Summing amplifier using op-amp

A summing amplifier (adder) is an op-amp circuit whose output is proportional to the (weighted) sum of several input voltages. The common form is the inverting summing amplifier.

Circuit

 V1 o--[R1]--+
             |        Rf
 V2 o--[R2]--+-----/\/\/\-----+
             |                |
 V3 o--[R3]--+   |\           |
             +---|-\          |
              V− |  >---------+--o Vo
           GND --|+/
                 |/

Assumptions (ideal op-amp)

1. V+ = 0 (grounded), and because of the virtual short V− = V+ = 0 (virtual ground).
2. No current enters the op-amp input.

Derivation

Input currents:

I1 = (V1 − 0)/R1 = V1/R1
I2 = V2/R2
I3 = V3/R3

No current enters the (−) terminal, so by KCL all these currents flow through Rf:

If = I1 + I2 + I3

The output voltage, with the inverting node at 0 V:

0 − Vo = If × Rf
Vo     = −Rf (V1/R1 + V2/R2 + V3/R3)

Special cases

• Equal resistors R1 = R2 = R3 = Rf = R: Vo = −(V1 + V2 + V3) (inverting adder).
• R1 = R2 = R3 = R, Rf = R/3: Vo = −(V1 + V2 + V3)/3 (averaging amplifier).
• Unequal resistors give a scaling (weighted) summer, Vo = −(a1V1 + a2V2 + a3V3) with ai = Rf/Ri.

Example: V1 = 1 V, V2 = 2 V, V3 = −0.5 V with all resistors 10 kΩ: Vo = −(1 + 2 − 0.5) = −2.5 V.

Because the inverting node is a virtual ground, the inputs do not interact with each other. Uses: audio mixers, digital-to-analog converters (weighted resistors), and adding a DC offset to a signal. A second inverting amplifier of gain −1 can be added to get a non-inverted sum.