Multi-phase System Supplied by PWM VSI. A NewTechnic to Compute the Duty Cycles.

Xavier Kestelyn, Eric Semail, Alain Bouscayrol, J.P Hautier.L2EP, Laboratoire d'Electrotechnique et d' Electronique de Puissance de Lille

Université de Lille Bât. P2 F-59655 Villeneuve d'Ascq cedex, Francee-mail: xavier.kestelyn@ed.univ-lille1.fr

http://www.univ-lille1.fr/l2ep/

Keywords: Multi-phase, VSI, Vectorial PWM, SVM Multi-machine.

AbstractWhen multi-phase machines with magnetic couplings between phases are supplied by

triangle intersection PWM, there are parasitic currents. To solve the problem, it's necessaryto use Space Vector Modulation. We propose a SVM extended to systems with arbitrarynumber of phases and with optimization of time computation and consideration of switchingconstraints.

1 IntroductionWhen multi-phase machines are supplied by VSI controlled by classical intersective PWM, largeparasitic currents appeared if the switching frequency is chosen with regard to the electrical timeconstant of the �rst-harmonic d-q equivalent machine [2], [8]. This phenomenon is observedeven in the steady state with sinusoidal references. In fact, it can be shown that a multi-phase machine is equivalent with a set of �ctitious one-phase or two-phase independent machinesmechanically and electrically coupled [1], [13]. The time constants of the di�erent machines areusually quite at all di�erent. The �rst-harmonic d-q equivalent machine, which produces thetorque for the most part, has the greatest time constant. If the switching frequency is chosenwith regard to this machine, large parasitic currents are induced in the other machines whosetime constants are smaller. To achieve good control with reasonable switching frequency it isimportant to choose right instantaneous voltages which reduce the parasitic currents. It is notpossible with classical intersective PWM but can be handle with pure numerical method [14] orwith a Space Vector Modulation extended to multi-phase systems. In particular cases, such SVMcontrols have already been implemented: [2] and [15] have chosen instantaneous vectors to controldual three-phase induction machines, [4] and [6] to control 5-phase machines. Nevertheless, theproposed technics need to determine the locus of the desired reference in a de�ned sector. Thisdetermination takes time of computation and can become prohibitive to systems with manyphases. Moreover switching constraints, as the reduction of the number of switchings in a PWMperiod, are not always taken into account [2], [4].

We propose a new algorithm to calculate the duty cycles of each leg of the VSI with opti-mization of time computation and consideration of switching constraints. This new algorithmis based on a vectorial approach of inverters developed in [3],[7],[5]. This algorithm permits to

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calculate the duty cycles of each leg with no need of �nd the location of the reference vector.Moreover the number of switchings in a period of PWM is taken into account.

Even if it is impossible to supply each �ctitious system independently, it is possible to imposethe right mean voltage to each one. In this paper,the duty cycles of VSI legs are computed fromthe references of �ctitious systems. To be able to illustrate this technic, we will apply it at a�rst time to a three-phase system. Then we propose the current control of a �ve-phase system.

2 Multi-phase system supplied by PWM VSI2.1 Presentation of the two used topologiesUsually to reduce the number of switches multi-phase systems are star connected and it's im-possible to generate zero-sequence current. In other words, there's a loss of one freedom degree.To have more reliability and maximum freedom degrees, then phases of the system can be notelectrically coupled. The �gure 1 shows the two topologies of supply.

Figure 1: Two of the possible topologies of VSI.

2.2 Vectorial modellingIn an orthonormal baseBn composed of the vectors {−→xn

1 ,−→xn

2 , ...,−→xn

n} it can be de�ned the voltagevector: −→v = v1

−→xn

1 + v2−→xn

2 + ... + vn−→xn

n

where vk is the voltage across the phase number k.If the VSI is a two-level one, it can be generated2n di�erent voltage vectors.In a great majority of cases, there are magnetic couplings between the phases of a multi-phase

system. The inductance matrix is then :

[Lns ] =

L11 L12 ... L1n

L21 L22 ... L2n... ... . . . ...

Ln1 Ln2 ... Lnn

Where Lkk is the self-inductance of the phase k and Ljk is the mutual inductance between thephases j and k.

The magnetic couplings between the phases make di�cult the realization of the commandand it's better to model the machine in a base where there are no magnetic couplings.

