Biomedical Engineering Reference
In-Depth Information
I
sc
=2
I
c0
cos
φ
1
(
t
)+
φ
2
(
t
)
2
sin
φ
1
(
t
)
−
φ
2
(
t
)
,
(3.27)
2
where the second term on the right-hand side of (3.27), which is the contri-
bution of quasi-particles, is neglected for simplicity.
Considering the flux quantization condition,
φ
1
(
t
)
−
φ
2
(
t
)
=
−π
Φ
t
(
t
)
Φ
0
(3.28)
2
is true. Substituting (3.28) into (3.26),
I
b
=2
I
c0
sin [
φ
(
t
)] cos
π
Φ
t
(
t
)
Φ
0
+
2
V
(
t
)
R
s
(3.29)
is derived. Since the flux caused by the screening current
Φ
sc
is written as
Φ
s
=
L
sq
I
sc
,
(3.30)
the following equation is yielded by rearranging (3.19), (3.26), and (3.28):
=
β
L
sin
π
Φ
t
(
t
)
Φ
0
cos [
φ
(
t
)]
,
Φ
ex
−
Φ
t
(3.31)
Φ
0
where 2
L
0
is replaced by
L
sq
, and the parameter
β
L
=
2
I
c0
L
sq
Φ
0
(3.32)
is introduced. If the screening current
I
sc
is neglected,
Φ
t
(
t
)
∼
Φ
ex
(
t
) because
β
L
−→
0. Equation (3.29) is simplified as
2
R
s
h
4
πe
d
φ
(
t
)
d
t
I
b
=
I
c
(
Φ
ex
)sin[
φ
(
t
)] +
,
(3.33)
where
I
c
(
Φ
ex
)=cos
π
Φ
ex
Φ
0
.
(3.34)
The
I
-
V
characteristics of SQUIDs are obtained by solving these nonli-
near equations with changing
I
b
for a constant flux. The
Φ
-
V
characteristics
are also obtained with changing
Φ
for a constant
I
b
. However, a simple ap-
proximation has been proposed for the detailed analysis, as follows:
0
≥|
I
b
|≥
I
c
(
Φ
ex
)
⇒
V
(
t
)=0
,
(3.35)
I
b
−
V
(
t
)=
R
s
2
|
I
b
|≥
I
c
(
Φ
ex
)
⇒
[
I
c
(
Φ
ex
)]
2
.
(3.36)
