decoupled vector control scheme for dual three-phase permanent magnet synchronous machines

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1

.

Abstract—Dual three-phase electric machines have many ad-

vantages compared with conventional three-phase machines.

However, the properties of such machines present challenges for

current control that make it difficult to produce sufficient per-

formance for the electric drive. This paper introduces an im-

proved vector control scheme for dual three-phase permanent

magnet synchronous machines. The study includes detailed solu-

tions for key parts of the control such as reference frame trans-

formations, decoupling of the current control loops, and modula-

tion. The performance of the suggested control scheme is evaluat-

ed using finite-element analyses and experimental results. The

results show that the scheme can produce desired dynamics for

the current control and guarantees balanced current sharing be-

tween the winding sets. In addition, the proposed solution is ca-

pable of reducing current harmonics produced by the internal

structure of the machine. This problem is, however, only partly

solved since complete elimination of harmonic components is not

achieved. Nevertheless, the suggested control scheme overcomes

many of the disadvantages found with other control solutions.

Improved control performance allows the full benefits of dual

three-phase drives to be utilized even in demanding applications.

Index Terms—current control, multiphase, modulation, voltage

source inverter

I. INTRODUCTION

ULTIPHASE electric machines have become a subject

of significant interest over the last two decades [1], [2].

The increased attention among researchers and industry to

machines with more than three phases results from the possi-

bility to achieve notable improvements in various aspects of

performance compared with the use of conventional three-

phase electric machine drives. Significant advantages of multi-

phase machines include: reduced torque pulsation [3], reduced

harmonic content of the DC link current [4], potentially higher

efficiency [3], [5], reduced power per phase, and greatly im-

proved reliability as a result of higher fault tolerance [6], [7].

.Manucscript received December 13, 2012; revised February 22, 2013 and

March 28, 2013; accepted May 21, 2013.

Copyright © 2013 IEEE. Personal use of this material is permitted. How-

ever, permission to use this material for any other purposes must be obtained

from the IEEE by sending a request to [email protected].

The authors are with the Department of Electrical Engineering, Lap-

peenranta University of Technology, 53850 Lappeenranta, Finland (e-mail:

[email protected]).

a1

a2b2

b1

c2

c1

30°

θr

Fig. 1. Dual three-phase permanent magnet synchronous machine. Two sets

of three-phase windings spatially shifted by 30 electrical degrees share the

same stator frame but are galvanically isolated from each other (separate

neutral points). Rotor angle θr refers to the angle between the direct axis of

the rotor and the magnetic axis of phase a1.

One of the most common multiphase machine structures is

the dual three-phase machine [2]. Dual three-phase machines

have two sets of three-phase stator windings spatially shifted

by 30 electrical degrees with isolated neutral points. Fig. 1

shows this configuration, which is also known in the literature

as an asymmetrical six-phase machine, a split-phase machine

or a double-star machine. Dual three-phase machines offer a

good compromise between additional complexity and potential

benefits, and make integration with conventional three-phase

technology relatively simple, which may explain the popularity

of this machine type.

In addition to scientific interest, dual three-phase machines

have also shown their effectiveness in practice. Dual three-

phase synchronous machines have been successfully used in

ship propulsion drives [8] and high-power turbo-compressors

[9]. It has been reported that dual three-phase permanent mag-

net synchronous machines (PMSM) are a particularly advanta-

geous alternative in wind power systems [10], [11] although

much lower power applications as found in more-electric air-

Decoupled Vector Control Scheme for Dual

Three-Phase Permanent Magnet Synchronous

Machines

Jussi Karttunen, Samuli Kallio, Pasi Peltoniemi, Pertti Silventoinen, and Olli Pyrhönen

M



2

craft [12] and electric vehicles [13] have also been proposed.

These wide ranging applications demonstrate the practical

value of research of dual three-phase PMSMs and their con-

trol.

An effective control scheme is an essential part of dual

three-phase PMSM drives and improvements in control can

greatly benefit the performance of the drive and consequently

the whole application. However, control of dual three-phase

machines is more challenging than control of conventional

three-phase machines because of the magnetic coupling be-

tween the winding sets, which complicates the dynamics of the

system. Further challenges are caused by unbalanced current

sharing between the winding sets [4], [14] and current harmon-

ics [15]–[17].

Various control schemes for dual three-phase machines have

been suggested in the literature. One approach for modeling

and control is a double d-q winding representation [2]. This

approach replaces both three-phase winding sets with equiva-

lent d-q windings, resulting in two pairs of d-q equations.

Many examples of vector control schemes using the double d-

q approach can be found for dual three-phase induction ma-

chines (IMs) [4], [16] and PMSMs [18], [19]. Our previous

study showed, however, that the double d-q approach causes

complex cross coupling between the two pairs of d-q stator

voltage equations that is very difficult to eliminate in PMSM

drives [20]. As a consequence, good dynamic performance is

hard to achieve using such vector control schemes.

An alternative modeling and control approach is the vector

space decomposition (VSD) method [15], which does not suf-

fer from such difficult cross coupling problems. Although

VSD itself has proven to be extremely useful, the original

VSD vector control scheme [15] has drawbacks such as the

inability to guarantee balanced current sharing between the

winding sets. Since its introduction, VSD based vector control

schemes with different modifications and improvements have

been suggested for dual three-phase IMs [14], [21] and

PMSMs [10]. In [10], for example, an additional pair of cur-

rent controllers is added to the original VSD control scheme,

which could in theory solve the imbalance problem [1]. How-

ever, in the case discussed, the solution may not work very

well in practice because the added current controllers (PI)

were placed in the stationary reference frame where simple PI

controllers cannot achieve zero steady state error for the fun-

damental component. Other suggested vector control schemes

demonstrate similar or other limitations.

Another potential way of realizing high performance drive

control is direct torque control (DTC). DTC schemes using the

VSD approach have been presented for dual three-phase IMs

[22], [23] and, using the double d-q approach, for synchronous

machines [8]. However, hysteresis based control can produce

low-frequency voltage harmonics, which makes it less suitable

for dual three-phase drives. Some DTC schemes have conse-

quently suffered from major problems, especially with current

harmonics [1]. Nevertheless, some versions of presented DTC

schemes have demonstrated feasibility even in commercial use

[8]. Thus, while DTC may not be the best alternative for all

dual three-phase machines, some versions clearly are usable.

