# 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, 2021 at 17:18
• May I know from where that K term came in the Acl equation and also could you please explain how you derived the value of k with the help of a diagram Apr 10 at 13:18
• @ Hari - Since the signal to be amplifies does not arrive directly at the opamps input (as it is the case for the non-inverting case) we must use superposition to find the input signal at the inverting input. This voltage is composed of two parts: One part resulting from the input voltage (called "k" in the contribution) and the second part feedback) comes from the output voltage (called "beta").
– LvW
Apr 10 at 13:33