So far we have dealt with operational amplifiers mostly in the infinite-gain approximation. This procedure is valid if we are interested only in the ideal response of circuits that are well designed and therefore presumably stable. In Chapter 8 we learned that the loop gain of an operational-amplifier circuit necessarily goes to zero at high frequencies; in some circuits—*RC* integrators, for example—it also goes to zero at low frequencies. We will now see that this behavior not only limits the bandwidth over which the response is ideal, but that it can also result in an unstable response.

We will first study direct amplifiers in order to determine which parameters of the loop gain control their dynamical behavior. We will then show that our conclusions can be extended to arbitrary operational-amplifier circuits; in doing so we will develop the *phase-margin criterion*, a simple rule for judging the degree of stability of a circuit. An important conclusion we will draw is that the loop gain of stable circuits looks much like an integrator in the vicinity of the upper transition frequency and, given the nature of things, that the best possible loop gain is an integrator with an infinite transition frequency rather than a (physically impossible) infinite constant gain.

For the purposes of studying their dynamical behavior, most linear feedback systems can be represented as operational-amplifier circuits; we ...

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