Introduction to Differential Pairs

Open any analog IC, and you’ll find a differential pair. More likely, you’ll find a half dozen. It has great advantages, even when amplifying a "difference" isn’t a goal.

The reasoning is simple: individual integrated components have large variations, but two or more of the same components match very well. If you can take advantage of the matching, you get better performance.

This isn't always true, of course. Noise, for example, can be smaller in a single-transistor stage, and some of the most ingenious designs are remarkably free of the common differential stage.

Still, the differential pair is a wondrous tool. Let’s take a look at it.

The Basic Differential Pair

In a differential pair, a current is divided by two transistors. This is illustrated in Figure 5-1.

Figure 5-1. In a differential pair, the current I₁ is divided by two transistors, Q1 and Q2.

Two transistors—in this case, bipolar—share a common emitter current. If the voltages at their bases are equal and the two transistors match perfectly, I₁ is split into two equal parts, I₂ and I₃, at the collectors.

If we increase V₁ relative to V₂, Q1 gets more of the current than Q2. If we decrease V₁, the opposite is true.

Non-Ideal Behavior of Differential Pairs

There are limitations and errors, however, in the operation of the differential pairs.

Nonlinear Operation

First of all, the current division between the two sides of the differential pair isn’t linear with respect to the input voltages. We’re dealing with two base-emitter diodes here—fundamentally exponential devices. Not counting stray effects, the emitter resistance is:

$$r_e ~=~ \frac {kT}{qI_e}$$

where:

k = the Boltzmann constant (1.38E–23 joules/kelvin)

T = the absolute temperature in kelvin

q = the electron charge (1.6E–19 coulombs)

I_e = the operating current through each emitter

At room temperature (27 °C or 298 K) and with a current of 1 mA, this expression amounts to about 26 Ω:

$$r_e ~=~ \frac {kT}{qI_e} ~=~ \frac{(1.38E-23) \times (298)}{(1.6E-19) \times (1E-3)} ~=~ 25.7 \text{ m}\Omega$$

If I_e drops to 100 μA, r_e becomes 260 Ω.

Since it’s very much a function of current, r_e is known as the dynamic emitter resistance. The conversion from base voltage to collector current—the transconductance—is:

$$g_m ~=~ \frac {1}{r_e ~+~R_e}$$

where R_e is the ohmic resistance of the emitter—in other words, the resistance between the emitter contact and the emitter-base junction. This is usually a few Ω.

As the current moves from one transistor to the other, both emitter resistances change. We then get some rather nonlinear behavior, as Figure 5-2 shows. Only a small portion of the curve in the middle, when the two currents are equal or nearly equal, could be called linear. In truth, though, that portion isn’t a straight line either.

Figure 5-2. The conversion from input voltage to current (the transconductance) is nonlinear.

Temperature Effects

The other variable in the equation is temperature. The emitter resistance is proportional to absolute temperature, so at high temperatures, you get less gain or transconductance. This is illustrated in Figure 5-3.

Figure 5-3. The differential pair transconductance is also temperature-dependent.

Errors in Differential Pair Operation

There are also three sources of error to be considered. First of all, a small portion of the emitter current comes from the bases, not the collectors. With a minimum h_FE of 100, this makes the sum of the collector current smaller than the emitter current by 1%.

Secondly, transistors only match well if they are operated identically. In this case, that specifically means the collector voltages have to be the same. If one is higher than the other, its transistor will have a higher gain because of the Early effect.

Finally, devices never match perfectly. There will be some differences in both V_BE and h_FE, and thus some uncertainty in the voltage at which I₂ and I₃ are equal. This shows up as an offset voltage.

MOS Differential Pairs

A differential pair using MOS transistors (Figure 5-4) behaves almost identically but for entirely different reasons. There’s no dynamic emitter resistance—the gain is determined directly by the transconductance.

Figure 5-4. MOS differential pair.

This transconductance is also nonlinear. As illustrated in Figure 5-5, it increases drastically with increasing gate voltage.

Figure 5-5. A CMOS differential pair is also non-linear.

In the bipolar transistor, size is only of second-order importance. In an MOS transistor, however, transconductance is directly proportional to gate width and decreases with increasing temperature (Figure 5-6).

Figure 5-6. The CMOS differential pair transconductance also has a temperature coefficient.

Notably absent in the MOS differential pair error sources is any kind of input current. Here, I₂ plus I₃ is indeed equal to I₁. There is, however, an offset voltage. For equal-sized devices, MOS transistors have a larger offset voltage (mismatch) than bipolar ones. The MOS offset is about 2:1 greater than the bipolar offset, but this depends greatly on the process.

Remember that you can always improve device matching by increasing size (total area). It’s best to do this by using multiple small devices.