The voltage gain is G = A/(1+AB). In the limit of infinite open-loop gain A, G = 1/B. For finite gain A, feedback acts to reduce the effects of variations of A (with frequency, temperature, amplitude, etc.). For instance, suppose A depends on frequency as in Figure 1. This will surely satisfy anyone’s definition of a poor amplifier (the gain varies over a factor of 10 with frequency). Now imagine we introduce feedback, with B = 0.1 (a simple voltage divider will do). The closed-loop voltage gain now varies from 1000/[1 +(1000×0.1)], or 9.90, to 10,000/[1+ (10,000×0.1)], or 9.99, a variation of just 1% over the same range of frequency! To put it in audio terms, the original amplifier is flat to ±10 dB, whereas the feedback amplifier is flat to ±0.04 dB. We can now recover the original gain of 1000 with nearly this linearity simply by cascading three such stages.
It was for just this reason (namely, the need for extremely flat-response telephone repeater amplifiers) that negative feedback in electronics was invented. As the inventor, Harold Black, described it in his first open publication on the invention [Elec. Eng., 53, 114, (1934)], “by building an amplifier whose gain is made deliberately, say 40 decibels higher than necessary (10,000-fold excess on energy basis) and then feeding the output back to the input in such a way as to throw away the excess gain, it has been found possible to effect extraordinary improvement in
constancy of amplification and freedom from nonlinearity.”
Black’s patent is spectacular, with dozens of elegant figures; we reproduce one of them here (Figure 2), which makes the point eloquently. It is easy to show, by taking the partial derivative of G with respect to A (i.e., ∂G/∂ A), that relative variations in
the open-loop gain are reduced by the desensitivity:
Thus, for good performance the loop gain AB should be much larger than 1. That’s equivalent to saying that the open-loop gain should be much larger than the closed-loop gain.
A very important consequence of this is that nonlinearities, which are simply gain variations that depend on signal level, are reduced in exactly the same way.
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