsymbol{\beta})} = R\_{s}\mathbf{i}\_{r}^{(a\boldsymbol{\beta})} + \frac{d}{dt}\boldsymbol{\Psi}\_{r}^{(a\boldsymbol{\beta})} - j\,\boldsymbol{\alpha}\_{r}\boldsymbol{\Psi}\_{r}^{(a\boldsymbol{\beta})} \tag{13}$$

The corresponding stator and rotor flux linkages given in Eqs. (7) and (8) are transformed into the αβ subspace as in Eqs. (14) and (15):

$$\left[\Psi\_{\mathfrak{s}}^{(a\beta)}\right] = \left[L\_{\mathfrak{s}}\right] \left[\Psi\_{\mathfrak{s}}^{(a\beta)}\right] + \left[L\_{\mathfrak{s}r}\right] \mathfrak{i}\_{\mathfrak{r}}^{(a\beta)}\tag{14}$$

$$\boldsymbol{\Psi}\_r^{(a\boldsymbol{\beta})} = \left[L\_r\right] \mathfrak{i}\_r^{(a\boldsymbol{\beta})} + \left[L\_{rs}\right] \mathfrak{i}\_s^{(a\boldsymbol{\beta})} \tag{15}$$

The presented methodology considers a n-phase winding with uniform distri- <sup>2</sup> <sup>π</sup> bution that is <sup>α</sup> <sup>¼</sup> ; in the case of machine with multiple groups of three-phase <sup>n</sup> windings, α is the shifting angle between each group of three-phase windings. It has to be noted that for each three-phase winding transformation, an additional row has to be included in Eq. (2).

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

$$T\_{\epsilon} = \frac{\partial}{\partial \theta\_r} \mathcal{W}\_{fld} \left( \dot{\mathbf{t}}\_s^{(a\beta)}, \boldsymbol{\Psi}\_s^{(a\beta)}, \theta\_r \right) \tag{16}$$

$$T\_e = \frac{1}{2} \left[ \dot{\mathbf{t}}^{(a\beta)} \right] \frac{\partial}{\partial \theta\_r} \begin{bmatrix} L \end{bmatrix} \begin{bmatrix} \dot{\mathbf{t}}^{(a\beta)} \end{bmatrix}^T \tag{17}$$

$$\begin{bmatrix} \mathbf{i}^{(a\boldsymbol{\beta})} \end{bmatrix} = \begin{bmatrix} \mathbf{i}\_{\boldsymbol{\mathfrak{s}}}^{(a\boldsymbol{\mathfrak{s}})} & \mathbf{i}\_{\boldsymbol{r}}^{(a\boldsymbol{\mathfrak{s}})} \end{bmatrix}^{T} \tag{18}$$

$$\begin{aligned} \begin{bmatrix} L \end{bmatrix} = \begin{bmatrix} \begin{bmatrix} L\_{\rm s} \end{bmatrix} & \begin{bmatrix} L\_{\rm sr} \end{bmatrix} \\ \begin{bmatrix} L\_{\rm sr} \end{bmatrix}^T & \begin{bmatrix} L\_{\rm r} \end{bmatrix} \end{bmatrix} \end{aligned} \tag{19}$$

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

$$T\_e = \frac{1}{2} \dot{\mathfrak{t}}\_s^{(a\beta)} \frac{\partial}{\partial \theta\_r} \left[ L\_{sr} \right] \dot{\mathfrak{t}}\_r^{(a\beta)^T} \tag{20}$$

The generalized machine representation given by Eqs. (12), (13) and (20) can be expressed in an arbitrary orthogonal synchronous reference frame dq, which rotates at synchronous speed ωk; thus the corresponding rotation into the dq state variables is given by Eq. (21):

$$
\theta\_k = \int a\_k \, dt \tag{21}
$$

Rotation into the dq subspace is given by the unitary rotation matrix U defined in Eq. (22):

$$U = \begin{bmatrix} \cos \theta\_k & \sin \theta\_k \\ -\sin \theta\_k & \cos \theta\_k \end{bmatrix} \tag{22}$$

