Information Technology Reference
In-Depth Information
[
U
e
][
U
e
]
=
[
U
e
][
U
e
]
=
[
I
]
,
[
U
h
][
U
h
]
=
[
U
h
][
U
h
]
=
[
I
]
,
(2
.
73)
where [
I
] is the identity matrix
10
01
[
I
]
=
.
This implies that [
U
e
] and [
U
h
] are unitary matrices: their conjugate matrices
coincide with their inverse matrices and they are commutative with their conjugate
matrices.
Now we shall derive impedance equation for the normalized eigenfields. Accord-
ing to (2.64),
a
E
1
+
a
H
1
+
b
E
1
e
i
E
b
H
1
e
i
H
h
1
E
τ
1
=
e
1
H
τ
1
=
a
E
2
+
a
H
2
+
b
E
2
e
i
E
e
2
b
H
2
e
i
H
h
2
.
E
τ
2
=
H
τ
2
=
Substituting these relations into the equation
E
τ
m
=
[
Z
]
H
τ
m
,
m
=
1
,
2, we
write
[
Z
]
h
1
=
1
e
1
,
[
Z
]
h
2
=
2
e
2
,
(2
.
74)
where
a
E
1
+
a
E
2
+
b
E
1
b
E
2
e
i
(
E
−
H
)
e
i
(
E
−
H
)
1
=
a
H
1
+
,
2
=
a
H
2
+
.
b
H
1
b
H
2
In matrix form
[
Z
][
U
h
]
=
[
U
e
][
]
,
(2
.
75)
where [
] is a diagonal matrix:
1
0
[
]
=
.
0
2
Multiplying (2.75) on the right by [
U
h
], we get
=
.
[
Z
]
[
U
e
][
] [
U
h
]
(2
76)
