Transistor Amplifier Design for Beginners. Part 6.
Negative Feedback. Here is the usual starting point of an explanation of negative feedback, a diagram of an amplifier with inverting (-) and non-inverting (+) inputs, and the output taken back to the inverting input via a resistive divider. The output voltage Vo = A (V+ - V-) where A is a usually large positive number, but in theory at least it can have an almost identical effect if it is a large negative number.

The easiest way to see why is maybe to look at the equation for the closed-loop gain, which is found in most treatments of feedback; for open-loop amplifier gain A, and feedback network gain B, the closed-loop gain is:
Gain = A / (1 + AB).
At low frequencies A will be a large positive number, something like 10,000, and suppose B = 0.1, chosen to give closed-loop gain about 10. Then substituting these values we get Gain = 10000 / (1 + 1000) = 9.990.
Suppose we make a mistake and use A = -10,000 instead of +10,000, then we get Gain = -10000 / (1 - 1000) = 10.01
So we get gain 10.01 instead of 9.99 so an error of only 0.2%, probably far less than typical component tolerances. For high values of AB as a first approximation we can ignore the 1 in the (1 + AB) and just get A / AB for the gain, so A cancels to leave Gain = 1/B, and so the exact value of A is unimportant, but also its sign has little effect, and for similar reasons its phase shift also has little effect. Also, nonlinearity can be thought of as variations in A at different signal levels, so this also cancels and so distortion reduction also works about as well with different phase feedback.
The 1 in (1+AB) does have a small effect, so we never get zero distortion with any finite value of A.
Suppose however we have smaller A, for example A = 20. Then the closed-loop gains for different signs of A become 20 / 3 = 6.67 and -20 /-1 = 20. The sign of A is now far more important, and if we go even lower with A = 10 we find the gains are 10 / 2 = 5, and -10 / 0 = -infinity. Infinite gain of course is impossible, in practice we would expect instability, with oscillation at some amplitude determined by the circuit linearity, generally because at some amplitude the gain will fall, maybe the result of clipping or slew-rate limiting.

Note that A can in principle be anything from minus to plus infinity and also any phase angle without stability problems (in theory at least) but only very close to A = -1/B do we get instability. So surely we would need to be extremely unlucky to make an amplifier with that exact magnitude and phase of A out of that whole range of possibilities, and yet we know amplifier instability is quite common, and often difficult to prevent. So why so much bad luck?
To see the problem remember that A is not just a single fixed value, it varies with frequency, it may be 10,000 at 20Hz but fall to 1 at 5MHz, so there will be some frequency where the magnitude of A is 1/B, but it would still need seriously bad luck to find exactly the right phase shift also at that same frequency.

That's true, but another way to look at the problem is to observe how the phase shift changes with frequency, here we will also find a frequency where A has the same phase as -1/B but is unlikely to also have exactly the same magnitude at that frequency. If A is then of greater magnitude than 1/B we would need to reduce the gain to create instability. This is usually known as 'conditional stability' and is important because there are situations where the gain can change, for example near clipping or slew-rate limiting, or just circuit nonlinearity, or even during switch-on or off. There are therefore plenty of possible triggers to start oscillation, and then various nonlinearities to stabilise the oscillation at some level. If we are very unlucky that level will be a full rail to rail oscillation with resulting serious damage. For my original MJR7 mosfet amplifier I found that reducing one component value to trigger instability with a capacitive load the oscillation was around 5MHz but at a low level which did no damage. The limiting factor I believe was a combination of output stage nonlinearity and slew-rate limiting. That's one of the reasons not to aim for extreme slew rate limits, if things go wrong they can go very badly wrong.

We do have some control over the open-loop gain A. This is where 'compensation capacitors' come into it, and usually an inductor is also involved. It is easy enough to get the magnitude of A down to less than 1/B before various phase shifts accumulate at higher frequencies sufficient to cause instability, but there are other conflicting requirements such as distortion and slew-rate which may need the compensation capacitor to be as small as possible. A big problem is that A depends to some extent on the load impedance added at the output, and generally this problem is more serious if the load is partly capacitive, which is where an added inductance in series with the output can help. Although this can make the resulting total load less capacitive the combination of capacitance and inductance will lead to resonances which need to be controlled, and that is why there are also various damping resistors added, at the very least a resistor of a few ohms in parallel with the inductor, and possibly a series resistor plus capacitor across the output before or after the inductor.

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