The generalized machine dynamic representation in the dq reference frame, as consequence of the rotation of Eqs. (12), (13) and (20), is given in Eqs. (23)–(25), representing the Kron's primitive machine model in a generalized synchronous reference frame:

$$\left(\boldsymbol{\sigma}\_{s}^{(dq)}\right) = R\_{s}\dot{\mathbf{i}}\_{s}^{(dq)} + \frac{d}{dt}\boldsymbol{\Psi}\_{s}^{(dq)} + j\alpha\_{k}\boldsymbol{\Psi}\_{s}^{(dq)}\tag{23}$$

$$\left(\boldsymbol{\nu}\_{r}^{(dq)}\right) = \boldsymbol{R}\_{s}\dot{\mathbf{i}}\_{r}^{(dq)} + \frac{d}{dt}\boldsymbol{\mu}\_{r}^{(dq)} + j\left(\boldsymbol{\alpha}\_{k} - \boldsymbol{\alpha}\_{r}\right)\boldsymbol{\Psi}\_{r}^{(dq)}\tag{24}$$

$$T\_e = p \frac{L\_m}{L\_r} \left( \psi\_r^d i\_s^q - \psi\_r^q i\_s^d \right) \tag{25}$$

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):

$$
\begin{bmatrix} v\_t^d \\ v\_t^q \\ v\_t^d \\ v\_r^q \end{bmatrix} = \begin{bmatrix} R\_\varepsilon + L\_\varepsilon \frac{d}{dt} & -\alpha\_k L\_\varepsilon & L\_m \frac{d}{dt} & -\alpha\_k L\_m \\ & \alpha\_k L\_\varepsilon & R\_\varepsilon + L\_\varepsilon \frac{d}{dt} & \alpha\_k L\_m & L\_m \frac{d}{dt} \\ & \alpha\_k \frac{d}{dt} & -(\alpha\_k - \alpha\_r) L\_m & R\_r + L\_r \frac{d}{dt} & -(\alpha\_k - \alpha\_r) L\_r \\ & L\_m \frac{d}{dt} & -(\alpha\_k - \alpha\_r) L\_m & R\_r + L\_r \frac{d}{dt} & -(\alpha\_k - \alpha\_r) L\_r \\ & (\alpha\_k - \alpha\_r) L\_m & L\_m \frac{d}{dt} & (\alpha\_k - \alpha\_r) L\_r & R\_r + L\_r \frac{d}{dt} \end{bmatrix} \begin{bmatrix} i\_t^d \\ i\_t^d \\ i\_t^d \\ i\_r^d \\ i\_t^d \end{bmatrix} \tag{26}
$$
 
$$
\begin{bmatrix} \mathbf{v}^{(dq)} \end{bmatrix} = [\mathbf{z}] \begin{bmatrix} \mathbf{i}^{(dq)} \end{bmatrix} \tag{27}
$$

### 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:


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.

6 The linear transformation operator <sup>T</sup>h i is defined as in Eq. (28), with <sup>φ</sup> <sup>¼</sup> <sup>π</sup>:

$$T(\cdot) = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & \cos 4\rho & \cos 8\rho & \cos \rho & \cos 5\rho & \cos 9\rho \\ 0 & \sin 4\rho & \sin 8\rho & \sin \rho & \sin 5\rho & \sin 9\rho \\ 1 & \cos 8\rho & \cos 4\rho & \cos 5\rho & \cos 9\rho & \cos 9\rho \\ 0 & \sin 8\rho & \sin 4\rho & \sin 5\rho & \sin \rho & \sin 9\rho \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \end{bmatrix} \tag{28}$$

The equivalent model in the dq synchronous subspace can be derived from Eq. (26); in this case it has to be noted that the synchronous reference frame dq coincides with the rotor natural reference frame, thus ω<sup>k</sup> ¼ ωr, becoming Eq. (29) and (30) for the stator:

$$
\psi\_s^d = R\_s i\_s^d + \frac{d}{dt} \psi\_s^d - \alpha\_k \psi\_s^q \tag{29}
$$

$$\upsilon\_s^q = R\_s i\_s^q + \frac{d}{dt} \psi\_s^q + a \nu\_k \psi\_s^d \tag{30}$$

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

$$\mathbf{O} = R\_r \mathbf{i}\_r^d + \frac{d}{dt} \boldsymbol{\psi}\_r^d \tag{31}$$

$$\mathbf{O} = R\_r i\_r^q + \frac{d}{dt} \psi\_r^q \tag{32}$$

Figure 1. Dual three-phase stator winding configuration.

