1
Lecture 7
Matrix Manipulations
in Reference Frame Transformation
Let the equations in the reference frame-1 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
N-tuple
vectors
1
1
,
i
v
are not
underlined. The
NxN
resistance matrix
[R]
and the leakage inductance matrix
]
[
l
are
diagonal matrices. But
)]
(
[
θ
M
is a full matrix.
It is desired to transform (1) to the reference frame-2 using the
)]
(
[
ψ
C
matrix.
2
1
)]
(
[
v
C
v
ψ
=
(2)
2
1
)]
(
[
i
C
i
ψ
=
(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
ψ
θ
ψ
ψ
ψ
+
+
=
l
(4)
Pre-multiplying (4) by the inverse of
)]
(
[
ψ
C
,
2
1
2
1
2
1
2
)]
(
)][
(
[
)]
(
[
)]
(
[
]
[
)]
(
[
)]
(
][
[
)]
(
[
i
C
M
dt
d
C
dt
i
C
d
C
i
C
R
C
v
ψ
θ
ψ
ψ
ψ
ψ
ψ
−
−
−
+
+
=
l
(5)
The right-hand side of (5) is examined term by term.
Term of
2
1
)]
(
][
[
)]
(
[
i
C
R
C
ψ
ψ
−
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
=
=
=
=
−
−
−
ψ
ψ
ψ
ψ
ψ
ψ
(6)
One concludes that
]
[
R
is not changed by the transformation.
Term of
dt
i
C
d
C
2
1
)]
(
[
]
[
)]
(
[
ψ
ψ
l
−

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