Rombaut C Laboratoire D'electrotechnique

D ' Electronique

New method to calculate the conduction durations of the switches in a n-leg 2-level Voltage Source Inverter

Keywords: Converter control, EMC/EMI, Control, Matrix converter, Modulation strategies, Multilevel converters, Real time processing, Three phase systems, Vector control

The proposed method allows to calculate explicitly, using mathematical notions of barycenter and mixed product, the conduction duration switches of a n-leg 2-level Voltage Source Inverter (VSI) supplying a n-wire load. This method depends on a new geometrical and vectorial VSI characterization. Its geometrical feature enables optimization and graphic representation such as the Space Vector Method (SVM) for a 3-leg VSI. Its vectorial feature enables to generalize to n phase systems the properties found out with three phase systems. _________________________________________________________________________________________________________________ New method to calculate the conduction durations of the switches in a n-leg 2-level Voltage Source Inverter Semail E __________________________________________________________________________________________________________________

Introduction

When a VSI is controlled with a carrier-based PWM, the conduction durations of the switches have to be calculated to obtain the correct average values of the voltages applied to the load. The more classical way is the Suboscillation Method [START_REF] Holtz | Pulsewidth Modulation -A Survey[END_REF] which determines the intersections between a triangular shape wave and the desired average values of voltages. For a 3-leg 2-level VSI supplying a star connected load with isolated neutral, another method, more favorable to optimization [START_REF] Blasko | Analysis of a hybrid PWM based on modified space vector and triangle-comparison methods[END_REF], [START_REF] Dae-Woong | Unified Voltage Modulation technique for real Time Three-Phase Power Conversion[END_REF] is the Space Vector Method. Then, to obtain the conduction durations, projections of the desired vectors must be achieved (Fig 1). However, this last method can hardly be generalized to the study of n-leg VSI supplying n-wire loads [START_REF] Toliyat | Analysis and Simulation of Five-Phase Synchronous Reluctance Machines Including Third Harmonic of Airgap MMF[END_REF], [START_REF] Zhao | Space Vector PWM Control of Dual Three-Phase Induction Machine Using Space Vector Decomposition[END_REF] or even a 3-leg VSI supplying a star connected load with neutral not isolated [START_REF] Edelmoser | Common Mode Disturbances in Force Controlled Rectifier Systems[END_REF]. It is always possible to use the Suboscillation Method but in this case we have no more geometrical tools to analyze and optimize the performances of different control laws. A few works, [START_REF] Ali | Kazmierkowski " Current regulation of four-leg PWM-VSI[END_REF], [START_REF] Dastfan | Control of a New Active Power Filter Using 3-D Vector Control[END_REF], develop, for 4leg VSI a 3-dimensional approach by using vectors that belong to a 3-dimensional space. Other works [START_REF] François | Design and modelling of a five-phase voltage-source inverter for two induction motors[END_REF], [START_REF] Oriti | Elimination of Common Mode Voltage in three phase sinusoidal Power Converters[END_REF] use more general methods for n-leg VSI but with no geometrical approach. We propose in this paper a general geometrical vectorial characterization of VSI. So, a n-dimensional vectorial space F n is introduced for studying a n-leg VSI. This characterization enables to use geometrical tools for optimization of n-leg VSI control as the SVM for a 3-leg VSI. Then, we describe how to determine explicitly, in a such space F n , the conduction durations of the switches. From this point of view, it is a generalization of the way to achieve, by SVM, the conduction durations for a 3-leg VSI (Fig 1). At last, we explicit for a 3-leg 2-level VSI the relations of the method with SVM and with Suboscillation method.

