International Journal on Electrical Engineering and Informatics - Volume 5, Number 1, March 2013

A Vectorial modeling for the Permanent Magnet Synchronous Machine (polyphase) based on multimachine approach

Abdelkrim Sellam1, Boubakeur Dehiba1, Mohamed Badreddine Benabdallah2,

Mohamed Abid1, Nacéra Bachir Bouiadjra3, Boubakeur Bensaid1, and Mustapha Djouhri4

1Laboratoire ERICOM, Université UDL de Sidi Bel Abbes, ALGERIA 2Université des sciences et technologie d’Oran, ALGERIA

3Applied Micro Electronic Laboratory, Université UDL de Sidi Bel Abbes, ALGERIA 4Laboratoire ICEPS, Université UDL de Sidi Bel Abbes, ALGERIA

bbnas63@yahoo.fr

Abstract: Our goal in this article is to present a model based on the Vectorial modelling associated with the multimachine concept. And we finish this work by a simulation to understand the behavior of the machine pentaphase (polyphase) in normal mode. The polyphase machines are developed mainly in the field of variable speed drives of high power because increasing the number of phases on the one hand allows to reduce the dimensions of the components in power modulators energy and secondly to improve the operating safety. By a vector approach (vector space), it is possible to find a set of single-phase machine and / or two-phase fictitious equivalent to polyphase synchronous machine. These fictitious machines are coupled electrically and mechanically but decoupled magnetically. This approach leads to introduce the concept of the equivalent machine (multimachine multiconverter system MMS) which aims to analyze systems composed of multiple machines (or multiple converters) in electric drives. A first classification multimachine multiconverter system follows naturally from MMS formalism. We present an example of a pentaphasé synchronous machine. Keywords: Polyphase machines, multimachine concept, vector space, eigenvectors, eigenvalues, pentaphase machine.

1. Introduction: Through the many advances in technology, the power applications high and average at speed variable are increasingly made on the based on the whole electrical machinery-static converters. For applications of high power density, low rotor losses and reduced inertia, the permanent magnet synchronous machines [1] are best suited. However with the traditional structures of static converters and high power machines, the power transmitted between the power source and the mechanically receiver cannot be treated appropriately. The use of current switches associated with machine double-star [2] on the one hand allows to reduce the power transmitted by each converter and, secondly, to reduce the torque ripple of the machine. Despite this improvement, the torque ripples are important, especially for low speeds. Despite this improvement, the torque ripples are important, especially for low speeds. The polyphase machines are an interesting alternative to reducing constraints applied to the switches and coils. Indeed, the increase in the number of phases allows a fractionated of power, and therefore a reduction in switched voltages at a given current. In addition, these machines can reduce the amplitude and increasing the frequency of the torque ripple, which allows at the mechanical loading of filter them more easily. Finally, increasing the number of phases provides increased reliability by allowing run, one or more faulted phases [7]. The polyphase machines are found in areas such as marine, railway, petrochemical industry, avionics, automotive, etc. In our research we try to give a simple mathematical model based on the multimachine concept that will allow us to study the behavior of the polyphase machine (pentaphase) by simulation in Matlab Simulink. Received: January 18th, 2013. Accepted: March 29th, 2013

67

2. Principle of modelling Vector A. Assumptions and presentation of the machine

- The effects of skin, shock absorbers, saturation, and variation of reluctance of the magnetic circuit are neglected.

- The emf induced in the stator windings are solely due to the rotor magnets which have a shape that is due only to the magnets and the structure of the windings. Armature reaction magnetic (due to stator currents) does not change the form of the emf.

- The phases are the same and offset by an nπα 2= , n is the number of phase of the

machine.

The emf induced in-phase depends only on the speed of the rotor and structural parameters such as: ( ) Ω⋅= θkfke , ( )θkf is a function of the form dependent of the rotor position θ and rotational speed Ω . Figure 1 shows a bipolar machine where the magnitude g is a voltage, current or flux on the phase k is denoted kg .

