In this article, we will discuss what is CMRR. How does CMRR has affected the output voltage of Op-amp? We will also learn the function of CMRR with the frequency and ideal and practical application of CMRR in Op-amp.

## 1**What is CMRR?**

- The full form of
**CMRR**is a**Common Mode Rejection Ratio**. - The CMRR(Common Mode Rejection Ratio) is the most important specification and it indicates how much of the common mode signals will present to measure. The value of the CMMR frequently depends on the signal frequency and the function should be specified. The function of the CMMR is specifically used to reduce the noise on the transmission lines.
- The detailed discussion and equation of CMRR will be discussed in this section later.

When the same input voltage is applied to both input terminals of an op-amp, the op-amp is said to be operating in a common mode configuration.

Since the input voltage applied is common to both the inputs, it is referred to as a common-mode voltage v_{cm}, as shown in figure 1.

A common-mode voltage v_{cm} can be ac, dc, or a combination of ac and dc.

Ideally, an op-amp amplifies only differential input voltages, no common-mode output voltage v_{ocm} should appear at the output. However, due to imperfections within an actual op-amp, some common-mode voltage v_{ocm} will appear at the output.

The amplitude of this v_{ocm} is very small and often insignificant compared to v_{cm}.

Therefore, in practice, the ratio of the output common-mode voltage v_{ocm} to the input common-mode voltage v_{cm}, which is called the common-mode voltage gain A_{cm}, is generally much smaller than 1.

In equation form,

\boxed {A_{cm}= \frac{v_{ocm}}{v_{cm}}}…..eq.1

Ideal value of the common mode voltage gain A_{cm} is Zero.

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Although, A_{cm} is usually not specified on op-amp data sheets, A_{cm} can be calculated for a given op-amp by applying a known value of common-mode input voltage v_{cm} and measuring the resultant output common-mode voltage v_{ocm}.

Generally, CMRR can be defined as the ratio of the differential gain A_D to the common-mode gain A_{cm}, that is,

\boxed{CMRR = \frac{A_D}{A_{cm}}} …..eq.2

Note that, in figure-1 A_D is equal to the internal gain A of the op-amp.

The CMRR can also be expressed as the ratio of the change in the input offset voltage to the total change in common-mode voltage. Thus,

\boxed{CMRR = \frac{V_{io}}{v_{cm}}} …..eq.3

From eq.1 and eq.2, we can then establish the relationship between the v_{ocm} and CMRR:

CMRR = \frac{A_D}{A_{cm}} = \frac{A_D}{ v_{ocm} / v_{cm}}

= \frac{A_D.v_{cm}}{ v_{ocm}} …..eq.4

\boxed{v_{ocm}= \frac{A_D. v_{cm}}{CMRR}}…..eq.5

Equation 5 indicates that the higher the value of CMRR, the smaller will be the amplitude of the output common-mode voltage v_{ocm}.

Generally, the value of CMRR is very large and is therefore usually specified in decibels (dB), where

CMRR (dB)= 20\log\left(\frac{A_D}{A_{cm}}\right)…..eq.6

or from eq.3,

CMRR (dB)= 20\log\left(\frac{V_{io}}{v_{cm}}\right)…..eq.7

Whether the CMRR is defined as in eq.2 or eq.3, it is a measure of the degree of matching between the two input terminals; that is, the larger the value of CMRR(dB), the better is the matching between the two input terminals and the smaller is the output common-mode voltage v_{ocm}.

On the other hand, a large voltage v_{ocm} for a given common-mode input voltage v_{cm} is an indication of a large degree of imbalance between the two input terminals or of poor common-mode rejection.

Thus in practice, it is advantageous to use op-amps with higher CMRRs since these op-amps have a better ability to reject common-mode voltage.

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## 2**Relation between CMRR and Frequency **

The CMRR is a function of frequency and decreases as the frequency is an increase as shown in the figure below:

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