**CMRR of Op Amp Formula (Common Mode Rejection Ratio):**

One of the more common features of a differential amplifier is its ability to cancel out or reject certain types of undesired voltage signals. Such undesired signals are referred to as “noise” and may occur owing to voltages induced by stray magnetic fields in the ground or signal wires, as voltage variations in the voltage supply. Here the important point is that these noise signals are not the signals that are desired to be amplified in the differential amplifier. Their distinguishing feature is that the noise signal appears equally at both inputs of the circuit. It means that any undesired (or noise) signals that appear in polarity, or common to both input terminals, will be largely rejected, or cancelled out at the differential amplifier output. The signal that is to be amplified appears at only one input or opposite in polarity at both inputs. Now what is important to be considered is that, if undesirable noise does occur, up to what extent it is rejected out by the differential amplifier? A measure of this rejection of signals common to both inputs is referred to as the CMRR of Op Amp and a numerical value is assigned, which is called the common-mode rejection ratio (CMRR).

CMRR is defined as the ratio of differential voltage gain to common-mode voltage gain and it is given as

If a differential amplifier is perfect, CMRR would be infinite because in that case common-mode voltage gain A_{cm} would be zero.

Figure 33.21 represents a linear active device with two input signals V_{1} and V_{2} and one output signal V_{out}, each measured with respect to ground. In an ideal differential amplifier the output signal V_{out} should be given as

where A is the gain of the differential amplifier.

Common to both the inputs will have no effect on the output voltage. However, in a practical differential amplifier, the output not only depends upon the difference signal V_{d} of the two input signals, but also upon the average level, called the **common-mode signal** V_{cm} where

For example the output in the two cases having input signals of (V_{1} = +100 μV and V_{2} = -50 μV) and (V_{1} = +1,200 μV and V_{2} = +1050 μV) will not be exactly the same, even though the algebraic difference of two input signals V_{d} in both cases is the same i.e., 150 μV.

Now let the output voltage V_{out} in Fig. 33.21 be expressed as a linear combination of two input voltages as below

where A_{1} is the voltage gain for input V_{1 }with V_{2} grounded and A_{2} is the voltage gain for the input V_{2} with V_{1} grounded.

But from Eqs. (33.11) and (33.12)

Substituting the values of V_{1 }with V_{2} from above Eqs. (33.14) and (33.15) in Eq. (33.13) we have

Here A_{d} is the voltage gain for the difference signal while A_{cm} is the voltage gain for the common-mode signal.

The voltage gain A_{d} may be measured directly by selecting V_{1} = -V_{2} = 0.5 V so that V_{d} = 1 V and V_{cm} = 0 V. Under such conditions output voltage is A_{d} x (1 V) i.e., the output voltage equals A_{d}.

Similarly for measuring A_{cm}, select V_{1} = V_{2} = 1 V, so that V_{d} = 0 and V_{cm} = 1 V. Under such conditions output voltage is A_{cm }x (1 V) i.e., the output voltage equals A_{cm}.

Thus, values of differential voltage gain A_{d} and common-mode voltage gain A_{cm} can be determined simply by measurement of output voltage.

Having measured A_{d} and A_{cm} for the amplifier, CMRR of Op Amp formula can be determined from the following relation

The value of CMRR of Op Amp can also be expressed in logarithmic terms as

From Eqs. (33.17) and (33.20) we have an expression for the output voltage in the following form

From above Eq. (33.22) it is seen that the amplifier should be designed so that CMRR is large in comparison to the ratio of common-mode signal to the difference signal (V_{cm}/V_{d}).

Even if both components of voltage V_{cm} and V_{d} exist at the inputs, the value of (1/CMMR V_{cm}/V_{d}) will be very small, for CMRR very large, and the output voltage will be almost equal to A_{d}V_{d}. It means that the output will be almost completely due to the difference signal with the common-mode input signals rejected or cancelled out.

As seen above, higher the value of CMRR, better is the performance of differential amplifier. Hence in practice the efforts are always to improve the CMRR of Op Amp.

For improvement of CMRR of Op Amp, it is necessary to reduce common mode gain A_{cm}. The common mode gain A_{cm} approaches zero, as the emitter resistance R_{E} tends to ∞. This is because R_{E} introduces a negative feedback in the common mode operation which reduces the common mode gain A_{cm}. Thus higher the value of R_{E}, lower is the value of A_{cm} and higher is the value of CMRR of Op Amp. The differential gain A_{d} is not dependent on R_{E}. But it is not practical to increase R_{E} to a very high due to certain limitations such as (i) Large R_{E} means higher biasing voltage to set the operating point Q of the transistors (ii) increase in overall chip area.

Hence, in practice, instead of increasing R_{E}, various other methods are used which provide effect of increased R_{E} without limitations. Such methods are (i) **constant current bias method** (ii) **use of current mirror circuit**.

The other method used is to increase the differential gain A_{d} and A_{d} can be increased by using active load.