Editing Differential amplifier (section)

== Applications ==

Differential amplifiers are found in many circuits that utilize series [[negative feedback]] (op-amp follower, non-inverting amplifier, etc.), where one input is used for the input signal, the other for the feedback signal (usually implemented by [[operational amplifier]]s). For comparison, the old-fashioned inverting single-ended op-amps from the early 1940s could realize only parallel negative feedback by connecting additional resistor networks (an op-amp inverting amplifier is the most popular example). A common application is for the control of [[electric motor|motor]]s or [[Servomechanism|servo]]s, as well as for signal amplification applications. In discrete [[electronics]], a common arrangement for implementing a differential amplifier is the [[#Long-tailed pair|long-tailed pair]], which is also usually found as the differential element in most op-amp [[integrated circuit]]s. A long-tailed pair can be used as an analog multiplier with the differential voltage as one input and the biasing current as another. <!-- maybe mention [[gilbert cell]] here? -->

A differential amplifier is used as the input stage [[emitter coupled logic]] gates and as switch. When used as a switch, the "left" base/grid is used as signal input and the "right" base/grid is grounded; output is taken from the right collector/plate. When the input is zero or negative, the output is close to zero (but can be not saturated); when the input is positive, the output is most-positive, dynamic operation being the same as the amplifier use described above.

The differential amplifier is used in the [[cathode follower oscillator]]. The advantages are high impedance of the differential amplifier input and output and small phase shift between input and output. This application uses only one input and one output of the differential amplifier.

=== Symmetrical feedback network eliminates common-mode gain and common-mode bias ===
[[File:Op-Amp Differential Amplifier input impedence and common bias.svg|thumb|280px|Figure 6: Differential amplifier with non-ideal op-amp: input bias current and differential input impedance]]
In case the operational amplifier's (non-ideal) input bias current or differential input impedance are a significant effect, one can select a feedback network that improves the effect of common-mode input signal and bias. In Figure&nbsp;6, current generators model the input bias current at each terminal; ''I''<sup>+</sup><sub>b</sub> and ''I''<sup>−</sup><sub>b</sub> represent the input bias current at terminals ''V''<sup>+</sup> and ''V''<sup>−</sup> respectively.

The [[Thévenin's theorem|Thévenin equivalent]] for the network driving the ''V''<sup>+</sup> terminal has a voltage ''V''<sup>+</sup>' and impedance ''R''<sup>+</sup>':
: <math>{V^+}' = V^+_\text{in} R^+_\parallel / R^+_\text{i} - I^+_\text{b} R^+_\parallel; \quad \text{where} \quad {R^+}' = R^+_\parallel  = R^+_\text{i} \parallel R^+_\text{f},</math>
while for the network driving the ''V''<sup>−</sup> terminal:
: <math>{V^-}' =  V^-_\text{in} R^-_\parallel / R^-_\text{i} + V_\text{out} R^-_\parallel / R^-_\text{f} - I^-_\text{b} R^-_\parallel; \quad \text{where} \quad {R^-}' = R^-_\parallel = R^-_\text{i} \parallel R^-_\text{f}.</math>
The output of the op-amp is just the open-loop gain ''A''<sub>ol</sub> times the differential input current ''i'' times the differential input impedance 2''R''<sub>d</sub>, therefore
: <math> V_\text{out} = A_\text{ol} \cdot 2 R_\text{d} \frac{{V^+}' - {V^-}'}{2R_\parallel + 2R_\text{d}} = ({V^+}' - {V^-}') A_\text{ol} R_\parallel / (R_\parallel \parallel R_\text{d}),</math>
where ''R''<sub>||</sub> is the average of ''R''<sup>+</sup><sub>||</sub> and ''R''<sup>−</sup><sub>||</sub>.

These equations undergo a great simplification if 
: <math>R^+_\text{i} = R^-_\text{i}, \quad R^+_\text{f} = R^-_\text{f},</math>
resulting in the relation
: <math>V^+_\text{in} - V^-_\text{in} - R_\text{i} I^\Delta_\text{b} = V_\text{out} \left[ \frac{R_\text{i}}{R_\text{f}} + \frac{1}{A_\text{ol} \frac{R_\text{i}}{R_\text{i} \parallel R_\text{f} \parallel R_\text{d}}}\right],</math>
which implies that the closed-loop gain for the differential signal is ''V''<sup>+</sup><sub>in</sub>&nbsp;−&nbsp;''V''<sup>−</sup><sub>in</sub>, but the common-mode gain is identically zero. 

It also implies that the common-mode input bias current has cancelled out, leaving only the input offset current ''I''<sup>Δ</sup><sub>b</sub> = ''I''<sup>+</sup><sub>b</sub>&nbsp;−&nbsp;''I''<sup>−</sup><sub>b</sub> still present, and with a coefficient of ''R''<sub>i</sub>. It is as if the input offset current is equivalent to an input offset voltage acting across an input resistance ''R''<sub>i</sub>, which is the source resistance of the feedback network into the input terminals.

Finally, as long as the open-loop voltage gain ''A''<sub>ol</sub> is much larger than unity, the closed-loop voltage gain is ''R''<sub>f</sub>/''R''<sub>i</sub>, the value one would obtain through the rule-of-thumb analysis known as "virtual ground".<ref 
group="nb">For the closed-loop common-mode gain to be zero only requires that the ratio of resistances ''R''<sub>f</sub> / ''R''<sub>i</sub> be matched in the inverting and non-inverting legs. For the input bias currents to cancel, the stricter relation given here must obtain.</ref>