38 2. TUNABLE MATERIALS–CHARACTERISTICS AND CONSTITUTIVE PARAMETERS
e expressions for and k are given by Eqs. (2.22a) and (2.22b). e similarity of
Eq. (2.24) to (2.21) is obvious.
2.6.4 CIRCUMFERENTIAL MAGNETIZATION IN CYLINDRICAL
COORDINATES
Very interesting phenomena, enabling corresponding applications, occur when the ferrite is bi-
ased transversely to the direction of propagation (usually it is considered along the z-axis). A
well-known phenomenon is “birefringence,” e.g., Pozar [14], where the propagation of two
waves is observed. ese are: (1) the “ordinary wave,” which is unaffected by magnetization, and
(2) the “extraordinary wave,” which depends on magnetization. For antennas printed on ferrite
substrate, if the magnetization is perpendicular to the printed structure, the resonant frequency
is tunable while the radiation pattern is unaffected [33]. In contrast, if the DC bias is parallel
to the substrate but perpendicular to the direction of the propagating wave below the radiator,
then a tunable radiated beam steering is enabled, e.g., [34].
In cylindrical coordinates (, ', z), a very practical way to implement transverse magneti-
zation is the application of a DC biasing magnetic field in the azimuth direction
N
H
0
D H
0
O
. In
this case, the permeability tensor is quite similar to that of Oy biasing in Cartesian coordinates,
e.g., Baden Fuller [12]:
2
6
4
B
B
B
z
3
7
5
D
2
6
4
0 j k
0
0
0
jk 0
3
7
5
2
6
4
H
H
H
z
3
7
5
: (2.25)
2.6.5 MAGNETIZATION AT AN ARBITRARY DIRECTION IN
CARTESIAN COORDINATES
Suppose that we have a DC biasing magnetic field
N
H
0
(Figure 2.11) oriented along some arbi-
trary direction ('
0
,
0
), that is,
N
H
0
D H
0
f
sin
0
cos '
0
Ox C sin
0
sin '
0
Oy C cos
0
Oz
g
: (2.26)
e permeability tensor was originally given by Tyras [35], and repeated by Rahmat
Samii [36], as well as by Hsia [18]. Furthermore, for the case in which
0
D 90
ı
, which means
that
N
H
0
is parallel to the x–y plane, the permeability tensor has been given by Hsia and Alex-
opoulos [34]. Finally, for
N
H
0
parallel to the x–z plane or '
0
D 0
ı
, the permeability tensor,
appears in [12].
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