Common-mode rejection ratio

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The common-mode rejection ratio (CMRR) of an amplifier (or other device) measures the tendency of the device to reject input signals common to both input leads. A high CMRR is important in applications where the signal of interest is represented by a small voltage fluctuation superimposed on a (possibly large) voltage offset, or when relevant information is contained in the voltage difference between two signals. (An example is audio transmission over balanced lines.)

The CMRR, measured in positive decibels, is defined by the following equation:

$\mathrm{CMRR} = 20\log_{10} \left (\frac{A_\mathrm{d}}{A_\mathrm{s}} \right ),$

where it is assumed that the amplifier output $V o$ can be modeled as

$V_{\mathrm{o}} = A_\mathrm{d} (V_+ - V_-) + \frac{1}{2} A_\mathrm{s} (V_+ + V_-).$

The first term $A d$ represents the differential amplification (the desired mode of operation), while the second term $A s$ represents the undesired amplification of a common-mode voltage that is the same for both inputs.

This 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.

CMRR is often important in reducing noise on transmission lines. For example, when measuring 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.

[edit] Example: Operational Amplifiers

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