Digital Signal Processing Reference
In-Depth Information
R
f
i
(
t
)
i
(
t
)
R
i
v
i
(
t
)
v
o
(
t
)
Figure 1.6
Practical voltage amplifier.
For this discussion, we assume that the amplifier is ideal, which is sufficiently
accurate for most purposes. The ideal op amp has zero input currents
Additionally, the ideal amplifier operates in its linear range with infinite
gain, resulting in the input voltage being zero.
Because the op amp is a very high-gain device, feedback is usually added for
stabilization. The feedback is connected from the output terminal to the inverting
input terminal (the minus terminal). This connection results in negative, or stabiliz-
ing, feedback and tends to prevent saturation of the op amp.
An example of a practical op-amp circuit is given in Figure 1.6. In this circuit,
is the circuit input voltage and the circuit output voltage. Because in
Figure 1.5(b) is assumed to be zero, the equation for the input loop in Figure 1.6 is
given by
[i
-
(t) =
i
+
(t) = 0].
v
d
(t)
v
i
(t)
v
o
(t)
v
d
(t)
v
i
(t)
R
i
v
i
(t) - i(t)R
i
= 0 Q i(t) =
.
(1.7)
i
-
(t)
Also, because
in Figure 1.5(b) is zero, the current through
R
f
in Figure 1.6 is
equal to that through
R
i
.
The equation for the outer loop is then
v
i
(t) - i(t)R
i
- i(t)R
f
- v
o
(t) = 0.
Using (1.7), we express this equation as
R
f
R
i
.
v
i
(t)
R
i
v
o
(t)
v
i
(t)
=-
v
i
(t) - v
i
(t) -
R
f
- v
o
(t) = 0 Q
(1.8)
This circuit is then a
voltage amplifier
. The ratio
R
f
>
R
i
is a positive real number;
hence, the amplifier voltage gain
v
o
(t)
>
v
i
(t)
is a negative real number. The model
(1.8) is a
linear algebraic equation
.
A second practical op-amp circuit is given in Figure 1.7. We use the preceding
procedure to analyze this circuit. Because the input loop is unchanged, (1.7) applies,
with
R
i
= R.
The equation of the outer loop is given by
t
1
C
L
v
i
(t) - i(t)R -
i(t)dt - v
o
(t) = 0.
(1.9)
-
q
