For an ideal inverting op-amp with negative feedback, the magnitude of the closed-loop voltage gain A_cl is equal to which ratio of resistances?

Introduction:
The inverting amplifier is a foundational op-amp building block. Its closed-loop gain is set precisely by passive components, making the result largely independent of the very large but uncertain open-loop gain. Knowing the exact relationship between the feedback resistor and the input resistor is essential for predictable amplification in sensor interfaces, audio stages, and data acquisition.

Given Data / Assumptions:

• Ideal op-amp operation with negative feedback (linear region).
• Single input applied through input resistor Rin to the inverting input; feedback resistor Rf from output to inverting input.
• Non-inverting input tied to ground (or a reference) for the classic inverting stage.

Concept / Approach:
Virtual ground holds the inverting input near 0 V. The input current through Rin must equal the feedback current through Rf. This yields a direct algebraic relation between Vout and Vin. The sign indicates inversion; the magnitude depends only on the resistor ratio.

Step-by-Step Solution:
Apply KCL at the inverting node: (Vin − 0)/Rin + (Vout − 0)/Rf = 0Rearrange: Vout/Rf = −Vin/RinSolve for gain: A_cl = Vout/Vin = −Rf/RinMagnitude: |A_cl| = Rf/RinThus, the magnitude equals the feedback resistance divided by the input resistance.

Verification / Alternative check:
If Rin = 10 kΩ and Rf = 100 kΩ, then A_cl = −100 kΩ / 10 kΩ = −10, so |A_cl| = 10. SPICE simulations and lab measurements confirm that as long as A_ol is very large and the op-amp remains unsaturated, the closed-loop gain tracks Rf/Rin closely across frequency where loop gain is high.

Why Other Options Are Wrong:

• The ratio of input to feedback resistance: This gives 1/|A_cl|, not the magnitude.
• The open-loop voltage gain A_ol: Enormous and device-dependent; closed-loop gain is set by feedback, not by A_ol directly.
• The input resistance: A resistance value is not a dimensionless gain.
• The reciprocal of β: β depends on the feedback network topology; for an inverting stage, |A_cl| = Rf/Rin, not simply 1/β.

Common Pitfalls:

• Forgetting the negative sign (inversion) when quoting A_cl.
• Ignoring op-amp bandwidth and slew-rate limits that can reduce effective closed-loop gain at high frequency or large signals.
• Using tolerances poorly; mismatch in Rin and Rf changes the exact gain and noise performance.

Final Answer:
the feedback resistance divided by the input resistance