In a differential amplifier, the Common-Mode Rejection Ratio (CMRR) is a key performance metric that indicates how effectively the amplifier rejects signals that are common to both input terminals (common-mode signals) compared to signals that are different between the two input terminals (differential-mode signals).
Based on the provided reference, the CMRR in a differential amplifier is precisely defined as:
The ratio of the common-mode gain to differential-mode gain.
Mathematically, this can be expressed as:
$$ \text{CMRR} = \frac{A{dm}}{A{cm}} $$
Where:
- $A_{dm}$ is the differential-mode gain (the amplification of the desired signal difference).
- $A_{cm}$ is the common-mode gain (the amplification of the unwanted common-mode signal).
Understanding the Terms
To understand CMRR better, let's look at the two types of signals and gains involved:
- Differential-Mode Signal: This is the intended input signal, the difference between the two input voltages ($V{in1} - V{in2}$). A differential amplifier is designed to amplify this difference.
- Common-Mode Signal: This is a signal that is present simultaneously and equally on both input terminals (e.g., noise or interference picked up by both input wires). Ideally, a differential amplifier should not amplify this signal.
- Differential-Mode Gain ($A_{dm}$): How much the amplifier amplifies the differential-mode input signal.
- Common-Mode Gain ($A_{cm}$): How much the amplifier amplifies the common-mode input signal. Ideally, this gain should be zero.
The Significance of CMRR
A high CMRR is highly desirable in a differential amplifier. Here's why:
- Noise and Interference Rejection: Common-mode signals often represent unwanted noise or interference picked up from the environment. A high CMRR means the amplifier greatly attenuates (reduces the effect of) these common-mode signals while amplifying the desired differential signal.
- Improved Accuracy: By minimizing the amplification of common-mode noise, the output signal is a more accurate representation of the intended differential input.
- Better Signal Integrity: It helps maintain the quality of the signal in noisy environments, making differential amplifiers excellent for applications requiring high precision or operating over long distances where noise pickup is common.
Calculating CMRR: An Example
The reference provides a clear way to think about calculating CMRR:
- "For example, if a differential input change of Y volts produces a change of 1 V at the output, and a common-mode change of X volts produces a similar change of 1 V, then the CMRR is X/Y."
Let's break this example down:
- Scenario 1: Differential Input: A change of Y volts in the differential input results in a 1 V change at the output. This implies the differential-mode gain, $A_{dm}$, can be thought of as $1V / YV$.
- Scenario 2: Common-Mode Input: A change of X volts in the common-mode input results in the same 1 V change at the output. This implies the common-mode gain, $A_{cm}$, can be thought of as $1V / XV$.
- Calculating CMRR: Using the definition $\text{CMRR} = \frac{A{dm}}{A{cm}}$:
$$ \text{CMRR} = \frac{1V / YV}{1V / XV} = \frac{1V}{YV} \times \frac{XV}{1V} = \frac{X}{Y} $$
As stated in the reference, the CMRR is indeed X/Y.
This example shows that a larger X (meaning it takes a much larger common-mode voltage to produce the same output as a small differential voltage Y) results in a higher CMRR.
CMRR in Decibels (dB)
CMRR is often expressed in decibels (dB) using the formula:
$$ \text{CMRR}{dB} = 20 \log{10} \left( \frac{A{dm}}{A{cm}} \right) = 20 \log_{10}(\text{CMRR}) $$
A higher dB value corresponds to a better common-mode rejection. For instance, a CMRR of 100,000 is 100 dB (20 * log10(100,000)).
In summary, CMRR is a critical specification for differential amplifiers and operational amplifiers (op amps), quantifying their ability to suppress unwanted common-mode signals while effectively amplifying the desired differential signals.
