Multi-phase Vectorial Control of Synchronous Motorswith Currents and Voltages Saturations

Marco Fei and Roberto Zanasi

Abstract— This paper deals with the torque control of ms-phase synchronous machines where the first odd harmonicsbelow ms are injected. A new vectorial approach to describethe voltage and current limits is proposed. Starting from thetransformed dynamic equations and using the voltage andcurrent constraints, the optimal current references is obtained.It holds for an arbitrary number of star connected phases andan arbitrary shape of the rotor flux. Some simulation resultsfor a 7-phase motor validate the proposed control law.

I. INTRODUCTION

Multi-phase machines offer some advantages and greaternumber of degrees of freedom compared to three-phasemachines, see [2] and [3]. One of these advantages is thehigher torque-to-volume ratio due to the injection of higherorder current harmonics for the machines with concentratedwinding and nearly rectangular back-emf, see [1], [4], [5].In [6] and [7] the effects of the voltage and current limits onthe third harmonic injection are considered. Almost all theabovementioned papers consider specific motors with5 or 7phases where only the first and the third current harmonicsare injected. Moreover although the amplitude of the injectedharmonics is tied to the harmonic spectrum of the back-emf,it is not clear how the current references are obtained.This paper, which is an extension of [8] , uses a new vectorialapproach to obtain the optimal current references consideringthe voltage and current limits. The approach is as general aspossible and it is suitable for machines with an arbitrary oddnumber of star-connected phases and an arbitrary shape ofthe rotor flux.The paper is organized as follows. Sec. II shows the details ofthe dynamic model of the multi-phase synchronous motors.In Sec. III the current and voltage constraints are presentedand their effects onto the torque producing capability areshown in Sec. IV and Sec. V. The proposed torque controlis given in Sec. VI. Some simulation results are presented inSec. VII and conclusions are given in Sec. VIII.

A. Notations

The full and diagonal matrices will be denoted as follows:

i j

|[ Ri,j ]|1:n 1:m

=

R11 R12 · · · R1mR21 R22 · · · R2m

.

.

.

.

.

.

...

.

.

.Rn1 Rn2 · · · Rnm

,

i

|[ Ri ]|1:n

=

R1R2

...

Rn

M. Fei and R. Zanasi are with the Information EngineeringDepartment, University of Modena and Reggio Emilia, ViaVignolese 905, 41100 Modena, Italy, e-mail:marco.fei,roberto.zanasi@unimore.it.

LsIs1

RsV1

I1Ls

Is1

Rs

V2

I2

LsIs3Rs

V3

I3

Ls

Isi

Rs

ViIi

Ls

Isms

Rs

ViIms

Vs0

···

···

Stator

Jm

bmωm

τm τe

φ(θ)

Fig. 1. Basic structure of a star-connected multi-phase synchronous motor.

The symbolsi

|[ Ri ]|1:n

andi

|[ Ri ]|1:n

will denote the column and

row matrices. The symbol∑b

n=a:d cn=ca+ca+d+ca+2d+... willbe used to represent the sum of a succession of numberscn

where the indexn ranges froma to b with incrementd.

II. ELECTRICAL MOTORS MODELING

The basic structure of a permanent magnet synchronousmotor with anodd numberms of concentrated winding instar connection is shown in Fig. 1 and its parameters areshown in Tab. I. A complex and reduced model in the rotatingframeΣω can be obtained using the following reduced andcomplex transformation matrixtTωN ∈ C

ms×ms−1

2 :

tTωN =

√2

ms

h k∣∣[ ejk(θ −hγs)]∣∣

0:ms−1 1:2:ms−2

. (1)

Using this transformation, see [9], and the POG modelingtechnique, see [10] one obtains the dynamic model reportedin Fig. 2. The transformed systemSω expressed in thecomplex reduced rotating frameΣω has the following form:

[ωLs 0

0 Jm

][ωIs

ωm

]=−

[ωZs

ωKτN

− ωK

∗

τN bm

][ωIs

ωm

]+

[ωVs

−τe

]. (2)

The original ms-dimension model is transformed and re-duced to a(ms−1)/2-dimension complex modelSω in therotating frameΣω. In this frame thems-phase motor can beseen as a set of(ms−1)/2 independent electrical machines,rotating at different velocitykωm, each one working withina complex subspaceΣωk with k ∈ 1 : 2 : ms − 2. Thecomplex impedance matrixωZs = ω

Rs+ωLs

ωJs in (2) is

defined as follows:

