SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

flux linkage produced by the rotor magnets, and a term for each additional winding present
describing the flux linkage due each of those windings’ current. For the present analysis, only one
winding and the rotor magnets will be considered; inclusion of the other windings is deferred
until the next chapter.
Assuming that all of the flux links each turn of the winding, the flux linkage is given by Equation
(2.17), where Equation (2.14) has been used. The subscript ‘S’ could be added to λ to denote that
this is the flux linkage of a stator winding, but the is no rotor winding in a BPMS motor thus the
flux linkage is understood to be the stator’s flux linkage (see Appendix A).
T
2
N i
RRm M
N N (2.17)
By definition, an inductance is a constant of proportionality that indicates the amount of flux
produced in a magnetic circuit per unit of current. The reluctance of the circuit is constant (not a
function of rotor position) so a constant inductance can be defined as Equation (2.18). In this
ideal analysis where leakage flux is ignored, the inductance is a function of the total reluctance of
the circuit as seen by the MMF source of the coil. Since this reluctance is dominated by the
airgap, it is often called the airgap inductance. 3
2
N N
L
R R
2
Rm
(2.18)
Using Equation (2.18), the flux linkage expression can be written as Equation (2.19). The first
term in Equation (2.19) is by definition the self flux linkage of the winding. The second term is a
flux linkage due to rotor flux.
Li N
(2.19)
M
In a real machine, L may be a function of rotor position (due to saliency) and N may be a function
of the angle around the stator (a “stator winding distribution”), and M will always be a function
of both the angle around the rotor (a “rotor flux distribution” described relative to the stator) and
the rotor position with respect to the stator. These dependencies are shown explicitly in Equation
(2.20).
L( r ) i(
t)
N(
s ) (
r , s )
(2.20)
3 Comparing Equation (2.17) with Equation (2.6) shows why the electrical model of the simple apparatus in
Figure 2.8 had no inductance: because the B field was assumed to be held constant by an external source. In
a real motor there is a similar source (the magnet) but it is not ideal, thus the current creates a flux linkage
and therefore the coil has inductance.
32