Although vector control and DTC have traditionally been

the most commonly used control schemes in PMSM drives, it

is worth considering other alternatives. Recently, utilization of

predictive current control has been proposed for dual three-

phase IMs [24], [25]. Predictive strategies use a system model

to predict optimal switching states for the converter. The pre-

diction requires intensive computation and relies heavily on

the accuracy of the system model and its parameters. Lack of

tolerance for parameter errors and non-idealities not covered

by the system model (e.g., back-EMF harmonics) can therefore

cause problems in dual three-phase PMSM drives.

The above brief review of control schemes for dual three-

phase machines shows that improvements are required in the

areas of dynamic performance, robustness, and control of cur-

rent balance and harmonics. Thus, further investigation is

needed in this field and better control solutions should be de-

veloped.

The aim of this paper is to develop an improved vector con-

trol scheme for dual three-phase PMSMs. The objective of the

control scheme is to offer excellent dynamic performance and

guarantee balanced current sharing between the winding sets.

In addition, the control scheme should reduce (or preferably

eliminate) internally generated current harmonics caused by

the PMSM itself and no additional low frequency current har-

monics should be produced from the converter side. The study

includes detailed solutions for reference frame transfor-

mations, the machine model, model-based selection of parame-

ters for current control, decoupling of the current control

loops, and modulation.

The proposed control scheme is based on a recent machine

model [26] that has not previously been used for control pur-

poses. Thus, this paper takes into account the latest develop-

ments in modeling of dual three-phase PMSMs. The novelty of

the control scheme is to perform the current control and modu-

lation in different reference frames: the first reference frame

(adopted from [26]) resembles the VSD approach and the sec-

ond is a conventional d-q frame. This solution makes it possi-

ble to obtain high-performance current control using well-

established techniques for conventional three-phase PMSMs

and, at the same time, it provides the means for solving charac-

teristic problems of dual three-phase machines, such as bal-

anced current sharing and current harmonics. The performance

of the suggested vector control scheme is verified by compre-

hensive finite-element analysis (FEA) and experimental stud-

ies.

The rest of this paper is structured as follows: Section II

presents the machine model that forms the basis for the control

scheme. Section III describes the control scheme. Section IV

provides an analysis of the performance of the scheme using

the finite-element method (FEM). Finally, Section V presents

experimental results and Section VI offers conclusions.



3

D1 D2

Q2Q1

d-q reference frames

d1

q1

iq2

iq1

id1 id2

iQ2iQ1

iD1 iD2

LQ2LQ1

LD1 LD2

Lq

Ld Md

Mq

d2

q2

iq2

iq1

id1id2

Ld Md

Mq

Lq

D-Q reference frames

Fig. 2. Conventional d-q reference frames (d1-q1 for the first winding set and

d2-q2 for the second winding set) compared with D-Q reference frames used

in this paper. The D-Q approach provides much simpler representation of the

machine which can greatly benefit the control. Mutual inductance terms Md

and Mq describe the coupling between the frames in the d-q approach.

II. MACHINE MODEL

From the perspective of the control scheme, the machine

model should adequately describe the dynamics of the system

as simply as possible. As was noted in the introduction, VSD

type approaches satisfy this requirement. Previous VSD based

vector control schemes for dual three-phase PMSMs [10] have

used the original VSD approach [15] to model the machine.

Recently, however, improvements have been made in the field

of modeling of PMSMs [26] that can benefit the control.

Therefore, the machine model for salient pole dual three-phase

PMSMs introduced in [26] is used in this paper. The machine

model is based on the following transformation matrices:

0

1

1

0

1

0

0

1

3

1

2

3

21

21

2

3

2

3

21

21

2

3

21

2

3

2

3

21

21

2

3

2

3

21

rotDQ TT (1)

rr

rr

rr

rr

rot

cossin00

sincos00

00cossin

00sincos

T (2)

where θr is the rotor angle (see Fig. 1). Since zero sequence

components cannot flow, they are neglected from the trans-

formation. The transformation matrix (1) consists of the sta-

tionary frame decoupling matrix and the rotation matrix (2).

Using (1) transforms the original phase-variable presenta-

tion of the machine into two decoupled two-axis synchronous

reference frames. Thus, applying (1) to the phase currents

Tc22b2a1c1b1aDQ

Q2

D2

Q1

D1

iiiiii

i

i

i

i

T

(3)

gives four independent current components to be controlled

in two separate reference frames. These reference frames are

called D1-Q1 and D2-Q2 in this paper. Capital lettering is

used to distinguish them from conventional three-phase d-q

reference frames (i.e., rotor reference frames). Fig. 2 clari-

fies the difference between the reference frames. It can be

seen that the D-Q approach provides a much simpler repre-

sentation of the machine. The transformation matrix (11)

describes mathematically the connection between the d-q

and D-Q frames. Advantages of the D-Q approach are de-

tailed below. Further information about both approaches can

be found in [20] and [26].

The D1-Q1 and D2-Q2 reference frames are totally decou-

pled with respect to each other, which yields a very simple

form for the machine equations. It is assumed in the model-

ing that there is no saturation, that the machine structure is

symmetrical, and that the winding distributions are sinusoi-

dal. In addition, higher order harmonics and iron losses are

neglected. Under these assumptions, the stator flux linkage

equation is

0

0

0

3

000

000

000

000PM

Q2

D2

Q1

D1

Q2

D2

Q1

D1

Q2

D2

Q1

D1

i

i

i

i

L

L

L

L

(4)

where LD1 to LQ2 are the synchronous inductances of the

PMSM and ψPM is the PM flux linkage. A detailed description

of calculation of the parameters of the machine model can be

found in [26].

It can be seen from (4) that the inductance matrix of the ma-

chine has no off-diagonal inductance terms, which means that

no coupling exists between the reference frames. In addition,

coupling between the stator and rotor occurs only in the D1-Q1

reference frame. Thus, all the electromechanical energy con-

version must be described by this reference frame and current

components in the D2-Q2 reference frame cannot affect the air-

gap flux. On the other hand, Fig. 2 shows that modeling the

machine using the double d-q approach results in both refer-

ence frames (d1-q1 and d2-q2) having coupling between each

other and between the rotor and stator. These additional cou-

plings are the fundamental problem of the double d-q ap-

proach.

