Summing amplifier

A summing amplifier (weighted summer) is the inverting configuration with several input resistors feeding the same inverting node. Its output is a weighted sum of all the inputs, inverted, with each weight set independently by its own resistor:

$v_O=-\frac{R_f}{R_1}{v}_{I1}-\frac{R_f}{R_2}{v}_{I2}-\cdots$

Derivation

Bring input $v_{I1}$ in through $R_1$, input $v_{I2}$ in through $R_2$, and so on; all of them meet at the inverting node, with a single feedback resistor $R_f$ running to the output $v_O$. The non-inverting input is grounded.

Apply the Ideal op-amp model golden rules. Golden rule 2 makes the inverting node a virtual ground: $v_{-}=0\text{V}$. Golden rule 1 says no current enters the op-amp input. So write KCL at the inverting node — the sum of every input current must equal the current leaving through $R_f$:

$\underset{\text{branch 1}}{\underbrace{\frac{v_{I1}-0}{R_1}}}+\underset{\text{branch 2}}{\underbrace{\frac{v_{I2}-0}{R_2}}}+\cdots =\underset{\text{through }{R}_f}{\underbrace{\frac{0-v_O}{R_f}}}$

That is

$\frac{v_{I1}}{R_1}+\frac{v_{I2}}{R_2}+\cdots =-\frac{v_O}{R_f}$

Solve for $v_O$:

$\boxed{v_O=-R_f\left(\frac{v_{I1}}{R_1}+\frac{v_{I2}}{R_2}+\cdots \right)=-\frac{R_f}{R_1}{v}_{I1}-\frac{R_f}{R_2}{v}_{I2}-\cdots }$

Each input $v_{Ik}$ appears with its own weight $-R_f/R_k$. Adding another input is trivial: drop in another resistor to the same node and it contributes another term — nothing else in the circuit changes.

Each input summed at the inverting node with weight $-R_f/R_k$.

Why the weights are independent

This is the cleanest illustration of why the virtual ground is so powerful. Every input branch terminates at the same fixed $0\text{V}$ node that draws no current and that no branch can disturb. So the current in branch $k$ is $v_{Ik}/R_k$ — determined only by that input and that resistor, with zero coupling to the other branches. The virtual ground lets you superpose currents at the node without the voltages interacting. Change $R_2$ and only the $v_{I2}$ weight moves; the $v_{I1}$ weight is untouched. Without the virtual ground (e.g. tying the inputs to a plain resistor network) the branches would load each other and the weights would be a tangled coupled system. [Background from general knowledge, not the source PDF] This independence is why the summer is the heart of an audio mixer (each channel a separate $R_k$ fader) and of a binary-weighted Digital-to-analog converter (resistors in powers of two reconstruct a number from its bits).