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

$$
\mu \boldsymbol{\upmu}\_s^d = L\_{sd} \dot{\mathbf{i}}\_s^d + L\_{md} \dot{\mathbf{i}}\_r^d + \boldsymbol{\upmu}\_m \tag{33}
$$

$$
\mu \mathfrak{g}\_s^q = L\_{sq} \mathfrak{i}\_s^q + L\_{mq} \mathfrak{i}\_r^q \tag{34}
$$

$$
\mu \boldsymbol{\upmu}\_r^d = L\_{rd} \dot{\mathbf{i}}\_r^d + L\_{md} \dot{\mathbf{i}}\_s^d + \boldsymbol{\upmu}\_m \tag{35}
$$

$$
\mu \mathfrak{g}\_r^q = L\_{rq} \mathfrak{i}\_r^q + L\_{mq} \mathfrak{i}\_s^q \tag{36}
$$

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

$$T\_{\epsilon} = p\left[\boldsymbol{\mu}\_{m}\dot{\boldsymbol{i}}\_{s}^{q} + \left(\boldsymbol{L}\_{md}\dot{\boldsymbol{i}}\_{r}^{d}\dot{\boldsymbol{i}}\_{s}^{q} - \boldsymbol{L}\_{mq}\dot{\boldsymbol{i}}\_{r}^{q}\dot{\boldsymbol{i}}\_{s}^{d}\right)\right] + p\left[\boldsymbol{L}\_{md} - \boldsymbol{L}\_{mq}\right]\dot{\boldsymbol{i}}\_{s}^{q}\dot{\boldsymbol{i}}\_{s}^{d} \tag{37}$$

#### 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):

$$
\psi\_s^d = R\_s i\_s^d + \frac{d}{dt} \psi\_s^d - \alpha\_k \psi\_s^q \tag{38}
$$

$$
\omega \upsilon\_s^q = R\_s \dot{\imath}\_s^q + \frac{d}{dt} \psi\_s^q + a \wp\_k \psi\_s^d \tag{39}
$$

$$\mathbf{0} = R\_r \mathbf{i}\_r^d + \frac{d}{dt} \boldsymbol{\mu}\_r^d \tag{40}$$

$$\mathbf{0} = R\_r i\_r^q + \frac{d}{dt} \psi\_r^q \tag{41}$$

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):

$$
\mu \boldsymbol{\varphi}\_s^d = L\_s \dot{\mathbf{r}}\_s^d + L\_m \dot{\mathbf{r}}\_r^d \tag{42}
$$

$$
\mu\_s^q = L\_s i\_s^q + L\_m i\_r^q \tag{43}
$$

$$
\mu\_r^d = L\_r \dot{\mathbf{i}}\_r^d + L\_m \dot{\mathbf{i}}\_s^d \tag{44}
$$

$$
\mu \overline{\varphi}\_r^q = L\_r \overline{i}\_r^q + L\_m \overline{i}\_s^q \tag{45}
$$

The electromechanical torque expression can be derived from Eq. (37) by considering ψ <sup>m</sup> ¼ 0, Lmd ¼ Lmq ¼ Lm, becoming Eq. (46):

$$T\_e = pL\_m \left[ \dot{\imath}\_r^d \dot{\imath}\_s^q - \dot{\imath}\_r^q \dot{\imath}\_s^d \right] \tag{46}$$

### 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 n individual half-cells connected in parallel to a single dc-link, as shown in Figure 3.

Modulation of each individual half-cell circuit is implemented in such a way, to obtain n voltage output signals shifted by 2π=n. Generation of the required voltage signals is supported by classical carrier-based strategies, such as sinusoidal pulse width modulation (SPWM) and space vector pulse width modulation (SVPWM).

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 Sx, as given in Eqs. (46) and (48):

$$\mathcal{S}\_{\mathfrak{x}} = \begin{cases} \mathbf{1} & |u\_{\mathfrak{x}N}^{\*}| \ge |u\_{\mathfrak{c}}| \\ \mathbf{0} & |u\_{\mathfrak{x}N}^{\*}| \le |u\_{\mathfrak{c}}| \end{cases} \tag{47}$$

Figure 2.

Classical VSC topologies: (a) two-level bridge, (b) neutral-point clamped and (c) cascaded H-bridge.

Power Converter Topologies for Multiphase Drive Applications DOI: http://dx.doi.org/10.5772/intechopen.81977

$$\mu\_{\rm xN}^\* = v\_{\rm xN}^\* - \frac{1}{2} \left\{ \min \left( v\_{aN}^\*, v\_{bN}^\*, \dots, v\_{nN}^\* \right) + \max \left( v\_{aN}^\*, v\_{bN}^\*, \dots, v\_{nN}^\* \right) \right\} \tag{48}$$

where <sup>∗</sup> <sup>x</sup> stands for the corresponding phase <sup>x</sup> ¼ ½1; … ; <sup>n</sup>�, uxN is the reference to be synthesized for the corresponding phase, and uc stands for the carrier wave.

