Biomedical Engineering Reference
In-Depth Information
¢(
) +¢(
)
g
ps
H
ber
g
rj
bei
g
r
E
=-
0
a
0
0
a
e
a
(4.48)
jt
w
(
) +
(
)
j
4
+
j
we
ber
g
aj
bei
g
0
0
Then, the eddy current at distance
r
from the center of the conducting cylin-
der is based on the expression
IE
= s
(4.49)
Substituting expression (4.48) into (4.49) yields
¢(
) +¢(
)
g
ber
g
rj
bei
g
r
I
=-
0
H
0
0
a
e
a
j
(4.50)
jt
w
a
(
) +
(
)
4
p
ber
g
aj
bei
g
0
0
Since the eddy current flows in the tangential direction of the circumference
of the conducting cylinder, its amplitude at
r
is given by the expression
¢(
) +¢(
)
g
ber
g
rj
bei
g
r
I
=-
0
H
0
0
a
e
(4.51)
jt
w
r
a
(
) +
(
)
4
p
ber
g
aj
bei
g
0
0
With this, we can derive the expression of the electric power loss based on the
ohmic term of the above expression. Let us consider a small area formed by
a width
dr
at a radius
r
from the center of the cylinder and by a unit length in
this position in the axial
z
direction. The resistance
dR
of this area is then
calculated using the total current through the area, which is expressed by
¢(
) +
¢(
)
g
ber
g
rj
bei
g
r
0
0
0
dI
=-
H
a
dre
jt
w
(4.52)
r
a
(
) +
(
)
4
p
ber
g
aj
bei
g
0
0
Therefore, the ohmic loss associated with the elementary area with small width
dr
and unit length in the
z
direction is
2
r
dr
p
2
2
dP
=
dR dI
=
r
dI
r
r
r
¢ (
) +¢ (
)
gr
p
H
2
ber
2
g
r
j
bei
2
g
r
=
0
a
0
0
a
rdr
(4.53)
(
) +
(
)
8
ber
2
g
aj
bei
2
g
0
0
Then, the total power loss is obtained by integrating the expression in
dP
r
for
the whole region of the conducting cylinder:
¢ (
) +¢ (
)
gr
p
2
H
2
ber
2
g
r
j
bei
2
g
r
a
Ú
P
=
a
0
0
a
rdr
a
(
) +
(
)
8
ber
2
g
aj
bei
2
g
0
0
0
(
)
¢(
) +
(
)
¢(
)
gr
p
a
H
ber
g
a
ber
g
a
bei
g
a
bei
g
a
0
0
0
0
0
(4.54)
=
2
a
(
) +
(
)
8
ber
g
a
bei
g
a
2
2
0
0
using the relations
d
dr
d
dr
[
r
ber
¢
(
g
r
)
]
=-
g
r
bei
(
g
r
)
[
r
bei
¢
(
g
r
)
]
=-
g
r
ber
(
g
r
)
0
0
0
0
0
0
Expression (4.54) is the total power loss per unit length in the direction of the
z
axis. Accordingly, the power loss for the actual conducting cylinder with
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