A New Concept of PWM Duty Cycle Computation Using the Barycentric Coordinates in a Three–Dimensional Voltage Vectors Arrangement

Abstract

The paper presents a novel approach to the Pulse Width Modulation (PWM) duty cycle computing for complex or irregular voltage vector arrangements in the two (2D) and three–dimensional (3D) Cartesian coordinate systems. The given vectors arrangement can be built using at least three vectors or collections with variable number of involved vectors (i.e. virtual vectors). Graphically, these vectors form a convex figure, in particular, a triangle or a tetrahedron. The reference voltage vector position inside that figure can be expressed by the barycentric coordinates , which are calculated using the second (2D case) or the third–degree determinant (3D case) – without trigonometry and angles. Thus, the speed of the PWM duty cycle computation rises significantly. The use of the triangle area or the tetrahedron volume, instead of the standard vector projection also permits for a well–defined and universal approach to identifying the reference vector position, especially for converters with complex and/or deformed space–vector diagrams (i.e. floating DC–link, multisource DC–link). The proposed computation scheme is based on simple instructions without trigonometry thereby, the DSP processor, or digital solution for field–programmable gate array, can fast–perform this operation using atomic operations. The aim of the presented considerations is not a novel PWM modulation, but a computable idea of a general calculation scheme for cases in which the distribution of vectors is non-trivial. A detailed algebraic and geometric analysis, as well as mathematical proofs on the total consistency of the results with the standard projection method, are also included. Subsequently, the Three–Dimensional Space Vector Modulation (3D–SVM), is considered as a special background to present a novel approach.