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)