Digital Signal Processing Reference
In-Depth Information
v
1
L
11
+
−
i
1
v
2
L
12
L
22
+
−
i
2
L
13
L
33
L
23
i
3
+
v
3
−
Figure 4-3
Circuit with three coupled inductors.
As a final step, we can generalize equation (4-8) to
n
inductively coupled
lines,
v
1
v
2
.
v
n
L
11
L
12
···
L
1
n
di
1
/dt
di
2
/dt
.
di
n
/dt
=
L
n
1
L
22
L
2
n
(4-9)
.
.
.
.
.
L
n
1
L
n
2
···
L
nn
where
L
ii
are the self-inductances and
L
ij
are the mutual inductances.
4.1.2 Mutual Capacitance
Our discussion of mutual capacitance follows a similar line of reasoning as for
mutual inductance, beginning with the capacitive circuit shown in Figure 4-4.
Given the input signals
v
1
and
v
2
on lines 1 and 2, we write expressions for the
currents
i
1
and
i
2
:
dt
+
C
M
dv
1
=
C
g
dv
1
dv
2
dt
=
(C
g
+
C
M
)
dv
1
dt
−
C
M
dv
2
i
1
dt
−
(4-10)
dt
dt
+
C
M
dv
2
=
C
g
dv
2
dv
1
dt
=
(C
g
+
C
M
)
dv
2
dt
−
C
M
dv
1
i
2
dt
−
(4-11)
dt
In analyzing the behavior of the circuit, let us first look at the case where line 1 has
a transient signal
dv
/
dt
applied to it, whereas line 2 has no signal (
dv
2
/dt
=
0).
Applying these inputs to (4-10) and (4-11) gives
=
C
g
+
C
M
dv
dt
i
1
(4-12)
=−
C
M
dv
1
dt
i
2
(4-13)
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