102 3. FINITE FERRITE SAMPLES
e demagnetization factors, for certain useful shapes (sphere, rod, disk or plate) approx-
imated by suitable ellipsoids of revolution, are given in Eq. (3.8). According to [17], for prolate
or oblate spheroids we have:
N
t
D
ab
2c
2
b
a
a
c
sinh
1
c
a
for a < b (3.61a)
N
t
D
ab
2c
2
a
c
sinh
1
c
a
b
a
for a > b (3.61b)
N
Z
D 1 2N
t
; (3.61c)
where a and b are the relevant semi-axes of their elliptical cross-section and c D
ˇ
ˇ
a
2
b
2
ˇ
ˇ
1=2
.
e next step is to account for the non-uniformity of magnetization
N
M or the volume
magnetic charge (r
N
M ¤ 0), where the dipolar magnetic field becomes non-uniform. is is
actually a far more complicated situation, where a Laplace equation for the magnetostatic poten-
tial must be solved by imposing the appropriate continuity conditions on the specimen’s surface.
Equivalently, one could include Eq. (3.36) in a numerical simulation model. Herein, only cer-
tain canonical practical shapes (disk, film, sphere, and cylinder), for which classical analytical
solutions are available in the literature, will be considered. e primary aim is to identify the
characteristics of magnetostatic modes in order to develop the ability to exploit them in practi-
cal microwave applications.
3.18 THE MAGNETOSTATIC EQUATION OF A
UNIFORMLY BIASED SPECIMEN
e electromagnetic field of a uniformly biased specimen is effectively magnetostatic. erefore,
the resonances observed are just the free modes of oscillation of an array of magnetic dipoles
in a uniform magnetic field. Focusing on the magnetostatic modes of a spheroid, we must say
that these belong to its normal modes, called “Walker modes” after the researcher who origi-
nally studied them. Note that a spheroid is a sphere with one of its dimensions altered, either
squashed toward a disk shape (oblate) or extended like a cigar (prolate). e analysis below
will follow Walker only in a general form. Avoiding numerical details, we will clarify the main
characteristics of magnetostatic modes.
At this point, it is necessary to recast the main equations in order to adapt them to the
magnetostatic regime and to Walker’s analysis. Concerning the magnetic field, it contains a DC
term internal to the specimen,
N
H
i
D
N
H
DC
, and a time-harmonic alternating term .
N
he
j!t
/:
N
H D
N
H
i
C
N
he
j!t
D H
i
Oz C
N
he
j!t
: (3.62)
A Oz-biased ferrite magnetized to saturation has already been assumed above while the
alternating component may also have a z-component, but this is usually negligible under the
3.18. THE MAGNETOSTATIC EQUATION OF A UNIFORMLY BIASED SPECIMEN 103
small-signal approximation,
ˇ
ˇ
N
h
ˇ
ˇ
ˇ
ˇ
N
H
i
ˇ
ˇ
. Likewise, magnetization is divided into a static com-
ponent, the saturation magnetization along the z-axis,
N
M
DC
D
N
M
s
D M
s
Oz, and a time-harmonic
component Nme
j!t
:
N
M D
N
M
s
C Nme
j!t
D M
s
Oz C Nme
j!t
: (3.63)
Again, under the small-signal approximation, the z-component of Nm is m
z
<< M
s
. us,
Nm can be assumed transverse to the z-axis.
A very important simplification results from the practical observation that the DC compo-
nents (
N
H
i
;
N
M
s
) can be assumed to be uniform throughout the specimen. In turn, the related volume
magnetic-charge density vanishes (r
N
M
s
D 0). erefore, the internal magnetic field is affected
only by the surface magnetic charge, which can be accounted for through the demagnetization
factors ŒN as
N
H
i
D
N
H
0
ŒN
N
M . However, since the DC bias is assumed along the Oz-axis, we
have
N
H
i
D
N
H
0
N
z
M
s
Oz: (3.64)
e external microwave field will be considered through the boundary conditions at the
specimen’s surface to be imposed on Maxwell equations for ( Nm;
N
h). Likewise, the dipolar field
contribution is inherently included within these equations. e only remaining term is the
anisotropy field
N
H
anis
. As far as crystalline samples are concerned and, in particular, single crys-
tals magnetized along the “easy” or “hard” direction, the anisotropy field has already been shown
to be inversely proportional to M
s
. at is,
N
H
anis
D
2k
M
2
S
M
z
Oz
2k
M
S
Oz: (3.65)
Consequently, assuming the above,
N
H
anis
acts exactly as an additional term of the DC
magnetic field and just like in the previous cases, it can be absorbed in
N
H
0
. at is,
N
H
0
N
H
0
C
N
H
anis
or
N
H
i
D
N
H
0
C
N
H
anis
D
.
H
0
C H
anis
/
Oz: (3.66)
In contrast to the above assumed DC behavior, the alternating quantities ( Nm;
N
h) are
non-uniform throughout the specimen, giving rise to volume (r Nm ¤ 0) as well as to surface
. On Nm ¤ 0/ magnetic-charge densities. e worst is that the field distribution depends on the
specimen’s shape, requiring the solution of a magnetostatic problem subject to the boundary
conditions on the specimen’s surface (sample-air interface). Sometimes, even plane-wave ap-
proximation may be employed to get some qualitative results, but these are very rough and far
from accurate. However, it has proved very useful to seek the expansion of the magnetostatic so-
lution in the superposition of a large number of plane waves. In addition, keep in mind that the
magnetostatic solution is expected to be size independent since it ignores propagation phenom-
ena. Returning to Walker’s modes, the alternating components ( Nm;
N
h) should satisfy Maxwell’s
equations in their reduced magnetostatic form. We define the corresponding alternating flux
density
N
b as
N
b D
0
N
h C Nm
: (3.67)
104 3. FINITE FERRITE SAMPLES
Once again, following the reasoning of magnetostatics, Maxwell’s equations within the
specimen read as follows:
r
N
b D 0 $ r
N
h D r Nm D
m
(3.68a)
r
N
h D 0 $
N
h D r (3.68b)
and r
2
.
Nr
/
D
m
.
Nr
/
(3.68c)
or r
2
.
Nr
/
r Nm
.
Nr
/
D 0: (3.68d)
Besides the above, the magnetization or LLG equation should be satisfied as well. We
have
d
N
M
dt
D
0
N
M
N
H : (3.69)
Let us consider (3.69) through (3.62) and (3.63) under small-signal approximation. It is
obvious that Nm can be related to
N
h through the same susceptibility tensor by just substituting
H
0
! H
i
or !
0
D
0
H
0
! !
i
D
0
H
i
. us, for the present case, (2.17) reads
Nm D ŒX
N
h D
2
6
4
X
xx
X
xy
0
X
yx
X
yy
0
0 0 0
3
7
5
N
h (3.70)
and
X
xx
D X
yy
D X D
!
i
!
m
!
2
i
!
2
D
H
2
H
2
(3.71a)
X
xy
D X
yx
D j k
1
D j
!!
m
!
2
i
!
2
D j
2
H
2
(3.71b)
where
where D !=!
m
and
H
D !
i
=!
m
: (3.71c)
Likewise, the flux density vector
N
b is related to
N
h through the permeability tensor:
N
b D
0
f
ŒI CŒX
g
N
h D Œ
N
h (3.72)
and
ΠD
0
2
6
4
1
C
X
jk
1
0
jk
1
1 C X 0
0 0 1
:
3
7
5
(3.73)
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