ωZs =

k∣∣∣∣[

ωZsk

]∣∣∣∣1:2:ms−2

=

k∣∣∣∣[Rs+jpkωmLsk

]∣∣∣∣1:2:ms−2

(3)

ms number of motor phasesp number of polar expansions

θ, θm electric and rotor angular positions:θ = p θm

ω, ωm electric and rotor angular velocities:ω = p ωm

Rs i-th stator phase resistanceLs i-th stator phase self induction coefficient

Ms0 maximum value of the stator mutual inductance

Mhi

mutual induction coefficient of coupledh-th andi-th phases:

Mhi = Ms0

ms−2X

n=1:2

aMn cos(n(h−i)γs)

Jm rotor moment of inertiabm rotor linear friction coefficientτm electromotive torque acting on the rotorτe external load torque acting on the rotorγs basic angular displacement (γs = 2π/ms)

φc(θ) total rotor flux chained with stator phase 1ϕc maximum value of functionφc(θ)

φ(θ)

normalized total rotor flux:

φ(θ) =φc(θ)

ϕc

=∞

X

n=1:2

ai cos(n θ)

TABLE I

PARAMETERS OF THE MULTI-PHASE SYNCHRONOUS MOTOR.

where:Lsk = Ls + Ms0

(ms

2 aMk − 1).

Note that, according to [4], in (3) the firstms − 2 oddcomponentsaMkMs0 of the mutual inductance (see Tab. I)are considered. The transformed torque vectorω

KτN isfunction of the electric angleθ and the coefficientsan ofthe flux Fourier series shown in Tab. I. It can be shown, see[11], that when the normalized rotor flux has the structure:

φ(θ) =

ms−2∑

n=1:2

an cos(i θ) (4)

then the transformed torque vectorωKτ is constant:

ωKτN =

k∣∣∣[Kτk

]∣∣∣1:2:ms−2

=

k∣∣∣[Kdk+jKqk

]∣∣∣1:2:ms−2

= jpϕc

√ms

2

k∣∣∣[kak

]∣∣∣1:2:ms−2

.

(5)According to the magnetic co-energy method, the motortorqueτm and the transformed back-electromotive forceω

E

are:

τm = Re

(ωK

∗

τNωIs

), ω

E = ωKτNωm (6)

In [1] and [4] it is shown that it is possible to increasethe motor torque of a multi phase motor by injecting oddharmonics with order below toms. Let us now consider thecase of balanced voltage and current stator vectorst

Vs andtIs composed by the first(ms − 1)/2 harmonics:

tVs =

h

|[Vsh ]|1:ms

=

ms−2∑

k=1:2

h

|[Vmk cos(kθ−k(θvk−(h−1)γs)) ]|1:ms

(7)

tIs =

h

|[Ish ]|1:ms

=

ms−2∑

k=1:2

h

|[Imk cos(kθ−k(θik − (h−1)γs))]|1:ms

, (8)

where Vmk and Imk are the amplitude of the balancedharmonics components of orderk and θvk, θik are proper

tVs

tIs

1

ComplexTransformation

Re

-Re-

tTωN

-tT

∗ωN

-ωVs

1Electrical part

-

1ωZs+

ωLss

?

?

ωIs

-

2

Energy Conversion

-ωK

∗τN

-

ωE ω

KτN

-Re-τm

Re

3

Mechanicalpart

-

1

bm+Jms

6

6

ωm

- τe

4

Fig. 2. POG scheme of a multi-phase electrical motor in the reducedcomplex rotating frameΣω .

initial phase shifts. The transformed current and voltagevectors ω

Is = tT

∗

ωNtIs and ω

Vs = tT

∗

ωNtVs have the

following structure:

ωVs =

k∣∣∣[

V sk

]∣∣∣1:2:ms−2

=

k∣∣∣[Vdk+jVqk

]∣∣∣1:2:ms−2

=

√ms

2

k∣∣∣∣[

Vmk ej k θvk

]∣∣∣∣1:2:ms−2

(9)

ωIs =

k∣∣∣[

Isk

]∣∣∣1:2:ms−2

=

k∣∣∣[Idk+jIqk

]∣∣∣1:2:ms−2

=

√ms

2

k∣∣∣∣[

Imk ej k θik

]∣∣∣∣1:2:ms−2

(10)whereIdk, Iqk, Vdk andVqk are, respectively, thedirect andquadraturecomponents of the current and voltage vectorsIsk and V sk. Expressions (9) and (10) show that eachharmonic of orderk and amplitudeVmk andImk in (7) and(8) is transformed into vectorsV sk and Isk with modulusVMk =

√ms

2 Vmk and IMk =√

ms

2 Imk which move inthe complex subspaceΣωk. In other words each harmonic oforderk is related to the fictitious electrical machine rotatingat the same frequencykωm. Substituting (5) and (10) in (6)the motor torque can be rewritten as:

τm = pϕc

√ms

2

ms−2∑

k=1:2

kakIqk (11)

This equation shows the dependence of torqueτm on thequadrature componentsIqk of the current vectorsIsk. From(11) it is clear that the torque production capability of themachine increases injecting the first odd harmonics withorder below toms. Moreover the amplitude of the injectedharmonics is tied to the harmonic spectrum of the rotorflux (it is useless to apply highIqk in subspaces with lowcomponentskak).

