Digital Signal Processing Reference
In-Depth Information
Fig. 3.1. Series RLC circuit used
in Example 3.1.
v
(
t
)
−
+
L
+
−
w
(
t
)
C
x
(
t
)
i
(
t
)
+
y
(
t
)
R
where coefficients
a
k
, for 0
≤
k
≤
(
n
−
1), and
b
k
, for 0
≤
k
≤
m
, are parameters
characterized by the linear system. If the linear system is also time-invariant,
then the
a
k
and
b
k
coefficients are constants. We will use the compact notation
y
to denote the first derivative of
y
(
t
) with respect to
t
. Thus
y
=
d
y
/
d
t
,
y
=
d
2
y
/
d
t
2
, and so on for the higher derivatives. We now consider an electrical
circuit that is modeled by a differential equation.
Example 3.1
Determine the input-output representations of the series RLC circuit shown in
Fig. 3.1 for the three outputs
v
(
t
),
w
(
t
), and
y
(
t
).
Solution
Figure 3.1 illustrates an electrical circuit consisting of three passive compo-
nents: resistor
R
, inductor
L
, and capacitor
C
. Applying Kirchhoff's voltage
law, the relationship between the input voltage
x
(
t
) and the loop current
i
(
t
)is
given by
t
L
d
i
d
t
+
Ri
(
t
)
+
1
C
x
(
t
)
=
i
(
t
)d
t
.
(3.2)
−∞
Differentiating Eq. (3.2) with respect to
t
yields
L
d
2
i
d
t
2
+
R
d
i
d
t
+
1
C
i
(
t
)
=
d
x
d
t
.
(3.3)
We consider three different outputs of the RLC circuit in the following dis-
cussion, and for each output we derive the differential equation modeling the
input-output relationship of the LTIC system.
Relationship between
x(t)
and
v(t)
The output voltage
v
(
t
) is measured across
inductor
L
. Expressed in terms of the loop current
i
(
t
), the voltage
v
(
t
)isgiven
by
L
d
i
v
(
t
)
=
d
t
.
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