# Nested loops and Error Feedback.

### Updated Nov.28.2006.

Two effective methods of reducing distortion in audio amplifiers are overall negative feedback and error feedforward. Occasionally alternative methods are proposed, and two of these are considered here. There is a very simple method of understanding the circuit operation, which I first encountered many years ago in the letters pages of Wireless World (R.G.Mellish July 1973 p334) used there to explain the operation of a distortion reduction design, and used here to explain both nested feedback loops and error feedback.

### 1. Nested Feedback Loops.

First is the use of 'nested feedback loops'. This term appears to be applied to several different arrangements, but all versions I have seen can be analysed in a similar way to the following example.

Here negative feedback is taken from the output back to several points in the circuit. To see why there is no necessary difference compared to a single feedback loop it is helpful to simplify the analysis by putting part of the circuit in a separate box, drawn in red.

In the next diagram the contents of the red box are not shown. From the point of view of the output stage it makes no difference what is inside that box provided its output Vx is the same function of Vi and Vo. The effect on output stage distortion alone can be considered if we assume the contents of the red box to be linear. What this means is that Vx is a function of Vo plus a function of Vi, and the two can be separated as in the third diagram, into two boxes A and B.

What we now have is just a single overall feedback loop reducing the distortion of the output stage, differing in no fundamental way from any other overall feedback amplifier. The contents of A and B may be more complex than is usually the case, but the same limitations of feedback loop gain and stability margins apply as for a normal overall negative feedback loop. If it is assumed that the primary purpose of the feedback from the output is to reduce output stage distortion then in this respect there is therefore no necessary advantage in using the original multi-loop circuit. Distortion in other parts of the circuit can more readily be reduced by local feedback, so output stage distortion is generally the problem to be solved by feedback from the output.

### 2. Error Feedback.

The 'error feedback' method is regularly rediscovered, though the general principle has been known for many years. The first published version I am aware of was 'Distortion Cancellation in Audio Amplifiers' by W.Baggally, in The Wireless Engineer & Experimental Wireless, Aug. 1933, p.413-419, the article being submitted for publication in Oct.1932. (I have not personally read this, so if anyone can locate it I would be grateful for a copy.) The first time I became aware of this sort of technique was through reading 'Distortion Reducer' by D.Bollen in Wireless World, Feb 1973, p54-57. The variation shown below is a little different, using subtraction of output stage input and output signals rather than subtraction of the whole amplifier input and output, but the method of understanding the operation proposed by Mellish applies equally well to all the variations I have seen.
This version can be analysed to show that the output error can be reduced to zero. As with error feedforward exact component values or gains are needed to achieve zero distortion, and in practice distortion reduction by typically 20dB to 40dB is achieved.

If we again put part of the circuit in a red box and assume linearity of the enclosed circuit, then it again has an output Vx which is a linear function of the two inputs Vi and Vo, so we can again arrive at the same equivalent circuit as before with a single overall feedback loop. This at first appears to be impossible because it is known that overall negative feedback will at best only reduce distortion in proportion to the loop gain, not to zero. The explanation is shown by following the loop shown in blue. This is a positive feedback loop, and the gain round this loop is unity. A unity gain positive feedback loop in theory gives infinite gain, and so a negative feedback loop including this infinite gain element will reduce distortion by a factor of infinity, to zero. The box B in the equivalent circuit must include such a positive feedback loop and have infinite gain. (It is also necessary to have infinite gain for A otherwise there would be zero output from the amplifier, and so this is not a very practical equivalent. Putting a single positive feedback loop after the summation of A and B would be better, but the important point is that zero distortion is only achieved by the use of infinite gain in the feedback loop.)

To see that infinite gain really is needed for perfect error cancellation, consider what happens with zero input signal Vi. With perfect error nulling the output Vo will also be zero. Suppose the output stage without error correction picks up an error signal D from supply line breakthrough, or some other source independant of the input signal. For the output to be zero the input to the output stage, Vx, must be the non-zero voltage needed to cancel the effect of D. The big question is, if we consider the circuit in the red box, with two inputs both zero how does it produce an output? The output for an ideal circuit will only be a function of the two input voltages, so where does Vx come from? Only with infinite gain can a non-zero output be produced by an input theoretically zero.

Using positive feedback to give infinite loop gain requires exactly unity loop gain, which is impossible in practice, and close to unity gain small component value drift can give wide gain variations. It is possible to use positive feedback to increase gain in a more predictable way, and an example of how to do this is my MJR-7 mosfet amplifier, which surprisingly can be interpreted as an error feedback circuit. Starting with the basic circuit above, shown again here, with distortion added by the output stage represented by D:

Instead of the differential amplifier used to extract the error D we could use two amplifiers, one inverting and the other non-inverting, and add their outputs to give the differential function. The addition could be done with two resistors or two current sources, or as shown next with a resistor and a current source.

It can be seen that the input and output of the output stage are again compared, the positive and negative paths being shown by arrows, and with suitable choice of R and gm they can be subtracted and the difference extracted. Input Vi is applied through the inverting amplifier, but this will not affect the error cancellation effect. Working out Vo as a function of Vi and output stage distortion D, the non-inverting feedback amplifier is taken to have gain (1 - e) rather than 1 to demonstrate what happens as the gain approaches 1. The result is:

Vo = -( gmRVi + eD)/(gmR + e)
Let gmR = K, and assume e is very small compared to K, then:
Vo = -Vi + eD/K

From this we can see that if e is reduced to zero, or if K is increased to infinity, then Vo = -Vi, and so output distortion is zero. Increasing K is the conventional way of increasing negative feedback loop gain, and infinite loop gain is needed to reduce distortion to zero. The error feedback method however merely requires an accurate unity gain stage to reduce e to zero and achieve zero distortion, which at least appears to be possible in principle.

Compared to the original error feedback circuit something different is the fact that the differential function is not accurate. The non-inverting part still needs to be accurately unity gain, but there is no requirement for any particular gain (K) for the inverting path. This shows that it is the unity gain positive feedback loop which is essential for zero distortion, not the exact subtraction of two voltages. This is equally true of the original circuit with the differential feedback amplifier, and putting more or less gain in the inverting feedback path does not prevent error nulling. (See footnote.)

The next diagrams show why this particular arrangement is interesting. If we use a common-emitter stage Tr1 for the input amplifier and inverting feedback path, and emitter-follower Tr2 as the unity gain buffer stage, then we arrive at the first of the next diagrams:

The resistor R has virtually zero signal across it, so if we connect it direct to the emitter of Tr2 without the unity gain stage, as in the second diagram, it has little loading effect on Tr2, so a buffer stage is not needed. The two differential feedback paths are shown with arrows, and they sum at the point P. Tr1 alone is not a very high impedance current source, but can be improved considerably by using a cascode stage. With this addition to the second diagram, and using a current source as emitter load for Tr2, and a series CR for high frequency compensation, we arrive at the MJR-7 circuit, and have shown an equivalent circuit and analysis which demonstrate that it is actually an 'error-feedback amplifier', provided we ignore the requirement for unity gain in the inverting feedback path, which appears to be unnecessary. (Again, see footnote.)
In contrast, the MJR-6 circuit has no positive feedback path, and is just a conventional negative feedback amplifier.

This is another demonstration that error feedback is nothing more than negative feedback with high loop gain achieved by the use of positive feedback. The positive feedback in this case is what is commonly known as 'bootstrapping', a well known method of increasing gain, which effectively increases the resistance of resistor R. To see how well this works suppose there was no positive feedback, and R was instead connected to signal earth. In the MJR-7 circuit the loop gain would then only be about 10dB. With R bootstrapped the loop gain at low frequencies becomes 80dB. The distortion reduction is therefore 70dB, far more than usually claimed for error feedback circuits, and achieved with a relatively simple circuit requiring no accurate component values or adjustment, and with predictable stability. The resistor R could be replaced by a high impedance current source instead of using bootstrapping, and similar results could be achieved. The positive feedback is therefore largely irrelevant, it is the negative feedback together with high loop gain, however achieved, which reduce the distortion so effectively.

