SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

As aforementioned a spatial analysis is required, but we can generalize the spatial relationships to
a degree in order to gain an understanding of the basic electrical model. First, this report is only
concerned with non-salient machines and therefore L will not be a function of rotor position. With
this simplification the expression for stator flux linkage becomes Equation (2.21).
L i(
t)
N(
s ) (
r , s )
(2.21)
Second, regardless of the nature of the last two terms (the distributions of stator winding and rotor
flux) their interaction defines a flux linkage that will be a cyclic function of rotor position. From
an electrical standpoint it makes sense to disregard the details of the two distributions and
represent only the resulting flux linkage, which will be denoted by a new variable, R( r)
. This
new variable represents the component of total stator flux linkage that is due to the rotor flux.
Since it is flux linkage due to the magnets it is sometimes called the “magnet flux linkage,” but
this report will use “rotor-stator flux linkage” because the former term sounds as if we are trying
to describe the flux linking the magnet as opposed to the coil. Using this new variable, the
expression for stator flux linkage becomes Equation (2.22). 4
Li ( )
(2.22)
R r
To recapitulate, Equation (2.22) is the flux linkage of a stator winding. It has two components:
flux linkage due to stator self inductance and flux linkage due to the rotor magnets.
Now, to finish the development of the general electrical model the dependence on rotor angle in
R( r)
will be ignored until the next section. In the meantime we will use the generalized stator
flux linkage expression given by Equation (2.23).
Li
(2.23)
R
Faraday’s law can be applied to Equation (2.23) to find an expression for the voltage induced in
the coil, Equation (2.24).
ddi d
R
gt () L (2.24)
dt dt dt
Equation (2.24) suggests that the induced voltage in the winding of a general BPMS motor could
be modeled as an inductor in series with a voltage source, as shown in Figure 2.17. (To
4 Unfortunately, the variable identified here as R
is sometimes referred to as the “rotor flux linkage” or
the “rotor flux” in the popular literature. It is neither, thus these terminologies are incorrect and misleading.
The reader unfamiliar with the distinction may find Appendix A helpful.
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