Common-mode rejection ratio

Ideally, a differential amplifier takes the voltages, ${\displaystyle V_{+}}$ and ${\displaystyle V_{-}}$ on its two inputs and produces an output voltage ${\displaystyle V_{\mathrm {o} }=A_{\mathrm {d} }(V_{+}-V_{-})}$, where ${\displaystyle A_{\mathrm {d} }}$ is the differential gain. However, the output of a real differential amplifier is better described as :

${\displaystyle V_{\mathrm {o} }=A_{\mathrm {d} }(V_{+}-V_{-})+{\tfrac {1}{2}}A_{\mathrm {cm} }(V_{+}+V_{-})}$

where ${\displaystyle A_{\mathrm {cm} }}$ is the "common-mode gain", which is typically much smaller than the differential gain.

The CMRR is defined as the ratio of the powers of the differential gain over the common-mode gain, measured in positive decibels (thus using the 20 log rule):

${\displaystyle \mathrm {CMRR} =\left({\frac {A_{\mathrm {d} }}{|A_{\mathrm {cm} }|}}\right)=10\log _{10}\left({\frac {A_{\mathrm {d} }}{A_{\mathrm {cm} }}}\right)^{2}{\text{dB}}=20\log _{10}\left({\frac {A_{\mathrm {d} }}{|A_{\mathrm {cm} }|}}\right){\text{dB}}}$

As differential gain should exceed common-mode gain, this will be a positive number, and the higher the better.

The CMRR is a very important specification, as it indicates how much of the common-mode signal will appear in your measurement. The value of the CMRR often depends on signal frequency as well, and must be specified as a function thereof.

It is often important in reducing noise on transmission lines. For example, when measuring the resistance of a thermocouple in a noisy environment, the noise from the environment appears as an offset on both input leads, making it a common-mode voltage signal. The CMRR of the measurement instrument determines the attenuation applied to the offset or noise.

CMRR is an important feature of operational amplifiers, difference amplifiers and instrumentation amplifiers, and can be found in the datasheet. The CMRR often varies depending upon the frequency of the common-mode signal. CMRR is often much higher at higher gain settings. The key to achieving a high CMRR is usually the use of very precisely matched resistors (better than 0.1%) to minimise any difference in the amplification of the negative and positive sides of the signal. Single-chip instrumentation amplifiers typically have laser-trimmed resistors to achieve a CMRR in excess of 100 dB, sometimes even 130 dB.

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