## Friday, November 13, 2020

### Audio Amplifier Feedback - Basics

This post is a part of the series on audio amplifier feedback. The contents of the series can be found here.

To make sure everyone is on the same page, here is a super simplified feedback loop that can be found in an audio amplifier:

In the diagram:
• $A$ is the gain of the amplifier with no feedback applied (its open loop gain)
• $B$ is the gain of the feedback network (typically, the feedback network attenuates the signal, so $|B|<1$)
• $x$ is the input signal
• $\epsilon$ is the error (noise, distortion) that the amplifier adds to the signal
• $y$ is the resulting output signal
That is, the amplifier receives the input signal $x$, amplifies it by $A$, adds some noise and distortion $\epsilon$, resulting in the output signal $y$. A portion $B$ of the output signal $y$ is fed back to the input by adding it to the input signal (hence 'feedback').

Working from the right side of the diagram to the left, we can write:
$$y=\epsilon+A(x+y*B)$$
Solving for $y$:
$$y={A \over (1-A*B)}*x+{1 \over (1-A*B)}*\epsilon$$
The output signal $y$ has two components:
• Input signal $x$ amplified by ${A \over (1-A*B)}$
• Distortion $\epsilon$ amplified by ${1 \over (1-A*B)}$
Let us call the input signal amplification factor ${A \over (1-A*B)}$ the Signal Transfer Function, or $STF$, and distortion amplification factor ${1 \over (1-A*B)}$, the Error Transfer Function, or $ETF$:
$$STF={A \over (1-A*B)}$$ $$ETF={1 \over (1-A*B)}$$
The promise of feedback is that, as the open loop gain $A$ increases, the $ETF$ approaches zero, while the $STF$ approaches $-{1 \over B}$. In other words, the contribution of noise and distortion in the output signal becomes small, and the closed loop gain of the amplifier becomes independent of the amplifier's open loop gain $A$.

The sum of the input signal $x$ and the feedback $y*B$ that the amplifier sees at its input is normally small and gets smaller as $A$ increases: $$x+y*B={1 \over (1-A*B)}*x+{B \over (1-A*B)}*\epsilon$$
The quantity $A*B$ is called loop gain, and is the correct term for the "amount of feedback". We can rewrite $STF$ and $ETF$ with the loop gain $LG$:
$$STF={LG \over (1-LG)}*{1 \over B}$$ $$ETF={1 \over (1-LG)}$$
In the next post, I will show how the above math applies to an opamp.