# Common-mode rejection ratio

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The common-mode rejection ratio (CMRR) of a differential amplifier (or other device) is a metric used to quantify the ability of the device to reject common-mode signals, i.e., those that appear simultaneously and in-phase on both inputs. An ideal differential amplifier would have infinite CMRR, however this is not achievable in practice. A high CMRR is required when a differential signal must be amplified in the presence of a possibly large common-mode input, such as strong electromagnetic interference (EMI). An example is audio transmission over balanced lines in sound reinforcement or recording.

## Theory

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.[citation needed] 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.