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1
Lecture 7
Matrix Manipulations
in Reference Frame Transformation
Let the equations in the reference frame1 be:
1
1
1
1
)]
(
[
]
[
]
[
i
M
dt
d
dt
di
i
R
v
θ
+
+
=
l
(1)
In order to reduce the labour of word processing, the
Ntuple
vectors
1
1
,
i
v
are not
underlined. The
NxN
resistance matrix
[R]
and the leakage inductance matrix
]
[
l
are
diagonal matrices. But
)]
(
[
M
is a full matrix.
It is desired to transform (1) to the reference frame2 using the
)]
(
[
ψ
C
matrix.
2
1
)]
(
[
v
C
v
=
(2)
2
1
)]
(
[
i
C
i
=
(3)
Substituting (2) and (3) in (1)
2
2
2
2
)]
(
)][
(
[
)]
(
[
]
[
)]
(
][
[
)]
(
[
i
C
M
dt
d
dt
i
C
d
i
C
R
v
C
+
+
=
l
(4)
Premultiplying (4) by the inverse of
)]
(
[
C
,
2
1
2
1
2
1
2
)]
(
)][
(
[
)]
(
[
)]
(
[
]
[
)]
(
[
)]
(
][
[
)]
(
[
i
C
M
dt
d
C
dt
i
C
d
C
i
C
R
C
v
−
−
−
+
+
=
l
(5)
The righthand side of (5) is examined term by term.
Term of
2
1
)]
(
][
[
)]
(
[
i
C
R
C
−
Consider the case where the resistances are all equal, that is
]
[
]
[
I
R
R
R
=
, where
R
R
is a
scalar and
]
[
I
is the identity matrix. Therefore
2
2
2
1
2
1
2
1
]
[
]
[
)]
(
[
)]
(
[
)]
(
][
[
)]
(
[
)]
(
][
[
)]
(
[
i
R
i
I
R
i
C
C
R
i
C
I
C
R
i
C
R
C
R
R
R
=
=
=
=
−
−
−
(
6
)
One concludes that
]
[
R
is not changed by the transformation.
Term of
dt
i
C
d
C
2
1
)]
(
[
]
[
)]
(
[
l
−
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One follows the chain rule in the differentiation with respect to t.
}
)]
(
[
)]
(
[
]{
[
)]
(
[
)]
(
[
]
[
)]
(
[
2
2
1
2
1
dt
di
C
i
dt
C
d
C
dt
i
C
d
C
ψ
+
=
−
−
l
l
}
)]
(
][
[
)]
(
[
)]
(
[
]
[
)]
(
[
}
)]
(
[
)]
(
[
]{
[
)]
(
[
2
1
2
1
2
2
1
dt
di
C
C
i
C
C
dt
d
dt
di
C
i
dt
C
d
C
l
l
l
−
−
−
+
∂
∂
=
+
=
The leakage inductance matrix
]
[
l
is assumed to be the product of a scalar
l
and an
identity matrix
]
[
I
. Therefore
2
1
2
1
)]
(
[
)]
(
[
)]
(
[
]
[
)]
(
[
i
C
C
dt
d
i
C
C
dt
d
∂
∂
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 Spring '09
 BonTekOoi

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