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):

$$\mathcal{S}\_{\mathbf{x}m} = \begin{cases} \mathbf{1} & |u\_{\mathbf{x}N}^{\*}| \ge |u\_{cm}| \\ \mathbf{0} & |u\_{\mathbf{x}N}^{\*}| \le |u\_{cm}| \end{cases} \tag{49}$$

where x stands for the corresponding phase x ¼ ½1; … ; n�, m to the corresponding level and, ucm is the carrier for the matched m level.

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 ðn � 1Þ dimensional spaces, which can be decomposed into ðn � 1Þ=2 subspaces, ensuring redundant switching states.

Figure 3. Multiphase multicell topology.

Figure 4. Common connected load arrangement.

#### Electric Power Conversion

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 N as presented in Figure 4; the main drawback with this configuration is the circulation of zero sequence currents, since the neutral points N � n are not insulated, thus establishing a common mode voltage vNn 6¼ 0 in the dq synchronous reference frame.

Split connected load arrangement is possible, if n ¼ 3k∣k ¼ ½1; 2; … �, so the n phase system can be divided into k three-phase insulated independent subsystems, as shown in Figure 5, with symmetric or asymmetric electrical shifting δ. It has to be noted that independently on the symmetric or asymmetric configuration, the

Figure 5. Split connected load arrangements. (a) Single-channel topology and (b) multichannel topology.

Figure 6. mxn matrix converter equivalent model.

Power Converter Topologies for Multiphase Drive Applications DOI: http://dx.doi.org/10.5772/intechopen.81977

electrical shifting between each phase on a single three-phase group is still 2π=3, constituting each individual symmetrical three-phase system.

Figure 6a shows a split-phase single-channel arrangement where the corresponding phase for each half-cell is derived as <sup>y</sup>ℓ, yℓþk, yℓþ2<sup>k</sup> with <sup>ℓ</sup> ¼ ½1; … ; <sup>k</sup>�. A multichannel arrangement is shown in Figure 6b, which introduces some advantages, like full load dynamic decoupling and lower dc-link power rating. On the other hand, additional dc-link capacitors are required.

For both previously described topologies, SPWM is achieved by implementing the modulation as given in Eqs. (47) and (48) considering the asymmetry (if <sup>∗</sup> required) <sup>φ</sup> for each reference as <sup>u</sup> ð Þ <sup>φ</sup> . xN

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 n=2 orthogonal subspaces for n even and into ðn � 1Þ=2 orthogonal subspaces and single-dimensional quantity (common mode component) for n odd. Under this formulation, the reference voltage space vector to be synthesized can be expressed as in Eqs. (50) and (51) for n even and odd, respectively:

$$\mathfrak{u}\_{\mathfrak{x}}^{\*} = \left[ \mathfrak{u}^{(1)} \mathfrak{u}^{(2)} \dots , \mathfrak{u}^{(n)} \right]^{T} \tag{50}$$

$$\mathfrak{u}\_{\mathfrak{x}}^{\*} = \left[ \mathfrak{u}^{(1)} \mathfrak{u}^{(2)} \dots , \mathfrak{u}^{(n-1)} \right]^{T} \tag{51}$$

For each orthogonal subspace, there exits two active vectors, vuk and vvk, corresponding to k orthogonal subspace. Each active vector is applied for a duty-cycle δ to synthesize the corresponding reference space vector, as in Eqs. (52) and (53):

$$\mathfrak{u}\_{\mathbf{x}}^{\*} = \delta\_{\mathfrak{u}1}\overline{\boldsymbol{\nu}}^{(\mathfrak{u}1)} + \delta\_{\mathfrak{v}1}\overline{\boldsymbol{\nu}}^{(\mathfrak{v}2)} + \delta\_{\mathfrak{u}2}\overline{\boldsymbol{\nu}}^{(\mathfrak{u}2)} + \delta\_{\mathfrak{v}2}\overline{\boldsymbol{\nu}}^{(\mathfrak{v}2)} + \dots + \delta\_{\mathfrak{u}(\mathfrak{u}/2)}\overline{\boldsymbol{\nu}}^{(\mathfrak{u}(\mathfrak{u}/2))} + \delta\_{\mathfrak{v}(\mathfrak{u}/2)}\overline{\boldsymbol{\nu}}^{(\mathfrak{v}(\mathfrak{u}/2))} \tag{52}$$

$$\mathfrak{u}\_{\mathbf{x}}^{\*} = \sum\_{j=1}^{n/2} \delta\_{\mathfrak{u}j} \overline{\nu}^{(\mathfrak{u}j)} + \sum\_{j=1}^{n/2} \delta\_{\mathfrak{v}j} \overline{\nu}^{(\mathfrak{v}j)} \tag{53}$$