III. MULTI HARMONICS CONSTRAINTS

The amplitudes of componentsVsh andIsh of the voltageand current vectorstVs and t

Is in (7) and (8) are bounded,respectively, by the maximum voltageVmax of the inverterDC link and the maximum rated currentImax, therefore thefollowing constraints hold:

ms−2∑

k=1:2

Vmk ≤ Vmax ,

ms−2∑

k=1:2

Imk ≤ Imax. (12)

−70 −60 −50 −40 −30 −20 −10 0 10 20 30

−50

−40

−30

−20

−10

0

10

20

30 IV k

Ivk

IZk

Izk

IIk

Iik

ICk

IckR0k

C0k

X0k

Y 0k

CVk

CIk

Ck IMk

Iqk

Idk

Σωk

Geometrical analysis of the control vectors

Fig. 3. Maximum current circlesCVk andCIk in subspaceΣωk.

Defining the vectorsVM ,IM andKτ ∈ Rms−1

2 as:

VM =

k∣∣∣∣[|V sk|

]∣∣∣∣1:2:ms−2

, IM =

k∣∣∣∣[|Isk|

]∣∣∣∣1:2:ms−2

, Kτ =

k∣∣∣∣[|Kτk|

]∣∣∣∣1:2:ms−2

the voltage and current constraints (12), multiplied by con-stant

√ms

2 , can be rewritten as a 1-norm constraint onvectorsVM andIM :

‖ VM ‖1=

ms−2∑

k=1:2

VMk ≤√

ms

2Vmax = VM , (13)

‖ IM ‖1=

ms−2∑

k=1:2

IMk ≤√

ms

2Imax = IM , (14)

where VMk =√

ms

2 Vmk and IMk =√

ms

2 Imk are themoduli of vectorsV sk andIsk, respectively.In the design of the control law there are some degrees offreedom that will be used to decide how to distribute themaximum voltageVM and currentIM into the componentsVMk and IMk to satisfy the constraints (13) and (14). Inthe next sections it will be shown how this defines in eachsubspaceΣωk a specific operative zone that modifies thetorque producing capability of the subspace.

IV. VECTORIAL CONTROL

In steady-state condition, the dynamic equation of theelectrical part is:ω

Vs = ωZs

ωIs + ω

Kτ ωm which isequivalent to the following(ms − 1)/2 equations of thecomplex subspacesΣωk :

V sk =ZskIsk+Kτk ωm = ZskIsk+jKqkωm (15)

where:Zsk =Rs+j k p ωmLsk.The voltage constraint

√V 2

dk + V 2qk ≤ VMk in the subspace

Σωk can be rewritten as follows:

(Idk − X0k)2

+ (Iqk − Y0k)2 ≤ R2

0k (16)

where:

R0k(ωm) =∣∣R0k (ωm)

∣∣|V sk|=VMk

=VMk

|Zsk|(17)

X0k(ωm) = Re(C0k (ωm)) =−Kqk k pω2

mLsk

|Zsk|2(18)

Y0k(ωm) = Im(C0k (ωm)) =−Kqk ωmRs

|Zsk|2. (19)

Relation (16) is the mathematical expression of the maximumvoltage circleCVk corresponding to the valueVMk thatsatisfies the voltage constraint (13). The termsC0k(ωm) =X0k + j Y0k and R0k (ωm) represent the center and theradius of the maximum voltage circleCVk. These terms arefunction of the parameterωm. When velocityωm increasesthe radiusR0k of circle CVk decreases and its centerC0k

moves in the complex planeΣωk on a circle with centerin

(−Kqk

2 k p Lsk, 0

)and radius Kqk

2 k p Lsk. The current vectorIsk

satisfies the voltage constraint only if its modulus is insidethe maximum voltage circleCVk. The current vectorIsk

must stay inside the maximum current circleCIk havingcenter in the origin and radiusIMk and satisfying the currentconstraint (14):

I2dk + I2

qk ≤ I2Mk. (20)

A graphical representation of the voltage and current circlesCVk and CIk on the complex planeΣωk for a particularvalue of ωm is shown in blue and violet in Fig. 3. Theintersection zoneCk between the two circles, shown in greyin Fig. 3, represents the area in which both the voltage andcurrent constraints are satisfied. Subtracting equation (20)from equation (16) one obtains the following relation:

−2X0kIdk − 2Y0kIqk + X20k + Y 2

0k − R20k + I2

Mk = 0.