The transformation matrix (1) also has the property that the

fundamental component and the harmonics of the order (12n ±

1, n = 1, 2, 3…) are mapped into the D1-Q1 reference frame

and the harmonics of the order (6n ± 1, n = 1, 3, 5…) are

mapped into the D2-Q2 reference frame. This condition, how-

ever, holds only in a perfectly symmetrical situation. Any im-

balance between the winding sets causes the fundamental

components and all the harmonics to appear in both reference

frames.



4

For current control, the most important equations of the ma-

chine are the stator voltage equations. Based on (4), the stator

voltage equations in the D1-Q1 reference frame are

PMD1D1

Q1

Q1Q1sQ1

Q1Q1D1

D1D1sD1

3d

d

d

d

iLt

iLiRu

iLt

iLiRu

(5)

and in the D2-Q2 reference frame

D2D2

Q2

Q2Q2sQ2

Q2Q2D2

D2D2sD2

d

d

d

d

iLt

iLiRu

iLt

iLiRu

(6)

where ω is the electrical angular speed of the rotor and Rs is

the stator resistance. The first set (5) is similar to the corre-

sponding pair of stator voltage equations for a three-phase

PMSM. The second set (6) differs from (5) only by the miss-

ing back-EMF term. Thus, the used modeling approach offers

simple dynamic equations for dual three-phase PMSMs. Fur-

thermore, the similarities with conventional three-phase equa-

tions suggest that existing control solutions for conventional

three-phase PMSM drives may also be successfully used with

this machine type.

Last part of the machine modeling is the torque equation

which can be straightforwardly derived using a similar proce-

dure as for conventional three-phase machines [26]. The elec-

tromagnetic torque Te produced by dual three-phase PMSMs

can be calculated from

Q1D1Q1D1Q1PMe 3 iiLLipT (7)

where p is the number of pole pairs.

III. VECTOR CONTROL

A. Control challenges

Similarly to conventional three-phase machines, the main

idea of vector control for dual three-phase PMSMs is that the

D1-axis stator current iD1 and the Q1-axis stator current iQ1 can

be controlled independently (providing independent control of

flux and torque). In voltage source inverter (VSI) fed drives,

the stator current components are controlled by controlling the

stator voltage components of the machine. However, cross

coupling exists between the stator voltage equations (5), and

consequently, the D1-axis voltage component uD1 also affects

iQ1 and the Q1-axis component uQ1 affects iD1. The D2-Q2 refer-

ence frame equations (6) are similarly coupled. Therefore, the

stator voltage components uD1, uQ1, uD2, and uQ2 and the corre-

sponding stator current components cannot be considered as

simple single-input single-output systems.

The stator currents iD1 and iQ1 can be independently con-

trolled (so called decoupled current control) by decoupling the

stator voltage equations. As a result of the choice of the cur-

rent control reference frame (1), this cross coupling problem is

identical with conventional three-phase PMSMs and can be

easily solved [27].

Some problems of dual three-phase machines are not en-

countered with conventional three-phase drives. A good exam-

ple is the challenging feature of easily occurring large stator

current harmonics [1], [14], [28]. The reason for this major

drawback is that the stator current harmonics of the order (6n

± 1, n = 1, 3, 5…) are limited only by the very small stator

leakage impedance [15], [16]. Therefore, even small excitation

of these components can lead to significant corresponding cur-

rent harmonics. In an ideal case, these harmonics do not affect

the air-gap flux and thus have no impact on torque pulsation.

However, they cause additional losses and consequently

should be avoided [16], [28].

Dual three-phase PMSMs have two main sources for stator

current harmonics. The first source is the supplying inverters

[15], [16], [28], [29]. VSIs can cause large stator current har-

monics if the supplied voltage contains harmonic components

of the order (6n ± 1, n = 1, 3, 5…). Thus, the modulated volt-

age waveform should not contain unwanted low frequency

voltage harmonics. Such harmonics are usually negligible if

the modulation method is working properly.

The second source is the internal structure of the PMSM it-

self [17], [20]. Permanent magnets may not produce pure si-

nusoidal flux distribution, and rotor saliency, pole shape and

possible magnetic saturation can cause harmonics in the air-

gap flux [17]. These non-idealities can be a major problem in

dual three-phase PMSMs because they can easily produce

large internally generated current harmonics. This issue was

clearly demonstrated in our previous study [20].

Current harmonics caused by non-sinusoidal air-gap flux

can be reduced by intentionally feeding the right combination

of supply voltage harmonics to the machine, as shown later in

this paper. This action differs from the usual desire for perfect-

ly sinusoidal line-to-line supply voltage and requires that the

modulation scheme can produce harmonic voltage components

in the D2-Q2 reference frame.

Voltage components in the D2-Q2 reference frame are also

necessary for controlling the current sharing between the wind-

ing sets. The properties of the winding configuration mean that

the best torque quality, stator MMF waveform and efficiency

of a dual three-phase PMSM are achieved when equal currents

flow through both winding sets. However, small inherent

asymmetries between the winding sets and between the VSIs

can cause unbalanced current sharing [4], [18]. This problem

is well-known in the literature and its established solution is to

use as many current controllers in the control scheme as there

are independent current components in the machine (four in

this case) [1].

To gain the desired advantage from four current controllers,

the modulation method has to be able to produce independent

voltage vectors in both reference frames (D1-Q1 and D2-Q2). In

this way, the winding sets can be supplied with voltages of

different magnitudes and phase shift other than 30 degrees, a

requirement needed to compensate the asymmetries.

Most of the challenges in control of dual three-phase ma-



5

chines are related to issues with current. Currents are directly

controlled variables in vector control schemes and consequent-

ly can be effectively manipulated. This property provides a

very effective way to solve current-related problems and

strongly supports vector control as the recommended control

method for dual three-phase machines.

B. Control scheme and decoupling

The control scheme is based on a conventional solution of

synchronous frame current control using simple PI controllers.

Fig. 3 presents the block diagram of the control scheme. Four

current controllers are used since there are four independent

current components mapped into two decoupled reference

frames: the main reference frame D1-Q1, where torque produc-

tion of the machine is controlled, and the harmonic reference

frame D2-Q2, where the balance between the winding sets and

current harmonics is controlled. Since D2-Q2 is a synchronous

reference frame, any steady state imbalance will be completely

eliminated.

