Digital Signal Processing Reference
In-Depth Information
Integrating the above equation with respect to
t
yields
i
(
t
)
=
1
L
v
(
t
)d
t
.
By substituting the value of
i
(
t
) into Eq. (3.3), we obtain
d
v
d
t
+
R
1
LC
=
d
x
d
t
.
The above input-output relationship includes both differentiation and integra-
tion operations. The integral operator can be eliminated by calculating the
derivative of both sides of the equation with respect to
t
. This results in the
following equation:
L
v
(
t
)
+
v
(
t
)d
t
d
2
v
d
t
2
LC
v
(
t
)
=
d
2
x
+
R
L
d
v
d
t
1
+
d
t
2
,
(3.4)
which models the input-output relationship between the input voltage
x
(
t
) and
the output voltage
v
(
t
) measured across inductor
L
. Equation (3.4) is a linear,
second-order differential equation with constant coefficients. In fact, it can
be shown that
an LTIC system can always be modeled by a linear, constant-
coefficient differential equation with the appropriate initial conditions.
Relationship between
x(t)
and
w(t)
The output voltage
w
(
t
), measured across
capacitor
C
,
is given by
t
w
(
t
)
=
1
C
i
(
t
)d
t
,
−∞
which is expressed as follows:
i
(
t
)
=
C
d
w
d
t
.
Substituting the value of
i
(
t
) into Eq. (3.3) yields
LC
d
3
w
d
t
3
+
RC
d
2
w
d
t
2
+
d
w
d
t
=
d
x
d
t
,
(3.5)
which specifies the relationship between the input voltage
x
(
t
) and the output
voltage
w
(
t
) measured across capacitor
C
. Equation (3.5) can be further sim-
plified by integrating both sides with respect to
t
. The resulting equation is
simplified to
LC
d
2
w
d
t
2
+
RC
d
w
d
t
+
w
(
t
)
=
x
(
t
)
,
(3.6)
which is a linear, second-order, constant-coefficient differential equation.
Relationship between
x(t)
and
y(t)
Finally, we measure the output voltage
y
(
t
) across resistor
R
. Using Ohm's law, the output voltage
y
(
t
)isgivenby
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