5.2 Critical State Model
83
Fig. 5.13 Magnetic flux
distribution in the
superconductor in the
process of decreasing the
external magnetic field from
the initial state
Here, we estimate the velocity v from (5.34). Since v and B are directed along the xand z-axes, respectively, (5.34) is reduced to
∂
∂x
Bv = −
∂B
∂t
.
(5.36)
Since magnetic flux lines move in the direction of the negative x-axis from the
branching point x = x b , v is zero at x = x b . The variation with time on the right-hand
side is given by −μ 0 ∂H 0 /∂t from the magnetic flux distribution given by (5.17).
Hence, we have
Bv(x) = −
x
x b
μ 0
∂H 0
∂t
dx = −μ 0
∂H 0
∂t
(x − x b ).
(5.37)
Note that Bv(x) is negative, independently of the sign of B. That is, v > 0 in the
region where B < 0 and v < 0 in the region where B > 0. Since the loss occurs in
the region 0 ≤ x ≤ x b in which magnetic flux lines move, the pinning loss power
density is written as
P p =
1
d
x b
0
|J c Bv|dx =
μ 0 J c
d
·
∂H 0
∂t
x b
0
(x − x b )dx
= −
μ 0 (H m − H 0 )
2
8H p
·
∂H 0
∂t
.
(5.38)
Note that ∂H 0 /∂t < 0. Then, the AC loss energy density is calculated as