182
Appendix
Thus, (4.32) is derived with Bvi y = B × v.
A.4 Derivation of (5.78)
In the case where a static approximation holds, we can set
= δ =
a f
f p − 3f pt
4f pt
(A.4.1)
at t = 0 and
= vt 1 + δ =
a f
f p + f pt
4f pt
(A.4.2)
at t = t 1 . Then, we obtain
vt 1 = a f .
(A.4.3)
If the second small term proportional to v
2 is neglected in (5.77), the pinning loss
power density is given by
P p =
N p η
∗
T
⎧
⎨
⎩
f pt
f p + f pt
2
t 1
0
v +
K 1
τ 1
exp
−
t
τ 1
2
dt
+
f pt
f p − f pt
2
T
t 1
v +
K 2
τ 2
exp
t − t 1
τ 2
2
dt
⎫
⎬
⎭
.
(A.4.4)
Using the condition (5.75), the first integral in the braces is calculated as
v
2 t 1 + 2K 1 v
1 − exp
−
t 1
τ 1
+
K
2
1
2τ 1
1 − exp
−
2t 1
τ 1
=
a f
f p + f pt
v
f p
− v
2 t 1 +
K
2
1
2τ 1
−
1
2τ 1
vt 1 −
a f
f p + f pt
2f pt
+ K 1
2
.
(A.4.5)
Using (A.4.2) and (A.4.3) and neglecting small terms proportional to v, the quantity
in (A.4.5) is reduced to
a f
f p + f pt
f p − f pt
2
2η ∗ f
2
pt
.
(A.4.6)