In the case of n odd, an additional degree of freedom is introduced by the common mode voltage vcm which represents a single quantity and not a space vector as in Eq. (54):

$$\mathfrak{u}\_{\mathbf{x}}^{\*} = \sum\_{j=1}^{(n-1)/2} \delta\_{\mathfrak{u}j} \overline{\boldsymbol{\nu}}^{(\boldsymbol{u}j)} + \sum\_{j=1}^{(n-1)/2} \delta\_{\mathfrak{v}j} \overline{\boldsymbol{\nu}}^{(\boldsymbol{v}j)} + \delta\_{\boldsymbol{cm}} \boldsymbol{v}\_{\mathbf{cm}} \tag{54}$$

Eqs (53) and (54) can be interpreted as the partitioning of the αβ stationary reference frame, into n! or ðn � 1Þ! adjacent sectors, and an additional common mode component, orthogonal to the αβ subspace. Sector identification can then be achieved by implementing Eq. (55) where S corresponds to the given sector within the αβ subspace:

$$S = \left(\frac{\theta}{M!}\right) + \mathbf{1} \quad M = \begin{cases} n & \text{even number of subspaces.}\\ (n-1) & \text{odd number of subspaces.} \end{cases} \tag{55}$$

#### 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 mxn power switches, where m is the number of input phases and n the number of output phases. In Figure 6, the equivalent model of a mxn matrix converter is shown.

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 Sjk, such that

$$\mathbf{S}\_{jk} = \begin{cases} 1 & \text{switch} \mathbf{S}\_{jk} \text{ is in on state.}\\ 0 & \text{switch} \mathbf{S}\_{jk} \text{ is in off state.} \end{cases} \quad \forall j = 1, 2, \dots, m \quad k = 1, 2, \dots, n \tag{56}$$

For every instant t, the switching state Sjk must comply with both conditions stated in Eqs (57) and (58), thus meaning that for all t only one input phase is connected to one output phase, avoiding short-circuit condition, and also all output phases are connected to at least one input phase, ensuring no open-circuit condition. This last condition is intended to protect the bidirectional switches that cannot handle reverse current flow, due to the inductive energy discharge:

$$\sum\_{j=1}^{n} \mathbf{S}\_{jk} = \mathbf{1} \tag{57}$$

$$S\_{jk} = \mathbf{1} \quad \forall k = \mathbf{1}, \mathbf{2}, \dots, n \tag{58}$$

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):

$$\begin{aligned} \begin{bmatrix} \mathbf{V\_{o}} \end{bmatrix} = \begin{bmatrix} M \end{bmatrix} \begin{bmatrix} \mathbf{V\_{i}} \end{bmatrix} & M = \begin{bmatrix} \delta\_{11} & \delta\_{12} & \dots & \delta\_{1m} \\ \delta\_{21} & \delta\_{22} & \dots & \delta\_{2m} \\ \vdots & \vdots & \dots & \vdots \\ \delta\_{n1} & \delta\_{n2} & \dots & \delta\_{nm} \end{bmatrix} \end{aligned} \tag{59}$$

$$\begin{bmatrix} \mathbf{I}\_{\mathbf{i}} \end{bmatrix} = \begin{bmatrix} \mathbf{M} \end{bmatrix}^{T} \begin{bmatrix} \mathbf{I}\_{\mathbf{o}} \end{bmatrix} \tag{60}$$

$$P\_i = P\_o \tag{61}$$

where ½Vo � and ½Vi � stand for the output and input voltage space vectors, respectively, ½ � Io and ½ � Ii for the output and input current space vectors, ½ � M is the low-frequency matrix or modulation index transfer matrix, and Po and Pi correspond to the output and input active power.

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 ð Þ possible switching states is 2 nm , from which the forbidden conditions have to be considered. Space vector implementation is largely explained in literature [9].

Power Converter Topologies for Multiphase Drive Applications DOI: http://dx.doi.org/10.5772/intechopen.81977

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 nineswitch topology, because of the fact that Sx, Sy and Sz switches are shared by both converter stages A and B. Hence 3 switches are eliminated of the 12 needed for

Figure 7. (a) Classical arrangement with 12 switches. (b) Nine-switch converter topology.

