SPACE VECTOR MODULATION –An Introduction

== Tutorial at IECON2001==

Dorin O. Neacsu

Correspondence Address: Satcon Corporation, 161 First Street, Cambridge, MA 02142, Email neacsu@earthlink.net I. INTRODUCTION Space Vector Modulation became a standard for the switching power converters and important research effort has been dedicated to this topic. Tens of papers, research reports and patents were developed in the last ten years and the theory of Space Vector Modulation is already wellestablished. Diverse implementation methods were tried and some dedicated hardware pieces were developed based on this principle. The initial use of Space Vector Modulation at three-phase voltage-source inverters has been expanded by application to novel three-phase topologies as AC/DC Voltage Source Converter, AC/DC or DC/AC Current Source Converters, Resonant Three-Phase Converters, B4-inverter, Multilevel Converters, AC/AC Matrix Converters, and so on). This tutorial presents the base theory of SVM when applied to a 3-phase voltage source inverter. II. REVIEW OF SPACE VECTOR THEORY A. History The roots of vectorial representation of three-phase systems are presented in the research contributions of Park [1] and Kron [2], but the decisive step on systematically using the Space Vectors was done by Kovacs and Racz [3]. They provided both mathematical treatment and a physical description and understanding of the drive transients even in the cases when machines are fed through electronic converters. In early seventies, Space Vector theory was already widely used by industry and presented in numerous books. Stepina [4] and Serrano-Iribarnegaray [5] suggested that the correct designation for the analytical tool to analyzing electrical machines has to be “Space Phasor” instead of “Space Vector”. “Space Phasor” concept is nowadays mainly used for current and flux in analysis of electrical machines. B. Theory Any three-phase system (defined by ax(t), ay(t) az(t)) can be represented uniquely by a rotating vector aS: where ( Aα , Aβ ) are forming an orthogonal 2-phase system and aS = Aα + j ⋅ Aβ . A vector can be uniquely defined in the complex plane by these components.

Fig.1 Equiv. between the 3-ph system and vectorial presentation

The reverse transformation (2/3 Transformation) is given by

a X (t ) = Re a S + a 0 (t ) 2 (3) where aY (t ) = Re a ⋅ a S + a 0 (t ) a (t ) = Re a ⋅ a + a (t ) S 0 Z 1 a 0 (t ) = ⋅ [a X (t ) + aY (t ) + a Z (t )] represents the homopolar 3

[

[ ]

[

]

]

component. It results an unique correspondence between a Space Vector in the complex plane and a three-phase system. The main advantages of this mathematical representation are: • Analysis of three-phase systems as a whole instead of looking at each phase; • It allows to use the properties of the vectorial rotation. Using rotation with ωt leads to an analysis in DC components by withdrawing the rotational effect. This vectorial representation is the basis for control algorithms in : • electrical drives (Induction Machine or Synchronous Machine drives); • AC/DC converters: • Active filtering systems based on the instantaneous power components (p-q) theory. All these control methods are based on vectorial mathematical models and are using PWM algorithms in the final control stage before the power converter. III. VECTORIAL ANALYSIS OF THE THREE-PHASE INVERTER A. Theory

aS =

2 ⋅ a X (t ) + a ⋅ aY (t ) + a 2 ⋅ aZ (t ) 3

j⋅ 2π j 4π

[

]

(1)

where a = e 3 and a2 = e 3 Given a three-phase system, the vectorial representation is achieved by the following 3/2 transformation:

Aα 2 1 = ⋅ Aβ 3 0

−1 2 3 2

− 1 a X 2 ⋅ a Y − 3 a 2 Z

(2)

The three-phase inverter presented in Fig.2 is herein considered. Fig.3 presents the appropriate output voltages without PWM (six-step).

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