Characterization of a n-leg 2-level VSI

Vectorial space F n associated to VSI

The n-leg inverter represented in Fig 2 imposes n voltages v ck . So, we associate to this converter a vectorial space F n with an orthonormal base of vectors (x c1 , x c2 ,…, x cn ). We can define then a voltage vector:

v c = v c1 x c1 + v c2 x c2 +…+ v cn x cn .
This voltage vector characterizes the different voltages that the inverter can impose to the load. Besides, it is easy to find out the voltage v ck from v c : we have only to achieve the scalar product of the two vectors v c and x ck : x ck = v c . x ck . From this point of view the vector v c is more convenient than complex vector c v . For this last one, it is not so easy to obtain v ck from c v . For a 3-level VSI, this kind of characterization can be useful for studying the ground current escaping through stray capacitors [START_REF] Edelmoser | Common Mode Disturbances in Force Controlled Rectifier Systems[END_REF] but also the bearing current and shaft voltage [START_REF] Erdman | Effect of PWM Inverters on AC Motor Bearing Currents and Shaft Voltages[END_REF] or the mean to reduce common mode emissions [START_REF] Cacciato | A digital current control reducing common mode emissions[END_REF]. Elimination of common mode voltage in three phase converters by use of 4-leg VSI ([10], [START_REF] Ogasawara | Circuit Configurations and Performance of the Active Common-Noise Canceler for reduction of Common-Mode Voltage Generated by Voltage-Source PWM Inverters[END_REF]) could also take advantage of this characterization.

Geometrical characterization of VSI

        + + + = - + - = + - + = - + + = + - - = - - + = + + - = - - - = c3 x c2 x c1 x OM c3 x c2 x c1 x OM c3 x c2 x c1 x OM c3 x c2 x c1 x OM c3 x c2 x c1 x OM c3 x c2 x c1 x 1 OM c3 x c2 x c1 x OM c3 x c2 x c1 x OM 7 3 6 2 5 4 0 E E E E E E E E E E E E E E E E E E E E E E E E 1 3 2 j e π 3 4 j e π (E,-E,-E) (E,E,-E) (-E,E,-E) (-E,E,E) (-E,-E,E) (E,-E,E) (-E,-E,-E) (E,E,E)         + + = π π

Average control and barycenter notion

We consider a carrier-based PWM with a period carrier T. At the kT instant, <v c >, the mean value of the vector v c (t) can be expressed by the formula:

∑ ∑ ∫ - = = - = = - = = = = 1 P r 0 r r 1 P r 0 r r kT T ) 1 k ( T t T t dt ) t ( T 1 ) kT ( OM OM v v v r cr c c (1)
In this expression, t r is the activation duration of the vector v cr . , can be considered as the barycenter of the P points M r , with barycentric coordinates t r /T. Moreover, as t r is positive, M is inside the polyhedron B. So, when an average control is adopted, it is the entire volume of the polyhedron B that characterizes the inverter and not only its vertex.

How to find the barycentric coordinates

Let us suppose that we have found a M point which allows to obtain the desired voltages for the load. We know that M belongs to the polyhedron B and can be considered as the barycenter of P vertex of the polyhedron B. The problem is to find the barycentric coordinates. In fact, it is possible to find less than P vertex whose M is the barycenter. The most practical interesting case consists in taking n+1 vertex which generate the vectorial space F n . First, because it will be always possible to find such n+1 vertex of the polyhedron B whose M will be the barycenter. The reason is that the dimension of the vectorial space F n associated to the n-leg VSI is E,E,) . So we can adapt the formula (1) to our particular case to obtain:

(E,E,E) (-E,-E,-E) (E,-E,-E) Μ 0 Μ 1 Μ 7 Μ 2 M 2 M 0 M 1 M 7
∑ + = = = 1 n k 1 k k T t k ON OM (2)
As

∑ + = = = 1 n k 1 k k t T , we have ∑ + = = = 1 n k 2 k k T t 1 1 ON ON
. The expression ( 2) is then equivalent to:

∑ + = = = 1 n k 2 k k T t k 1 1 N N M N . ( 3
)
On the basis of this formula we will find, by use of classic vectorial properties, the barycentric coordinates t k /T.

