SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

variables are independent, but the resulting axes (A&B, A&C, or B&C) are not orthogonal. Thus
the purpose of the Clarke transform is to form an orthogonal basis such that a SV is easily defined
by components along the α and β axes as shown in Figure 3.33. This also allows other reference
frames to be developed more easily as shown next.
Figure 3.33 – Space vector in stationary reference frame showing phase axes.
When a quantity is described as a SV ( x ) or in terms of the αβ plane ( x ) it is said to be in the
stationary or stator reference frame because the angle of the SV is measured from the real (α)
axis fixed to the stator. A vector can be described in different reference frames and there are a
number of them defined in the literature. It has been seen that SV theory can be used to describe
purely electrical quantities or physical quantities such as those present in a machine; it follows
that a reference frame could be defined relative to an electrical or physical reference. Examples
include the voltage at a power grid bus, the rotor position in a motor, the rotor flux (which is not
coincident with the rotor position in induction motors), or a flux linkage. The only difference
between these various reference frames is how they are defined with respect to the stationary
reference frame, thus many texts base discussions on the general reference frame; the position of
this frame (call it θΔ) could then be specialized to whatever frame is appropriate [87], [35]. Since
in the synchronous machine the rotor flux and rotor-stator flux linkage SVs are aligned with the
rotor position, only the stationary reference frame and the rotor reference frame will be used in
this report. The rotor reference frame is defined using the position of the rotor ( r
) as
shown in Figure 3.34. The direct axis (d) is defined as being in the center of the rotor’s North
pole and the quadrature axis (q) axis is defined to be 90° electrical ahead of the direct axis in the
positive direction of rotation. The rotor position is measured from α to d. These choices are not
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