Any current control strategy such as maximum torque per

ampere (MTPA, which is the same as iD1 = 0 for non-salient

pole machines) can be used to calculate the reference currents

in the D1-Q1 frame. The situation is equivalent with the calcu-

lation of reference currents in vector controlled three-phase

PMSM drives. In this paper, the applied MTPA strategy is

based on a direct analytical calculation of optimal reference

currents for given Te using (7) and assuming constant induct-

ance values.

The current components in the D2-Q2 frame indicate either

imbalanced operation or specific current harmonics. Thus, the

reference currents for the D2-Q2 frame should always be zero,

as shown in Fig. 3, since the aim is usually balanced current

sharing and elimination of current harmonics. However, noth-

ing in the control scheme prevents the creation of desired im-

balance and harmonic content by using non-zero references if

some advantage is thereby obtained.

Performance of a simple PI controller based current control

can be greatly improved by eliminating the cross coupling ef-

fects in the stator voltage equations. This decoupling proce-

dure requires that the following decoupling voltage terms are

added to the output of the current controllers

D2D2

decoupling

Q2

Q2Q2

decoupling

D2

PMD1D1

decoupling

Q1

Q1Q1

decoupling

D1

3

iLu

iLu

iLu

iLu

. (8)

Decoupling can be done by feeding the decoupling terms (8)

as positive feedback (measured quantities are used to calculate

the decoupling voltage terms) or as feed-forward (reference

values are used to calculate the decoupling voltage terms) to

the output of the current controllers. In both approaches, the

system should behave from the perspective of the current con-

trol like four independent linear first order systems, thus sim-

plifying the design of the current controllers. An alternative

way of decoupling the current control is multivariable PI con-

trol [30] (also known as vector PI control) which uses integra-

tors to calculate the decoupling terms. This method theoreti-

cally produces the same result as (8) but is more complex, and

hence (8) is used here.

Based on the machine model, the decoupled system in the

D1-Q1 reference frame has transfer functions

sLRsU

sI

sLRsU

sI

Q1Q1Q1

Q1

D1D1D1

D1

1

)(

)(

1

)(

)(

. (9)

Note that Rs is replaced in (9) by apparent D1- and Q1-axis

resistance parameters RD1 and RQ1 to give better match be-

tween the actual voltage-current behavior of the machine and

the transfer functions.

In this paper, the resistance and inductance parameters are

identified from D1- and Q1-axis step responses. For example,

applying UD1 step to the machine while iQ1 current is kept at

zero should theoretically produce a first order response in iD1

current, as can be seen from (5) and (9). The desired parame-

ters can be determined by noting the time constant and steady

state value of the response. Such a test can be easily performed

using the VSIs, the control scheme and current measurements

of the control. Alternatively, an identification method based on

cross coupling voltage terms could be used. Thus, no special

VSI

S7-12

uα2

uβ2

ud2

uq2

PI

PI

uD2*

uQ2*

MTPATe,ref

iD2,ref = 0

a1b1c1

a2b2c2

ωia1, ib1, ic1

iD1

SV-PWM

VSI

S1-6

uα1

uβ1

d-q

α-β

ud1

uq1

θr

De-coupling

D1-Q1

D2-Q2

d-q

α-βd1-q1

d2-q2

PI

PI

uD1*

uQ1*

uD1

uQ1

uD2

uQ2

iD1,ref

iQ1,ref

SV-PWMiQ2,ref = 0

a1

a2

b1

b2

c1

c2

θr

+_

+_

+_

+_

ia2, ib2, ic2 D1-Q1

D2-Q2

θr

iD2

iQ1

iQ2

Fig. 3. Block diagram of the vector control scheme used to control the VSIs. The current control is made simultaneously in two decoupled reference frames

(D1-Q1 and D2-Q2) using PI controllers. MTPA refers to the maximum torque per ampere operating point. SVPWM is a conventional three-phase space

vector modulator. The electrical angular speed ω and the rotor angle θr are obtained from an angle encoder.



6

knowledge about the machine design is needed to obtain the

parameters.

Assuming that (9) adequately describes the behavior of the

system and machine parameters are accurate, current controller

parameters can be easily designed to give the desired closed

loop response. The model based method in [31] is used here.

Parameters for the Q1-axis PI current controller are calculated

from

Q1

r

iQ1

r

p

9ln,

9lnR

tKL

tK (10)

where Kp is the proportional gain, Ki is the integral gain, and tr

is the desired rise time (from 10 % to 90 %) of the resulting

first order closed loop system. The parameters for the other

current controllers are calculated similarly.

The suggested control scheme offers totally decoupled cur-

rent control where the controllers see the machine as simple

first order systems. Thus, very good dynamic performance and

simple current controller design can be achieved. It should be

noted that achieving the desired dynamic performance is

strongly dependent on the accuracy of the machine parameters

as can be expected for such a model based method. Inaccuracy

in the parameters causes a decrease in the dynamic perfor-

mance but has no effect on steady state operation. Because of

the synchronous frame current control, the control scheme can

provide zero steady state error in terms of constant current

reference tracking even without any knowledge of the parame-

ters of the machine since the integral action eliminates the er-

ror. Overall, the significance of the accuracy of the machine

parameters is similar to that in vector controlled three-phase

PMSM drives, where it appears not to be a major problem.

C. Modulation

The aim of the modulation is to synthesize the required ref-

erence voltage vectors in the D1-Q1 and D2-Q2 reference

frames using a VSI. Thus, four different voltage vector com-

ponents (uD1, uQ1, uD2, and uQ2) are controlled simultaneously

in two decoupled reference frames.

One possibility to achieve this objective is a VSD-type ap-

proach. The original VSD space vector PWM (SVPWM)

strategy was introduced in [15]. Over the last decade, im-

proved versions have been proposed, leading to different con-

tinuous and discontinuous 12-sector [28] and 24-sector [32]

SVPWM strategies. Although some of these VSD SVPWM

strategies have shown good performance [33], they all assume

that a zero average voltage vector is always produced in the

harmonic reference frame. As noted at the beginning of this

section, such a limitation clearly cannot be accepted in dual

three-phase PMSM drives. Thus, these methods cannot be

used in the case discussed in this paper.

Another solution is that both winding sets are supplied by

separate three-phase VSIs which are independently modulated

by conventional three-phase modulation methods. For this

approach, the literature contains examples using carrier [34]

and SVPWM [35] based methods. Unfortunately, these papers

suggest that reference voltage vectors for the modulators al-

ways have equal magnitude and phase shift of 30 degrees. This

restriction is analogous to the major limitation of VSD

SVPWM strategies that a zero average voltage vector is al-

ways produced in the harmonic reference frame. Thus, such a

modulation would lead to an inability to reduce current har-

monics and guarantee balanced current sharing between the

winding sets.