Using this relation together with (20), one obtains the inter-section pointsIIk andIik of circle CVk with circle CIk:

II,ik =Y0k∣∣C0k

∣∣

X0kPk∣∣C0k

∣∣ ±

√√√√I2Mk−

Y 20kP 2

k∣∣C0k

∣∣2

(1−j

X0k

Y0k

)+jPk

(21)

where:Pk =

∣∣C0k

∣∣2 − R20k + IMk2

2Y0k

.

The coordinates of the other points shown in Fig. 3 are:

IV k(ωm) = X0k(ωm) + jY0k(ωm) + jR0k(ωm) (22)

IZk(ωm) = jY0k(ωm) + j√

R20k(ωm) − X2

0k(ωm) (23)

ICk = jIMk (24)

Ick = −jIMk (25)

Izk(ωm) = jY0k(ωm) − j√

R20k(ωm) − X2

0k(ωm) (26)

Ivk(ωm) = jY0k(ωm) − jR0k(ωm) (27)

Note that the coordinates of all points shown in Fig. 3 arefunction of the componentsVMk and IMk. The torqueτmk

generated in the subspaceΣωk can be provided using only

Kτ

IM

IM

VM

VM

Kq1

Kq3

IMaIMd

IMc

IMdVMc

Σω1 Σω1

Σω3Σω3

a) b)

Fig. 4. Minimum dissipation a) current and b) voltage constraintsdistribution for a5-phase machine.

the current vectorIsk ∈ Ck. Two main control laws can beused:

1) field oriented controlwhere the direct componentcurrent vector is zeroIsk = max

ICk, IZk

∈ Ck;

2) flux weakening controlwhere also the direct componentIdk is used,Isk = max

Ivk, IIk

∈ Ck.

In Fig. 3 the pointsICk and Izk cannot be used in thecontrol because they do not satisfy the constraints (ICk andIzk /∈ Ck); using the field oriented control the operationpoint is Isk = IZk, while using the flux weakening controlthe operation point isIsk = IIk.

V. CONSTRAINTS DISTRIBUTION

The torque can be provided using only the current vectorsIsk inside the intersection zoneCk of the maximum voltageand current circlesCVk and CIk therefore the voltage andcurrent limits determine the torque producing capability ofthe subspaceΣωk. For a givenωm, it is possible to modulatethe componentsVMk andIMk in each subspaceΣωk in orderto increase or decrease the maximum voltage and currentcirclesCVk andCIk satisfying the constraints (13),(14). Fora 5-phase motor, the current and voltage 1-norm constraintscan be represented as planes inR

2 as shown in Fig. 4 andFig. 5. From (6) it is clear that the torqueτm is tied to thescalar product ofKτ andIM and for this reason the vectorKτ has been reported in the current 1-norm constraint ofFig. 4.a and Fig. 5.a. The torque control law described inthe next section is a generalization, for ams-phase machine,of the following two cases of constraints distribution for a5-phase machine:

Case 1)The optimal distribution of the current constraintminimizing the power dissipation is reported in Fig. 4.a. ThevectorIMd does not satisfy the current constraint, thereforethis distribution cannot be used. The scalar product of thethree vectorsIMa, IMb and IMc with the vectorKτ isthe same but the vectorsIMa and IMb do not minimizethe power dissipation because their moduli are greater thanthe modulus ofIMc. Therefore the current constraint vectorwhich minimizes the power dissipation is the vectorIMc withthe minimum modulus parallel toKτ . The voltage vectorVMc related to the current vectorIMc is reported in Fig. 4.b.

Case 2)The optimal distribution of the current constraint

Kτ

IM

IM

VM

VM

Kq1

Kq3

IMe

IMf

IMc

IMg

VMe

Σω1 Σω1

Σω3Σω3

a) b)

Fig. 5. Maximum torque a) current and b) voltage constraintsdistribution for a5-phase machine.

