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

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. 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 Suboscill ation Method [9] 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 [2],[4] 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 [13], [14] or even a 3-leg VSI supplying a star connected load with neutral not isolated [6]. It is always possible to use the Suboscill ation Method but in this case we have no more geometrical tools to analyze and optimize the performances of different control laws. A few works, [1], [5], develop, for 4leg VSI a 3-dimensional approach by using vectors that belong to a 3-dimensional space. Other works [8], [10] 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 Fn 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 Fn, 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 Suboscill ation method. _________________________________________________________________________________________________________________ New method to calculate the conduction durations of the switches in a n-leg 2-level Voltage Source Inverter Semail E __________________________________________________________________________________________________________________ EPE 2001 Graz P. 1 Characterization of a n-leg 2-level VSI Vectorial space Fn associated to VSI The n-leg inverter represented in Fig 2 imposes n voltages vck. So, we associate to this converter a vectorial space Fn with an orthonormal base of vectors (xc1, xc2,., xcn). We can define then a voltage vector: vc = vc1 xc1 + vc2 xc2 +. + vcn xcn. This voltage vector characterizes the different voltages that the inverter can impose to the load. Besides, it is easy to find out the voltage vck from vc: we have only to achieve the scalar product of the two vectors vc and xck : xck = vc . xck. From this point of view the vector vc is more convenient than complex vector c v . For this last one, it is not so easy to obtain vck from c v . Geometrical characterization of VSI Each coordinate of vc can accept two values, +E and –E. Consequently, a family of P = 2 n vectors vcr characterizes the inverter. Let us consider, for geometrical representation, the points O and Mr such as OM r = vcr. The P points Mr are the vertex of a polyhedron B. For a 3-leg 2-level VSI (n=3), we obtain thus the 8 following vectors: