## Feedback Effects

It is well known that if we start with an amplifier with an open-loop square-law response it will produce only second harmonic distortion, but with overall negative feedback added higher order harmonics will then be produced. This is perfectly true, but almost entirely misleading for several reasons. For a start there are no perfect square-law amplifiers. The closest we can get is probably the field-effect transistor, (fet), but measurements confirm that these already add low levels of high order harmonics even with no feedback. The following graph shows approximately what happens to the harmonics of a typical fet as feedback is applied. The distortion levels at zero and low feedback may differ widely for different fets, but as feedback is increased the results will become more similar to the example shown:

Note: the results can be very different depending on how the feedback is increased. The above version, similar to that presented by Baxandall and others, kept a constant signal level at the input of the square-law device, and changed feedback level by changing the gain through the feedback network. If instead we use a square-law input device, and increase the gain of a following stage, while keeping a fixed overall feedback network, then with high loop gain the signal at the input of the nonlinear input device is reduced, and this can be the dominant effect reducing distortion, and the high order harmonics fall faster. Instead of the parallel lines for falling harmonics seen above they will diverge, the second harmonic will fall faster than shown, now following the red line, and the higher harmonics fall even faster. If instead the square-law device is used as the output stage it no longer benefits from a falling signal level, and assuming a constant output level the above plots then apply however the loop gain is increased.

I have added a page: Square-Law Transfer Function With Feedback which looks at this from a different direction, plotting the shape of the DC and low frequency transfer function to determine the peak error with different loop gains. In one example a loop gain of 500 reduced the original open-loop distortion by a factor of 500,000 primarily because of the reduced input stage signal level.The signal level for the above plots is chosen to give 10% second harmonic at zero feedback. The harmonics are shown as H2 (second) to H7 (seventh). With zero feedback harmonics up to the 5th are above the -120dB level. Applying a low level of feedback the 6th and 7th harmonics increase a little above -120dB. Increasing feedback further the higher harmonics one by one fall down below the -120dB level until at a little over 80dB feedback we are left with only the fundamental plus second harmonic.

This shows the misleading aspect of the initial assertion that feedback adds high order harmonics. In any single fet amplifier stage the third and higher order harmonics are already there without feedback, become worse with low levels of feedback, but all disappear below -120dB at high levels of feedback. It is only at sufficiently high feedback levels that the 'audible' level distortion is purely second harmonic, and in the present example at 80dB feedback that second harmonic level is at -100dB (0.001%) rather than the -20dB (10%) of the zero feedback amplifier.

This however is not the end of the story. We have only looked at what happens to a single sine-wave input. A music signal includes many frequencies, and even a single note from one musical instrument such as the flute will have a fundamental plus harmonics. If we use a very simple approximation of a musical instrument with just a fundamental plus second harmonic at 20% and third at 10%, and we apply this to a square law amplifier at a level such that the second harmonic distortion of the amplifier is at 5%, then we will find intermodulation products being produced. The second and third harmonic of the input signal will combine to generate a 5th harmonic component. Working out all the added harmonics there will be 4th at 1.2% (-38dB), 5th at 0.2% (-54dB) and 6th at 0.05% (-66dB). These levels are shown in red as h4 to h6 in the graph. Our zero feedback amplifier is now adding even worse higher order harmonics, and would still do so even if a perfect square-law device was used.

There are other problems with a square-law amplifier, such as the addition of a d.c. component at the output varying with signal amplitude. For a typical square-law amplifier with output voltage:

Vo = 100 Vin + 100 Vin^{2}

then using input Vin = A sin(wt) and substituting in the equation for Vo we find:

Vo = 100A sin(wt) - 50A^{2}cos(2wt) + 50A^{2}.

The first term is no problem, it is the undistorted amplified sine wave. The second term is the second harmonic, which many believe is subjectively unobjectionable. The third term however is a d.c. component proportional to signal amplitude squared. For a constant signal amplitude this would be a constant voltage which could be removed by an output coupling capacitor or transformer, but with a music signal where the amplitude is constantly changing this is an a.c. distortion signal. For the example above an input of 0.2V peak will produce a 2V peak error voltage at the output, the same level as the 10% second harmonic.The added higher harmonics when using a square-law amplifier is sometimes used as an argument against feedback 'because feedback adds high order harmonics'. This is however an unjustified generalisation, and these added harmonics are not a general property of feedback, it is more accurate to say that it is a property of square-law amplifiers, and some other types. The point is that there are other open-loop transfer functions for which there is no increase in high order harmonics at any level of feedback, and the relative levels of all harmonics remains unchanged as feedback is increased, while all harmonics reduce in proportion to the feedback loop gain.

