SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Torque Production
As with all other forms of equations, the torque can be derived using numerous methods; only
two are discussed here: the vector cross product in terms of space vectors and the cross product in
terms of the dq components.
The torque of any nonsalient or singly-salient machine can be expressed as the cross product of
the stator flux-linkage and the stator current [87, p.12], whether space vectors or scalar vectors
are used. For simplicity the SV is chosen and the torque can be written as Equation (3.156).
T ci (3.156)
Substituting Equation (3.139) in for the flux linkage gives Equation (3.157), where the derivation
has relied on the fact that a vector crossed with itself is zero.
T cLi s Ri
cLi s i cRi T cRi
(3.157)
Using the notation introduced earlier, the current SV is expressed as ˆ j
i Ie
and the rotor-stator
jr
flux linkage SV as e . This corresponds to the situation shown in Figure 3.45, where
R R
the rotor is at position r .
Figure 3.45 – Torque production; stationary reference frame.
The cross product can be defined as the product of the magnitude of each vector multiplied by the
sine of the angle measured from the first toward the second vector. Using this definition the
torque can be expressed as Equation (3.158). The angle δ is shown in Figure 3.45 as well and it is
clearly the same torque angle defined in Part I. The equation is valid only as long as the stator
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