Figure 8. Nine-switch converter allowed switching states.

multiphase operation as in multichannel topology, some forbidden switching states are introduced, thus remaining 27 allowed states, and also the maximum output voltage gain is limited [13]. In Figure 8, the allowed switching states per leg are shown.

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 S4, S<sup>5</sup> and S6, its switching pattern can be obtained as in Eq. (62):

$$\mathbf{S}\_{k+3} = \overline{\mathbf{S}\_k \cdot \mathbf{S}\_{k+6}} \quad k = \mathbf{1}, \dots, \mathbf{3} \tag{62}$$

The implementation of SPWM considers the use of two voltage reference space vectors x<sup>∗</sup> and y<sup>∗</sup>, for each three-phase winding group. The NSC can operate either in constant (CFM) or variable frequency (VFM) modes, depending on each space vector angular frequency. However, due to the nature of the particular application in multiphase drives, the NSC is to be operated in the constant frequency mode, as presented in Figure 9.

Figure 9. CFM modulation.

Power Converter Topologies for Multiphase Drive Applications DOI: http://dx.doi.org/10.5772/intechopen.81977

Figure 10. Space vector decomposition. (a) Single carrier, (b) shifted multi-carrier and (c) line-line voltage.

For the constant frequency mode operation (CFM), both reference space vectors x<sup>∗</sup> and y<sup>∗</sup> have the same angular frequency ωk, preserving its spatial shifting δ, so they are to be found at least in contiguous sectors of the voltage hexagon, as shown in Figure 10a, being the maximum voltage gain g given in Eq. (63):

$$\mathbf{g} = \frac{1}{\sqrt{3}\left(\frac{\delta}{2} + \frac{\pi}{3}\right)}\tag{63}$$

By introducing a phase shifting of π=6 between both PWM carriers, a second group of active vectors is introduced as shown in Figure 10b, which results in a voltage gain as in Eq. (64):

Figure 11.

Simulation results for a back-to-back NSC for wind energy conversion.

Figure 13. 11-switch converter.

$$\mathbf{g} = \frac{1}{\sqrt{3}\left(\delta + \frac{\pi}{6}\right)}\tag{64}$$

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 Sx, Sy, whose main propose is to mitigate the common mode voltage during the zero switching states.

### 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 αβ subspace, thus meaning that the other subspaces do not contribute to the energy conversion process but only losses. This fact makes possible the implementation of oriented control schemes, such as field-oriented control (FOC), direct torque control (DTC) and model-based predictive control (MBPC), in a synchronous reference frame dq, by rotating the state variables in αβ subspace.

Power Converter Topologies for Multiphase Drive Applications DOI: http://dx.doi.org/10.5772/intechopen.81977

#### Figure 14.

dq synchronous reference frame rotation.

Figure 16.

Dual three-phase winding induction machine FOC scheme simulation results.

An additional control loop in the dq reference frame can be implemented, to compensate the losses in the remaining subspaces. Control goals for the multiphase drive can be summarized as maximum torque per Ampere operation, control of nominal flux and control of rotor speed/torque.

#### Figure 17.

Direct torque control scheme of a dual three-phase winding PM synchronous machine.

#### 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. Fieldoriented control is based in the decoupling of the current space vector, into a fluxproducing component and a torque-producing component. This is achieved by rotating the current space vector from αβ subspace, into the synchronous rotating reference frame dq, oriented with respect to the rotor flux linkage space vector, as shown in Figure 14.

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 i q ,s because ψ<sup>q</sup> <sup>r</sup> ¼ 0, due to the orientation of ψ<sup>r</sup> which only exists in the direct axis orientation. Implementation of the FOC scheme for a multiphase AC drive is shown in Figure 15.

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.

#### Power Converter Topologies for Multiphase Drive Applications DOI: http://dx.doi.org/10.5772/intechopen.81977

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).

Electric Power Conversion

## Author details

Carlos A. Reusser Department of Electronics, Universidad Tecnica Federico Santa Maria, Valparaiso, Chile

\*Address all correspondence to: carlos.reusser@usm.cl

© 2018 The Author(s). Licensee IntechOpen. This chapteris distributed underthe terms oftheCreative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Power Converter Topologies for Multiphase Drive Applications DOI: http://dx.doi.org/10.5772/intechopen.81977

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Section 3
