Common-mode rejection ratio

The common-mode rejection ratio (CMRR) measures how well a differential amplifier amplifies the differential signal while ignoring the common-mode signal. It is the ratio of differential gain to common-mode gain:

$\text{CMRR}=\frac{A_d}{A_{cm}}{\text{CMRR}}_{\text{dB}}=20{\log }_{10}\frac{A_d}{A_{cm}}$

$A_d$ is the gain the amplifier applies to the differential input $v_{Id}=v_2-v_1$; $A_{cm}$ is the (unwanted) gain it applies to the common-mode input $v_{Icm}=(v_1+v_2)/2$. An ideal op-amp has $A_{cm}=0$, so its CMRR is infinite. A real op-amp leaks a little common-mode through, so CMRR is large but finite.

For a 741, CMRR $\approx 90\text{dB}$. Convert: $90\text{dB}$ means $A_d/A_{cm}=10^{90/20}=10^{4.5}\approx 30,000$. The differential gain is about thirty thousand times the common-mode gain. [Background from general knowledge, not the source PDF] The decibel form is standard because CMRR spans many orders of magnitude; a precision Instrumentation amplifier reaches well over $100\text{dB}$ ($>10^5$).

Why it matters

The whole reason to use a differential front end is to pull a small wanted difference out of a large unwanted common pedestal. Picture an ECG electrode pair: the heart signal is about $1\text{mV}$ differential, but both leads pick up roughly $1\text{V}$ of 60 Hz mains hum in common. Suppose $A_d=1000$. The wanted output is $1000\times 1\text{mV}=1\text{V}$. With CMRR $=90\text{dB}$, the common-mode gain is $A_{cm}=A_d/30000\approx 0.033$, so the hum at the output is $0.033\times 1\text{V}\approx 33\text{mV}$ — small enough to live with. Drop CMRR to $20\text{dB}$ ($A_d/A_{cm}=10$) and the same hum becomes $100\text{V}$-equivalent at the input: the signal is buried. High CMRR is the single specification that makes sensor measurement — strain gauges, thermocouples, biopotentials — possible.

For a single-op-amp Difference amplifier, CMRR is not set by the op-amp at all but by how precisely the four external resistors are matched; any mismatch converts common-mode into a differential error. That fragility is exactly why the Instrumentation amplifier exists: its input stage amplifies only the differential signal before any resistor matching is relied upon, so realistic resistor tolerances still yield CMRR above $100\text{dB}$.