SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

In addition to the forms above, the torque is often described in terms of dq components. As with
the SV case, the dq derivation [80, p.678-688] is built up from first principles but we can begin at
the cross product. (Normally the cross product is not shown and only Equation (3.162) is given—
one must know the identity below to see how it is actually a cross product.)
3
T λ dq idq
2
3
det
2 ,
3 d
2
id
i
λdq i dq
q q
3
T diqqi d
2 (3.162)
Substituting Equation (3.150) into Equation (3.162) and simplifying gives Equation (3.163).
3
T R iq
(3.163)
2
This result can also be arrived at using the conservation of energy equation [107].
Electromechanical Models
The equations required to construct the BPMS motor model in the stationary and rotor reference
frames follow directly from the equations in the previous two sections. It should be noted that
everything said here regarding the rotor frame applies to the general frame with only a
modification of the rotor angle.
Stationary Frame
In the stationary frame the voltages are given by Equations (3.138) and (3.139).
d
(3.138): v Ri
dt
Li
(3.139): s R
Substitution of Equation (3.139) into (3.138) yields Equation (3.164), which can be split into its
α- and β- components as Equation (3.165) and rearranged into state space variable as Equation
(3.166).
v Ri Ls d d
i R
dt dt
v Ri Ls d
i e
dt
(3.164)
154