SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

to use SV theory. It may be helpful to review the relationships summarized in Equations (3.106)
and (3.115) (which were summarized in Equation 3.109). An attempt is made to give the simplest
derivation here. One alternate derivation is [87, pp.61-62,156-157], and another in terms of the
general reference frame is [87, p.62-66].
(3.106): R
x x jx
R
x x e
(3.115): R x xe d q
j r
j r
As before, the voltage and stator flux linkage are given by Equations (3.138) and (3.139).
d
(3.138): v Ri
dt
Li
(3.139): s R
The rotor-stator flux linkage is given by Equation (3.143), where R represents the peak value.
jr
e (3.143)
R R
Similar to the stationary case, the voltage, current, and flux linkage are written in the stationary
frame for direct substitution.
R j r v v e
(3.144)
R j r i i e
(3.145)
R jr
e (3.146)
Substitution of Equations (3.144) – (3.146) into Equations (3.138) yields Equation (3.147).
R jr v e
R j d
r R jr
Ri e e dt
R jr Ri e
d
R j d
r jr
R
e e dt dt
R jr Ri e
d
R jr jr
R
e
je dt
R
v
R d
R R
Ri j
dt
(3.147)
At the same time, the stator flux linkage is written in the rotor frame, Equation (3.148). In the first
line the variables are simply transformed to the rotor frame, as was done above. But it becomes
clear that since the common exponential term cancels everywhere, we could have simply written
the quantities expressed in the rotor frame. This is an advantage of the SV theory: an expression
147