Biomedical Engineering Reference
In-Depth Information
e
e
-
+
1
2
MN
m
2
Ê
Ë
ˆ
¯
P
()
=
w
r
=
a
+
(4.29)
m
r
3
e
3
kT
r
0
0
On the other hand, when applying an alternating electric field and provided
expression (4.29) holds, the polarization is given by what is called a Debye dis-
persion expression:
e
e
-
+
1
2
MN
m
2
1
Ê
Ë
ˆ
¯
P
()
=
w
r
=
a
+
(4.30)
m
r
3
e
3
kT
1
+
j
wt
r
0
0
Defining e
0
as the value of e
r
when w =
0 in this expression yields
e
e
-
+
1
2
MN
m
2
Ê
Ë
ˆ
¯
r
=
a
+
(4.31)
r
3
e
3
kT
r
0
0
Similarly, defining e
•
as the value of e
r
when w is very high yields
e
e
-
+
1
1
MN
a
e
•
=
(4.32)
r
3
0
•
The permittivity e
r
between the extreme frequency regions defined by (4.31)
and (4.32) can be expressed as follows using (4.30), (4.31), and (4.32):
(
) +
[
(
)
]
[
(
)
(
)
]
ee
+
2
j
j
wte e
+
2
e
+
j
te
+
2
e
+
2
e
e
=
0
0
••
=
0
0
•
•
(4.33)
r
(
) +
[
(
)
]
[
(
)
(
)
]
1
e
+
2
wt
1
e
+
2
1
+
j
wt
e
+
2
e
+
2
0
•
0
•
Defining
x
as
e
e
+
+
2
2
0
x
=
wt
(4.34)
•
the expression (4.34) can be written as:
e
+
+
jx
jx
e
=+
-
+
ee
0
•
0
•
e
=
e
r
•
1
1
jx
(
)
=+
-
+
ee
ee
-
x
0
•
0
•
(4.35)
e
-
j
•
1
x
1
+
x
2
2
Finally, defining a generalized relaxation time t
0
= t(e
0
+
2)/(e
•
+
2) yields the
following equation when using permittivitye
r
from Eqn. (4.33):
e te
wt
+
+
j
j
=+
-
+
ee
wt
0
0
•
0
•
e
=
e
r
•
1
1
j
0
0
=+
-
+
ee
wt
(
eewt
wt
-
)
e
0
•
-
j
0
•
0
(4.36)
•
1
1
+
2
2
2
2
With this, we have derived the final expression for the complex permittivity
based on the orientation polarization. As a consequence, the microwave
dielectric heat quantity per unit time and unit volume is obtained when sub-
stituting the imaginary part of Eqn. (4.36) into Eqn. (4.3).
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