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 idea that feedback phase is unimportant at high loop gains can be misleading, it depends on how the phase shift is achieved. A single inversion can be represented by a phase shift of 180 deg, but if we had two inversions and applied overall feedback we would get something like a bistable circuit, with only two stable states. With a single inversion plus additional phase shift from resistors plus capacitors in the circuit, then stable linear operation is possible even when the feedback is exactly positive phase at some frequencies, provided the added phase shift falls at high frequencies before the loop gain falls to unity. I have added a page about Positive Phase Negative 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.
Footnote:
When we apply negative feedback the reduction in percentage distortion is equal to the gain round the feedback loop at the frequency of the distortion, not at the test frequency. Given that our hearing is most sensitive around 3kHz this suggests feedback needs to be maximised at 3kHz and can be allowed to fall at higher or lower frequencies. That is a justification for the common but much maligned specification of 'THD at 1kHz' which nevertheless tells us about distortion in the sensitive 2kHz to 5kHz range, and can be expected to tell us whether there is likely to be audible distorton.
The often quoted 'distortion at 20kHz' obviously detects only distortion at 40kHz and above, far beyond human hearing, and where loop gain is almost invariably falling. One justification I have seen is that this specification tells us about linearity at high frequencies, which is responsible for intermodulation distortion further down the frequency range. The problem with this is that the 20kHz distortion depends on both the 20kHz open-loop linearity and also the feedback level at 40kHz plus. This can encourage designers to aim for high loop gain at these high frequencies which may compromise stability, while contributing little or nothing to audio frequency distortion reduction.
Suppose we do the common intermodulation test with 19kHz + 20kHz components, and detect a distortion component at 1kHz. The level of this 1kHz component is reduced by feedback loop gain at 1kHz. Looking at my MJR7 distortion tests, the 20kHz distortion is around 0.002%, but in the 19kHz + 20kHz test the 1kHz component is down under 0.0001%, reflecting the far higher feedback loop gain at 1kHz. So does that mean high frequency distortion doesn't matter? No, the feedback level only determines the closed-loop reduction in distortion, but the open-loop distortion we start from is also important.
A good example is to compare my old MJR6 with the current MJR7. The MJR6 had relatively poor high frequency linearity because of the nonlinear mosfet input capacitance, and 19 + 20kHz IMD at 1kHz was -90dB. Adding an emitter-follower in the MJR7-Mk1 reduced that source of nonlinearity and the 1kHz IMD is then -109dB, a big improvement even though both amplifiers have similar high feedback at 1kHz.
The conclusion is that reducing 20kHz distortion by improved open-loop linearity at 20kHz is a good approach, but aiming for a low 20kHz THD by increasing high frequency loop gain above 20kHz achieves little beyond impressive distortion figures. It's a little more complex than this, e.g. those 40kHz + distortion components can cause intermodulation products down in the audio range, but such effects can usually be assumed to be very small.
