A differential amplifier is a type of electronic amplifier that amplifies the difference between two input voltages but suppresses any voltage common to the two inputs. It is an analog circuit with two inputs and and one output , in which the output is ideally proportional to the difference between the two voltages:
where is the gain of the amplifier.
Single amplifiers are usually implemented by either adding the appropriate feedback resistors to a standard op-amp, or with a dedicated integrated circuit containing internal feedback resistors. It is also a common sub-component of larger integrated circuits handling analog signals.
Mathematics of the amplifier
where and are the input voltages, and is the differential gain.
In practice, however, the gain is not quite equal for the two inputs. This means, for instance, that if and are equal, the output will not be zero, as it would be in the ideal case. A more realistic expression for the output of a differential amplifier thus includes a second term:
where is called the common-mode gain of the amplifier.
As differential amplifiers are often used to null out noise or bias voltages that appear at both inputs, a low common-mode gain is usually desired.
The common-mode rejection ratio (CMRR), usually defined as the ratio between differential-mode gain and common-mode gain, indicates the ability of the amplifier to accurately cancel voltages that are common to both inputs. The common-mode rejection ratio is defined as
In a perfectly symmetric differential amplifier, is zero, and the CMRR is infinite. Note that a differential amplifier is a more general form of amplifier than one with a single input; by grounding one input of a differential amplifier, a single-ended amplifier results.
Long-tailed pair
Historical background
Modern differential amplifiers are usually implemented with a basic two-transistor circuit called a “long-tailed” pair or differential pair. This circuit was originally implemented using a pair of vacuum tubes. The circuit works the same way for all three-terminal devices with current gain. The bias points of “long-tail” resistor circuit are largely determined by Ohm's law and less so by active-component characteristics.
The long-tailed pair was developed from earlier knowledge of push–pull circuit techniques and measurement bridges. An early circuit which closely resembles a long-tailed pair was published by British neurophysiologist Bryan Matthews in 1934, and it seems likely that this was intended to be a true long-tailed pair but was published with a drawing error. The earliest definite long-tailed pair circuit appears in a patent submitted by Alan Blumlein in 1936. By the end of the 1930s the topology was well established and had been described by various authors, including Frank Offner (1937), Otto Schmitt (1937) and Jan Friedrich Toennies (1938), and it was particularly used for detection and measurement of physiological impulses.Geddes, L. A. Who Invented the Differential Amplifier?. IEEE Engineering in Medicine and Biology, May/June 1996, p. 116–117.
The long-tailed pair was very successfully used in early British computing, most notably the Pilot ACE model and descendants,Details of the long-tailed pair circuitry used in early computing can be found in Alan Turing’s Automatic Computing Engine (Oxford University Press, 2005, ) in Part IV, "ELECTRONICS". Maurice Wilkes’ EDSAC, and probably others designed by people who worked with Blumlein or his peers. The long-tailed pair has many favorable attributes if used as a switch: largely immune to tube (transistor) variations (of great importance when machines contained 1,000 tubes or more), high gain, gain stability, high input impedance, medium/low output impedance, good clipper (with a not-too-long tail), non-inverting (EDSAC contained no inverters!) and large output voltage swings. One disadvantage is that the output voltage swing (typically ±10–20 V) was imposed upon a high DC voltage (200 V or so), requiring care in signal coupling, usually some form of wide-band DC coupling. Many computers of this time tried to avoid this problem by using only AC-coupled pulse logic, which made them very large and overly complex (ENIAC: 18,000 tubes for a 20-digit calculator) or unreliable. DC-coupled circuitry became the norm after the first generation of vacuum-tube computers.
Configurations
A differential (long-tailed,Long-tail is a figurative name of high resistance that represents the high emitter resistance at common mode with a common long tail with a proportional length (at differential mode this tail shortens up to zero). If additional emitter resistors with small resistances are included between the emitters and the common node (to introduce a small negative feedback at differential mode), they can be figuratively represented by short tails. emitter-coupled) pair amplifier consists of two amplifying stages with common (emitter, source or cathode) degeneration.
Differential output
With two inputs and two outputs, this forms a differential amplifier stage (Figure 2). The two bases (or grids or gates) are inputs which are differentially amplified (subtracted and multiplied) by the transistor pair; they can be fed with a differential (balanced) input signal, or one input could be grounded to form a phase splitter circuit. An amplifier with differential output can drive a floating load or another stage with differential input.
