Problem 2
Simplify these equations into a relation between
V
out,
2
and
V
in,
1
:
˙
V
out,
1
+
1
R
1
C
1
V
out,
1
=
1
R
1
C
1
V
in,
1
(10)
R
2
C
2
˙
V
out,
2
+
V
out,
2
=
R
2
C
2
˙
V
in,
2
=
R
2
C
2
˙
V
out,
1
(11)
Differentiate (10),
¨
V
out,
1
+
1
R
1
C
1
˙
V
out,
1
=
1
R
1
C
1
˙
V
in,
1
.
(12)
From (11),
˙
V
out,
1
=
˙
V
out,
2
+
1
R
2
C
2
V
out,
2
,
(13)
which is differentiated,
¨
V
out,
1
=
¨
V
out,
2
+
1
R
2
C
2
˙
V
out,
2
.
(14)
Substitute (13) and (14) into (12),
R
1
C
1
¨
V
out,
2
+
1
R
2
C
2
˙
V
out,
2
+
˙
V
out,
2
+
1
R
2
C
2
V
out,
2
=
˙
V
in,
1
which is rearranged to,
R
1
C
1
¨
V
out,
2
+
1 +
R
1
C
1
R
2
C
2
˙
V
out,
2
+
1
R