Op-amp gain relationships Which equation correctly expresses the closed-loop voltage gain for a standard inverting operational amplifier configuration?

Introduction / Context:
Operational amplifiers are the workhorse of analog design. The most common closed-loop configuration is the inverting amplifier, whose gain is set by two resistors. Recognizing the correct gain relationship is essential for sizing components and predicting circuit behavior.

Given Data / Assumptions:

• Ideal op-amp assumptions: infinite open-loop gain, infinite input impedance at the inverting node via feedback, and zero output impedance.
• Standard inverting topology with Rin from source to inverting input and Rf in feedback from output to inverting input; noninverting input at ground.

Concept / Approach:
Under ideal assumptions, the inverting input is at virtual ground. The input current is approximately Vin / Rin and flows through Rf, generating output voltage Vout = −(Rf/Rin) * Vin. The negative sign indicates 180° phase inversion.

Step-by-Step Solution:
Assume virtual ground at inverting node.Compute current: I = Vin / Rin.Voltage across Rf = I * Rf; output is negative of that (due to inversion).Therefore, Vout/Vin = −Rf/Rin.

Verification / Alternative check:
Dimensional check and limiting cases (Rf = Rin → gain −1) agree with standard results.

Why Other Options Are Wrong:

• Vout = Vin/Av: Relates to open-loop gain, not closed-loop.
• Ratios with Rout: Output resistance does not set closed-loop gain here.

Common Pitfalls:

• Forgetting the sign inversion.
• Mixing open-loop and closed-loop gain formulas.

Final Answer:
Vout/Vin = –Rf/Rin