In an inverting op-amp averaging amplifier, if the feedback resistor Rf equals the value of each input resistor divided by the number of inputs (Rf = Rin/N), what is the resulting relationship between output and the individual input voltages?

Introduction:
Summing and averaging amplifiers are standard op-amp applications. By choosing resistor ratios appropriately, the same circuit can output a weighted sum, a true average, or a scaled/inverted version of either. This question checks if you can map the resistor relationship onto the mathematical function realized.

Given Data / Assumptions:

• Inverting summing topology with N inputs, each through Rin to the inverting node.
• Feedback resistor Rf set to Rin/N.
• Ideal op-amp, linear region (negative feedback), non-inverting input at ground/reference.

Concept / Approach:
For the inverting summer, Vout = −Rf * Σ (Vi / Rin). With Rf = Rin/N, this becomes Vout = −(1/N) * Σ Vi, which is the inverted average of the inputs. If a non-inverted average is desired, a second inverting stage (gain −1) can be added.

Step-by-Step Solution:
Start from KCL at virtual ground: Σ (Vi − 0)/Rin = (0 − Vout)/Rf.Rearrange: Vout = −Rf * Σ (Vi / Rin).Substitute Rf = Rin/N: Vout = −(Rin/N) * Σ (Vi / Rin) = −(1/N) * Σ Vi.Interpretation: output equals negative of the arithmetic mean of the inputs.

Verification / Alternative check:
Example with N = 4, inputs {1, 2, 3, 4} V gives Σ Vi = 10 V, average = 2.5 V. The circuit outputs −2.5 V, matching the formula. A unity-gain inverter afterward produces +2.5 V if a non-inverted average is required.

Why Other Options Are Wrong:

• Average (non-inverted) / sums: sign or scaling does not match given Rf.
• Inverted sum: would require Rf = Rin (not divided by N).
• RMS: Requires squaring, averaging, and square-root operations, not a simple linear op-amp network.

Common Pitfalls:

• Forgetting the minus sign associated with the inverting input.
• Using unequal Rin values unintentionally, which creates a weighted (not simple) average.

Final Answer:
the inverted average of the individual inputs