The following is Part 1 of a four-part series.

The basic *one-op-amp differential amplifier* consists of one op-amp and four resistors. Yet few have contemplated the depth of its complexity. It is studied in undergraduate engineering school and is familiar to analog-circuit engineers. Yet familiarity can easily be mistaken for understanding. Most of us have used it multiple times in designs. This article is intended take some readers deeper into diff-amp theory than they might have thought even existed, yet what is presented can reveal otherwise inexplicable design problems with this simple circuit.

A *differential amplifier*, or *diff-amp* for short, is a two-port circuit that inputs two quantities and amplifies their difference. The most familiar example is the op-amp, with inputs marked + and – and with a voltage gain that is ideally infinite. Op-amps can be used to construct finite-gain differential amplifiers or *diff-amps* for short. The most commonly used and simplest is the *one-op-amp diff-amp*, shown below, with input-range offset and common-mode (CM) voltage extension; the third voltage, *V*_{OS}, has been included along with two offset resistors, *R*_{OS+} and *R*_{OS–} .

The differential input voltage of a diff-amp is defined as:

The difference of two voltages as *v*_{I} tells us nothing about their actual values, which can be far removed from 0 V while their difference can be within a specified input range. The *common-mode* (CM) voltage is defined as

It is the voltage added to both of the differential inputs relative to input ground (0 V). The CM voltage is simply the average voltage of the two input terminals. The input voltages straddle *v*_{CM} and from the above two equations are

Both + and − inputs are *floating* above ground at the CM voltage. Set *v*_{I} = 0 (shorted input terminals) so that *v*_{I+} = *v*_{I-} and *v*_{CM} is applied to both input terminals. It is simply the input offset of *v*_{I}.

The output voltage of the amplifier can be expressed in either of the two input voltages, or as their differential and common-mode combinations:

(Offset voltage *V*_{OS} as a third input voltage is included later.) The CM voltage gain *A*_{CM} and differential voltage gain *A*_{v} can be expressed in the voltage gains of each input to the output as

and

For an ideal differential amplifier, the gains of the two inputs are of the same magnitude and *A*_{CM} = 0. These basic equations apply to diff-amps generally.

The amplifier gains can be expressed in circuit component values by solving the one-op-amp diff-amp circuit in various ways. Here, a path less traveled will illustrate a simplifying method based on finding the node resistances at the op-amp inputs. This can be accomplished by inspection, assuming the inputs are driven by voltage sources. Then

where || is the parallel math operation. These node resistances can be used in divider formulas as follows:

where *R*_{out} and *R*_{in} are the divider output and input resistances. Applying divider formulas and voltage superposition to the circuit,

The op-amp output is

where *K* is the op-amp open-loop differential gain. When *K* approaches infinity, then

The usefulness of this rather involved form of the op-amp output equation that includes each resistance is that it allows us to later analyze the effects of resistance tolerance on accuracy. Ideally,

Under these ideal conditions,

and the input CM voltage range is

where *V*_{+max} is the maximum *v*_{+} voltage of the op-amp input range. For rail-to-rail-input (RRI) op-amps, this is the op-amp positive supply voltage.

Adding *R*_{OS+} and *R*_{OS–} to the circuit do not change either of the above two formulas. The *R*_{OS} increase the CM input voltage range by increasing the divider input attenuation (by making *R*_{+} smaller) and at the same time increase the op-amp feedback divider attenuation which increases the non-inverting closed-loop gain. The effects of the two dividers cancel so that *A*_{v} is unaffected. However, there is an effect on the circuit in that by dividing down more from the input and then amplifying more, the output is noisier and the op-amp input offset voltage, *V*_{IOS}, is also amplified more. The advantage is that the CM range is extended by the increased input attenuation but at the design cost of a more precise and less noisy op-amp.

This amplifier can be used before the ADC in data-acquisition systems to provide a wider range for the input voltage than the ADC allows. For instance, suppose a single-supply, 10-bit ADC is driven by it, and the ADC has an input range of 0 V to 5 V. It is desired that the differential input span a range of ±2.5 V with a common-mode range of 0 V to over 30 V. By adding the *R*_{OS} resistors, a given differential gain, *A*_{v}, can be chosen while extending the CM input voltage range. Otherwise, omitting the *R*_{OS} resistors, *R*_{f+} can be offset (as shown) by *V*_{OS} with the same gain equation, but for a given *A*_{v}, the CM range will also be determined. The above formula for *v*_{CM} is valid, but *R*_{i+}/R_{+} is determined by *R*_{f+} and *R*_{i+}.

The amplifier shown above has three inputs and one output and can be represented by a block diagram based on the above *v*_{O} equation, as shown below. These are the equations in graphical form.