Figure 1. Representation of the polyphase synchronous machine

B. Definition of a natural basis: If we associate the n-phase machines [3] Euclidean vector space nE of dimension n, an

orthonormal basis of the space Bn: ⎭⎬⎫

⎩⎨⎧

= nnxnxnxnB .....,,2,1

It is called natural when the vector g can be written as:

nnxngnxgnxgg ⋅++⋅+⋅= ...2211 (1)

nggg ,...,2,1 : Measurable magnitudes of the stator phases. Consequently in this space can therefore be defined vectors:

- Voltage: nnxnvnxvnxvv ⋅++⋅+⋅= ...2211

- Current: nnxninxinxii ⋅++⋅+⋅= ...2211

V1

i1

I2 InV2 α α

(k-2)α

N

SkV

ki

nV

Abdelkrim Sellam, et al.

68

The voltage vector of the machine is:

enBdt

sdisRv +Φ+⋅=⎥⎥⎦

⎤

⎢⎢⎣

⎡

/ (2)

The projection of the machine voltage v to a vector k of the voltage of a phase stator gives:

ke

nBdt

skdkisRn

kxvkv +Φ

+⋅=⋅=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

/ (3)

This is the equation of the stator voltage with a phase: - skΦ : the flux in the phase k created by the stator currents.

- ke : is the emf induced in the phase k created by rotor magnets. Assumptions of

unsaturation and non reluctance variation can define a linear relationship ( )is λ=Φ between the current vector and the stator flux more usually written in the form of a matrix with constant coefficients:

[ ] [ ]ninsLs *=Φ

[ ]⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

nsnsLsnsLsnsL

nssLssLssLnssLssLssL

nsL

K

MOMM

K

K

21

2221212111

(4)

ksks

L Is the inductance of a stator phase

ksjsL Mutual inductance between stator phases.

The instantaneous power is transiting in the machine:

ivn

k kikvp ⋅=∑=

⋅=1

(5)

By replacing the expression vector voltage (2), we obtain the following equation:

( ) ieinBdt

sdisRp ⋅+⋅Φ+=⎥⎥⎦

⎤

⎢⎢⎣

⎡

/

2 (6)

- Electrical power lost by Joule effect are: 2

isRjp =

- The magnetic power is: inBdt

sdwp ⋅Φ=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

/

- Electromagnetic power is: ieemp ⋅=

- The electromagnetic torque is:Ω⋅= ie

emC

Ω is the instantaneous speed of the rotor.

A Vectorial modeling for the Permanent Magnet Synchronous Machine

69

C. Modelling of the machine with n phases in a base ensuring decoupling magnetic The relation ( )is λ=Φ , which is one of the morphism, between the current vector and stator

flux remains true whatever the base of the space nE chosen [4]. The base where exist the magnetic decoupling is that in which one coordinated stator flux vector can be expressed as a function of a single coordinate of the current vector (matrix diagonal inductance). Diagonalization of an inductance matrix requires research of the eigenvalues and eigenvectors associated with them. We define the eigenvalues kΛ the morphism λ as being

solutions of the characteristic equation: [ ] [ ]( ) 0det =−Λ nsLnI [ ]nI is the identity matrix of dimension n. The eigenvalues are real because the inductance matrix is symmetric. The hypothesis of regularity spatial of phases construction, allows us to affirm that inductance matrix is circulant. Circularity property allows us to calculate analytically the eigenvalues by using the formula for circulant determinant. These two conditions are respected; the complex eigenvalues are given by the solutions of the equation:

( )( )0

1 1

112

1=∏

=∑=

−−⋅−Λ

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛n

l

n

kn

klje

kssLπ

(7)

j is the complex operator. Equation (7) is divided into n equations each having a specific value as a solution of the morphismλ . These n eigenvalues are given in complex forms which are associated eigenvectors. These complex coordinate vectors form an orthonormal basis of the Hermitian space associated to machine. We want, as with the transform Concordia, work with real coordinates eigenvectors associated with real eigenvalues. The inductance matrix is symmetrical, therefore the values kΛ are real. We notice that: 2+−Λ=Λ knk then there exists an eigenvector

associated with the eigenvalue kΛ . It is therefore in the plane spanned by the vectors of an infinite orthonormal bases generated by the eigenvectors.