A free family

As the n+1 points N k generate the vectorial space F n , the family of n vectors {N 1 N k , 1<k<n+2} is free. Consequently the determinant, ) ,..., , det(

1 Nn N N3 N N2 N 1 1 1 + , is different from zero: 0 ) ,..., , det( ≠ +1 Nn N N3 N N2 N 1 1 1 (4)

Mixed product

In a n-dimensional vectorial space F n , the mixed product of n vectors w k is simply the determinant of a matrix, elaborated by concatenation of the n vectors w k :

( ) ) ,..., , det( ... The following property will be used : if exist k and r, k ≠r, as w k = w r than ( ) 0 ...

1 n 1 3 1 2 1 N N ... N N N N
+ to give the announced result.

Analysis of results

The expression of t k is easy to implement because, for each t k , , there is only one vector, N 1 M, whose n coordinates change. The others vectors N 1 N k are constant and have to be calculated only one time.

The relation [START_REF] Edelmoser | Common Mode Disturbances in Force Controlled Rectifier Systems[END_REF] shows that to obtain t k we have, in general, to solve an algebraic system of n equations with n unknowns:

(t) = A (<v c >) + (b), with (b) a constant vector, ( ) ( ) ( ) ( ) t cn 2 c 1 c t 1 n 3 2 v v v and t t t > > + < > < < = > < = c v t
. The matrix A characterizes the polyhedron B n+1 . _________________________________________________________________________________________________________________ New method to calculate the conduction durations of the switches in a n-leg 2-level Voltage Source Inverter Semail E

Taking account the properties of the polyhedron B n+1 can lead to reductions. For example, in a few particular cases the matrix A is triangular. Then it is possible to obtain simply the solution t k because A -1 , the inverse matrix of A, is also triangular. We explicit this last case for a 3-leg inverter in a next paragraph. An explicit research of the duration t k is then no more useful because they can be considered as the intersection between a triangular shape wave and 3 horizontal lines (we find out the Suboscillation Method which is one way to implement the results).

Other cases 1. More than n+1 vertex

When the number k of vertex is higher than n+1, the decomposition of OM is not single. This will lead to more commutations and is not used for this reason.

Less than n+1 vertex

When the number k of vertex is less than n+1, the barycentric resolution is not always possible. It depends on the position of M in the polyhedron B. M must belong to a polyhedron generated by the k vertex considered. For example, for a 3-leg inverter M must be inside a triangle defined by three vertex, or inside the line segment defined by two vertex. These cases are used to optimize the number of commutations [START_REF] Blasko | Analysis of a hybrid PWM based on modified space vector and triangle-comparison methods[END_REF], [START_REF] Semail | Tools and studying method of polyphase electrical systems. Generalisation of the space vector theory[END_REF].

Study of a 3-level VSI

We consider a carrier-based PWM with a period carrier T.

Comparison with Suboscillation Method

Let us prove that the Suboscillation Method is one way to determine the instants of commutation without calculate them explicitly. We suppose that the desired average voltages at kT instant are those represented on Fig 4 . The busy points of the polyhedron are then M 0 , M 1 , M 2 , M 7 . We have also defined a corresponding point M by

) kT ( c v OM = =<v c1 > x c1 + <v c2 > x c2 + <v c3 > x c3 .
Let us identify (N 1 , N 2 , N 3 , N 4 ) to (M 7 , M 0 , M 1 , M 2 ). Then, our method gives us the durations t 0 , t 1 , t 2 corresponding to M 0 , M 1 and M 2 vertex: ( ) ( )

t 7 = T-t 0 -t 1 -t 2
The development of these expressions gives:

( ) 3 1 c 2 0 E 8 E v E 4 T t - - > < = ; ( ) 3 1 c 2 c 2 1 E 8 v v E 4 T t - > < - > < = ;
( )

3 2 c 3 c 2 2 E 8 v v E 4 T t - > < - > < = .
We can rewrite these expressions:

