# Common Mode Rejection Ratio (CMRR) and The Operational Amplifier

The CMRR(Common Mode Rejection Ratio) is the most important specification and it indicates the 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. For an example, when we measure the resistance of a thermocouple in the noisy environment the noise from the environment appears as an offset on both input leads and making it as a common mode voltage signal. The CMRR instrument determines the attenuation applied to the noise.

## What is a CMRR?

The CMRR in an operational amplifier is a common mode rejection ratio. Generally, the op amp as two input terminals which are positive and negative terminals and the two inputs are applied at the same point. This will give the opposite polarity signals at the output. Hence the positive and the negative voltage of the terminals will cancel out and it will give the resultant output voltage. The ideal op amp will have the infinite CMRR and with the finite differential gain and zero common mode gain.

CMMR = Differential mode gain / Common-mode gain

## Common-mode Rejection Ratio Formula

The common mode rejection ratio is formed by the two inputs which will have the same sign of DC voltage. If we assume one input voltage is 8v and the other 9v here the 8v is common and the input voltage should be calculated through the equation of V+ – V- . Hence the result will be 1v but the common DC voltage between the two inputs has a non-zero gain.

The differential gain Ad magnifies the difference between the two input voltages. But the common mode gain Ac magnifies the common mode DC voltage between the two inputs. The ratio of two gains is said to be as a common mode rejection ratio. The value of the format is in dB. The formula of a common mode rejection ratio is calculated by the following equation.

CMRR = 20log|Ao/Ac| dB

### Power Supply Rejection Ratio

The power supply rejection ratio is defined as the changes in input offset voltage per unit changes in the DC supply voltage. The power supply is also calculated in the format of dB. The mathematical equation of the power supply rejection ratio is given below.

PSRR= 20log|ΔVDc/ΔVio| dB

### Common Mode Rejection Ration of Op Amp

The common mode rejection ratio is a differential amplifier and the op amps are amplified in with the differential input. Hence the CMMR ratio can be applied to the operational amplifier. By using the condition of common mode rejection ratio, i.e. when both the input of the amplifier has same voltages, then the output of the amplifier should be zero or the amplifier should be rejecting the signal. The following image shows the amplifier of MCP601 of common mode rejection ratio.

### Offset Error of a CMRR of the Op-Amp

The CMRR can build parallel out offset voltage in op amps configured in the non-inverting amplifier which is shown in the below figure. The non-inverting operating amplifier will have a small amount of CMRR error because both the inputs are connected to ground, there is no presence of CM dynamic voltage.

Error (RTI) = Vcm / CMRR = Vin / CMRR

Vout = [1 + R2/R1] [ Vin + Vin/ CMRR]

Error (RTO) = [1+R2/R1] [Vin/CMRR]

#### Measuring Common Mode Rejection Ratio

There are different ways to measure the common mode rejection ratio. In the below figure we will discuss the four precision resistor to configure the op amp as a differential amplifier. A signal is applied to the both inputs, changes in the output are measured and an amplifier with infinite CMRR also no changes in the output. The inherent difficulties of this circuit are that the ratio match of the resistors is important as the CMRR of the op amp. The 0.1% mismatch is between resistor pair and the result will be in CMR of 66 dB. Hence the most of the amplifiers will have a low frequency of CMR is between the 80dB to 120Db. In this circuit, it is clear that there is only marginally useful for measuring the CMRR.