The Inverting amplifier

To determination of gain formula assume v_o=A(v₂-v₁). Where the positive input is v₂ and the negative input is v₁.

Since v₂=0, v₁ = -v_o/A

The current through R₁ is i₁=(V_i-V₁)/R₁.

Since no current goes into the input the current in R_f is the same and is i_f= (V₁-V_o)/R_f. Thus

(V_i-V₁)/R₁=(V₁-V_o)/R_f

this gives

v_o/v_i = -R_f/R₁/(1+(1+R_f/R₁)/A)

This is true for any gain and gain that changes with frequency. If the gain A is very large, we get the standard formula.

v_o/v_i = -R_f/R₁