Define differential gain, common mode gain and CMRR. Derive the relationship between them.

Q.) Define differential gain, common mode gain and CMRR. Derive the relationship between them.

 Subject: electronic devices and circuits

Differential Gain (Ad):

Differential gain is the ratio of the differential output voltage to the differential input voltage. It is a measure of the amplifier's ability to amplify the difference between two input signals. Mathematically, it is expressed as:

Ad = Vout_diff / Vin_diff

where:

  • Vout_diff is the differential output voltage, which is the voltage difference between the two output terminals.
  • Vin_diff is the differential input voltage, which is the voltage difference between the two input terminals.

Common Mode Gain (Acm):

Common mode gain is the ratio of the output voltage to the common mode input voltage. It is a measure of the amplifier's ability to amplify the average of two input signals. Mathematically, it is expressed as:

Acm = Vout / Vin_cm

where:

  • Vout is the output voltage.
  • Vin_cm is the common mode input voltage, which is the average of the two input voltages.

Common Mode Rejection Ratio (CMRR):

Common mode rejection ratio is the ratio of the differential gain to the common mode gain. It is a measure of the amplifier's ability to reject common mode noise and amplify only the differential signal. Mathematically, it is expressed as:

CMRR = Ad / Acm

or

CMRR = 20 log(Ad / Acm)

The CMRR is typically expressed in decibels (dB) and higher values indicate better common mode rejection.

Relationship between Differential Gain, Common Mode Gain, and CMRR:

The differential gain, common mode gain, and CMRR are related by the following equation:

CMRR = Ad / (1 + Acm)

This equation shows that the CMRR is directly proportional to the differential gain and inversely proportional to the common mode gain. In other words, a higher differential gain and a lower common mode gain result in a higher CMRR.

Derivation of the Relationship:

To derive the relationship between differential gain, common mode gain, and CMRR, we can start with the differential and common mode gain equations:

Ad = Vout_diff / Vin_diff Acm = Vout / Vin_cm

We can rearrange the differential gain equation to obtain:

Vin_diff = Vout_diff / Ad

Substituting this expression for Vin_diff into the common mode gain equation, we get:

Acm = Vout / [(Vout_diff / Ad) + (Vout_diff / Ad)]

Simplifying this equation, we get:

Acm = Vout / (2Vout_diff / Ad)

Multiplying both sides by Ad, we get:

Ad * Acm = Vout * Ad / (2Vout_diff)

Rearranging this equation, we get:

CMRR = Ad / (2Acm)

or

CMRR = Ad / (1 + Acm)

which is the desired relationship.