An example is a class-B design published by Peter Blomley in Wireless World, March 1971. In this design the signal is split into two parts at the zero-crossing point, and the positive half is applied to one highly linear output sub-amplifier while the negative half is applied to another linear output sub-amplifier. The two outputs are then added. Unless the two output sub-amplifiers are exactly matched in gain there will be a change in the slope of the transfer function at the zero-crossing point. If we extract the distortion resulting from this it will look like a rectified sinewave, and this has an infinite series of even order harmonics. The 4th is at 1/5 the level of the 2nd, the 6th is at 1/7 the level of the 4th, and so on. Adding any level of feedback will reduce the difference in level of the two halves of the signal, but the transfer function will still be two linear parts with a slope change at zero crossing, and the distortion will still be the same rectified sinewave at a lower level. All harmonics fall in proportion to the feedback level, and in this case at least there is no increase in high order harmonics at any feedback level.If some amplifiers with feedback have added high order harmonics and some do not, then is there some simple way to determine which do or don't? One approach is to look at two different ways of specifying distortion. The usual method is to apply a sinewave input and observe the distortion at the output. An alternative is to find out what distortion must be applied at the input to achieve an undistorted sinewave output. In the case of the square-law amplifier the output distortion with an undistorted sinewave input is just the second harmonic, but to get an undistorted sinewave output the input signal must have an infinite series of harmonics, and this infinite series has the same relative levels we will approach at the output when used as a feedback amplifier, as the loop gain is increased towards very high values. For the amplifier with a slope change at zero-crossing the input distortion for an undistorted output has exactly the same relative levels of harmonics as the output distortion for an undistorted input, and high feedback leaves the relative levels unchanged. Looking at this 'input distortion' is a useful clue about what will happen as feedback is increased to high levels. We can conclude that the effect of feedback will be different for different amplifiers, with the square-law at one extreme, and the zero-crossing slope change at the other extreme.

That is not actually the other extreme, we could go further for example using the inverse of the square-law response. I derived and illustrated that in my page: Inverse Transfer Functions. If that was the open-loop response then increasing overall feedback will change the distortion spectrum starting from a theoretically infinite series and approaching the spectrum of the inverse of the inverse, which gets us back to the square-law spectrum with just the second harmonic. It should work in theory, but generating the necessary square-root function with sufficient accuracy may not be easy compared to just linearising the open-loop response with a little local feedback.

## Phase Shift Effects

The generally accepted definition is that feedback is negative if when it is applied to an amplifier the gain is reduced, and positive if it results in an increased gain. It is sometimes suggested that feedback is only fully effective in reducing distortion if it is accurately negative phase, but at high feedback levels the phase of the feedback is almost completely irrelevant, and the analysis needed to demonstrate this is included here. The following is an inverting amplifier, approaching -1 closed-loop gain as the open-loop gain is increased.

The input stage has gain -A, where A is a positive real number, e.g. 100. All the phase shift is included as a single element P, where P can conveniently be a complex number with magnitude 1, and in practice will be a function of frequency. The output stage is shown with a gain of 2 to compensate for a gain of 1/2 from the feedback network, so that -A is the gain round the feedback loop. Distortion is represented by a single input D, added to the output stage. Output stage distortion is usually the most important component for the overall feedback to reduce, and although just adding D at this point is not an entirely accurate representation it is close enough for the present purpose.

There are two equations we can derive, for Vo as a function of Vx, and for Vx as a function of Vi and Vo, assuming the input stage to have a very high input impedance

:

Vo = 2(D - APVx)

and Vx = Vi/2 + Vo/2

If we combine these equations to eliminate Vx this gives an equation for Vo:

Vo = 2D / (1 + AP) - APVi / (1 + AP)

If A is large, e.g. 100 or more, then (1 + AP) differs little in magnitude from AP,

so to a good approximation Vo = (2D / AP) - Vi

If D was zero we would have Vo = -Vi as expected, but of more interest is the distortion term 2D / AP. Without the overall feedback the output distortion would be 2D, so the magnitude of the distortion is reduced by a factor A, i.e. the gain round the feedback loop, and the 1/P factor means that its phase is shifted by the same amount but in the opposite direction to the open-loop amplifier phase shift.The approximation used when assuming that A was large does hide a small effect, and if we do the exact calculation for A=100 then for exactly negative phase feedback distortion is reduced by a factor of 101. For positive phase feedback the factor reduces to 99, so the effect of feedback loop phase is small, and becomes even smaller as loop gain is increased further. In the case of exactly positive phase feedback of course the phase would need to be reduced before high frequency loop gain fell to unity to achieve stability.

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.

