■ FIGURE 5. Inverting
amplifier configuration (power
connections not shown).
A = vout = −Rf MB vin R1
non-inverting amplifier (not shown). Thus, the non-inverting
amplifier provides twice the bandwidth when used as a
unity gain voltage follower or buffer compared with a
buffer constructed using an inverting amplifier.
Now that we have a basic understanding of how
feedback circuits allow us to trade op-amp gain and
bandwidth, let’s apply this knowledge by considering the
following scenario: If I need an audio amplifier with a
midband gain of 1,000 V/V and a 3 dB or half-power
bandwidth of at least 20 kHz, how do I build it? The GBW
required is 20 MHz and I simply can’t achieve that with a
single amplifier. The solution is to cascade two or more
amplifiers and divide up the gain requirement between
■ FIGURE 6. Effect of feedback on the bandwidth and
gain of the inverting amplifier.
Rf = ∞
Rf = 500
f0 = 5.2 Hz f0 = 2.0 kHz
1. 0Hz 10Hz 100Hz 1.0 KHz 10KHz
DB(V(U1:OUT)/V(U1:+)) DB(V(U2:OUT)/V(V6:+)) DB(V(U3:OUT)/V(V9:+))
Fre qu e ncy
DC = 15V
them. Either inverting or non-inverting amplifiers
can be placed in cascade (series) with each other to
provide increased gain compared to a single amplifier.
The very high input impedance and very low output
impedance of the op-amp in the non-inverting
configuration allow us to ignore the loading effect
of the second amplifier on the first in the cascade.
Essentially, this allows us to design a single amplifier
stage and then iterate the same amplifier multiple times
to attain the desired result. Figure 7 shows how a cascade
of non-inverting amplifiers solves a 20 MHz GBW
dilemma. For this amplifier, the overall gain is merely the
product of all of the individual stage gains. You can see
that the ratio of the two feedback resistors for each stage
is set so that the gain is equal to the square root of
1,000 minus 1 or 30. 6. The resulting bandwidth is found
by dividing the 1 MHz GBW of the uA741 by 31. 6,
resulting in a bandwidth of 31. 6 kHz — which seems more
than sufficient to meet the requirement.
Unfortunately, the bandwidth of the cascade is reduced
from the bandwidth of a single amplifier. This phenomenon
is known as bandwidth shrinkage and is calculated using
the formula shown in Figure 7 and discretely in Table 2.
The BWout in the figure represents the overall bandwidth
of the cascade accounting for the shrinkage — where n is
equal to the number of amplifiers in the cascade and BW0
is the bandwidth of an individual amplifier.
Table 2 lists the shrinkage for
cascades of two to five amplifiers.
As can be seen in the table, each
additional stage significantly
reduces overall BW — by the fifth
stage, roughly 39% of the original
bandwidth is all that remains. To
offset this reduction, simply increase
the bandwidth of the individual
amplifiers by the reciprocal of the
shrinkage factor. For the two stage
cascaded example, the shrinkage
f0 = 182 kHz factor is 0.644, so increase the
100KHz 1. 0MHz bandwidth of the individual amplifiers by 1.55 times over the required
bandwidth to achieve the requirement. In this case, the bandwidth
of a single amplifier is just sufficient
since 20 kHz times 1.55 equals
31 kHz and there is 31. 6 kHz of
BWout = BW0
2n - 1
1k AC Magnitude = 1uV
Phase = 0
DC = 15
■ FIGURE 7. Cascade of
Stages BW Shrinkage Factor
■ TABLE 2. BW Shrinkage.