The two inputs of a diff-amp can either be considered independent or be combined as differential and common-mode (CM) quantities. A diff-amp functionally amplifies only the differential input voltage, *v*_{I}, while ideally not amplifying the CM voltage, *v*_{CM}, at all: *A*_{CM} = 0. Actual op-amps and the diff-amps constructed from them have a non-zero *A*_{CM}. A “figure of merit” or *performance parameter* for diff-amps is the *common-mode rejection ratio*,

The *CMRR* is the ratio of the differential and CM gains -- the desired over the undesired. Thus it is desirable that *CMRR* be maximized in a diff-amp. In the previous parts of this article, it was assumed that the op-amp *CMRR* is infinite because it was assumed that *A*_{CM} of the op-amp, or *K*_{CM} = 0, where by definition

and the op-amp differential gain, as included in the original circuit equation for *v*_{O} (in Part 1) is

Op-amp specifications usually include the *CMRR*, and it is typically 3 to 5 decades (10^{3} to 10^{5}), or in the more awkward scaling units of dB, 60 to 80 dB. By including *K*_{CM}, the original three diff-amp circuit equations are modified by adding the CM term to *v*_{O}:

As before, the other two circuit equations are

The three circuit coefficients, *T*_{i+}, *T*_{i-}, and *A*_{0} are part of the one-op-amp diff-amp model as developed in Part 1 of this article. By substituting for *v*_{+} and *v*_{-} in the *v*_{O} equation and simplifying,

This somewhat complicated expression can be unraveled into meaningful quantities when expressed in terms of differential and CM input voltage. From Part 1,

Alternatively, it can be expressed in the diff-amp input voltages as

When the coefficients of this equation are equated with those in the complicated expression for vO (second prior equation), then the diff-amp gains with *K*_{CM} included become

The *CMRR* of the diff-amp is thus

When the op-amp *CMRR* is ideal, *K*_{CM} = 0 and

The non-ideal op-amp has

Then factoring *K*_{CM} out of numerator and denominator of the full *CMRR* expression, it simplifies to

The 1/4 in the numerator is negligible whenever

Then the *CMRR* expression collapses to the parallel formula:

This is the classic formula for *CMRR* with op-amp *CMRR* = *K*/*K*_{CM} included.

Finally, the effect of diff-amp resistor mismatch on *CMRR* (with infinite op-amp *CMRR*) is

or for R mismatch only,

Then solving for *A*_{CM},

which agrees with the derivation for *A*_{CM} in Part 3 of this article.

The relevance of these equations can be demonstrated by a mundane diff-amp example. An LM324 quad op-amp is used to make a ×4 one-op-amp diff-amp as shown below, extracted from the circuit diagram of a commodity-grade instrument, the Innovatia TPA242 two-port analyzer (curve tracer), showing the input-port SMU.

An LM324 op-amp, U25C forms a one-op-amp diff-amp with *A*_{v} = 4. R48, R89, R90 constitute a trimmed resistance that nulls the common-mode gain by matching the two divider ratios as close as a manual trimpot will allow. Digipots are not yet comparable in resolution, which is typically 3 to 4 decades. The CM ADJ trim range is somewhat larger than +/- 1 %. The input offset voltage of the commodity op-amp is typically within about 5 mV. IISN is the input-port current sense output to an ADC with an input range of +/- 4 V. The diff-amp input has a CM voltage range determined by the output range of the port, specified as 24 V per polarity. The derived design formulas lead to a quick assessment of the amount of error at the diff-amp output.

First, the error caused by the input offset voltage, *V*_{IOS} = 5 mV, is about (*A*_{v} + 1)•(5 mV) = 25 mV. U25D contributes another 25 mV. Both are nulled by the R49 trimpot. The remaining concern is the offset-voltage drift which is specified as typically 7μV/^{O}C. The Innovatia standard instrument temperature range is 20 /^{O}C +/- 15 /^{O}C. The maximum typical offset voltage error from drift is thus 0.105 mV. At the maximum gain of 400 (including the second-stage PGA), that amounts to an error of 42 mV out of a fs 4 V output (per polarity), or about 1 % error.

For this grade of instrument, this is an acceptable maximum error that cannot (easily) be compensated with analog circuitry. It can easily be improved by using a better op-amp, almost all of which are at a higher price. The TPA242 goal was to build as good a TPA as is possible with commodity-grade parts. With a better op-amp, however, it might be possible to eliminate the offset trim, but probably not the CM trim. The desirable elimination of both trimpots is not feasible. The use of 0.1 % gain-setting resistors in the diff-amp is calculated later and shows this.

Second, the differential gain error can be found from

where *A*_{v}0 = 4 and the resistor gain tolerance is ε = 1 % = 0.01. Then *A*_{v} = 3.937 = 4 V – 63 mV, or –63 mV of error for a full-scale 4 V output. This is about –1 % of error and is within the μC two-point calibration range of the ADC.

Third, the CM error is calculated from

and is essentially nulled by the CM trim. However, if 0.1 % resistors were used in an effort to eliminate the trimpot, *A*_{CM} ≈ 1/312.5. Over the input range (for a given output polarity) of 24 V, the CM output, which is an error voltage that adds to the others, is 24 V/312.5 = 77 mV. The CM error is 12 times the differential-gain error with 1 % gain-setting resistors. The CM voltage error is only that of the diff-amp first stage. The PGA, with a maximum gain range of 100, results in an output (to the ADC) voltage error of (77 mV)•(100) = 7.7 V, which is beyond the fs output range. Even 0.1 % resistors do not result in sufficient nulling of the CM gain. This example shows how important it is to not overlook the common-mode effect of *A*_{CM} in the design of diff-amps.

Another (fourth) error not previously considered is from finite op-amp gain, *K* ≈ 25 k. The closed-loop gain is

and the gain error is 3.999 – 4 ≈ –640 ppm. This is a μC-compensated error for this circuit. For some precision applications, op-amps do not always have sufficiently high open-loop gain.

It is not a formality for op-amp manufacturers to include in the specifications all those seemingly irrelevant parameters such as *CMRR*. From the derived formulas, the above diff-amp *CMRR* ignoring the op-amp *CMRR* of *K*/*K*_{CM} is

The specified LM324 (with the same amplifier as the dual LM358) *CMRR* = *K*/*K*_{CM} = 65 dB = 10^{65/20} = 1778. This is not much larger than the diff-amp R-mismatch *CMRR*_{0} for 0.1 % resistors, and the combined diff-amp

The CM trim is even more necessary with finite (and in this case, somewhat low) op-amp *CMRR*.

**Closure**

Some algebra was required to derive the important design equations for the one-op-amp diff-amp when resistor mismatch and finite op-amp *K* and *K*_{CM} are included. It is not uncommon that these imperfections cause significant degradation in amplifier performance and must be taken into account in circuit design.

Op-amps have been around since at least the 1960s, yet it was not until 1991 that the paper presenting some of these essential formulas was first published, titled “Common-Mode Rejection Ratio in Differential Amplifiers,” in the *IEEE Transactions on Instrumentation and Measurement*, vol. 40, no. 4, Aug. 91, pp. 669-676. The authors were Ramón Pallás-Areny from Barcelona and John G. Webster. Does technological progress move swiftly? Sometimes important design equations are not derived and disseminated for decades!

Hopefully, this multipart diff-amp article filled in possible “holes” in your knowledge of diff-amps, if they existed. If so, then it also served the purpose of alerting you to the possibility of more invisible “holes” in other areas of seemingly simple circuit theory.