In the case of two independent three-phase modulators,

there is no compelling technical reason to artificially restrict

the modulation this way. Therefore, three-phase modulators

can be used to produce any combination of voltage vectors

within the limits of available DC voltage, thus providing a very

effective modulation strategy, as also noted in [36].

It is suggested in this paper that two conventional three-

phase SVPWM modulators are used for modulation. This solu-

tion requires that the reference voltage vectors in the D1-Q1

and D2-Q2 reference frames are transformed into conventional

three-phase d-q reference frames. The variables in the D1-Q1

and D2-Q2 reference frames have a straightforward connection

to corresponding variables in conventional d-q reference

frames (here d1-q1 for the first winding set and d2-q2 for the

second winding set)

Q2

D2

Q1

D1

q2

d2

q1

d1

0110

1001

0110

1001

3

1

u

u

u

u

u

u

u

u

. (11)

The voltage vectors can easily be transformed from one refer-

ence frame to another using (11). In this way, modulation be-

comes a simple task since voltage vectors in conventional d-q

reference frames are trivial to synthesize using any well-known

three-phase modulation method. The suggested modulation

strategy together with the whole control scheme is illustrated

in Fig. 3.

The combination of the transformation (11) and the two in-

dependent three-phase SVPWM modulators has many advan-

tageous properties. The most obvious one is the ease of im-

plementation. Existing algorithms and tested three-phase mod-

ulation methods can be effectively utilized, which can save

time and trouble. It also makes the method computationally

efficient since years of extensive study and wide usage have

made space vector modulation a very simple task. Further-

more, the suggested control scheme does not require much

computing power because, in addition to modulation, the

scheme contains only simple PI controllers and transfor-

mations between reference frames. Most importantly though,

the proposed modulation method allows fully independent and

exact control of all the required voltage vector components

(uD1, uQ1, uD2, and uQ2) unlike many other solutions presented

in the literature.

IV. FEM RESULTS

Performance of the suggested control scheme was first ana-

lyzed using FEM-Simulink cosimulation. In this cosimulation,

FEM (Cedrat Flux 2-D including skewing) was used to model

the dual three-phase PMSM and the control scheme was im-

plemented in Simulink. The input variables of the FEM based

machine model were the rotor speed and line-to-neutral volt-



7

ages given by Simulink. From these values, FEM calculated

the electromagnetic torque and stator currents, which were

returned to the control scheme in Simulink, thus closing the

control loops. VSIs were modeled in this case as ideal voltage

sources because of the sample time limitations caused by

FEM.

The FEM machine model itself was based on the same de-

sign as the test machine of the experimental setup used in this

paper (see Table I for parameters). The FEM model was thor-

oughly verified against the real machine. Thus, the perfor-

mance of the control scheme can be reliably evaluated taking

into account the real behavior of the machine but avoiding all

the practical problems in the implementation of the control

scheme. Note that the FEM model was used only as a verifica-

tion tool and treated from the perspective of the control

scheme similarly as a real machine. No information about the

machine design which cannot be known without the FEM

model was utilized in this paper.

In the following cases, the rotational speed of the machine is

the nominal speed 350 r/min unless otherwise stated. The ma-

chine is operated in most of the cases as a generator because of

the limitations of the experimental setup. However, the control

scheme operates equally well in both (motor and generator)

modes since they differ only by the sign of the reference cur-

rents. Thus, all the results presented hold for motor and gener-

ator operation of the machine. Operating conditions in all the

cases were selected to correspond with the experimental tests

presented in the next section.

A. Current harmonics

As mentioned earlier, the internal structure of the dual

three-phase PMSM can be a major source of current harmon-

ics. These harmonics can be reduced by producing compensat-

ing harmonic voltage components in the D2-Q2 reference

frame. The introduced idea is that current harmonics produced

by the machine itself can be eliminated using the D2-Q2 frame

current control. To demonstrate the issue, the upper curve of

Fig. 4 shows the case where current control in the D2-Q2 refer-

ence frame is omitted. In this case, a zero average voltage vec-

tor is always produced in the D2-Q2 reference frame, as pre-

sented in many papers. Large current harmonic components

can be seen because of the inability of the control to compen-

sate the effect of the non-sinusoidal air-gap flux.

As a comparison, Fig. 4 also shows the case (lower curve)

where current control in the D2-Q2 reference frame is operating

as suggested in this paper. A major reduction can be seen in

the magnitude of the current harmonic components (fifth har-

monic reduces 72 % and seventh 77 %). The control scheme is

producing substantial voltage components in the D2-Q2 refer-

ence frame to reduce the current harmonics. The result shows

that intentional feeding of supply voltage harmonics to the

machine can be used to compensate the effect of the non-

sinusoidal air-gap flux, and the basic concept of harmonic

elimination is thus verified.

0 10 20 30 40 50 60 70 80

-30

-20

-10

0

10

20

30

T ime [ms]

Curr

ent

[A]

0 10 20 30 40 50 60 70 80

-30

-20

-10

0

10

20

30

T ime [ms]

Curr

ent

[A]

Fig. 4. Phase current with (lower curve) and without (upper curve) current

control in the D2-Q2 reference frame. Large current harmonic components can

be seen when omitting current control in the D2-Q2 reference frame. Using the

suggested control gives a major reduction in harmonic components. Both

curves have equal fundamental component. The D1-Q1 frame reference cur-

rents are in this case iD1,ref = 0 A and iQ1,ref = -40 A.

It is evident that the current is not pure sinusoidal in the

lower curve of Fig. 4. Thus, the current controllers are able to

reduce the effect of internal non-idealities but not completely

remove them. This outcome is as expected since fixed gain PI

controllers have limited ability to compensate the effect of

sinusoidal disturbances. Without this limitation (e.g., using

resonance controllers), pure sinusoidal phase current could be

produced with this control scheme. An example of using reso-

nance controllers for stator current harmonic control of double

fed induction generator can be found in [37]. Since the fre-

quencies of the major harmonics (5th

and 7th

) are always

known, a similar approach could be applied in the case of this

paper. However, extensive analysis of the solution would be

required, and thus it is left for a future work.