1 c 0 v E T 2 / t E 4 = + - ; ( ) 2 c 1 0 v E T 2 / t t E 4 = + + - ;
( )

3 c 2 1 0 v E T 2 / t t t E 4 = + + + - .
We find out the intersections of three horizontal lines <v c1 >, <v c2 > and <v c3 > with a triangular shape varying between -E and E :

Comparison with SVM Method

For a symetric three phase load which is star connected with neutral isolated (Fig 6 ) we have the classic relations between voltages:

• u c1 = v c1 -v cN = 3 1 ( 2 v c1 -v c2 -v c3 ) ; • u c2 = v c2 -v cN = 3 1 ( -v c1 + 2 v c2 -v c3 ) ; • u c3 = v c3 -v cN = 3 1 ( -v c1 -v c2 + 2 v c3 ) ;
The image of the vector v c = v c1 x c1 + v c2 x c2 + v c3 x c3 is then the vector u c = u c1 x c1 + u c2 x c2 + u c3 x c3 .

As u c1 + u c2 + u c3 = 0, this vector belongs always to a plane which can be considered as a complex plane.

If we consider u c as a complex vector c u , we find the following expression:

[ ]

a v a v 1 v 3 2 2 c3 c2 c1 + + + = c u with 3 2 j e a π = .
This vector is effectively proportionnal to the classic phasor [ ]

a v a v 1 v 3 2 2 c3 c2 c1 + + + .
It is possible to characterize the inverter for this kind of load by the image of the cube. We find that the image of the vextex of the cube are the vertex M kp of the well known hexagon in complex plane represented Fig 7.

Consequently, the SVM formulation is effectively a particular case of our formulation.

Conclusion

We prove that, even with a characterization of VSI in a n-dimensional vector space F n , it is possible to calculate explicitly, as the SVM allows it in the complex plane, the conduction durations of the VSI switches. The method that we propose relies on a geometrical vectorial VSI characterization which enables, as the SVM, optimization, because of using geometrical tools as Euclidian distance, barycenter and scalar or vectorial products. But, contrary to the SVM, the desired vector v c does not have to belong to a 2-dimensionnal vectorial space. We have explicited a few aspects of this approach in the well kwown case of 3-leg 2-level VSI. So we have showed the relations with the SVM method and with the Suboscillation method. The vectorial feature of the method allows to use it also for multilevel VSI or instaneous control as the Direct Torque Control of electrical machines.

Fig 1 :

 1 Fig 1: Characterization by Space Vector Method of a 3-leg 2-level VSI connected to a star load with isolated neutral.

Fig 2 :

 2 Fig 2: Representation of a n-leg 2-level inverter.

Fig 3 :Fig 4 :

 34 Fig 3: Characterization of a 3-leg 2-level Voltage Source Inverter (VSI)

Fig 5 :

 5 Fig 5: Representation of busy M k points (M 0 , M 1 , M 2 , M 7 ) in the sequence of symetric Intersective Pulse Width Modulation chosen in Fig 4.

Fig 6 :Fig 7 :

 67 Fig 6: Representation of a 3-leg 2-level VSI

  Each coordinate of v c can accept two values, +E and -E. Consequently, a family of P = 2 n vectors v cr characterizes the inverter. Let us consider, for geometrical representation, the points O and M r such as OM r = v cr . The P points M r are the vertex of a polyhedron B.

	For a 3-leg 2-level VSI (n=3), we obtain thus the 8 following vectors:

Barycentric coordinates

, we can express t k with the n vectors N 1 N j : ( ) ( )

Then t 1 is found by the following expression: t 1 = T-t 2 -t 3 -…-t n+1 .

Proof:

To obtain this formula we have only to consider each term of the equation ( 3) and apply to them the following application φ defined by: φ (w) = ( )

The property (5) implies the nullity of n terms of the equation (3) since:

if r≠k The equation (3) becomes then:

The property (4) allows us to divide by ( )