Bandwidth Modification
The next example illustrates the effect of negative feedback on the
bandwidth. Suppose the feedforward amplifier has a one-pole transfer
function:
where A0 denotes the low-frequency gain and ω0 is the 3-dB bandwidth.
What is the transfer function of the closed-loop system? From (8.5), we have
The numerator of (8.21) is simply the closed-loop gain at low frequencies—as
predicted by (8.5)—and the denominator reveals a pole at (1 + β A0)ω0.
Thus, the 3-dB bandwidth has increased by a factor of 1 + β A0, albeit at the
cost of a proportional reduction in the gain (Fig. 8.10).
The increase in the bandwidth fundamentally originates from the gain
desensitization property of feedback. Recall from (8.6) that, if A is large
enough, the closed-loop gain remains approximately equal to 1/β even if A
experiences substantial variations. In the example of Fig. 8.10, A varies with
frequency rather than process or temperature, but negative feedback still
suppresses the effect of this variation. Of course, at high frequencies, A
drops to such low levels that β A becomes comparable with unity and the
closed-loop gain falls below 1/β.
Equation (8.21) suggests that the “gain-bandwidth product” of a one-pole
system is equal to A0ω0 and does not change much with feedback, making
the reader wonder how feedback improves the speed if a high gain is
required. Suppose we need to amplify a 20-MHz square wave by a factor of
100 and maximum bandwidth, but we have only a single-pole amplifier with
an open-loop gain of 100 and 3-dB bandwidth of 10 MHz. If the input is
applied to the open-loop amplifier, the response appears as shown in Fig.
8.11(a), exhibiting a long risetime and falltime because the time constant is
equal to 1/(2π f3-dB) ≈ 16 ns.
Now suppose we apply feedback to the amplifier such that the gain and
bandwidth are modified to 10 and 100 MHz, respectively. Placing two of
these amplifiers in a cascade [Fig. 8.11(b)], we obtain a much faster
response with an overall gain of 100. Of course, the cascade consumes twice
as much power, but it would be quite difficult to achieve this performance
with the original amplifier even if its power dissipation were doubled.