B. Balance

Current control in the D2-Q2 reference frame is also respon-

sible for controlling the balance between the winding sets. The

suggested control scheme can in theory guarantee balanced

current sharing between the winding sets. Performance of the

balance control was tested by intentionally creating asymmetry

in the system by increasing the resistance of one winding set.

A resistance of 4.4 Ω was added, which created very severe

asymmetry.

Fig. 5 presents the currents of phases a1 and a2 with and

without current control in the D2-Q2 reference frame. It can be

seen that the created asymmetry leads to severe imbalance if

current control in the D2-Q2 reference frame is not used. Use

of current control in the D2-Q2 reference frame, on the other

hand, results in symmetrical loading.



8

0 10 20 30 40 50 60 70 80

-30

-20

-10

0

10

20

30

T ime [ms]

Curr

ent

[A]

0 10 20 30 40 50 60 70 80

-20

-10

0

10

20

T ime [ms]

Curr

ent

[A]

Fig. 5. Currents of phases a1 and a2 when system has asymmetry. Current

control in the D2-Q2 reference frame is omitted in the upper curve. The lower

curve shows the result using the control scheme. The suggested method pro-

vides balanced current sharing. Both cases produce same average electromag-

netic torque. Reference currents are iD1,ref = -3.5 A and iQ1,ref = -30 A.

A further test of the control scheme under asymmetrical con-

ditions is presented in Fig. 6. In this case, the constant torque

reference giving nominal current of the machine is requested

from the control. Rotational speed of the machine is changed

from 350 r/min to -350 r/min linearly during a 1 s period. The

machine operates the first half of the test as a motor and the

second half as a generator. Fig. 6 shows that constant torque is

produced under the continuous speed transient state despite

asymmetries. It can be seen that balanced current sharing is

achieved through the whole speed range. Voltage commands

for modulators show that the VSIs supply the first and second

three-phase winding sets with significantly different voltages

to compensate the asymmetry. The tests show that the suggest-

ed control scheme can offer balanced current sharing under a

wide range of operating conditions.

C. Dynamic performance

The dynamic performance of the suggested control scheme

was tested using step responses of the iQ1 current. During this

test, the iD1 current was kept at zero and the iQ1 current refer-

ence was changed from zero to -30 A. Parameters of the cur-

rent control were dimensioned using (10) to give first order

responses with rise times tr from 5ms to 20 ms. It can be seen

from the results given in Fig. 7 that the closed loop step re-

sponses are as desired. Responses have the shape of the first

order system and they accurately follow the rise time specifica-

tion of the current control. Thus, the decoupling scheme with

model based design of the control parameters clearly works as

intended.

It should be noted that the DC link voltage level of the VSIs,

0 0.2 0.4 0.6 0.8 1

-400

-200

0

200

400

T ime [s]

Speed [

rpm

]

0 0.2 0.4 0.6 0.8 1-20

0

20

40

60

T ime [s]

Curr

ent

[A]

iD1

iQ1

0 0.2 0.4 0.6 0.8 1-40

-20

0

20

40

T ime [s]

Curr

ent

[A]

ia1

ia2

0 0.2 0.4 0.6 0.8 1

-200

0

200

400

T ime [s]

Volt

age [

V]

u

d1 u

q1 u

d2 u

q2

Fig. 6. Rotational speed, D1 and Q1-axis currents, a1 and a2 phase currents,

and voltage commands in conventional d-q reference frames given to the

SVPWM modulators when the system has asymmetry. The suggested method

provides balanced current sharing through whole speed range. Note that bal-

anced current sharing does not mean that currents of phases a1 and a2 should

be equal at standstill (0 rpm, 0.5 s).

the switching frequency of the VSIs, and the sample time of

the control loops set limitations for the speed of the response.

Nevertheless, the results with different rise time specifications

show that when these limitations are not exceeded, the control

scheme can produce the desired dynamics for the current con-

trol.

Using a maximum torque per ampere (MTPA) current con-

trol strategy with salient pole machines as presented in Fig. 3

causes both currents (iD1 and iQ1) to change simultaneously

during torque changes. Because of the decoupling between the

current control axes, iD1- and iQ1-currents can in theory be con-

trolled independently, thus enabling fast simultaneous current

changes without disturbances between axes. Good dynamic

performance also includes that changes in rotational speed do

not cause substantial changes in currents and torque.

Operation of the control scheme under these conditions is

presented in Fig. 8, which shows the iD1- and iQ1-currents and



9

0 10 20 30 40 50 600

5

10

15

20

25

30

T ime [ms]

Curr

ent

[A]

tr = 5 ms

tr = 10 ms

tr = 15 ms

tr = 20 ms

Fig. 7. Step response of the iQ1 current with different rise time tr specifica-

tions of current control. Actual rise times of the currents (4.8 ms, 10.2 ms,

15.2 ms, and 20.4 ms) correspond well with the specifications. In addition,

responses have the desired shape of the first order system. The dashed line

represents the reference current. In the figure, the sign of the currents is re-

versed from the actual values (generator operation).

0 50 100 1500

10

20

30

40

50

T ime [ms]

Curr

ent

[A]

iD1

iQ1

0 50 100 1500

200

400

600

800

T ime [ms]

Torq

ue [

Nm

]

Te

Fig. 8. Torque step giving nominal current of the machine. Presented torque

is given by FEM. The control scheme uses the MTPA current control strategy.

Dynamic performance remains good despite simultaneous change of both

currents. Torque quite accurately has the dimensioned 10 ms rise time (actual

value 9.7 ms). After the step (t = 100 ms), the rotational speed of the machine

is halved from the nominal value. This major speed change causes only minor

changes in currents and torque. The dashed lines represent the reference cur-

rents. In the figure, the sign of the currents and torque is reversed from the

actual values (generator operation).

torque of the machine. In Fig. 8, the control scheme is first fed

by the torque reference step giving the nominal current of the

machine. After the step, the rotational speed of the machine is

halved (linearly during a 10 ms period) from the nominal val-

ue. The control scheme again shows desired performance since

the torque response using the MTPA current control strategy is

only slightly faster (tr = 9.7 ms) than the dimensioned value (tr

= 10 ms) and only minor disturbances can be seen when the

rotational speed changes. Note that different control parame-

ters were used compared with Fig. 7 because of changes in the

machine parameters caused by saturation.

