6.4 Negative Feedback Amplifier
Op amp circuits can achieve controlled voltage gain while avoiding saturation by employing negative feedback. The basic idea is to provide a path between the op amp output node, which has voltage Vo, and the inverting terminal input node, which has voltage V–. In the example circuit shown in figure 6.22, this is accomplished via the feedback resistor Rf.
This feedback path serves to subtract part of the output voltage from the differential input voltage as a way to avoid saturation by the very high gain amplifier, A. The impact of feedback is to drive the differential input voltage (V+-V–) toward a very small value (ideally, zero). When multiplied by the very large (ideally, infinite) open-loop gain, , the output voltage remains finite and the op amp avoids saturation. Read the last three sentences again and think about what is written. Repeat.
How negative feedback works. We begin by considering the closed-loop feedback amplifier block diagram shown in figure 6.23. This circuit has the same function as the op amp circuit shown above, but it is implemented more simply with three functional blocks (when we say “functional” we mean that we will concern ourselves with what the block does, rather than what the block is or how the functionality is achieved; we don’t care what’s inside the block): a difference block that subtracts one signal from another, and two amplifier blocks, a forward amplifier having gain A and a reverse, or backward amplifier, having gain f. (Forward and backward just refer to the input-output flow direction of the amplifiers: left to right in the forward case and right to left in the backward case.) The amplifier blocks simplify multiply their respective inputs by A and f to produce output signals. Given these definitions, we can readily derive the input-output relationship, vs. for this circuit by inspecting the block diagram:
, defined as the “error voltage”, is the difference between the input signal and the output signal , having been amplified by backward gain f. This is called an error voltage in control theory because it often represents the difference between a desired value and an achieved value. As we shall see, this voltage is driven to zero by the feedback circuit.
The error voltage is given by
(1)
is an amplified version of the error voltage,
(2)
Substituting, eq. (1) into eq. (2) we have
(3)
from which we obtain, via algebraic manipulation:
(4)
and then
(5)
We know that op amps have very large (ideally, infinite) values of A, so we want to examine the behavior of this input-output relationship for very large A. To do this, we divide numerator and denominator by A, achieving
(6)
The closed-loop gain is given by
(7)
when we let A become very large, becomes negligible and we have
(8)
For the ideal op amp with A=∞, eq. (8) is an equality rather than an approximation since .
Note that the closed-loop gain of the circuit, , is independent of gain . All that was required was that be very