## Feedback Works. Part 3.

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.

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

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 minimum 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.

An interesting point arises from the above analysis of a feedback amplifier with phase shift P. The closed-loop distortion appearing at the output was shown to be shifted forward in phase, and this is fed back, then inverted, amplified and shifted back in phase by the amplifier phase shift P, and so arrives at the output stage with the amplitude and phase needed to give good cancellation of the distortion D, leaving just a small remainder sufficient to account for the closed-loop output distortion we started from. Representing our time delay by a phase lag proportional to frequency, we could conclude that the distortion will be given a shift forward in phase proportional to frequency, equivalent to a shift forward in time when it reaches the output. A time advance would enable us to predict the future, and is almost certainly impossible. This suggests that although there is some similarity between a time delay and a phase shift they are fundamentally different and not always interchangeable.

For 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 at the sort of level expected from sine wave tests suggests that any real time delay is small enough, 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.

HOME.