If we adjusted the positive feedback in some way to increase the loop gain this would increase gain only at low frequencies, the CR compensation would still be needed to limit gain at high frequencies to keep the overall feedback stable. The next diagram shows what happens as the positive feedback loop gain approaches unity. There is virtually no possibility of the gain being exactly unity, but it could easily be made to exceed unity, and then instead of a high positive resistance R would become a high negative resistance, and the overall feedback loop of the amplifier would have high positive phase feedback. (Not the same thing as positive feedback, which in addition requires the loop gain to be close to unity so that the feedback increases the amplifier gain, - see part 3 of the 'feedback works' series. High positive phase feedback is still negative feedback, it reduces gain and distortion, but also inverts the distortion.) As the positive feedback loop increases beyond unity the low frequency gain falls again, so it is only close to unity where high gain and low distortion are achieved.

Our hearing has maximum sensitivity around 3kHz, so if we have achieved sufficient overall loop gain at this frequency to reduce distortion to inaudible levels there is probably nothing to be gained by increasing lower frequency loop gain much further. The final diagram shows a simple way to add adjustable positive feedback to the MJR-7 circuit (shown again in simplified form) for anyone who wants to experiment with this. The addition could make it possible to null out low frequency distortion, but measuring distortion already under 0.0002% at 1kHz to adjust for a minimum is not easy. The MJR-7 already has a small negative output impedance at low frequencies, so I think it is probably not a good idea to add positive feedback. There are other ways to reduce the distortion if required, some of these were mentioned near the end of the MJR-7 article.

FOOTNOTE 1:

The unity gain differential amplifier in the conventional error feedback circuit has unity gain for both inverting and non-inverting feedback paths, and as shown this is not essential for error cancellation, it is only the positive feedback loop which needs unity gain. With gain (1-e) non-inverting, and gain G inverting we get output:
Vo = Vi/G + eD/G,
so then distortion nulling requires only e=0. Using G=1 does not therefore contribute directly to minimising distortion. Without G=1 however there is a signal component in the differential amplifier output in addition to the distortion extracted, and I have seen it suggested that this is relevant to stability. I am not sure about this, but find it unconvincing. Any phase shift in the output stage is itself an error, and will contribute to D, so this will be fed back and potentially cause stability problems whatever the value of G. Adding a balancing phase shift to the non-inverting feedback path could cancel this effect, but then this reduces the positive feedback loop gain, and error nulling is no longer accurate. Perhaps the only real point of using G=1 is that we are then only feeding back the error, and so this justifies the name 'error feedback' and makes it appear to be something different to conventional and unfashionable negative feedback.

There was a fascinating discussion on DiyAudio about error feedback, which had reached 109 pages the last time I looked, and so goes deeper into the subject than I could in this one page. I have not had the time to go through it all, but the design which started that discussion used no less than 29 transistors, which personally I would regard as unnecessarily complex for an amplifier with a single pair of power mosfets in the output stage. My own MJR-7 amplifier has only slightly higher distortion level using only 7 transistors and conventional negative feedback. To be fair, I did cheat a little by making my design an inverting amplifier and using a capacitor coupled output, which avoids some of the problems of a non-inverting direct-coupled design. I did however use the lower gm lateral mosfets and lower quiescent current, which cause higher distortion.

My MJR-7 uses about 60dB overall feedback loop gain at 20kHz to reduce distortion to around 0.002% (latest measurement of Mk5 version) using lateral mosfets at Iq = 100mA, and this is certainly not the upper limit for conventional overall feedback. Using parallel devices or higher quiescent current, or just substituting higher gm vertical mosfets, could take the distortion down even further. Feedforward can be even more effective, and a recent example is the MJR9 for which initial tests reveal only audio frequency distortion components under -120dB (0.0001%)

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