# Temperature Compensation of BJT Diff-Amps

The bipolar junction transistor (BJT)-pair emitter-coupled differential-amplifier circuit is a familiar amplifier stage in the repertoire of analog designers, but has an interesting complication. This article examines the emitter-circuit current source, I0 , of BJT diff-amps and the effects on amplifier gain of different implementations for it.

The widespread belief that a BJT current source can temperature-compensate the BJT-pair diff-amp is true, but the conditions for it do not appear to be widely known. The typical circuit is shown below.

This is a differential-input, differential-output voltage amplifier. With both input and output quantities differential, the incremental voltage gain of the circuit is

The condition for differential amplification is that A ν 1 = A ν 2 . The circuit is made symmetrical whenever

Then the voltage gain becomes

where rM is the transresistance , the resistance across which the input voltage develops the (pre-α ) output current.

A goal of good design is to make Aν a fixed value. The choice of resistors with a low temperature coefficient (TC) and sufficiently tight accuracy is one factor. This is usually easy to achieve, though for high-precision design, the change in resistance caused by a change in ambient temperature is a factor to be considered. Even more so are “thermals”, dynamic, waveform-related changes in resistance caused by changes in power dissipation with νi . For very precise design, the change in resistance with applied voltage must be considered too.

Other transistor parameters than the two (re and β ) of the BJT T model used here – namely, ro – also need to be included for precision design. We will assume that the BJTs have a sufficiently high Early voltage that ro need not come into our list of considerations – at least not here. In practice, this assumption is often valid.

BJTs are typically the least ideal elements of the circuit. From the gain formula, it is evident that two BJT parameters affect gain, the incremental emitter resistance, re , and β . For high β – that is, for β >> 1 – the gain factor,

approaches 1. For a typical β value of 200, then α = 0.995, contributing a gain error of 0.5 %. If that is too much error, α -compensation techniques are required. Usually, this error can be compensated by including it in the gain formula, as we have done. What is more important is how much it drifts with temperature. Typical

Then for large β , α has a TC of around 50 ppm; α is usually not much of a problem.

The rM transresistance expression of Av – the denominator – is the resistance across which the input voltage develops the (emitter) current that is common to input and output loops. The output current is modified by α , which accounts for loss along the way from the emitter circuit. This transresistance, rM , also includes β in the RB term. If RB is kept small, and the inputs are driven by voltage sources, then this β is of no concern. If the sources are high in resistance, then the RB term will affect gain by β variation with temperature. Its 1 %/o C variation is scaled down by the extent to which RB /(β + 1) is not dominant in rM . Keeping RB small is another design factor.

The most troublesome term in rM is re , for it varies with temperature and emitter current, IE , according to

With IE constant, re varies with the thermal voltage, VT , which varies in proportion with absolute temperature;

At 300 K (about 80o F), this is 1/300 K or about 0.33 %/K = 0.33 %/o C. For laboratory-quality instrument design, let us suppose, the temperature range over which the equipment ought to be able to operate within its specifications is over 25o C +/- 15o C which is 10o C to 40o C. Over a 15o C maximum change from ambient, VT changes about 5 % – far too much for most precision designs. Therefore, VT variation in gain needs to be compensated.

The simplest compensation for re is to make it a negligible term (along with the RB term) in rM . This is accomplished by making RE dominant. For RE >> re , the drift in re affects gain far less than 5 %. In many cases, dominant external emitter resistance solves the drift problem, but at the expense of gain and power dissipation. By increasing I0 , then re is reduced proportionally, though circuit power increases. This is not only undesirable for power-limited equipment but it also exacerbates thermals by increasing ΔPD (νi ) in the BJTs.

In some cases, re cannot be made negligible, and some compensation for it is desired. One of the most common schemes is to make I0 track re and cancel its effect, at least approximately. To make I0 have the TC of VT , the simplest scheme is to use a BJT current source implementation of I0 . The b-e junction voltage of the current-source BJT then decreases with temperature, I0 increases, and decreases re .

Current-Source Circuits

The first circuit sourcing I0 that we will consider is simply a resistor, R0 , returned to a negative supply. This “long-tail” current source approaches an ideal current source as the supply voltage, –V , approaches negative infinity or the value of R0 approaches infinity. It does nothing to compensate for the TC of re .

The second implementation to consider is shown below.

This simple circuit has a voltage across R0 of V – VBE (Q0). As temperature increases, VBE decreases but not by the TC of VT . The other major BJT parameter affecting VBE is the saturation current, IS , as found in the p-n junction (b-e junction) voltage equation,

For a typical BJT, such as a PN3904, IS ≈ 10 fA. Then 1 mA of current produces a VBE ≅ 0.65 V.

Both VT and IS contribute to the TC(VBE ). IS has a greater effect than VT and of opposite polarity on VBE , resulting in a combined effect of about –2 mV/o C for VBE . (For more on BJT TC effects, see the volume, Signal-Processing Circuits of Analog Circuit Design , by the author; check at www.innovatia.com.) It is therefore more important to cancel IS effects than those of VT .

Depending on the relative values of V and VBE , the effect of the TC(VBE ) can be scaled by choice of RE and supply voltage, V , which is often constrained by system-level design. By adding a resistor network between the emitter and ground, a Thevenin equivalent supply voltage and value of R0 can be independently set. If scaled properly, as T increases, VBE decreases and I0 increases. If the increase is made to be such that the reduction in re caused by it cancels the increase in re caused by VT , then BJT-pair re and gain remain constant.