V. EXPERIMENTAL RESULTS

The experimental setup consisted of two three-phase VSIs with

a common DC link, a commercial active front end (AFE), a 25

kW salient pole dual three-phase PMSM, and a 180 kW DC

machine drive for loading the PMSM. Table I presents the

parameters of the PMSM. The VSIs operate at a 5 kHz switch-

ing frequency. The setup includes an encoder for the rotor an-

gle feedback. Control of the VSIs is implemented in a

dSPACE platform. In addition, a DSP/FPGA card is used for

VSI

VSI

G MAFE

PC dSPACEDSP/

FPGA

User

interface S7-12

S1-6

S7-12

S1-6

DC link θr

ia2,b2,c2 ia1,b1,c1

Fig. 9. Schematic of the experimental setup. The dual three-phase PMSM

operates as a generator rotated by the DC machine drive. Two three-phase

VSIs supply the dual three-phase PMSM. The VSIs have a common DC link

and are controlled by the dSPACE platform. The AFE feeds the generated

power back to the grid. TABLE I

MACHINE PARAMETERS

Nominal power Pn 25 kW

Nominal current In 22.5 A

Nominal voltage Un 380 V

Nominal speed nn 350 rpm

Number of pole pairs p 4

Slots per pole per phase q 1

Stator skewing 1 slot

Stator resistance Rs 0.53 Ω

PM flux linkage ψPM 2.06 Wb

LD1, LQ1, LD2, LQ2, 31 mH, 42 mH, 7 mH, 8 mH

signal processing. Fig. 9 presents the schematic of the experi-

mental setup.

The performance of the suggested control scheme was ex-

perimentally verified by repeating the FEM-Simulink cosimu-

lation cases (current harmonics, balanced current sharing, and

dynamic performance) with the setup. In addition, it was inves-

tigated how asymmetry in the system affects decoupling of the

current control loops. A test with speed control covering a

wider range of operating points was also conducted.

Reduction of current harmonics is presented in Fig. 10. The

results are again shown with and without current control in the

D2-Q2 reference frame to clearly demonstrate the difference.

Large current harmonic components can be seen when a zero

average voltage vector is produced in the D2-Q2 reference

frame, and a notable reduction (fifth harmonic reduces 60 %

and seventh remains the same) is achieved by using the current

control. The reduction is much smaller than with FEM since

the current controllers have lower gains at the harmonic fre-

quencies because of limitations in the experimental setup.

To verify the balance control, asymmetry had to be inten-

tionally created because the experimental setup does not inher-

ently have asymmetries that would lead to significantly unbal-

anced current sharing between the winding sets. The asym-

metry needed was created by adding resistances (4.4 Ω, the

lowest available option in the laboratory) between the machine

and the VSI of the second winding set. Fig. 11 presents the

currents of phases a1 and a2 with and without current control in

the D2-Q2 reference frame. It can be seen that even severe im-

balance can be corrected with the suggested control scheme.



10

0 10 20 30 40 50 60 70 80

-30

-20

-10

0

10

20

30

T ime [ms]

Curr

ent

[A]

0 10 20 30 40 50 60 70 80

-30

-20

-10

0

10

20

30

T ime [ms]

Curr

ent

[A]

Fig. 10. Phase current with (lower curve) and without (upper curve) current

control in the D2-Q2 reference frame. Large current harmonic components can

be seen when omitting current control in the D2-Q2 reference frame. Use of

the suggested control scheme results in a notable reduction in harmonic com-

ponents. Both curves have equal fundamental component. The D1-Q1 frame

reference currents are in this case iD1,ref = 0 A and iQ1,ref = -40 A.

0 10 20 30 40 50 60 70 80

-30

-20

-10

0

10

20

30

T ime [ms]

Curr

ent

[A]

0 10 20 30 40 50 60 70 80

-20

-10

0

10

20

T ime [ms]

Curr

ent

[A]

Fig. 11. Currents of phases a1 and a2 when the system has asymmetry. Cur-

rent control in the D2-Q2 reference frame is omitted in the upper curve. The

lower curve shows the result using the control. The suggested method pro-

vides balanced current sharing. Both cases produce the same average electro-

magnetic torque. Reference currents are iD1,ref = -3.5 A and iQ1,ref = -30 A.

The dynamic performance was evaluated using the step re-

sponse of the iQ1 current. A zero reference was given for the

iD1 current and -30 A for iQ1. Current controllers were again

dimensioned to give first order responses with rise times tr

0 10 20 30 40 50 600

5

10

15

20

25

30

T ime [ms]

Curr

ent

[A]

tr = 5 ms

tr = 10 ms

tr = 15 ms

tr = 20 ms

Fig. 12. Step response of the iQ1 current with different rise time tr specifica-

tions of current control. Actual rise times of the currents (3.2 ms, 8.3 ms, 13.0

ms, and 19.1 ms) correspond adequately with the specifications and responses

have the desired shape of the first order system. The dashed line represents

the reference current. In the figure, the sign of the currents is reversed from

the actual values (generator operation).

0 0.5 1 1.5

-30

-20

-10

0

T ime [s]

Curr

ent

[A]

iD1

iQ1

iD2

iQ2

Fig. 13. Step responses of the iD1, iQ1, iD2, and iQ2 currents when the system

has asymmetry. The dashed line represents the reference current. Severe

asymmetry causes minor coupling between the current control axes. The

disturbances between the axes are quickly eliminated by the current control

thus enabling nearly independent control of the currents even in asymmetrical

conditions.

0 0.5 1 1.5 2 2.5 30

100

200

300

T ime [s]

Rota

tional

speed [

rpm

]

0 0.5 1 1.5 2 2.5 3-20

0

20

40

60

T ime [s]

Curr

ent

[A]

i

D1

iQ1

Fig. 14. Rotational speed and D1-Q1 reference frame currents of the machine

in the speed control test case. The dashed line shows the reference speed

curve. Acceleration speed is limited by the current limit as can be seen. Load

torque is 170 Nm. In this case, the machine operates most of the time as a

motor. A short period of generator operation can be seen during the decelera-

tion phase. The control scheme provides fast and accurate current control

independent from rotational speed.



11

from 5 ms to 20 ms. Fig. 12 shows the results. The control

scheme presents good dynamic performance, although the ac-

tual rise times of the currents are somewhat shorter than the

specified values. Deviation from the desired performance is

caused by issues such as the increase in the DC link voltage

level (AFE is not able to maintain constant DC voltage be-

cause of the fast increase in power generation) which are not

taken into account in the modulation. In addition, the DC ma-

chine drive is not capable of maintaining constant rotational

speed during rapid torque changes. As a result, the speed de-

creases 10 % – 20 % during the steps in Fig. 12. These prob-

lems impair performance compared with the perfect result in

the FEM-Simulink cosimulation and are also the reason why

the size of the step is limited to 30 A.

