82
5 Flux Pinning Phenomena
When the current exceeds the critical current (I c ), the force balance given by (5.6)
no longer holds, and the flux flow state sets in.
(4) Pinning loss energy
Next, we will estimate the loss energy in the superconductor. Sometimes the loss
energy is calculated from the area of the closed magnetization curve in the case of
AC loss energy. This method cannot be used, however, in the case of the loss energy in
a superconducting magnet during the energizing process or that in a superconducting
power transmission line during the process of increasing the current. On the other
hand, the loss energy in any case can be estimated using the critical state model. The
total loss power density is [3]
P = J · E = J · (B × v) = (J × B) · v.
(5.32)
The reason why the loss power can be estimated is guaranteed by the assumption of
irreversibility in the critical state model. The loss power density given by (5.32) can
be regarded as the power generated by the Lorentz force when it drives flux lines
with velocity v. This agrees with a model of dynamics. From (5.6)–(5.9) the pinning
loss power density is given by
P p = |J c Bv|,
(5.33)
where we used the fact that δ and v point in the same direction. Hence, it is necessary
to estimate the velocity v to calculate the loss power. The electromotive induction is
caused by a time variation in the magnetic flux density, and such a variation comes
from the motion of flux lines. Substitution of (4.41) into (2.49) leads to
∇ × (B × v) = −
∂B
∂t
,
(5.34)
which is called the continuity equation for flux lines [3]. This equation is useful to
determine v.
Here, we treat the AC loss energy density when an AC magnetic field of amplitude
H m is applied parallel to a superconducting slab of thickness 2d (0 ≤ x ≤ 2d ).
From symmetry, we have only to treat the half region 0 ≤ x ≤ d . We needed the
classification of the value of H m for the calculation of the magnetization in (2). The
same thing occurs again. We treat the case of H m < H p here.
Since the AC loss energy density is the same between the process of changing the
external magnetic field from H m to −H m and that from −H m to H m , it is enough to
double the loss energy density in the former process. The magnetic flux distribution
when the external magnetic field is decreased from H 0 = H m is shown in Fig. 5.13.
The branching point of the magnetic flux distribution is
x b =
H m − H 0
2J c
.
(5.35)
5 Flux Pinning Phenomena
When the current exceeds the critical current (I c ), the force balance given by (5.6)
no longer holds, and the flux flow state sets in.
(4) Pinning loss energy
Next, we will estimate the loss energy in the superconductor. Sometimes the loss
energy is calculated from the area of the closed magnetization curve in the case of
AC loss energy. This method cannot be used, however, in the case of the loss energy in
a superconducting magnet during the energizing process or that in a superconducting
power transmission line during the process of increasing the current. On the other
hand, the loss energy in any case can be estimated using the critical state model. The
total loss power density is [3]
P = J · E = J · (B × v) = (J × B) · v.
(5.32)
The reason why the loss power can be estimated is guaranteed by the assumption of
irreversibility in the critical state model. The loss power density given by (5.32) can
be regarded as the power generated by the Lorentz force when it drives flux lines
with velocity v. This agrees with a model of dynamics. From (5.6)–(5.9) the pinning
loss power density is given by
P p = |J c Bv|,
(5.33)
where we used the fact that δ and v point in the same direction. Hence, it is necessary
to estimate the velocity v to calculate the loss power. The electromotive induction is
caused by a time variation in the magnetic flux density, and such a variation comes
from the motion of flux lines. Substitution of (4.41) into (2.49) leads to
∇ × (B × v) = −
∂B
∂t
,
(5.34)
which is called the continuity equation for flux lines [3]. This equation is useful to
determine v.
Here, we treat the AC loss energy density when an AC magnetic field of amplitude
H m is applied parallel to a superconducting slab of thickness 2d (0 ≤ x ≤ 2d ).
From symmetry, we have only to treat the half region 0 ≤ x ≤ d . We needed the
classification of the value of H m for the calculation of the magnetization in (2). The
same thing occurs again. We treat the case of H m < H p here.
Since the AC loss energy density is the same between the process of changing the
external magnetic field from H m to −H m and that from −H m to H m , it is enough to
double the loss energy density in the former process. The magnetic flux distribution
when the external magnetic field is decreased from H 0 = H m is shown in Fig. 5.13.
The branching point of the magnetic flux distribution is
x b =
H m − H 0
2J c
.
(5.35)