The new matrix inductance [Lds ], in a base Bd = {

−→xd

1,−→xd

2, ...,−→xd

n} where there are no magneticcouplings, is :

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[Lds ] =

Λ1 0 ... 00 Λ2 ... 0... ... . . . ...0 0 ... Λn

The linear relations between, Λ and L,−→xd and −→xn, are obtained by the use of an extended

Concordia transformation.In the base Bd, we can de�ned a set of �ctitious magnetically independent systems.To �nd the new voltage vectors generated by the �ctitious VSI, it is enough to apply the

same transformation to the initial voltage vectors.Instead of generating the initial mean voltage vector reference of the initial system, we prefer

to generate the mean voltage vector reference of each �ctitious system. These voltage referencesare obtained by vectorial projections onto subspaces associated to the vectors of the baseBd.

2.3 Generation of a mean voltage vector with consideration of switching con-straints

To reduce the number of switchings in a period of PWM, the following switching constraints areimposed:

• A minimum number of switchings are used to generate any initial mean voltage reference,

• Only one switching is possible at the same time.

A simple way to respect these two switching constraints is to make the same choice as a triangleintersection PWM:

• Only n+1 di�erent voltage vectors are used to generate any initial mean voltage reference(A su�cient condition to generate any voltage reference),

• Two voltage vectors consecutively generated by the VSI have only one di�erent coordinate(There is only one switching between two consecutive voltage vectors),

• Each sequence of generated vectors begins and �nishes with the same vector. A simplemanner to do that is to repeat the sequence in the reverse order after its generation (Thereis no switching between two periods of PWM).

The generated patterns are the same as a triangle intersection PWM but each conductiontime of the vectors is computed with a extended SVM. Compared to classical technics [11], [10],[9], [12], there's no need �nding the location of the desired vector onto a de�ned sector. At thebeginning of each period of PWM, each duty cycle (2αn

kT ) is computed and held during all theperiod. Figure 2 shows the implemented PWM pattern.

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Figure 2: Implemented PWM pattern

3 Case of a three-phase system3.1 IntroductionFor example, let us choice a three-phase machine with no-couplings. In an orthonormal baseBn

composed of the vectors {−→x1,−→x2,

−→x3} the vectorial voltage equation of the machine is:

−→vs = Rs−→is +

[d−→φs

dt

]

/Bn

where−→vs ,−→is and−→φs are respectively the statoric voltage, statoric current and statoric �ux vectors.

Classically, the command of the machine is done in a Concordia frame, in other words thereis a changement of the natural baseBn to a base Bd = {−→xz,

−→xa,−→xb} where there are no magnetic

couplings.We remind the Concordia's transformationC−1

33 :

−→xz−→xa−→xb

=

1√3

1√3

1√3√

23 − 1√

6− 1√

3

0 − 1√2

− 1√2

−→x1−→x2−→x3

3.2 Modelling of the VSIDue to the two-level command (vk = ±Vbus), the voltage vector −→v = v1

−→x1 + v2−→x2 + v3

−→x3 cantake 23 = 8 di�erent values. Figure 3 shows the representation of the di�erent values of−→v (bluecrosses) in an orthonormal frame {O,−→x1,

−→x2,−→x3}. Two values separated by only one switching

are linked by a line. In this case, we obtain a cube.To easily spot the di�erent values of the vector −→v , each value is associated to a number

between 0 and 7. This number is directly associated to the switching mode. When each decimalnumber is converted to a three-digit binary number, the 1's in this number indicates that thecorresponding phase is supplied by+Vbus and the 0's indicates that it is supplied by−Vbus. Themost signi�cant bit (�rst bit on the left) represents the switching state of phase 1 and so on.

In the base Bd, the statoric inductance matrix only has two di�erent values. One associatedto the vector −→xz, called the leakage inductance, and the other associated to the vectors−→xa and−→xb, called cyclic inductance.

Then in the new base Bd it can be de�ned two particular orthogonal subspaces : a linesupported by the vector−→xz and a plane supported by the vectors−→xa and −→xb.

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Figure 3: 3D representation of the voltage vector and its projections onto particular subspaces

Figure 3 shows the line (in green) and the plane (in red) and the projections of the23 valuesof the voltage vector onto.

It's important to notice that usually machines are star connected and only the plane isconsidered.

3.3 Generation of a mean voltage in a period of PWMThe generation of the initial mean voltage reference is now become generations of two references−−−−−−→< vzref

> and−−−−−−−→< vmref>, projections of−−−−−−→< vref > respectively belonging to the line and the plane,

such as : −−−−−−→< vref > = −−−−−−→

< vzref> +−−−−−−−→

< vmref>

There are many solutions to respect the switching constraints previously de�ned but a naturalchoice consists in using the family (0,7) composed by four vectors as a triangle intersection PWMdoes. In this family there are3! = 6 di�erent combinations which respect the constraints (formingsix identical trihedrons in the cube which are projected in six triangles onto the plane).

The classical technic needs to �nd the location of the vector into the 6 trihedrons (or thesix triangles if there is a star coupling) and to calculate the conduction time of the four vectorscomposing the chosen volume (or sector).

Due to the symmetry of the six trihedrons (or triangles), it is no more necessary to �ndthis location. Indeed, due to the two levels command, voltage vectors are opposite by two(−→vk = −−−−−−−→v(2n−1)−k) and a vector associated to a negative duty cycle is the same as its oppositeassociated to a positive duty cycle.

If we take for example the combination of the vectors (0 1 3 7), the vector−−−−−−→< vref > (whereverit is located onto the cube) is given by the relation :

−−−−−−→< vref > = αd

0−→v0 + αd

1−→v1 + αd

3−→v3 + αd

7−→v7 (1)

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Where the αdk ∈ [−1, 1] can be positive or negative.

The aim is to write the vector−−−−−−→< vref > in a manner :−−−−−−→< vref > = Vbus((2αn

1 − 1)−→x1 + (2αn2 − 1)−→x2 + (2αn

3 − 1)−→x3) (2)

where αnk ∈ [0, 1] are the duty cycles of the legs of the VSI.

If we give the vectors −→v0 , −→v1 , −→v3 and −→v7 as an expression of Vbus, −→x1, −→x2 and −→x3 we obtain :−→v0 = −Vbus

−→x1 − Vbus−→x2 − Vbus

−→x3−→v1 = −Vbus

−→x1 − Vbus−→x2 + Vbus

−→x3

−→v3 = −Vbus−→x1 + Vbus

−→x2 + Vbus−→x3

−→v7 = +Vbus−→x1 + Vbus

−→x2 + Vbus−→x3

These four equations placed in equation (1) gives:−−−−−−→< vref > = Vbus((−αd

0−αd1−αd

3 +αd7)−→x1 +(−αd

0−αd1 +αd

3 +αd7)−→x2 +(−αd

0 +αd1 +αd

3 +αd7)−→x3) (3)

As the four vectors −→v0 , −→v1 , −→v3 and −→v7 are used in a period of PWM, we have the relation:

αd0 + αd

1 + αd3 + αd

7 = 1 (4)

With the use of relation (4) in equation (3) we obtain :−−−−−−→< vref > = Vbus((2αd

7 − 1)−→x1 + (2(αd3 + αd

7)− 1)−→x2 + (2(αd1 + αd

3 + αd7)− 1)−→x3) (5)

By identi�cation of the equations (2) and (5) we can �nally �nd:

αn1 = αd

7

αn2 = αd

3 + αd7

αn3 = αd

1 + αd3 + αd

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3.4 Computation of the αdk's

To calculate an αdk it's enough to make successive cross products between the vectors of the

family. Indeed the cross product of a vector by itself is null: −→g ∧ −→g =−→0 . To use this technic

of calculation, it's impossible to have two colinear vectors. The easier solution to respect thisconstraint is to add a fourth dimension to each vector.

As an example, let calculate αd7:

−−−−−−→< vref > ∧ −→v0 = (αd

0−→v0 + αd

1−→v1 + αd

3−→v3 + αd

7−→v7) ∧ −→v0 = αd

1−→v1 ∧ −→v0 + αd

3−→v3 ∧ −→v0 + αd

7−→v7 ∧ −→v0

−−−−−−→< vref > ∧ −→v0 ∧ −→v1 = αd

3−→v3 ∧ −→v0 ∧ −→v1 + αd

7−→v7 ∧ −→v0 ∧ −→v1

−−−−−−→< vref > ∧ −→v0 ∧ −→v1 ∧ −→v3 = αd

7−→v7 ∧ −→v0 ∧ −→v1 ∧ −→v3

αd7 =

−−−−−−→< vref > ∧ −→v0 ∧ −→v1 ∧ −→v3−→v7 ∧ −→v0 ∧ −→v1 ∧ −→v3

After rearrangement it appears the mixed product:

αd7 =

(−→v0 |−→v1 |−→v3 |−−−−−−→< vref >)(−→v0 |−→v1 |−→v3 |−→v7)

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By the same way, it is possible to calculate the othersαdk's:

αd3 =

(−→v0 |−→v1 |−−−−−−→< vref >|−→v7)(−→v0 |−→v1 |−→v3 |−→v7)

αd1 =

(−→v0 |−−−−−−→< vref >|−→v3 |−→v7)(−→v0 |−→v1 |−→v3 |−→v7)

αd0 =

(−−−−−−→< vref >|−→v1 |−→v3 |−→v7)(−→v0 |−→v1 |−→v3 |−→v7)

The calculation of a mixed product is the same as a determinant and requires only somesums and products. The denominator is constant and can be calculated o�-line. The numeratorhas only one variable and can be reduced to few calculations.