Obviously, the desired rise time specifications could have

been achieved precisely, despite the practical problems, by

fine-tuning the control parameters. However, this was not done

here (or with the FEM results) because the main aim is to

demonstrate direct use of (10) in the design of current control-

lers. The results clearly show that the suggested method works

satisfactorily.

Previous results have shown that the control scheme pro-

vides fully decoupled current control with the desired dynamic

performance when the machine parameters are accurately

known and the system has no asymmetry. However, an asym-

metry caused by, for example, the difference in the stator re-

sistances of the winding sets results in additional cross cou-

plings between the current control axes. The machine model

does not take into account such coupling which must thus be

actively eliminated by the current controllers. The conse-

quences of this issue are illustrated in Fig. 13 which presents

the dynamic performance of the control scheme when system

has asymmetry. Note that the asymmetry was very severe (see

Fig. 11) and control parameters of a symmetrical case were

used. It can be seen in Fig. 13 that a rapid change in the cur-

rent of one axis causes a minor change in the current of anoth-

er axis. The disturbances between the axes are relatively small

and they are quickly eliminated by the current control. In addi-

tion, it can be seen that the asymmetry causes a decrease in the

dynamic performance: the actual rise times of the currents are

approximately doubled from their specified values (5 ms)

which is not a major concern. Thus, sufficient performance is

achieved and the control scheme provides nearly independent

control of the currents even in asymmetrical conditions.

For the final test presented in Fig. 14, an outer speed control

loop (PI controller) was added to the control scheme. The ref-

erence speed curve shown was given for the speed controller

to test the dynamic performance of the scheme under a wider

range of operating points. The speed reference tracking is ac-

curate but somewhat slow because of the large moment of iner-

tia of the axis compared to the maximum available torque.

Nevertheless, the results demonstrate again the good dynamic

performance of the torque control, which is the main objective.

It can be seen that the current control loops provide fast and

accurate control independent of the rotational speed or the

operation mode of the machine.

VI. CONCLUSION

This paper presented a vector control scheme for dual three-

phase PMSMs. Discussion of the scheme covered reference

frame transformations, the machine model, decoupling of the

current control loops, model-based selection of current control

parameters, and modulation.

The performance of the suggested control scheme was eval-

uated using FEM and an experimental setup. The results clear-

ly demonstrated that the dynamics of the current control can be

accurately specified (within the limits of the system) using the

scheme and the presented model based methods. Consequent-

ly, the control scheme can be easily applied to a variety of

applications having different requirements and limitations for

the rise time of torque.

Comparative tests with and without the suggested synchro-

nous frame current control in the D2-Q2 reference frame clearly

showed the importance of this feature neglected in many other

papers. The results verified that this control can completely

eliminate steady state imbalance even under very severe

asymmetry. Thus, balanced current sharing can be achieved

despite the differences between the winding sets and the VSIs.

In the proposed scheme, current control in the D2-Q2 reference

frame is responsible also for reducing the current harmonics.

Only a moderate result was obtained in this case. The control

scheme was able to reduce the magnitude of the current har-

monics but not completely eliminate them. This finding is to

be expected since fixed gain PI controllers are not capable of

completely compensating the effect of sinusoidal disturbances.

Nevertheless, the suggested control scheme provides a basis

for approaching this issue using more advanced control solu-

tions. Future work aims to show that when the limitation of the

PI controllers is removed, pure sinusoidal phase current can be

produced with the proposed control scheme.

All in all, the suggested scheme provides an excellent con-

trol solution for dual three-phase PMSM drives. Overall per-

formance is good, although the scheme still has problems with

current harmonics. Other well-known issues in control of dual

three-phase PMSMs appear to have been adequately ad-

dressed. Furthermore, the simple structure using conventional

three-phase SVPWM modulators makes implementation easy

and computationally efficient, thus minimizing the required

effort to utilize this high-performance control scheme.

ACKNOWLEDGMENT

The authors would like to thank Dr. R. Pöllänen and Mr. T.

Knuutila from The Switch Drive Systems Oy for their valuable

ideas and comments.

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Jussi Karttunen received the B.Sc. and M.Sc. de-

grees in electrical engineering from the Lappeenran-

ta University of Technology, Lappeenranta, Finland,

in 2010 and 2011, respectively, where he is currently

working as a Researcher toward the D.Sc. degree.

His research interests include power electronics

and electrical drives.



13

Samuli Kallio received the M.Sc. degree in electri-

cal engineering from the Lappeenranta University of

Technology, Lappeenranta, Finland, in 2008. He is

currently working toward the D.Sc. degree at the

same university.

His research interests include modeling of electri-

cal machines and power electronic converters espe-

cially in the field of renewable energy.

Pasi Peltoniemi (M’09) was born in 1978 in Fin-

land. He received his M.Sc and D.Sc degrees in

electrical engineering from Lappeenranta University

of Technology,Finland in 2005 and 2010, respec-

tively.

His areas of interests include distributed genera-

tion in microgrids, control of grid-connected con-

verters and electric drives and passive filtering solu-

tions.

Pertti Silventoinen is a Professor (Electronics) with

Lappeenranta University of Technology (LUT). He

received his Doctoral in 2001, Lic. Tech in 1997,

and M.Sc. in 1993 in Electrical Engineering.

During his career at LUT, he has had several re-

search and teaching positions since 1990, funded by

the National Technology Agency of Finland

(TEKES) and several Finnish companies.

Olli Pyrhönen received the M.Sc. and D.Sc. degrees

in Electrical Engineering in 1990 and 1998 from

Lappeenranta University of Technology (LUT),

Finland. He has been Professor in Applied Control

Engineering since 2000 at LUT. In 2010 he received

further teaching and research responsibility in the

wind power technology at LUT. He has gained

industrial experience as a R&D Engineer at ABB

Helsinki in 1990-1993 and as a CTO of The Switch

in 2007-2010. He has published about 80 papers in

the control of electrical drives, power electronics and other industrial and

renewable energy applications.

decoupled vector control scheme for dual three-phase permanent magnet synchronous machines

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