SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

In a symmetrical machine the self inductance of each phase is equal and the mutual inductance
between any two phases is equal,
LAA LBB LCC L
LAB LBA LAC LCA LBC LCB M
thus the set of flux linkages can be written as Equation (B.8).
A LiA MiB MiC AR
B MiA LiB MiC BR
Mi Mi Li C A B C CR
(B.8)
When the phase resistance is considered the voltage equation can be written for each phase as
Equation (B.9).
d
vAN RiA A
dt
d
vBN RiB B
(B.9)
dt
d
vCN RiC C
dt
Substituting in Equation (B.8) yields the full voltage equations, Equation (B.10). Using the bEMF
and matrix form this is given as Equation (B.11).
vAN RiA Li A Mi B Mi C AR
vBN RiBMi A Li B Mi C BR
v Ri Mi Mi Li
CN C A B C CR
vAN R iA L M M iA eA d
v
R
i
M L M
i
e
BN B B B
dt
v CN R i C M M L i C e C
(B.10)
(B.11)
This report considers only isolated neutral motors thus the currents must sum to zero. The linear
dependence of the currents is expressed by 0 the expression; this can be arranged in the
three different ways shown in Equation (B.12).
i i i
i i i
i i i
B C A
A C B
A B C
(B.12)
Writing out Equation (B.11) by hand, substituting Equation (B.12) and simplifying gives
Equation (B.13). The result is that although the mutual inductances still exists, the isolation of the
neutral decouples the circuits.
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