A is the "open loop" amplification or gain. (A is complex to represent both magnitude |A| and phase shift arg A.)V_{out} = A V_{in}.
Usually we want:
the ability to control the gain; independence of frequency (over some range or bandwidth); high input impedance (to avoid drawing too much current from the source of V_{in}); low output impedance (so that a load does not affect V_{out} too much).
V' = V_{in} + beta V_{out} and V_{out} = A V',where we have introduced the (complex) feedback factor, beta, the proportion of the output added to the input. Eliminating V', one gets
V_{out} = V_{in} A / (1 - beta A).Thus the "overall" gain of the circuit is A' = V_{out}/V_{in} = A / (1 - beta A).
If |beta A| >> 1, A' = - 1 / beta (approx.). It usually turns out that beta is much easier to manufacture reproducibly than A. But by makine |A| large, we can produce an overall gain which is independent of A. It is only stable if Re beta < 0 (negative feedback). Negative feedback increases the overall input impedance, decreases the overall output impedance and increases the overall bandwidth relative to their open loop values. In other words, negative feedback improves all the important characteristics of an amplifier.
V_{out} = A (V_{+} - V_{-}) very high open loop gain (typically, |A| ~ 10^{5);} good bandwidth (typically 1 MHz); very high input impedance (typically 10 MOhms); low output impedance (typically 100 Ohms - don't connect loads with lower impedance than this!).
V_{-} = V_{out}R_{1} / (R_{1} + R_{2}).From above we also have
V_{out} = A (V_{in} - V_{-})and eliminating V_{-}, we obtain the overall gain,
V_{out}/V_{in} = A' = A / (1 - beta A),where, in this case, beta = -R_{1} / (R_{1} + R_{2}).
The differential amplifier has allowed us to add - or in this case subtract - some portion of the output from the input. This circuit is an example of negative feedback.
If |A| is large,
A' = -1/beta (approx.) = (R_{1} + R_{2}) / R_{1}.Thus, it is easy to control the overall gain by choosing the resistors. Resistors are easy to make, stable and reproducible. Even if the open loop gain, A, has characteristics that are somewhat unknown or variable, as long as |A| is large, the overall gain is well defined. Note that this approximation is good even if A is complex, i.e., introduces a phase shift; the overall gain is real, i.e., no phase shift, since beta is real (in this case).
Although the overall gain depends only on the ratio (R_{1} + R_{2}) / R_{1}, it is important to choose R_{1}_{, }R_{2} large enough that they do not constitute a serious load on the output and not too large that they cannot supply enough current to the input. This means they must have values comfortably between the open loop input impedance, 10 MOhms, and the open loop output impedance, 100 Ohms. Let us say between 1 kOhm and 100 kOhms.
John Allison
13th Sep 2001