Understanding Ideal Op-Amp Characteristics: Common Mode Gain

Operational amplifiers (op-amps) are fundamental building blocks in analog electronics. They are designed to amplify the voltage difference between their two input terminals. However, op-amps also respond, to some extent, to signals that are common to both input terminals. This response is characterized by the common-mode gain. For an ideal op-amp, certain parameters are assumed to have specific theoretical values that simplify circuit analysis and represent optimal performance.

Differential Gain vs. Common Mode Gain

The output voltage ($V_{out}$) of an op-amp can be generally expressed using its differential gain ($A_d$) and common-mode gain ($A_{cm}$):

$V_{out} = A_d \cdot (V_{in+} - V_{in-}) + A_{cm} \cdot \frac{V_{in+} + V_{in-}}{2}$

Where:

• $V_{in+}$ is the voltage at the non-inverting input terminal.
• $V_{in-}$ is the voltage at the inverting input terminal.
• $(V_{in+} - V_{in-})$ is the differential input voltage.
• $\frac{V_{in+} + V_{in-}}{2}$ is the common-mode input voltage ($V_{cm}$).
• $A_d$ is the differential gain.
• $A_{cm}$ is the common-mode gain.

Ideal Op-Amp Assumptions

An ideal op-amp is a theoretical model defined by the following characteristics:

• Infinite differential gain ($A_d \to \infty$).
• Zero common-mode gain ($A_{cm} = 0$).
• Infinite input impedance.
• Zero output impedance.
• Infinite bandwidth.
• Zero input offset voltage.

The assumption of a zero common-mode gain for an ideal op-amp means that the op-amp does not amplify any signal that is present on both input terminals simultaneously. It only amplifies the difference between the two inputs.

Common Mode Rejection Ratio (CMRR)

The ability of an op-amp to reject common-mode signals is quantified by the Common Mode Rejection Ratio (CMRR). It is defined as the ratio of the differential gain to the common-mode gain:

$CMRR = \frac{A_d}{A_{cm}}$

For an ideal op-amp, since $A_d$ is infinite and $A_{cm}$ is zero, the CMRR is theoretically infinite:

$CMRR_{ideal} = \frac{\infty}{0} \to \infty$

An infinite CMRR signifies perfect rejection of common-mode signals, which aligns with the assumption that the common-mode gain is zero.

Conclusion

Based on the definition of an ideal op-amp, the common-mode gain, which represents the amplification of signals common to both inputs, is assumed to be always zero. This characteristic is crucial for the ideal behavior where the op-amp exclusively amplifies the differential signal.