The TC(re ) is calculated as follows, by differentiating re with respect to T ;

where TC% is the fractional change in TC.

Setting TC%(I0 ) = TC%(IE ) can be constructed as follows. The only change across R0 is from VBE . Therefore, the fractional change in I0 with T is

Setting TC%(I0 ) = TC%(VT ) = 1/T ≅ 0.33 %/o C, the voltage across R0 , V – VBE = 0.6 V. With −V = −1.25 V, this is not an attractive compensation scheme. The polarity of TC(I0 ) is correct for compensation, but not its magnitude, which leads to the next scheme, shown below.

This implementation of I0 is more versatile and more common in occurrence than the previous scheme. The base divider provides extra freedom for setting TC%(I0 ), which, for ignored TC(β ), is now

The divider ratio that gives the correct compensation can now be found. When TC%(I0 ) is set equal to TC%(VT ), then

or

for VBE = 0.65 V. This result is interesting; whatever the value of V , the unloaded divider voltage must be 1.25 V for gain compensation. This is also the voltage of bandgap references, as well it should be. Bandgap circuits use the negative TC(VBE ) and scale it to cancel the positive TC(VT ). Then the resulting bandgap voltage always comes out to be close to 1.25 V, and varies slightly with BJT doping levels.

Another current-source variation that is often used to provide rough temperature compensation is to insert a diode in series with R2 , as shown below.

The usual explanation is that the TC of the diode compensates for the TC of the BJT ¬b-e ¬ junction, resulting in a more stable I0 . A typical example is to use a 1N4152 diode to compensate a PN3904. The diode and BJT b-e junctions are quite different, however. The junction gradients are different and diode doping levels are far less than the BJT base, to achieve a higher breakdown voltage. The emitter minority carrier concentration is made intentionally large for good emitter injection efficiency into the base, at the expense of VBE reverse breakdown, which is typically around 7 V, much below the 40 V of the diode. The point is that although both junctions are silicon, they are rather unmatched.

Suppose, however, that a similar BJT b-e ¬ junction is used as a diode, with base connected to collector. Then the junction matching is much better (though not as good as adjacent integrated BJTs), and allowing for α ≅ 1, then applying KVL around the BJT input loop (with IS of the two BJT junctions cancelling),

or

where ID is the diode current. If the junction currents are equal, the TC of VT is removed and the TC%(I0 ) ≅ 0 %/o C. This is useful for applications where a stable current source is needed, but it does not compensate re of the diff-amp. The currents must deliberately be set unequal to achieve the desired TC, and for a compensating polarity of TC, it must be positive. Consequently, we must have ID > I0 .

The TC%(I0 ) is found through implicit differentiation of I0 in the above equation;

With additional algebraic manipulation,

Then to compensate, set TC%(I0 ) = TC%(VT) = 1/T and solve;

Because of the exponential function, practical current ratios require that the voltage across R0 be not much larger than VT . For I0 = 2 mA, R0 = 22 Ω, and VT = 26 mV, then the voltage across R0 is 44 mV, or 1.69xVT , and ID = 14.77xI0 = 29.5 mA, larger than is desired in most designs. Such small values of R0 are required to keep R0 from dominating the emitter circuit so that the TC of VBE can be expressed. Yet in many designs, R0 is relatively large and its voltage drop far exceeds VT . As a consequence, the TC%(VT ) of re is not correctly compensated and a TC drift in gain exists.

The previous scheme, which omitted the base diode, was only slightly better in allowing for larger R0 voltage. Perhaps we should go in the opposite direction and add a diode or two in the emitter. The TC of the combined junctions would be multiplied by the number of them, and that would allow RE to be made proportionally larger. It is usually not desirable to add a large number of series diodes because the static stability of I0 is degraded. Consequently, use of the diff-amp current source to temperature-compensate re results in a circuit requiring careful static design. I0 is then made sensitive to junction parameters, and these parameters, such as IS , have a somewhat wide tolerance among discrete transistors, even of the same part number. Expect as much as 50 mV of variation among PN3904 BJTs at the same current and temperature from different suppliers or manufacturing lots. This compensation method is best suited for monolithic integration.

Closure

The widespread belief that a BJT current source can temperature-compensate a BJT diff-amp is true, but it often does not. Temperature compensation of I0 for a constant re results in low voltage across the current-source external emitter resistance, R0 – so low that it can make accurate setting of I0 infeasible.

Consequently, except for more elaborate schemes which amplify VT , the dominant-RE approach to diff-amp gain stability appears to be acceptable in some designs. Another scheme, with multiple stages, is to use a successive complementary (PNP) stage to cancel the gain TC of the first stage.

Acknowledgement

Credit goes to Robert A. Pease of National Semiconductor Corp. who, when he was still with us, found an error in my math, leading to improvements. A related article by him is in his compendium, Analog Circuits: World Class Designs (Newnes-Elsevier, 2008; www.newnespress.com) in the chapter titled “What’s All This VBE Stuff, Anyhow?” (pp. 383 – 391) where he gives a bench-oriented explanation of some of the subtleties of the b-e junction.

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