−40 −35 −30 −25 −20 −15 −10 −5 0 5 10

−20

−10

0

10

20

30

Iqk

Idk

−40 −35 −30 −25 −20 −15 −10 −5 0 5 10

−20

−10

0

10

20

30

Iqk

Idk

−40 −35 −30 −25 −20 −15 −10 −5 0 5 10

−20

−10

0

10

20

30

Iqk

Idk

−40 −35 −30 −25 −20 −15 −10 −5 0 5 10

−20

−10

0

10

20

30

Iqk

Idk

a) b)

c) d)

Fig. 6. Maximum current circlesCVk andCIk in subspaceΣωk

obtained modulating the componentsVMk andIMk .

maximizing the torqueτm is reported in Fig. 5.a. The vectorsIMc, IMf and IMg do not maximize the torque becausetheir projections ontoKτ are smaller than the projection ofvectorIMe. The vectorIM = IMe that maximizes the scalarproductKT

τ IM is obtained giving the maximum valueIM

to the componentIM3 related to the maximum componentKq3 of vector Kτ . The voltage vectorVMe related to thecurrent vectorIMe is reported in Fig. 5.b.

Generalizing these two cases, four different constraints distri-bution into the subspacesΣωk will be considered, see Fig. 6.The dashed lines are the voltage and current circlesCVk andCIk showed in Fig. 3, while the solid lines are the new circlesobtained modulating the componentsVMk andIMk.In Fig. 6.a the operation point isIsk = IZk ≡ ICk ≡ IIk,so the torque can be generated using only the quadraturecomponentIqk. Given the current constraintIMk and usingequations (23) and (24), one obtains the componentVMk:

VMk = |Zsk|√

X20k + (IMk − Y0k)2. (28)

In Fig. 6.b the operation point is the originIsk = 0 becauseIMk = 0, so the torque generated by subspaceΣωk is zero.The componentVMk has the following structure:

VMk = |Zsk|∣∣C0k

∣∣ = Kqk ωm. (29)

In Fig. 6.c the operation point isIsk = C0k becauseVMk =0, so the torque generated by subspaceΣωk is negative. ThecomponentIMk has the following structure:

IMk =∣∣C0k

∣∣ =Kqk ωm

|Zsk|. (30)

In Fig. 6.d the operation point isIsk = IIk ≡ Iik. Also inthis case the torque generated by subspaceΣωk is negative.Given the current constraintIMk, from (21) one obtainsR0k + IMk = |C0k| and the componentVMk is:

VMk = |Zsk| (|C0k| − IMk) = Kqkωm − |Zsk|IMk. (31)

These four cases show that when the operation point isdefined, the componentsVMk and IMk are bounded by theequations (20) and (16).

VI. TORQUE CONTROL

Torque τm can be controlled by using current vectorsωId in frame Σω not exceeding the constraints on the

maximum input voltage and current. WhenωId is constant,

the conditionωIs = ω

Id can be achieved using the followingcontrol law:

ωVs =ω

ZsωIs+ ω

Kτ ωm−Kc(ωIs−ω

Id) (32)

where Kc > 0 is a diagonal matrix used for the controldesign andωId as follows:

ωId =

ωIM if τd ≥ τM (ωm)

ωIcc if τMd(ωm) < τd < τM (ωm)

ωImd if 0 ≤ τd ≤ τMd(ωm)

(33)

where τd is the desired torque.τMd(ωm) is the maximumtorque with minimum dissipation whileτM (ωm) is themaximum torque. These two limit torques are function of themotor parameters, the voltage and current constraints and thecontrol law. In Fig. 7 the two curvesτMd(ωm) andτM (ωm)define three zones in the (τm, ωm) torque plane: the greenzone represents the region where the desired torqueτd can beprovided using theminimum dissipation torque control; thered zone represents the region where the torqueτd is obtainedusing theconvex combination torque control; the white zoneis not allowed becauseτd cannot exceed the maximum torqueτM , in this area themaximum torque control is used.This control law, used together with relation (32), providesthe desired torqueτd satisfying the constraints (13),(14) andminimizing, when possible, the current dissipation.

ωm

τm

τM (ωm)

τMd(ωm)

τM (0)

τMd(0)

ωrMdωrM

Fig. 7. Limit torques for a multi-phase synchronous motors.

A. Minimum dissipation torque control

The current constraint vectorIM which minimizes thepower dissipation is the vector with the minimum modulusparallel to vectorKτ (see Fig. 4.a):

IM =τd

|Kτ |Kτ =

Kτ

|Kτ |2τd =

k∣∣∣[

τd Kk

]∣∣∣1:2:ms−2

(34)

where:Kk =Kqk

|ωKτ |2=

kak

pϕc

√ms

2

ms−2∑

k=1:2

(kak)2

are the distribution coefficients of the current constraintIM

into the subspacesΣωk. Only the quadrature componentsIqk

of the current vectorsIsk are used to generate torque (seeFig. 6.a), therefore the current vectorω

Imd which minimizesthe power dissipation is:

ωImd =

k∣∣∣[

jτd Kk

]∣∣∣1:2:ms−2

=ωKτN

|ωKτN |2τd (35)

