Differential amplifiers ideally amplify only the voltage difference across their input terminals - the *differential input voltage*, *v*_{I} - and not the average voltage across them to ground, the *common-mode* (CM) *input voltage*, *v*_{CM}. However, the output voltage, *v*_{O}, to ground of a real voltage diff-amp will be affected to some extent by *v*_{CM}. This article explains causes for CM amplification and how it can be measured with an instrument designed for this purpose, the *Floating Differential Source*.

**Diff-Amp Basics**

The basic one-op-amp diff-amp is shown below.

Its input voltage is *v*_{I}, the difference between the voltage to ground of the + input, *v*_{I+}, and the – input, *v*_{I-}, which is the *differential input voltage*,

The *common-mode input voltage* is defined as the average of the two input voltages to ground;

Ground is the – terminal of the output port of the amplifier. As *v*_{CM} changes, no change should occur in the output of a true differential amplifier. It should amplify only the difference voltage across its input - hence the name *differential* amplifier.

The one-op-amp diff-amp has two voltage dividers, each associated with one of the op-amp inputs. Their voltage transfer functions are

;

From them, the circuit equations are

;

These equations can be expressed as a simplified block diagram of the amplifier, shown below, for infinite op-amp open-loop gain: *K* →
∞
.

For matched resistors: *R*_{i}+ = *R*_{i}- = *R*_{i} and *R*_{f}+ = *R*_{f}- = *R*_{f} , then the familiar gain formulas result from the cascaded blocks:

Real op-amps not only have a finite *K*; they also have a non-zero CM gain, *K*_{CM}. The op-amp itself has a differential (open-loop) voltage gain of

where its differential input voltage is *v*_{+} – *v*_{-} , and a non-zero common-mode voltage gain of

When the input circuit equations are substituted into the equation for *v*_{O} ,

Solving for the output voltage,

The voltage gain of the diff-amp is defined as

and its CM voltage gain is

Then the output voltage can be expressed in them as

When the coefficients of *v*_{+} and *v*_{-} are equated to the corresponding coefficients in the circuit equation for *v*_{O} above and solved for the differential and CM gains, they are

For an ideal diff-amp with *A*_{CM} = 0, the conditions from the above equation are that *K*_{CM} = 0 and that the dividers are matched: *T*_{i+} = *T*_{i-}. Resistor mismatch in the dividers can increase *A*_{CM} which decreases a performance parameter defined as the *common-mode rejection ratio*,

The CMRR is the ratio of the differential to CM gains of the diff-amp and it expresses numerically how much more of the desired gain (*A*_{v}) the amplifier has than undesired gain (*A*_{CM}).

**Diff-Amp Gain Errors**

In 1991, a classic paper was published in the *IEEE Transactions on Instrumentation and Measurement* (Vol. 40, No. 4, AUG 1991), titled “Common Mode Rejection Ratio in Differential Amplifiers” by Ramón Pallás-Areny and John G. Webster. It includes the previous derivation of one-op-amp diff-amp differential and CM gains. They went further, to include in these gains divider mismatch as expressed by their resistor tolerances. The voltage gains can be expressed from each of the inputs as

;

The output voltage can be expressed in these gains as

The new quantities,

,

are the node resistances at the inputs of the op-amp. They simplify the voltage divider expressions. For instance,

which is the non-inverting diff-amp voltage gain. Similarly,

and

It is more convenient to use R+ and R– to simplify the divider expressions.

The diff-amp gains can be expressed in the input gains as

Solving for the input gains,

;

The gains that include the resistors for an op-amp with *K* →
∞
with balanced inputs

Resistor error tolerance is assumed the same for all the resistors. In other words, all have the same tolerance rating of, for instance,
*ε*
= +/-1 % or
*ε*
= +/-0.1 %. With tolerance included, each resistor value has the form, *R*(1 +/-
*ε*
) and whether *ε* is + or – depends on which sign will maximize error for a worst-case analysis. With tolerances included, the gains are

where the ideal diff-amp gain is

The CM gain is

,

The boxed equations are the design formulas.

With the above expressions for the gains, substituting and simplifying,

The *CMRR* can include resistor mismatch by separating the effects of op-amp and resistor-mismatch. Given that

and assuming that the op-amp *K*_{CM} = 0, then the diff-amp *CMRR* is only a result of resistor mismatch:

Then dividing the *CMRR* expression through by (*T*_{i}+ – *T*_{i}- ), and with op-amp *K*_{CM} included,

,

The diff-amp *CMRR* with an op-amp having a *CMRR* as given is approximately the parallel combination of the resistor-mismatch and op-amp *CMRR*s. For resistor mismatch alone,

and therefore

A diff-amp with an ideal gain of 1 and resistor tolerance of 1 % can be expected to have a *CMRR* of no less than

if the op-amp CMRR, *K/K*_{CM} >> 50 (≈ 34 dB). Typical op-amp *CMRR* ranges from 1000 (60 dB) to 10^{5} (100 dB). Consequently, resistor mismatch can easily dominate *CMRR* performance. The use of 0.1 % resistors increases *CMRR* to about 500 (54 dB). In many cases, a trimpot is used for *R*_{f}+ to cause the *T*_{i} to become matched, thereby increasing the diff-amp *CMRR* immensely so that the op-amp *CMRR* becomes a factor.