Averaging amplifier design rule — In a multi-input op-amp averaging amplifier, how should the input resistances compare to the feedback resistance to obtain a true arithmetic average at the output?

Introduction:
An averaging amplifier outputs the arithmetic mean of several input voltages. It is implemented as a special case of the inverting summing amplifier, with resistor choices that scale the sum by 1/N (N = number of inputs) so the result equals the average rather than the simple sum.

Given Data / Assumptions:

• N input channels, each through an equal resistor Rin to the inverting node.
• Feedback resistor Rf from output to the inverting node.
• Noninverting input grounded; op-amp operates linearly.

Concept / Approach:

The inverting summer gives Vout = −(Rf / Rin) * Σ Vi. To obtain the average, you require Vout = −(1/N) * Σ Vi, which is achieved by choosing Rf = Rin / N. Therefore, Rin must be N times larger than Rf, meaning each input resistance is greater than the feedback resistance whenever N > 1.

Step-by-Step Solution:

Verification / Alternative check:

Example: N = 4, choose Rin = 40 kΩ, then Rf = 10 kΩ. Evaluate with any four inputs; the output equals the negative of their average, confirming the design rule.

Why Other Options Are Wrong:

• equal to the feedback resistance: Produces a sum (not an average) when all Rin are equal.
• less than the feedback resistance: Would yield magnitude greater than an average.
• unequal / randomly chosen: Breaks symmetry and corrupts averaging unless precisely compensated.

Common Pitfalls:

• Forgetting the negative sign in the inverting configuration; a noninverting averaging stage needs a different topology.
• Mixing Rin values; mismatch skews weighting and degrades accuracy.

Final Answer:

greater than the feedback resistance