4 Case of a �ve-phase system4.1 IntroductionTo reduce the number of legs, we choose to star connect the phases of the machine. The baseBd

where there are no magnetic couplings is deduced with the extended Concordia's transformationbellows :

−→xz−→xa−→xb−→xc−→xd

=

1√5

1√5

1√5

1√5

1√5√

25

√25 cos 2π

5

√25 cos 4π

5

√25 cos 6π

5

√25 cos 8π

5

0√

25 sin 2π

5

√25 sin 4π

5

√25 sin 6π

5

√25 sin 8π

5√25

√25 cos 4π

5

√25 cos 8π

5

√25 cos 12π

5

√25 cos 16π

5

0√

25 sin 4π

5

√25 sin 8π

5

√25 sin 12π

5

√25 sin 16π

5

−→x1−→x2−→x3−→x4−→x5

Due to the 5 dimensions, it is no more possible to represent graphically the di�erent valuesof the voltage vectors in the natural base. Three orthogonal subspaces can be de�ned:

• a line associated to the vector−→xz

• a plane called main associated to the vectors−→xa and −→xb

• a plane called secondary associated to the vectors−→xc and −→xd

The 25 projections of the values of the voltage vector onto these subspaces are shown in �gures4 and 5. The line is not considered due to the star connection.

The voltage vector reference is then:−−−−−−→< vref > = −−−−−−→

< vzref> +−−−−−−−→

< vmref> +−−−−−−→

< vsref>

where −−−−−−→< vzref> =

−→0 , −−−−−−−→< vmref

> belongs to the main plane and −−−−−−→< vsref> belongs to the sec-

ondary plane.

4.2 Computation of the duty cyclesLike for the three-phase system, a natural choice consists in using the family (0, 31) as a triangleintersection PWM does. In this family, there are 5! = 120 di�erent combinations of six vectorswhich respect the constraints.

If we take for example the combination of the vectors (0 1 3 7 15 31) shown in red in the�gures 4 and 5, the vector−−−−−−→< vref > to be synthetized is given by the relation :

−−−−−−→< vref > = αd

0−→v0 + αd

1−→v1 + αd

3−→v3 + αd

7−→v7 + αd

15−→v15 + αd

31−→v31

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Figure 4: Projections of the voltage vector onthe main plane.

Figure 5: Projections of the voltage vector onthe secondary plane.

By the same way as previously, we can �nd the relations :

αn1 = αd

31

αn2 = αd

15 + αd31

αn3 = αd

7 + αd15 + αd

31

αn4 = αd

3 + αd7 + αd

15 + αd31

αn5 = αd

1 + αd3 + αd

7 + αd15 + αd

31

4.3 Practical resultsThis algorithm of computation has been used in a current control of a �ve-phase synchronousmachine. The switches are controlled with a DSPACE board programmed under MATLAB-SIMULINK. The experimental bench is shown in the �gure 6 and the structure of the controlsystem is shown in the �gure 7.

The �gure 8 and 9 shows the currents in the real and the �ctitious machines with the followingreferences: −−−−−−−→

< vmref> = Im(sin(2π10t)−→xa + sin(2π10t +

π

2)−→xb)

−−−−−−→< vsref

> = Is(sin(2π30t)−→xc + sin(2π30t +3π

2)−→xd)

We can observe more noisy currents in the secondary �ctitious machine than in the main�ctitious machine because of the di�erence between the two time constants.

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Figure 6: Experimental bench.Figure 7: Structure of the proposed control.

Figure 8: Real currents. Figure 9: Fictitious currents.

5 ConclusionIn the paper we show a new method to compute the duty cycles of VSI's legs. This methoddoesn't need to locate the vector in a particular sector and can be implemented with a systemwith no great capability of computation.

Particular patterns of PWM are used to take into account switching constraints. Otherconstraints could be chosen for example to generate less parasitic currents in the secondarymachine.

This work is based on a vectorial modelling of multi-phase systems which permits easily theextension to an arbitrary number of phases and for three-level inverters.

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