# Feedback loop-gain(beta) derivation for the inverting amplifier?

During my study on the negetive feedback concept, i observed that the beta derivation for the non-inverting amplifier is so evident which is just a potential divider voltage.But i am unable to find the same for the inverting amplifier.

Could somebody get the beta(feedback gain) with the derivation exempting the approximations for the below shown simple inverting amplifier. Definition: The feedback factor beta is defined as the portion of the output voltage that is fed back to the differential opamp input (input source grounded). Therefore, regarding beta there is no difference between the non-inverting and inverting configuration. In both cases, we have

$$\beta = \frac{R_{in}}{R_{in} + R_f}$$

Using the general formula for the closed-loop voltage gain $A_{cl}$ of an amplifier with feedback

$$A_{cl} = k \cdot \frac{A_{ol}}{1 + A_{ol} \cdot \beta}$$

with open-loop gain $A_{ol}$ and the input damping

$$k = -\frac{R_f}{R_{in} + R_f}$$

we get (for $A_{ol} \to\infty$)

$$Acl = -\frac{k}{\beta} = -\frac{R_f}{R_{in}}$$

EDIT: Comment/explanation: The "input damping" factor $k$ is defined as "portion of the input voltage arriving at the diff. opamp input" (negative because $k \cdot V_{in}$ arrives at the inv. terminal).

• That's a super simple equation, yet not mentioned in any books. It makes opamp amplifier so much easier to understand. Apr 21 at 17:18