Note that the current vectorωImd is parallel with the torquevector ω

Kτ . Substituting (34) in (28) and using the voltageconstraint (13), one obtains the following equation:

ms−2∑

k=1:2

|Zsk|√

X20k + (τd Kk − Y0k)2

︸ ︷︷ ︸VMk

=VM . (36)

At low velocity the current constraint limits the torque. Using(34) and the current constraint (14) the maximum torque withminimum dissipation at low velocity is:

τMd(0) = IM/

ms−2∑

k=1:2

Kk

SobstitutingτMd(0) in (36) one obtains the rated velocityωrMd. When ωm > ωrMd, the limit torque decreases andit is limited by the voltage constraint. Givenωm > ωrMd,equation (36) can be numerically solved with respect toτd

in order to obtain the maximum torqueτMd(ωm) satisfyingminimum dissipation and the voltage and current constraints.The desired torqueτd can be obtained with minimum currentvector ω

Imd only if τd < τMd(ωm). In Fig. 7 the curveτMd(ωm) defines the green zone that represents the regionwhere the desired torqueτd can be provided minimizing thepower dissipation.

B. Maximum torque controlThe maximum torquesτM (ωm) can be obtained maximiz-

ing the projection of the vectorIM onto the torque vectorKτ , see Fig. 5.a. To reach this goal it is necessary to sort thecomponentsKqk of the vectorKτ and apply the followingcurrent and voltage constraints distribution:

8

>

>

>

>

>

<

>

>

>

>

>

:

IMk =IM , VMk = |ZsG|q

X2

0G + (IM − Y0G)2 if k=G

IMk = |C0k|, VMk =0 if k∈Ng

IMk =IMg, VMk =Kqk ωm − |Zsg| IMg if k=g

IMk =0, VMk =Kqk ωm if k∈Og

(37)where:IM =IM−

∑

k∈Ng

|C0k| − IMg

and:

- G is the index of the maximum component ofKτ ,- g is the index of the considered subspace univocally

defined from the motor velocityωm (the reasonwill be explained later),

- Ng is the set of the subspaces already considered,- Og is the set of the subspaces not jet considered.

The subspaceΣωG related to the maximum componentKqk

of the torque vector is shown in Fig. 3, the subspaceΣωg

is shown in Fig. 6.d and the subspaces not yet consideredand already considered are shown, respectively, in Fig. 6.band Fig. 6.c. Substituting the componentsVMk of (37) in thevoltage constraint (13) one obtains the following equation:

∑k∈[g,Og]

Kqk ωm +|ZsG|√

X20G

+(IM−Y0G)2 − |Zsg| IMg=VM (38)

Given ωm, equation (38) can be rewritten as:

v

u

u

u

u

t

X20G

+

0

B

@IM −

∑k∈Ng

|C0k|−IMg−Y0G

1

C

A

2

=

VM −∑

k∈[g,Og]

Kqk ωm

|ZsG|+

|Zsg|

|ZsG|IMg

This relation can be solved with respect toIMg obtaining thevoltageVMg of the considered subspaces. The componentsIsk of the maximum current vectorωIM are univocallydefined from the constraints distribution (see Sec. IV):

ωIM =

k

˛

˛

˛

»

Isk

–

˛

˛

˛

1:2:ms−2

, Isk =

8

>

>

>

>

>

<

>

>

>

>

>

:

max

Ivk, IIk, IZk

ff

∈ Ck if k=G

X0k + jY0k if k∈Ng

IIk if k=g

0 if k∈Og

(39)

Note thatΣωG is the only subspace that generates torquebecause the current components in the other subspaces arenegative or equal to zero.When ωm ≤ ωrM the current constraint limits the torqueand the maximum valueIM is given only by the subspaceΣωk related to the maximum componentKqk of vectorKτ

(see Case2 of Sec.V), thereforeg = 0, Ng = ∅ and in thiscase the equation (37) can be rewritten as:

IMk =IM , VMk = |ZsG|√

X20G + (IM − Y0G)2 if k=G

IMk =0, VMk =Kqk ωm if k∈Og

and the maximum torque at low velocity is:

τM (0) = pϕc

√ms

2GaGIM .