The property njkenknje /2/2)( ππ −=− allows determining an orthonormal basis composed of eigenvectors with real coefficients such that:

The new matrix inductance ⎥⎦⎤

⎢⎣⎡ dsL ¸ characteristic morphism in the new basis

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= dnxdxdxdB ,,2,1 K becomes:

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎥⎦⎤

⎢⎣⎡

Λ

ΛΛ

=

'00

020001

n

dsL

K

MOMM

K

K

(8)

This matrix is diagonal and we recall that the inductances of the matrix are at least equal in pairs (eigenvalue of multiplicity of order 2). This research eigenvalues associated to the inductance matrix can formulate a generalized Concordia transformation with transition matrix as the natural base to base decoupling [5]. Bases of departure and arrival are orthonormal,

Abdelkrim Sellam, et al.

70

This transformation has the property to preserve the instantaneous power regardless of the base in which it is expressed. D. The equations of the machine in a base decoupling A vector g the initial space decomposes into:

∑=

==

Ng

gggg

1 (9)

Let N be subspaces each associated to an eigenvalues Λg, gg is the projection of the vector

g on the subspace gE . The new equation of the flux vector and current:

∑=

=Λ=∑

=

=Φ=Φ

Ng

ggig

Ng

gsgs

11 (10)

Allows writing in each sub-space, a new voltage equation:

genE

dtsgd

gisRgv +Φ

+=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

/ (11)

Using the property (10) into (11):( )

gegE

dtgigd

gisRgv +Λ

+=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

/

gegE

dtgid

ggisRgv +Λ+=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

/

(12) The electrical power which transits into the real machine is expressed by [6]:

∑=

⋅=⋅=N

ggigvivp

1 (13)

By replacing the expression of voltage (12) in the power equation (13) we obtain:

( )∑=

⋅+⋅Λ+=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡N

ggigegi

gEdtgid

ggisRp1 /

2

(14)

The previous equation shows that the energy transits through N fictitious machines, independent magnetically associated with N eigenspaces. Consequently, the actual torque is written:

A Vectorial modeling for the Permanent Magnet Synchronous Machine

71

∑=

=N

ggCC

1 (15)

With: gigegC ⋅=Ω⋅ . Remark: Equation (14) shows that each fictive machine produces a torque participant in the creation of a total torque. These N fictitious machines are mechanically coupled: they rotate at the same speed and are rigidly coupled to the same mechanical shaft. 3. Pentaphase machine application A. presentation of the machine We model [8], for the application, a synchronous machine with permanent magnets pentaphase. This machine is represented symbolically in Figure 2.

Figure 2. Representation of the pentaphase machine

B. Modelling of the machine in the natural basis

We associate the five phases a Euclidean vector space 5E of dimension 5. We write in an orthonormal base Bn the voltage equation of the machine:

enBdt

sdisRv +Φ+⋅=⎥⎥⎦

⎤

⎢⎢⎣

⎡

/ (16)

{ }5,4,3,2,1 xxxxxnB = Orthonormal basis In linear mode, there exists morphism between vectors stator flux and current, such as:

( )is λ=Φ (17)

N

S

V1

V4

V3

V5

V2I2

I3 I4

I5

720

I1

Abdelkrim Sellam, et al.

72

[ ]⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

LMMMMMLMMMMMLMMMMMLMMMMML

nsL

12211122211222111221

(18)

With: - L the inductance of a phase ⎟

⎠⎞⎜

⎝⎛ += flpLL ;

- 1M The mutual inductance between two phases shifted from5

2π± ;

- 2M The mutual inductance between two phases shifted from5

4π± .

emf vector :

5544332211 xexexexexee ⋅+⋅+⋅+⋅+⋅= With:

=ke 5,...,1),5

)1(2sin(max =−−= kktEke πω

Ω⋅= keEmax is the maximum value of the emf with k emf coefficient and Ω the

rotational speed of the rotor. C. Modelling in a base decoupling

There exists an orthonormal basis ⎭⎬⎫

⎩⎨⎧= βαβα sxsxpxpxzxdB ,,,, in which the inductance

matrix is diagonal:

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎥⎦⎤

⎢⎣⎡

ΛΛ

ΛΛ

Λ

=

4000003000005000002000001

dsL (19)

It appears then double values:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛

⋅+⋅⋅−=Λ=Λ

⋅+⋅⋅−=Λ=Λ

++=Λ

53cos25

cos1243

5cos25

3cos1252

)21(21

ππ

ππ

MML

MML

MML

Inductors are associated with eigenvectors. There is a single eigenvalue and two double eigenvalues. This property allows us to decompose the vector space 5E at three orthogonal subspaces, namely: - A subspace zE generated by the eigenvector ( )zx associated with the eigenvalue 1Λ=Λ z .

This subspace is a straight line called homopolar

A Vectorial modeling for the Permanent Magnet Synchronous Machine

73

- A subspace pE generated by the eigenvectors ⎟⎠⎞⎜

⎝⎛

βα pxpx , associated to the eigenvalue:

52 Λ=Λ=Λ p . This subspace is called the primary plan.

- A subspace sE generated by the eigenvectors ⎟⎠⎞⎜

⎝⎛

βα sxsx , associated with the eigenvalue

43 Λ=Λ=Λs . This subspace is called the secondary plan.

the vector sgpghgg ++= D. Equivalence between real and fictitious machine 1: Equation electrical of a real machine. 2: Equation electrical of the fictitious machines A fictive machine may be associated with each subspace, respectively - A machine associated with the two-phase principal plan possessing the time constant and

emf induced the most important. I The fictitious principal machine is magnetically decoupled and depend

of his own current principal. - A machine associated with the plan secondary at two-phase possessing the electrical constant time lowest and emf induced the less important. The fictitious secondary machine is magnetically decoupled and depends of his own current secondary. - A machine phase associated with the straight line with the homopolar electric time constant

and emf induced weaker. The fictitious homopolar machine is magnetically decoupled and depend of his own current homopolar. 4. Simulation of the pentaphase machine We implement the model of the machine on the software numerical simulation Matlab Simulink. Us apply a load at startup resistant Cr=1N.m we take the speed loop fixed to the rated speed 157 rad / s. emf :

5,...,1),5

)1(2sin(max =−−= kktEke πω

The currents are sinusoidal and in phase with the emf

5,...,1),5

)1(2sin(max =−−= kktIkiπω

k means the number of the phase. For our machine there are 5 phases (k phases).

1 2

Real machine fictitious machines

Abdelkrim Sellam, et al.

74

A. emf in the basis of Concordia for fictitious machines: Figure 3

Figure 3. emf in fictitious machines (The basis of Concordia).

emf in the principal machine:

)sin(max2/5 tEpe ωα =

)cos(max2/5 tEpe ωβ −=

emf in the secondary machine :

0=seα

0=seβ emf in homopolar machine:

0=he Interpretation (figure 3): We note that the emf in the basis of Concordia for fictitious machines that only that emf of the fictitious principal machine is different from zero. B. Currents in the basis of Concordia for fictitious machines: Figure4

Figure 4. Current in fictitious machines (The basis of Concordia).

Current in the principal machine :

)sin(max2/5 tIpi ωα =

)cos(max2/5 tIpi ωβ −=

A Vectorial modeling for the Permanent Magnet Synchronous Machine

75

Current in the secondary machine: 0=siα , 0=siβ

Current in the homopolar machine : 0=hi Interpretation (figure 4): We note that the current in the basis of Concordia for fictitious machines that only that current of the fictitious principal machine is different from zero. C. The emf in the fictitious machine (the base of Park): Figure 5

0=he , 0=pde , 0=sde et 0=sqe

Figure 5. emf in the fictitious machine (Base of Park)

Interpretation (figure 5): We notice (Figure 5) in the basis of Park, that the components of the emf are zero except the principal emf quadratic component which is a constant ( 0 . D. Current in the fictitious machine (the base of Park): Figure6 0=sdi 0=sqi

Figure 6. Current in the fictitious machines

(Basis of Park).

Abdelkrim Sellam, et al.

