# Common-mode rejection ratio

The common-mode rejection ratio (CMRR) of a differential amplifier (or other device) is the rejection by the device of unwanted input signals common to both input leads, relative to the wanted difference signal. An ideal differential amplifier would have infinite CMRR; 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. An example is audio transmission over balanced lines.

## Theory

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

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

where $A_{{\mathrm {cm}}}$ 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):

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

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.

## Example: operational amplifiers

Typical instrumentation amplifier implementation, designed to have a high CMRR.

An operational amplifier (op-amp) has two inputs, V+ and V-, and an open-loop gain G. In the ideal case, the output of an ideal op-amp behaves according to the equation

$V_{{\mathrm {out}}}=(V_{+}-V_{-})\cdot G_{{\mathrm {openloop}}}$

This equation represents an infinite CMRR: if both inputs fluctuate by the same amount (while maintaining a constant difference V+ - V-), this change will have no bearing on the output. In real applications, this is not always the case: the lower the CMRR, the larger the effect on the output signal, following the general equation

$V_{{\mathrm {out}}}=(V_{+}-V_{-})\cdot G_{{\mathrm {openloop}}}\pm {\frac {V_{{\mathrm {cm}}}}{10^{{{\frac {CMRR}{20}}}}}}$

Where VCM represents the common-mode voltage at the inputs, or (V+ + V-)/2.

The 741 (a common op-amp chip) has a CMRR of 90 dB, which is reasonable in most cases. A value of 70 dB may be adequate for applications which are insensitive to the effects on amplifier output;some high-end devices may use op-amps with a CMRR of 120 dB or more.

So for example, an op-amp with 90dB CMRR operating with 10V of common-mode will have an output error of ±316uV.