Differential and common-mode signals

An op-amp is a differential device, so it pays to split any pair of input voltages $(v_1,v_2)$ into the part it cares about and the part it should ignore. The differential input signal is the difference between the inputs, the common-mode input signal is their average:

$v_{Id}=v_2-v_1{v}_{Icm}=\frac{v_1+v_2}{2}$

Here $v_1$ is the inverting input voltage and $v_2$ the non-inverting input voltage, both measured to ground. $v_{Id}$ is the genuine signal of interest; $v_{Icm}$ is whatever offset both inputs ride on together.

Recovering the originals

The decomposition is invertible, no information lost. Add and subtract:

$v_2=v_{Icm}+\frac{v_{Id}}{2}{v}_1=v_{Icm}-\frac{v_{Id}}{2}$

Check: $v_2-v_1=v_{Id}$ and $(v_1+v_2)/2=v_{Icm}$. So any input pair is “the average, plus or minus half the difference.” The differential part separates the inputs; the common-mode part is the pedestal they both stand on.

Any input pair = differential part (difference) + common-mode part (average).

Why split it this way

An ideal op-amp output depends only on $v_{Id}$: $v_O=A{v}_{Id}$, with the common-mode part rejected entirely. A real op-amp has a small unwanted dependence on $v_{Icm}$, so its output is more like $v_O=A_d{v}_{Id}+A_{cm}{v}_{Icm}$, where $A_d$ is the differential gain and $A_{cm}$ the (tiny) common-mode gain. Their ratio is the Common-mode rejection ratio.

Real measurements live in this split. A strain gauge or thermocouple produces a few millivolts of differential signal sitting on top of several volts of common-mode (noise picked up on the wires, ground-loop offsets, the bridge excitation). You want the difference and want the average thrown away, which is what a Difference amplifier, and far better an Instrumentation amplifier, is built to do. Thinking in $(v_{Id},v_{Icm})$ instead of $(v_1,v_2)$ makes “amplify the signal, reject the interference” a precise, designable statement.