Voltage-Sourced Converters in Power Systems

APPENDIX A

Space-Phasor Representation of Symmetrical Three-Phase Electric Machines

A.1 INTRODUCTION

This appendix presents a space-phasor domain dynamic model for a symmetrical three-phase electric machine. The developments mainly concern the (squirrel-cage) asynchronous machine, the doubly-fed asynchronous machine¹, and the nonsalient-pole permanent-magnet synchronous machine (PMSM).

A.2 STRUCTURE OF SYMMETRICAL THREE-PHASE MACHINE

Figure A.1 illustrates a simplified electrical structure of a symmetrical three-phase machine. The rotor and the stator each has three star-connected windings. The stator windings are interfaced with either a three-phase voltage source or a three-phase current source. However, the rotor windings can be either short circuited or excited by a three-phase voltage/current source. In the former case, the machine is called the asynchronous machine (or the squirrel-cage asynchronous machine), whereas in the latter case, the machine is the doubly-fed asynchronous machine. For each phase, the positive current enters the corresponding winding (motoring convention). For each winding set, we assume that the voltages (currents) of phases b and c lag the voltage (current) of phase a, respectively, by −120° and −240°.

With reference to Figure A.1, the mechanical angular position is defined with respect to the magnetic axis of the stator phase a winding. Thus, the windings of phases b and c are, respectively, located at positions 120° and 240°. We then define the ...