When the velocityωm increases the componentsVMk in-crease and there is a velocityωrM for which the voltageconstraint is exactly satisfied. The rated velocityωrM canbe obtained substituting the componentsVMk in (13). Whenωm > ωrM the voltage constraint limits the torque then it isnecessary to redistribute the current constraintIM into theother subspaces to reduce the componentsVMk, see (31).Since this operation causes a reduction of the torque (seevector IMf in Fig. 5.a), the current constraintIM is redis-tributed only into the subspacesΣωk with k ∈ [Ng, g] thusminimizing the torque reduction. The current and voltageconstraints distribution (37) must be applied starting fromthe subspaceΣωk related to the minimum componentKqk

up to the subspaceΣωG. The indexg is univocally definedby motor velocityωm: it is the first subspaceΣωg whereVMg > 0. At the end, wheng = G andOg = ∅, the equation(37) can be rewritten as:

IMk =IM −∑

k 6=G

|C0k|, VMk =VM if k=G

IMk = |C0k|, VMk =0 if k∈Ng

The maximum torque control is obtained choosingωId =

ωIM when τd ≥ τM (ωm). In Fig. 7 the curveτM defines

the white region that represents the zone not allowed becauseτd cannot exceed the maximum torque.

C. Convex combination torque control

In Fig. 7 the curvesτMd and τM define a red zone thatrepresents the region where the two previous control lawscannot be used. In this region the optimal control law whichsatisfies the constraints (13), (14) and minimizes the currentdissipation is quite complex and difficult to be found. Inthis case we propose the following suboptimal control lawobtained as a convex combination of the maximum currentvector ω

IM , see (39), and the maximum current vector withminimum dissipationω

IMd =ωKτ

|ωKτ |2τMd, see (35). When

τMd < τd < τM the torqueτd is obtained using the followingcurrent vector:

ωIcc =ω

IMd+α ( ωIM− ω

IMd)

where coefficientα is defined as follows:

α =τd − τN (ωm)

τM (ωm) − τN (ωm).

VII. SIMULATION RESULTS

The simulation results described in this section have beenobtained in Matlab/Simulink environment considering a mo-tor with the following electrical and mechanical parameters:ms = 7, p = 1, Rs = 2Ω, Ls = 0.03 H, Ms0 = 0.025H, aM1 = 1, aM3 = 1/9, aM5 = 1/25, ϕr = 0.02 Wb,Jm = 1.6 kg m2, bm = 0.15 Nm s/rad, Vmax = 100V,Imax = 35A a1 = 0.40, a3 = 0.3, a5 = 0.25. The externaltorqueτe is zero until t = 10 s thenτe = 45 Nm (see the

−6 −5 −4 −3 −2 −1 0−3

−2

−1

0

1

2

3

4

5

Current in the subspaceΣω1

I q1

[A]

Id1 [A]

VA

B

C

D

E

−25 −20 −15 −10 −5 0

−10

−5

0

5

10

15

20

Current in the subspaceΣω3

I q3

[A]

Id3 [A]

A

B

C

D

E

−30 −20 −10 0

−10

0

10

20

30

40

Current in the subspaceΣω5

I q5

[A]

Id5 [A]

A

B

C

D

E

Fig. 12. Current and voltage circlesCVk, CIk and current vectorsIsk in the complex subspacesΣωk.

0 5 10 15 20 25 30 35 400

20

40

60

80

100

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

Motor velocity ωm

Motor torqueτm and desired torqueτd

Time [s]

ωm

[rad

/s]

τ m,τ d

[Nm

]

AB

C

DE

τMτN

Fig. 8. Time behaviors of motor velocityωm, motor torqueτm

(brown), desired torqueτd (red), external torqueτe (magenta) andmaximum torquesτM andτN (black).

0 50 100 150 200 2500

10

20

30

40

50

60

70

80

90

100

ωm [rad/s]

τ m,τ d

[Nm

]

Zoom of Torque plane

τM

τN

A

B

C

DE

Fig. 9. Motor torque τm (brown), desidered torqueτd (red)and maximum torquesτM and τN (black) as a function of motorvelocity ωm.

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120

140

160

180

Multi harmonics current constraint

Time [s]

‖V

M‖1

[V]

VM

VM1 + VM3 + VM5

0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

Multi harmonics voltage constraint

Time [s]

‖I

M‖1

[A]

IM

IM1 + IM3 + IM5

Fig. 10. Time behaviors of the sum of componentsVMk andIMk

of vectorsVM andIM .

