Page 244 - Electromagnetics

Because its permeability dyadic is that for a lossless gyrotropic material (2.33), we call
the ferrite gyromagnetic.
Since the ferrite is lossless, the dyadic permeability must be hermitian according to
(4.49). The specific form of (4.115) shows this explicitly. We also note that since the
sign of ω M is determined by that of H dc , the dyadic permittivity obeys the symmetry
relation

˜ µ ij (H dc ) = ˜µ ji (−H dc ),

which is the symmetry condition observed for a plasma in (4.87).
A lossy ferrite material can be modeled by adding a damping term to (4.111):

dM(r, t) M dc dM T (r, t)
=−γµ 0 [M T (r, t) × H dc + M dc × H T (r, t)] + α × ,
dt M dc dt
where α is the damping parameter [40, 204]. This term tends to reduce the angle of
precession. Fourier transformation gives

˜
˜
˜
˜
jωM T = ω 0 × M T − ω M × H T + α ω M × jωM T .
ω M
Remembering that ω 0 and ω M are aligned we can write this as

 ω 
ω 0 1 + jα 1
˜ ˜ ω 0 ˜
M T + M T ×   = − ω M × H T .
jω jω
This is identical to (4.112) with

ω
ω 0 → ω 0 1 + jα .
ω 0
Thus, we merely substitute this into (4.113) to find the susceptibility dyadic for a lossy
ferrite:
jω ¯ω M + ω M ω 0 (1 + jαω/ω 0 ) − ω M ω 0 (1 + jαω/ω 0 ) I ¯
˜ ¯ χ (ω) = .
m 2 2 2
ω (1 + α ) − ω − 2 jαωω 0
0
Making the same substitution into (4.115) we can write the dyadic permeability matrix
as
 
˜ µ xx ˜µ xy 0
[ ˜ ¯µ(ω)] =  ˜µ yx ˜µ yy 0  (4.118)
0 0 µ 0

where
2 2 2 2 2 2
ω 0 ω (1 − α ) − ω + jωα ω (1 + α ) + ω
0 0
˜ µ xx = ˜µ yy = µ 0 − µ 0 ω M (4.119)
2 2 2 2 2 2 2
ω (1 + α ) − ω + 4α ω ω
0 0
and
2
2

2
2µ 0 αω ω 0 ω M − jµ 0 ωω M ω (1 + α ) − ω 2 0
˜ µ xy =− ˜µ yx = . (4.120)
2 2 2
2
2
2
2
ω (1 + α ) − ω + 4α ω ω
0 0
In the case of a lossy ferrite, the hermitian nature of the permeability dyadic is lost.
© 2001 by CRC Press LLC