Phase shifts in amplifiers are in practice associated with high frequency gain reduction, and then these gain changes do have an effect on distortion reduction. If D in the above analysis was a single frequency then the factor A by which it is reduced is the loop gain at that frequency, so with a more realistic distortion signal with many components the higher frequency components can be expected to be reduced less than those at lower frequencies. Also, distortion components are not all reduced by the same factor as output stage distortion. Increasing A will also reduce Vx for a given Vo, which reduces some input stage distortion in addition to the feedback effect, so some distortion components can be reduced by far more than the factor A. In a non-inverting amplifier common-mode input distortion may not be reduced at all by increasing A.

## Time Delay Effects

The phase shift P is generally a function of frequency. A time delay is sometimes represented by a phase lag proportional to frequency, and from this we could easily conclude that a time delay also has little effect on distortion reduction. There is, however, an argument that any error signal sent back to the input can have no immediate effect on the output because it is delayed in passing through the amplifier, so its effect arrives at the output too late to correct any error. Try adding a ten second delay in the middle of your amplifier circuit to create an obvious effect, but for a normal amplifier it is important to consider the limited bandwidth of the signal, and the maximum time delay involved before concluding that there is a serious problem.

Suppose we start with an ideal step signal input shown below as A:

For the duration of the amplifier delay this signal alone detemines the input, there is no feedback to have any effect. What happens during that delay time is therefore likely to lead to a large error. What saves us from this large error is the limited bandwidth of audio signals. A perfect step would have an infinite bandwidth, and so will not occur in the input signal. The second diagram, B, has a gradual change of amplitude, but the instantaneous change in slew rate will also lead to an infinite bandwidth, and again cannot occur. Possible audio transients must start as in diagram C, with a gradual change. In a real amplifier, with noise added to the signal, the starting point of the transient is uncertain, and if the time delay in the amplifier is small enough any error will also be small, and could be buried under the noise.

The time delay is difficult to estimate or measure, being difficult to distinguish from frequency dependent phase shift. The time for a light velocity signal to travel the distance from one end of the circuit board to the other suggests a possible value for the delay of typically half a nanosec., which should have a negligible effect. Time delays, as opposed to phase shifts, are added by distributed reactance, as in a transmission line, and normal amplifier components will hopefully add very little further time delay.

Using normal music signals with my mosfet power amplifier designs the null test method would pick up any distortion caused by time delay errors in the feedback loop. The high feedback loop gain would be expected to cause serious problems if the time delay was significant, so the fact that distortion using a music signal was around the noise level suggests that any time delay effects are undetectable, while the open-loop phase shift, which is about 50 deg. at 5kHz, should make little difference. A distortion increase at higher frequencies is expected from a drop in loop gain plus increasing effect from non-linear capacitances rather than because of phase shifts or time delays.

For a band-limited audio signal the feedback can actually compensate for a small time delay, which seems counter-intuitive, that should involve a time advance, which we would expect to be impossible. I have added a page about Group Delay, Time Delay, and Phase Shift which explains how a simple phase advance circuit can cancel a small time delay over a limited frequency range. A negative feedback loop can create a phase advance sufficient to achieve this. Some interesting references about negative group delay are listed.

## References

1).Peter Baxandall, 'Audio power amplifier design-5', Wireless World Dec.1978 p53-56.

This shows the levels of the 2nd to 6th harmonics for a fet with 0 to 40dB feedback.

2).Peter Baxandall, 'Audio power amplifier design-6', Wireless World Feb.1979 p69-73.

This does the same for a bipolar junction transistor. The general conclusion is similar, but there are dips and peaks in the plots at low levels of feedback, and again at sufficiently high feedback all harmonics continue to fall. There is also a section about the method of specifying distortion as the input required for an undistorted output.

3).In 1961 M.G.Scroggie published what at first sight appears to be an analysis of almost exactly the same circuit arrangement used by Peter Baxandall in Ref.1. This was reprinted and updated in Wireless World, Oct 1978, p.47-50, the author writing under the name 'Cathode Ray'.

A square law device has overall feedback applied, but with 40dB feedback there is now over 7% third harmonic, 3% 4th, and so on. I have seen this result used as a justification for using only low levels of feedback, so the question of why this version was so much worse deserves some consideration. Reading the entire article the explanation becomes clear. The open-loop distortion is higher, at 20%, and one result of this is that the gain varies considerably with signal amplitude, and the feedback loop gain is 40dB only at zero input signal. For the range of signal voltage at which the distortion figures are quoted the loop gain actually falls to zero over part of the sine-wave, and what we are really looking at is the effect of asymmetric clipping. Of course it is impossible to reduce the distortion of an amplifier once it has been driven up to its output clipping level, whatever level of feedback we use, and it is well known that high feedback makes the clipping more abrupt, so that more unpleasant high harmonics are produced. The easiest cure for clipping is just to turn down the volume control, or if this is unacceptable use a more powerful amplifier, more efficient speakers, or use some form of 'soft clipping' ahead of the amplifier. The summing up at the end of the article made 7 points, 3 of these being about clipping.