Two Important Details
Feedback is a rich subject, which we’ve simplified shamelessly in this brief introduction. Here are two details that should not be overlooked, however, even at this somewhat superficial level of understanding.
Loading by the Feedback Network
In feedback computations, you usually assume that the beta network doesn’t load the amplifier’s output. If it does, that must be taken into account in computing the open-loop gain. Likewise, if the connection of the beta network at the amplifier’s input affects the open-loop gain (feedback removed, but network still connected), you must use the modified open-loop gain. Finally, the preceding expressions assume that the beta network is unidirectional, i.e., it does not couple any signal from the input to the output.
Phase Shifts, Stability, and “Compensation”
The open-loop amplifier gain A is central in the expressions we’ve found for closed-loop gain and the corresponding input and output impedances. By default one might reasonably assume that A is a real number – that is, that the output is in phase with the input. In real life things are more complex, 70 because of the effects of circuit capacitances (and Miller effect), and also the limited bandwidth (fT) of the active components themselves. The result is that the open-loop amplifier will exhibit lagging phase shifts that increase with frequency. This has several consequences for the closed-loop amplifier.
If the open-loop amplifier’s lagging phase shift reaches 180 degrees, then negative feedback becomes positive feedback, with the possibility of oscillation. This is not what you want! (The actual criterion for oscillation is that the phase shift be 180◦ at a frequency at which the loop gain AB equals 1.) This is a serious concern, particularly in amplifiers with plenty of gain (such as op-amps). The problem is only exacerbated if the feedback network contributes additional lagging phase shift (as it often will). The subject of frequency compensation in feedback amplifiers deals directly with this essential issue.
Gain and Phase Shift
The expressions we found for closed-loop gain and for the input and output impedances contain the open-loop gain A. For example, the voltage amplifier with series feedback has closed-loop gain GCL = A/(1+AB), where A = GOL, the amplifier’s open-loop gain. Let’s imagine that the open-loop gain A is 100, and that we’ve chosen B = 0.1 for a target closed-loop gain of GCL ≈ 10. Now, if the open-loop amplifier had no phase shifts, then GCL ≈ 9.09, also without phase shift. If instead the amplifier has a 90◦ lagging phase shift, then A is pure imaginary (A=−100 j), and the closed-loop gain becomes GCL = 9.90−0.99 j. That’s a magnitude |GCL|=9.95, with a lagging phase shift of approximately 6◦.
In other words, the effect of a pretty significant (halfway to oscillation!) open-loop phase shift turns out, in fact, to be favorable: the closed-loop gain is only 0.5% less than the target, compared with 9% for the case of the same amplifier without phase shift. The price you pay is some residual phase shift and, of course, an approach to instability.
As artificial as this example may seem, it in fact reflects a reality of op-amps, which usually have an ∼90◦ lagging phase shift over almost their entire bandwidth (typically from ∼10 Hz to 1MHz or more). Because of their much higher open-loop gain, the amplifier with feedback exhibits very little phase shift, and an accurate gain set almost entirely by the feedback network.
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