This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.
Negative feedback is often described with a simple rule:
Distortion generated inside the feedback loop is reduced by the loop gain.
This statement is true, but it is incomplete. It leads to a common confusion: if a nonlinear element is hidden inside the overall forward transfer function, shouldn't feedback reduce its effect? And if a nonlinear element at the summing junction is "inside the loop," why is it said that feedback cannot correct it?
The resolution comes from looking at disturbances as equivalent input errors.
A general feedback system
Consider a nonlinear amplifier with feedback:
- $x$ is the input signal,
- $y$ is the output,
- $\beta$ is the feedback factor,
- $S$ is the summing element with transfer function $S(x,-\beta y)$,
- $e=S(x,-\beta y)$ is the signal at the output of the summing element,
- $G$ is the forward amplifier with transfer function $G(e)$, so $y=G(e)$.
Note that for the usual ideal summer, the transfer function is simply $S(x,-\beta y)=x-\beta y$, but a real summing element may be nonlinear and/or add terms unrelated to either $x$ or $-\beta y$.
Now suppose the summer characteristic changes slightly:$$S\rightarrow S+\delta S.$$
Perhaps our summer is a differential amplifier (such as a long-tailed pair a.k.a. LTP), and its operating point changes slightly, or maybe we now have noise coming into the summer. How does the output change?
The perturbation of the summer creates an error $$\delta e=\delta S-\beta S_y\delta y,$$ where $$ S_y=\frac{\partial S}{\partial(-\beta y)}.$$ Let $g=G'(e)$ be the incremental (i.e. small signal) gain of the amplifier $G$. Then the output change is $$\delta y=g\delta e=g(\delta S-\beta S_y\delta y).$$
Solving for $\delta y$ gives us $$\boxed{\delta y=\frac{g\delta S}{1+\beta gS_y}}.$$The denominator contains the familiar loop-gain term, so at first glance, this appears to say that even summing-junction distortion is corrected by feedback. But this interpretation is incomplete.
The equivalent input error
Suppose instead that the input itself changes by a small amount $\delta x$. The corresponding output change is $$\delta y=\frac{gS_x}{1+\beta gS_y}\delta x,$$where $$S_x=\frac{\partial S}{\partial x}.$$Now define an equivalent input disturbance $\delta x_{\rm eq}$ that produces the same change in the summer output:$$S_x\delta x_{\rm eq}=\delta S.$$Therefore,$$\boxed{\delta x_{\rm eq}=\frac{\delta S}{S_x}},$$and the output disturbance becomes$$\delta y=\frac{gS_x}{1+\beta gS_y}\delta x_{\rm eq}.$$But this is exactly the same transfer function as for the desired input!
In other words, the feedback loop cannot distinguish between:
- a real change in the input signal, and
- a nonlinear error created in the summing element.
Both appear at the same location and are processed identically.
Why downstream distortion is different
Now consider distortion generated after the summing junction, for example in the output stage of an amplifier:$$y=G(e)+d.$$The disturbance $d$ is not present at the input of the forward amplifier. Linearizing,$$
\delta y=
\frac{d}{1+L},
$$where $L=\beta g {S_y}$ is the loop gain.
This looks like feedback has suppressed the distortion. However, to compare it with the input signal, we must refer it back to the summing point. The equivalent input disturbance is$$\delta x_{\rm eq}=\frac{d}{g {S_x}}.$$The large forward gain between the summing junction and the distortion source has already reduced the disturbance when expressed as an input error.
Feedback is not suppressing the equivalent input disturbance. It is simply reproducing it through the same closed-loop transfer function as the desired signal.
The general rule
Any disturbance can be converted into an equivalent input disturbance at the summing junction. The size of that equivalent input disturbance depends on where the original disturbance occurs.
For a disturbance generated after a forward gain $P$,
$$
\delta x_{\rm eq}
\approx
\frac{d}{P}.
$$
For a disturbance generated at the summing junction, $P=1$, so $$\delta x_{\rm eq}=d.$$
This is why the summing junction is special. Not because feedback somehow stops working there, but because there is no forward gain before the error is created.
A better way to state the feedback principle
Instead of saying:
Feedback reduces distortion generated inside the loop.a more precise statement is:
Feedback reduces the effect of disturbances according to their equivalent value when referred to the summer's input.
It follows that
Any disturbance already present at the summer's input is indistinguishable from the input signal and receives no correction.
This viewpoint resolves several apparent contradictions in feedback theory:
- amplifier gain errors are reduced by feedback;
- distortion generated in later stages is reduced according to the gain preceding the distortion source;
- summing-junction nonlinearity is not reduced relative to the input signal;
- feedback never knows whether a signal at its input is "wanted" or "unwanted."
The loop only acts on the error signal it sees. Everything else is bookkeeping.
