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Analysis of Negative Feedback

The above feedback connection actually determines a new amplifier, namely the one with input $\overline{v}_{in}$, and output v_out, and it will have a new gain, A_K, which depends in the scaling parameter K in the feedback path:

$$v_{out} = A_K \overline{v}_{in} .$$

This is illustrated explicitly in Figure 5.

  

Figure 5: Negative feedback amplifier.

Lets find the gain A_K. From the equations (2) we have

$$v_{out} = A\overline{v}_{in} - AKv_{out}$$

and solving for v_out we get

$$v_{out}= \frac{A}{1+AK} \overline{v}_{in}$$

This means that the new gain is

$$A_K = \frac{A}{1+AK}$$ (3)

This formula was discovered about 100 years ago and is very important. Note how it depends on the negative feedback parameter K.

An interesting and important feature of this is that when the original amplifier gain A is very large, i.e. $A \to \infty$, then A_K is approximately independent of A. To see this, rewrite A_K as

$$A_K = \frac{1}{1/A + K} \approx \frac{1}{K}$$ (4)

when A is large, since then $1/A \approx 0$. So for large A, we have $A_K \approx 1/K$. If A=10,000 and K=0.1, then $A_K = 9.9995 \approx 10 = 1/K$.

This is important for opamps and transistor amplifiers, since due to variations in component values, etc, the gain cannot be designed exactly, and in fact can vary substantially. Negative feedback offers a solution, and by using it designers can obtain closely the gains they want, having reduced sensitivity to device parameter variations.

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