1.      Permanent Magnet Synchronous Machine

Recently, permanent magnet synchronous machines (PMSMs) become an important class of high performance ac drives. PMSMs are special types of the synchronous machines. They have conventional three-phase stator windings but, instead of a field winding, permanent magnets produce field flux. Synchronous machines with electrically excited field winding require brushes and slip rings to transfer current to the rotor. The use of permanent magnets eliminates this requirement thus, problems related to the brushes and slip rings are overcome. Lack of brushes also results in a more robust mechanical construction. Moreover, the copper losses are eliminated therefore; higher efficiency and higher torque/inertia ratio can be achieved.

PMSM types are demonstrated in Figure 1.1. When the stator windings are concentrated, the machine is named as trapezoidal type or brushless dc (BLDC) machine. This machine has a trapezoidal back-emf waveform. When the stator windings are sinusoidally distributed, the back-emf waveform is also sinusoidal and the machine is named as permanent magnet ac (PMAC) machine. This type of PMSM is usually named as servo motor and widely used in high performance servo applications.

Figure 1.1: Permanent-magnet machine types.

The PMSMs are further classified according to their magnet mounting types. One of them is the surface-mounted PMSM where the magnets are mounted on the rotor surface as demonstrated in Figure 1.2. The other type is the interior permanent magnet (IPM) machine where the magnets are buried inside the rotor core.

Figure 1.2: Surface-Mounted PMSM rotor.

2.      Stator Voltage Equations in the Stationary abc Reference Frame

In a PMSM the rotor houses the permanent magnets which establish a dc magnetic field linking the surrounding three-phase stator windings placed spatially apart from each other as shown inFigure 2.1. Two-pole PMSM structure is depicted in the figure nevertheless, the analysis is applicable to machines with any number of poles. To derive the model, magnetic saturation, core losses and eddy currents are neglected.

Figure 2.1: Three-phase permanent-magnet synchronous machine structure.

Instantaneous stator voltages developed in these three windings can be written as:

(2.1)

(2.2)

(2.3)

where , ,  are stator voltages, , ,  are stator currents,  is the stator resistance and , ,  are the flux linkages of the corresponding phases. Here it is assumed that, all three-phase windings have an equal resistance.

The flux linkages with the stator winding for one of the phases consist of two components from stator and rotor,

(2.4)

Here  is the flux linkage with the stator phase winding due to the stator currents and  is the flux linkage with the stator phase winding due to the rotor flux. Therefore, the flux-linkages of the stator windings, ,  and  can be written as:

(2.5)

(2.6)

(2.7)

 

where , and  are the self-inductances of the stator phases-a, -b and –c respectively,  ,  and   are the mutual inductances between stator phases-a and –b; -a and –c; -b and -c, respectively. It is assumed that, all three stator phases have equal self-inductances such that,  . Also  is assumed to be equal to and .

is the flux linkage with the stator phase windings due to the flux produced by the permanent magnets placed on the rotor surface. is the electrical angle between the axis of the rotor flux and stator phase-a axis which is related with the spatial angle of the rotor as in (1.8) where  is the number of poles.

(2.8)

 

The self-inductance of a stator phase can be defined as:

(2.9)

 

where  is the leakage inductance which accounts for the flux produced by the stator winding which does not cross the air gap. the magnetizing inductance that can be formulated as,

(2.10)

 

 

where  is the permeability of air,  is the mean radius at the air gap,   is the length of the rotor along its shaft axis,  is the air gap length and   is the number of turns per phase in the stator winding. Since the surface-mounted PMSMs are almost always non-salient, it is assumed that, air gap length  does not vary with .

Mutual inductances between two stator phases can be found as,

(2.11)

 

3.      Transformation to Rotor Reference Frame

Each stator phase winding produces a sinusoidally distributed mmf  ,  and  in the air gap which can be represented by vectors along the axis of that phase. The resultant stator mmf, at any point in the air gap around the rotor periphery can be represented by a vector which is the sum of mmf vectors due to all three phases as shown inFigure 3.1.

 

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