Biomedical Engineering Reference
In-Depth Information
Current-Voltage Characteristics of dc SQUIDs.
If a dc SQUID is
biased with a current
I
b
, the voltage across the SQUID loop is modulated by
an applied flux because the critical current of the loop is changed as shown
for the threshold curves. For simplicity, suppose that the inductance of each
branch and the critical current of each junction are the same, respectively, or
L
sq1
=
L
sq2
=
L
0
,and
I
c1
=
I
c2
=
I
c0
. The bias current
I
b
and the screening
current
I
sc
are written in terms of the current flowing in each branch,
i
1
(
t
)
and
i
2
(
t
), as follows:
I
b
=
i
1
(
t
)+
i
2
(
t
)
,
(3.19)
I
sc
=
i
2
(
t
)
−
i
1
(
t
)
.
(3.20)
2
i
1
(
t
)and
i
2
(
t
) are expressed using the voltages across each junction,
v
1
and
v
2
, namely:
i
1
(
t
)=
I
c0
sin
φ
1
(
t
)+
v
1
R
s
i
2
(
t
)=
I
c0
sin
φ
2
(
t
)+
v
2
R
s
,
,
(3.21)
where the phase differences
φ
1
and
φ
2
are related to the Josephson equation:
d
φ
1
d
t
=
4
πe
h
d
φ
2
d
t
=
4
πe
h
v
1
,
v
2
.
(3.22)
The voltage across the SQUID loop
V
is the sum of the voltages across the
junctions and the inductance:
V
=
v
1
+
L
0
d
i
1
d
t
=
v
2
+
L
0
d
i
2
d
t
,
(3.23)
where the mutual inductance between two branches is neglected. Rearranging
(3.19), (3.21), and (3.23), for a constant bias current,
h
4
πe
d
φ
(
t
)
d
t
V
(
t
)=
(3.24)
is obtained, where
φ
(
t
) is written as
φ
(
t
)=
φ
1
(
t
)+
φ
2
(
t
)
2
.
(3.25)
Equation (3.24) indicates an “ac Josephson effect”, which means that the
phase difference varying in time across the junction results in a voltage out-
put.
Using (3.21) and (3.23), (3.19) and (3.20) are rewritten as
I
b
=2
I
c0
cos
φ
1
(
t
)+
φ
2
(
t
)
2
cos
φ
1
(
t
)
+
2
V
(
t
)
R
s
−
φ
2
(
t
)
,
(3.26)
2
