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

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

 Subject: Electronic Devices and Circuit

Differential Gain (Ad):

Differential gain (Ad) is defined as the ratio of the differential output voltage (Vout) to the differential input voltage (Vin) in a differential amplifier. It represents the amplification of the difference signal between the two inputs. Mathematically, it is expressed as:

Ad = Vout / Vin

Common Mode Gain (Acm):

Common mode gain (Acm) is defined as the ratio of the differential output voltage (Vout) to the common mode input voltage (Vcm) in a differential amplifier. It represents the amplification of the average value of the two input signals. Mathematically, it is expressed as:

Acm = Vout / Vcm

Common Mode Rejection Ratio (CMRR):

Common mode rejection ratio (CMRR) is defined as the ratio of the differential gain (Ad) to the common mode gain (Acm) in a differential amplifier. It represents the ability of the amplifier to reject or suppress common mode signals while amplifying differential signals. Mathematically, it is expressed as:

CMRR = Ad / Acm

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

The relationship between differential gain, common mode gain, and CMRR can be derived as follows:

  1. Differential Input Voltage:

The differential input voltage (Vin) is the difference between the two input signals, Vin+ and Vin-. Mathematically, it is expressed as:

Vin = Vin+ - Vin-

  1. Differential Output Voltage:

The differential output voltage (Vout) is the difference between the two output signals, Vout+ and Vout-. Mathematically, it is expressed as:

Vout = Vout+ - Vout-

  1. Common Mode Input Voltage:

The common mode input voltage (Vcm) is the average value of the two input signals, Vin+ and Vin-. Mathematically, it is expressed as:

Vcm = (Vin+ + Vin-) / 2

  1. Common Mode Output Voltage:

The common mode output voltage (Vcom) is the average value of the two output signals, Vout+ and Vout-. Mathematically, it is expressed as:

Vcom = (Vout+ + Vout-) / 2

  1. Derivation of Relationship:

Using the above equations, we can derive the relationship between differential gain, common mode gain, and CMRR as follows:

CMRR = Ad / Acm = (Vout / Vin) / (Vout / Vcm) = (Vout*Vcm) / (Vout*Vin) = Vcm / Vin

Therefore, the CMRR is equal to the ratio of the common mode input voltage to the differential input voltage. A higher CMRR value indicates a better ability of the amplifier to reject common mode signals.