SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

SPE model from Part I of this chapter), the stator voltage equations can be written as Equation
(3.138), where the stator flux linkage is given by Equation (3.139).
d
v Ri
dt
(3.138)
Li
(3.139)
s R
In the phasor analysis each quantity is associated with one phase (such as phase-A in the SPE
discussion). In SV analysis, the voltage, current, and bEMF for each phase are all taken into
account for by the respective space vectors. This is easy to see because each is an electrical
quantity and we’ve seen examples of them. What may not be so obvious is that the flux linkages
(which had to be specified per-phase as a
, Ra ,
, Sa ,
) are also accounted for by the space
vector ( , R
, S
) and are thus not associated with any phase. For comparison, the SV
equivalent of the phasor quantities in SPE analysis (Equations 3.29, 3.30, and 3.31) are given as
Equations (3.140), (3.141), and (3.142).
S
R
Li s R
(3.140)
d
g
dt
d d
Li s R
(3.141)
dt dt
d
Li s e
dt
d
e R
dt
(3.142)
In the SPE analysis, Figure 3.23 showed the relationship between the flux linkages and the
voltages induced by them. Just as Figure 3.17 from SPE was redrawn using SVs as Figure 3.41,
Figure 3.23 could be redrawn using SVs and the same comments regarding their similarity would
apply.
Rotor Frame
The phase-variable electrical model was transformed to an equivalent model in the stationary αβ
reference frame. Now we can apply the concept of reference frame theory to derive an equivalent
model in the dq reference frame of the rotor. In deriving the stationary model, the matrix form
had to be used to reduce the size of the impedance matrix; in deriving the rotor model it is easier
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