SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

d
R( r) ke( r) kt( r) sin( r)
(2.52)
d r
To recapitulate, the torque and bEMF were previously found via the BLi and BLv laws,
respectively. Then the rotor-stator flux linkage was derived (Equations 2.43-2.44) and its position
derivative was computed (Equation 2.49). By comparing the former expressions for torque and
bEMF it was seen that they could be rewritten in terms of the position derivative of rotor-stator
flux linkage (Equations 2.50 and 2.51). These two derivations demonstrate the two common
perspectives from which we usually think about motor operation. The BLi and BLv laws focus on
the conductors (of the coilsides) and how much flux they “cut” as the rotor moves, whereas use of
the flux linkage focuses on the entire winding. Obviously both interpretations are useful but it is
the flux linkage method that is most used in modern analysis (and thus, in this report).
That Equations (2.32) and (2.37) are identical to Equations (2.50) and (2.51) (with Equation 2.52)
is not a coincidence. When the circuit model was developed earlier, the bEMF was given in
Equation (2.25) as the time derivative of the rotor-stator flux linkage R , which was left general
at that stage in the development. Now that we have found an expression for R in a sine motor
(Equation 2.44), Faraday’s law can be invoked to yield Equation (2.53).
d
et () R( r())
t
dt
R( r) dr
dt
r
d
( )
(2.53)
d
() R r
et
r
Equation (2.53) is seen to be identical to Equation (2.51). This shows that the latter is valid and is
just a restatement of Faraday’s law. Similarly, Equation (2.50) is simply a restatement of another
result (although there is no associated surname). For the nonsalient machines considered here,
Equation (2.50) is the same as the torque determined by taking the partial derivative with respect
to rotor position of the energy (flux held constant) or of the coenergy (current held constant)
stored in the magnetic circuit [68, pp.59-60], [69, p.5.36]. The topic of stored energy and
coenergy is useful to study but is far beyond scope; it can be found in many older machine texts
such as [27], [42], [26], [38]. Aside from highlighting the duality between torque and bEMF that
we have come to expect, this derivation shows how each is related to the rotor-stator flux linkage,
which is a key idea in the SV model.
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