SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Figure 2.16 – Components of total flux (cf. Figure 2.15).
This simple analysis is not valid in practice but is good enough to illustrate some key points.
First, notice that the airgap flux due to the magnet is controlled by the properties of both the
magnetic circuit and the magnet—the magnetic circuit loads the magnet to an operating point.
Further modeling or understanding is not necessary but it is useful to understand the general idea
that the airgap flux due to the magnet is set by the magnetic circuit. Examining Figure 2.15 from
a circuit-theory point of view, if the magnet was an ideal flux source ( Rm ) the airgap flux
would be dominated by the “current source” of magnet flux. If the magnet was not magnetized
but the magnet material kept in place ( r 0 =open circuit) the airgap flux would be comprised
of only coil flux. Also examine these two cases in light of Figure 2.16: it is clear that if the
magnet is ideal the coil cannot influence airgap flux, or if the magnet is not magnetized the coil
completely controls the airgap flux. This is an oversimplified analysis but is useful to know that
real motors are somewhere in between these extremes.
Equally importantly, this model can be used to gain a better understanding of flux linkage. Flux
linkage describes the total flux linked by a winding. A winding in a magnetic circuit can be linked
by lines of flux that were produced by a means external to the winding (mutual flux linkage) and
it can also be linked by lines of flux produced by its own current (self flux linkage). Thus, flux
linkage is an expression that describes the total flux linked by the winding and this expression is
written in terms of the winding’s own current and in terms of the variables that describe the
external source(s) of flux. It follows that a winding’s flux linkage expression will take on a form
consistent with the type of magnetic equipment containing the winding. Two familiar examples
are the standard inductor and the transformer. For the inductor, the flux linkage expression
involves only the flux produced by its own current; there is only one term. For the transformer,
the simplest (ideal) flux linkage expression for a winding would contain a term describing the
flux linkage due to that winding’s current and one term for each additional winding of the
transformer. For a brushless permanent magnet motor, the expression for a winding’s flux linkage
will contain the term describing the flux produced by the winding current, a term describing the
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