SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

theory the time derivative operator operates on the inductance as well but only nonsalient
machines are considered here thus it can be placed after the inductance, as shown in Equation
(3.125).
vA RLsiA eA
v
RL
i
e
B s B B
v C RLs i C e C
The resulting matrix is called the impedance matrix in the literature, thus Equation (3.126).
(3.125)
vabc Zabciabc e
abc
(3.126)
Using any (Σ=0) Clarke transform (such as Equation 3.87) the three-phase voltages, currents, and
bEMFs may be transformed to their two-phase equivalents in the stator reference frame,
Equations (3.127)-(3.129).
(3.87):
3
0 0 x A
x 2
k
x
B
x
3
3 0 x C
2
abc
(Σ=0, k=1)
v Cv (3.127)
i Ci (3.128)
abc
e Ce (3.129)
abc
Using the corresponding “inverse” of C given earlier, Equations (3.127)-(3.129) are rewritten as
Equations (3.130)-(3.132).
v C v (3.130)
1
abc
i C i (3.131)
1
abc
e C e (3.132)
1
abc
Substituting these into Equation (3.126) yields Equation (3.133).
C v Z C i C
e (3.133)
1 1 1
abc
Premultiplying by C gives Equation (3.134) which simplifies to Equation (3.135) (the derivation
is done by hand but is not worth repeating because it gives no additional information).
CC v CZCiCC e (3.134)
1 1 1
abc
I v Z i I
e
2x2 2x2
v Zi e
(3.135)
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