Understanding the Common Mode Rejection Ratio (CMRR) of an Ideal OP-Amp

Let's analyze the question about the Common Mode Rejection Ratio (CMRR) of an ideal Operational Amplifier (OP-Amp).

First, let's define some key terms:

• OP-Amp (Operational Amplifier): A high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output.
• Differential Input: The input voltage is the difference between the voltages applied to the two input terminals (inverting and non-inverting).
• Common Mode Signal: A signal that is common to both input terminals. This is often unwanted noise picked up by both inputs equally.
• Differential Mode Signal: The actual signal we want to amplify, which is the difference between the voltages at the input terminals.
• Common Mode Rejection Ratio (CMRR): A measure of how well the OP-Amp rejects common mode signals compared to differential mode signals. A high CMRR means the OP-Amp amplifies the desired differential signal much more than the unwanted common mode signal.

The CMRR is mathematically defined as the magnitude of the differential gain (\(A_d\)) divided by the magnitude of the common mode gain (\(A_{cm}\)).

\[ \text{CMRR} = \left| \frac{A_d}{A_{cm}} \right| \]

Now, let's consider an ideal OP-Amp. An ideal OP-Amp has several characteristics, including:

• Infinite input impedance.
• Zero output impedance.
• Infinite open-loop differential gain (\(A_d \rightarrow \infty\)).
• Zero common mode gain (\(A_{cm} = 0\)).
• Infinite bandwidth.

For an ideal OP-Amp, the common mode gain (\(A_{cm}\)) is zero. This means that the ideal OP-Amp completely ignores and does not amplify any signal that appears equally on both of its input terminals. It only amplifies the difference between the two input signals.

Using the formula for CMRR:

\[ \text{CMRR} = \left| \frac{A_d}{A_{cm}} \right| \]

Substitute the values for an ideal OP-Amp:

• \(A_d = \infty\) (Infinite differential gain)
• \(A_{cm} = 0\) (Zero common mode gain)

Plugging these values into the formula gives:

\[ \text{CMRR} = \left| \frac{\infty}{0} \right| \]

Mathematically, division by zero is undefined, but in this context, as the common mode gain approaches zero while the differential gain is infinite, the ratio approaches infinity. An infinite CMRR signifies perfect rejection of common mode signals.

Therefore, the Common Mode Rejection Ratio (CMRR) of an ideal OP-Amp is infinite.

Evaluating the Options

Based on the analysis of an ideal OP-Amp's characteristics and the CMRR formula, let's look at the options:

• Zero: If CMRR were zero, it would mean \(A_d=0\) or \(A_{cm}=\infty\). Neither is true for an ideal OP-Amp; an ideal OP-Amp has infinite differential gain and zero common mode gain.
• Infinite: As explained above, with \(A_d=\infty\) and \(A_{cm}=0\), the ratio \(A_d/A_{cm}\) is infinite. This represents perfect common mode rejection.
• Low: A low CMRR means the OP-Amp amplifies common mode signals significantly relative to differential signals. This is undesirable and not characteristic of an ideal OP-Amp.
• Medium: Similar to low, a medium CMRR implies some common mode amplification, which is not the case for an ideal OP-Amp with zero common mode gain.

Thus, the only value consistent with the definition and characteristics of an ideal OP-Amp is infinite CMRR.

Revision Table: Ideal OP-Amp Characteristics & CMRR

Additional Information: Real-World OP-Amp CMRR

While an ideal OP-Amp has infinite CMRR, real-world OP-Amps have a finite, but typically very high, CMRR. The CMRR of a real OP-Amp is usually expressed in decibels (dB).

\[ \text{CMRR}_{\text{dB}} = 20 \log_{10} (\text{CMRR}) \]

A higher CMRR (or higher CMRR in dB) indicates a better OP-Amp in terms of rejecting common mode noise. Typical values for good OP-Amps might range from 70 dB to over 100 dB. A CMRR of 100 dB corresponds to a voltage ratio of \(10^5\), meaning the differential signal is amplified 100,000 times more than the common mode signal.

Understanding CMRR is crucial in applications where the desired signal is small and might be contaminated by larger common mode noise, such as in instrumentation amplifiers or differential sensors.