Single-ended output If the differential output is not desired, then only one output can be used (taken from just one of the collectors (or anodes or drains), disregarding the other output; this configuration is referred to as single-ended output. The gain is half that of the stage with differential output. To avoid sacrificing gain, a differential to single-ended converter can be utilized. This is often implemented as a current mirror (Figure 3, below).
Single-ended input
The differential pair can be used as an amplifier with a single-ended input if one of the inputs is grounded or fixed to a reference voltage (usually, the other collector is used as a single-ended output) This arrangement can be thought of as cascaded common-collector and common-base stages or as a buffered common-base stage.More generally, this arrangement can be considered as two interacting voltage followers with negative feedback: the output part of the differential pair acts as a voltage follower with constant input voltage (a voltage stabilizer) producing constant output voltage; the input part acts as a voltage follower with varying input voltage trying to change the steady output voltage of the stabilizer. The stabilizer reacts to this intervention by changing its output quantity (current, respectively voltage) that serves as a circuit output.
The emitter-coupled amplifier is compensated for temperature drifts, VBE is cancelled, and the Miller effect and transistor saturation are avoided. That is why it is used to form emitter-coupled amplifiers (avoiding Miller effect), phase splitter circuits (obtaining two inverse voltages), ECL gates and switches (avoiding transistor saturation), etc.
Operation
To explain the circuit operation, four particular modes are isolated below although, in practice, some of them act simultaneously and their effects are superimposed.
Biasing
In contrast with classic amplifying stages that are biased from the side of the base (and so they are highly β-dependent), the differential pair is directly biased from the side of the emitters by sinking/injecting the total quiescent current. The series negative feedback (the emitter degeneration) makes the transistors act as voltage stabilizers; it forces them to adjust their VBE voltages (base currents) to pass the quiescent current through their collector-emitter junctions.Interestingly, it is as though the negative feedback has reversed the transistor behavior - the collector current has become an input quantity while the base current serves as an output one. So, due to the negative feedback, the quiescent current depends only slightly on the transistor's β.
The biasing base currents needed to evoke the quiescent collector currents usually come from the ground, pass through the input sources and enter the bases. So, the sources have to be galvanic (DC) to ensure paths for the biasing current and low resistive enough to not create significant voltage drops across them. Otherwise, additional DC elements should be connected between the bases and the ground (or the positive power supply).
Common mode
In common mode (the two input voltages change in the same directions), the two voltage (emitter) followers cooperate with each other working together on the common high-resistive emitter load (the "long tail"). They all together increase or decrease the voltage of the common emitter point (figuratively speaking, they together "pull up" or "pull down" it so that it moves). In addition, the dynamic load "helps" them by changing its instant ohmic resistance in the same direction as the input voltages (it increases when the voltage increases and vice versa.) thus keeping up constant total resistance between the two supply rails. There is a full (100%) negative feedback; the two input base voltages and the emitter voltage change simultaneously while the collector currents and the total current do not change. As a result, the output collector voltages do not change as well.
Differential mode
Normal. In differential mode (the two input voltages change in opposite directions), the two voltage (emitter) followers oppose each other—while one of them tries to increase the voltage of the common emitter point, the other tries to decrease it (figuratively speaking, one of them "pulls up" the common point while the other "pulls down" it so that it stays immovable) and vice versa. So, the common point does not change its voltage; it behaves like a virtual ground with a magnitude determined by the common-mode input voltages. The high-resistance emitter element does not play any role—it is shunted by the other low-resistance emitter follower. There is no negative feedback, since the emitter voltage does not change at all when the input base voltages change. The common quiescent current vigorously steers between the two transistors and the output collector voltages vigorously change. The two transistors mutually ground their emitters; so, although they are common-collector stages, they actually act as common-emitter stages with maximum gain. Bias stability and independence from variations in device parameters can be improved by negative feedback introduced via cathode/emitter resistors with relatively small resistances.
Overdriven. If the input differential voltage changes significantly (more than about a hundred millivolts), the transistor driven by the lower input voltage turns off and its collector voltage reaches the positive supply rail. At high overdrive the base-emitter junction gets reversed. The other transistor (driven by the higher input voltage) drives all the current. If the resistor at the collector is relatively large, the transistor will saturate. With relatively small collector resistor and moderate overdrive, the emitter can still follow the input signal without saturation. This mode is used in differential switches and ECL gates.
Breakdown. If the input voltage continues increasing and exceeds the base-emitter breakdown voltage, the base-emitter junction of the transistor driven by the lower input voltage breaks down. If the input sources are low resistive, an unlimited current will flow directly through the "diode bridge" between the two input sources and will damage them.