The gain of an opamp circuit is basically $$A_v=-\frac{R_f}{R_i}$$
But how come the gain of the opamp itself does not appear in the equation? How is the opamp amplifying anything in the circuit? What is the opamp doing in the circuit if it does not appear in the formula?
Best Answer
From Texas Instruments "Stability Analysis of Voltage-Feedback Op Amps Including Compensation Techniques"
http://www.ti.com/lit/an/sloa020a/sloa020a.pdf
A is the open loop gain (the gain of the opamp itself), and \$ \beta\$ is the feedback resistors.
If A\$\beta\$ is very large relative to 1, then the closed loop gain approximates as \$\frac{A}{A\beta}\$ which simplifies down to
$$ \frac{V_{out}}{V_{in}} =\frac{1}{\beta} $$
That means that with very large open loop gain, you can use negative feedback to produce a closed loop gain that is virtually independent of the exact open loop gain of the op-amp (which is difficult to control). It only depends on the negative feedback (the resistors) which makes the circuit easier to design, more predictable, and less tied to the specific op-amp.
The white paper I linked goes into it in more detail. It goes into more detail about how to actually do this calculation with resistors instead of just \$\beta\$