76

Interpretation (figure 6): We notice (Figure 6) in the basis of Park, the currents electrical of the homopolar and secondary machine are zero but the currents of the principal machine are constants nonzero. E. The electromagnetic torque: Figure 7

max25maxmax

25)(1

)(1

kIIEpqipqenC

pipepipezizenC

=Ω

=Ω

=

++Ω

= αβαβαβαβ

Ω

= eEk max

Figure 7. electromagnetic torque Interpretation (figure 7): We remark (Figure 7) that the torque is constant is equal to the load torque after a transitional regime. F. The speed of the pentaphase machine: Figure8

Figure 8 The speed of the pentaphase machine Interpretation (figure 8): We find after a short transitional regime, the curve becomes a constant of 157rd / s which is the rated speed of the machine.

A Vectorial modeling for the Permanent Magnet Synchronous Machine

77

General interpretation of the simulation: The simulations allow us to observe the emf in fictitious machines and currents in the basis of Park. The transformation of Park applied to these magnitudes indicates that only the component (q) of the emf of the principal machine is a nonzero constant (Figure 5). One can observe that the homopolar emf is zero, which is coherent for a machine sinusoidal, balanced and non-electrically coupled. Accordingly, the homopolar machine produces no torque (Ch = 0) as well as for the secondary machine (Cs=0).So the principal machine (two-phase) alone produces torque. We find that the torque is constant (Figure 7). With: 0 0 Conclusion A polyphase machine is composed of n windings spatially 2π/n and powered by the voltage phase-shifted temporally 2π/n. These machines are characterized by a magnetic coupling between phases. The generalization of the method of space vector allows defining a base change of dimension n, implying a simplification of the study of the machine by diagonalization of the matrix inductance. This change leads to subspaces orthogonal base of dimension 2 and 1, each subspace can be independent. Association the vector space modelling and concept multimachine allows us to consider a polyphase machine as equivalent to a set of fictitious machines mechanically coupled. The study of a complex machine turns into several studies of simple machines. References [1] E. Semail, "Outils et méthodologie d’étude des systèmes électriques polyphasés.

Généralisation de la méthode des vecteurs d’espace," thèse de doctorat, Université des Sciences et Technologies de Lille (USTL), juin 2000.

[2] D. Hadiouche, "Contribution à l'étude de la machine asynchrone double étoile : modélisation, alimentation et structure", thèse de doctorat de l'UHP, Nancy 1, décembre 2001.

[3] E. Semail, A. Bouscayrol, J.P. Hautier, “Vectorial formalism for analysis and design of polyphase synchronous machines,”EPJ AP (European Physical Journal-Applied Physics), vol. 22, no 3, June 2003, pp. 207-220.

[4] H.A. Toliyat, T.A. Lipo, J. C. White, “ Analysis of a Concentrated Winding Induction Machine for Adjustable Speed Drive Application Part 1 (Motor Analysis),” IEEE Transactions on Energy Conversion, Vol. 6, no.4, pp. 679-683, 1991.

[5] R. Lyra, T. Lipo, “Torque Density Improvement in a Six-Phase Induction Motor with Third Harmonic Current Injection,” IEEEIAS' 01, Chicago, CD-ROM, September 2001.

[6] A. Bouscayrol, Ph. Delarue, E. Semail, J. P. Hautier, J. N. Verhille, "Application de la macro-modélisation à la représentation énergétique d’un système de traction multimachine," Revue Internationale de Génie Electrique, vol. 5, n°3, pp 431-453, octobre 2002.

[7] A. Mohammadpou, L. Parsa, "A Unified Fault-Tolerant Current Control Approach for Five-Phase PM Motors With Trapezoidal Back EMF Under Different Stator Winding Connections," Power Electronics, IEEE Transactions on , vol.28, no.7, pp.3517,3527, July 2013.

[8] N. Bianchi, S. Bolognani, M.D.PreÌ, "Strategies for the Fault-Tolerant Current Control of a Five-Phase Permanent-Magnet Motor," Industry Applications, IEEE Transactions on , vol.43, no.4, pp.960,970, July-Aug. 2007.

Abdelkrim Sellam, et al.

78

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Abdelkrim Sellam, et al.

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