black dashed line in Fig. 8). The time behaviors of motorvelocity ωm, motor torqueτm, desired torqueτd, externaltorque τe and maximum torquesτM and τN are shown inFig. 8 and the corresponding trajectories on the torque plane(τm, ωm) are reported in Fig. 9. The lettersA, B, C, Dand E refer to the critical points for the control:A whenτd = τMd, B and D when τd = τM , C when the externaltorqueτe is applied andE the final steady-state condition.Note that forτd ≤ τM , i.e. from point 0 to point B andfrom point D to point E, the control law (32) and (33)guaranteesτm = τd. The constraints (13) (14) are alwayssatisfied as it is shown in Fig. 10. Fig. 11 shows the phasecurrent and voltage waveforms in the steady-state conditionand their corresponding spectrum. It is clear that the phasevoltage and current are obtained injecting the1-st, the 3-rd and the5-th harmonics. Moreover the obtained currentand voltage waveforms satisfy (12). The current vectorsIsk

in the complex subspacesΣω1, Σω3 and Σω5 are shownin Fig. 12. The desired torqueτd is provided only by thequadrature componentsIqk of the current vectorsIsk untilit is τd = τN in point A. Note that fromA to B, where theconvex combination controlωIcc is used, the current vectorsIsk remain within the intersection zonesCk. From pointBto point D the maximum torque controlωIM is used, thedesired torqueτd is generated only by the current componentIq5M . FromD to E the desired torqueτd is provided by the

99.85 99.9 99.95

−30

−20

−10

0

10

20

30

0 100 200 300 400 5000

5

10

15

99.85 99.9 99.95

−100

−50

0

50

100

0 100 200 300 400 5000

10

20

30

40

50

60

70

w [rad/s] w [rad/s]

Time [s] Time [s]Phase voltage spectrum Phase current spectrum

ImaxVmax

1-st1-st

3-rd

3-rd

5-th 5-th

volta

geha

rmon

ics

[V]

curr

ent

harm

onic

s[A

]

Injected phase current

I s1

[A]

Injected phase voltage

Vs1

[A]

Fig. 11. Phase current and voltage waveform with their correspond-ing harmonic spectrum in steady-state condition.

convex combination torque control.

VIII. CONCLUSIONS

In this paper a new vectorial approach to obtain theoptimal current references considering the voltage and cur-rent constraints has been proposed. Since the number ofphases, the impedance matrix and the rotor flux have beenconsidered as design variables the current references are asgeneral as possible. The optimality of the control law isguaranteed when the minimum dissipation torque control orthe maximum torque control are applied. Simulation resultsobtained in Simulink for a7-phase motors validated theeffectiveness of the presented control law.

REFERENCES

[1] E. Semail, X. Kestelyn, A. Bouscayrol, “Right Harmonic Spectrumfor the Back-Electromotive Force of an-phase Synchronous Motor”,Industry Applications Conference, 2004, 39th IAS Annual Meeting,ISBN: 0-7803-8486-5.

[2] E. Levi, “Multiphase Electric Machines for Variable-Speed Applica-tions” IEEE Transactions on Industrial Electronics, , vol.55, no.5,pp.1893-1909, May 2008.

[3] L. Parsa, “On advantages of multi-phase machines” 31st AnnualConference of IEEE Industrial Electronics Society, IECON 2005.

[4] L. Parsa and H.A. Toliyat, “Five-Phase Permanent-MagnetMotorDrives”, IEEE Transactions on Industry Applications, 2005, Vol. 41,No. 1, pp. 30-37.

[5] Shan Xue, Xuhui Wen, Zhao Feng, “Multiphase Permanent Mag-net Motor Drive System Based on A Novel Multiphase SVPWM”,CES/IEEE 5th International Power Electronics and Motion ControlConference IPEMC 2006, vol.1, pp.1-5.

[6] F. Locment, E. Semail, X. Kestelyn, “Optimum use ofDC bus byfitting the back-electromotive force of a7-phase Permanent MagnetSynchronous machine”,European Conference on Power Electronicsand Applications, EPE 2005.

[7] L. Parsa, N. Kim, H.A. Toliyat, “Field Weakening Operation of a HighTorque Density Five Phase Permanent Magnet Motor Drive”,IEEEInternational Conference on Electric Machines and Drives,15-15 May2005

[8] R. Zanasi, M. Fei, “Saturated Vectorial Control of Multi-phaseSynchronous Motors”, NOLCOS 2010 - 8th IFAC Symposium onNonlinear Control Systems, Bologna, Italy, 1-3 September 2010.

[9] R.Zanasi, F.Grossi, M.Fei, “ Complex Dynamic Models of Multi-phasePermanent Magnet Synchronous Motors” accepted for IFAC 201118thWorld Congress of the International Federation of AutomaticControl,Milano, Italy, 28 August - 2 September 2011.

[10] R. Zanasi, “The Power-Oriented Graphs Technique: system modelingand basic properties”, Vehicular Power and Propulsion Conference,Lille, France, Sept 2010.

[11] R. Zanasi, F. Grossi, “Multi-phase Synchronous Motors: POG Model-ing and Optimal Shaping of the Rotor Flux”, ELECTRIMACS 2008,Quebec, Canada, 2008.