What a pair cancels, and what it only halves

Two devices sharing a tail current, driven in opposition, produce an output current proportional to tanh(v/2V_T). That characteristic is the whole subject of this page, and one property of it does all the work: it is an odd function.

An odd function has no even terms in its power series. Drive it with a sinusoid and no even harmonic can appear in the output, because there is nothing in the series that could produce one. The cancellation is structural — a consequence of the shape of the curve rather than of matching, trimming or biasing — and that is a much stronger kind of statement than any number.

Zero, and how it is different from small #

The measured second harmonic of the pair is 1.5×10⁻¹⁶ of its fundamental at twenty millivolts of drive, and 4.1×10⁻¹⁷ at fifty. The single stage’s is 9.6% and 42.2%.

The pair’s numbers are not measurements of a physical quantity. They are the residue of summing five hundred and twelve floating-point samples, and they move about with the drive in a way that has no pattern because there is nothing underneath them to have a pattern. That is worth saying plainly, because a figure that reported “second harmonic: 1.5×10⁻¹⁶” without comment would invite the question of what physical effect produces 10⁻¹⁶ of second harmonic, and the answer is none.

The distinction between small and zero is the reason this topology is used in almost every amplifier input stage ever built. A small second harmonic could be made smaller by driving the stage less hard, and would have to be re-checked every time anything changed. A zero one is zero at every drive, every bias current and every temperature, because it follows from the symmetry of the circuit rather than from the values in it.

What is not guaranteed is that a real pair achieves it. The cancellation requires the two halves to be identical, and two transistors on a real die are not. A mismatch in saturation current appears as an offset in the characteristic, the characteristic is then no longer odd about the drive’s zero, and the even harmonics reappear in proportion to the mismatch. That is a genuine and well-known effect, and it is worth being clear that this page does not measure it: the model here has two identical halves, so what is being shown is the ceiling that matching approaches, not what a real pair delivers.

The third harmonic, and the factor of exactly two #

The interesting result is not the even harmonics — they are zero and that is the end of it — but what happens to the odd ones.

Measured at the same drive, the pair’s third harmonic is exactly half the single stage’s. Not approximately: at two, five, ten and twenty millivolts the ratio is 0.500000 to six figures. At forty millivolts it is 0.499920, at eighty 0.497686, and at a hundred and fifty 0.480333.

Where the half comes from is visible in the two series. The exponential’s cubic coefficient relative to its linear one is 1/6V_T², since every derivative of exp(v/V_T) is the function over a power of V_T. The hyperbolic tangent’s expansion is tanh u = u − u³/3 + …, and with u = v/2V_T the cubic coefficient relative to the linear one is 1/12V_T². Half.

The departure at large drive is equally explicable: the halving is a statement about the leading cubic terms, and once the fifth-order terms contribute, the two curves stop being related that simply. The measured ratio holds to six figures up to about a thermal voltage of drive and then drifts, which is exactly the range in which a third-order truncation is a good description of either curve.

So the pair’s benefit is asymmetric in a way the usual summary — “a differential pair cancels even harmonics” — states correctly and incompletely. Even harmonics: gone. Odd harmonics: halved, until the drive is large, and then less than halved.

Where the tail current goes #

The hyperbolic tangent is bounded, and that boundedness is a physical statement rather than a mathematical convenience: the two devices share a fixed tail current, so the most either of them can carry is all of it.

The measured split is worth setting out because it is much steeper than most readers expect. At ten millivolts of differential drive one side carries 59.6% of the tail and the other 40.4%, a difference of 19.1%. At twenty-six millivolts — one thermal voltage — the difference is 46.4%. At fifty it is 74.7%, and by a hundred and thirty-seven millivolts one device carries 99.4% and the other has effectively switched off.

A hundred and thirty-seven millivolts is 5.29 thermal voltages, and it is the whole dynamic range of the pair as an amplifier. Below about a tenth of that it is linear enough to be useful; above it, the pair is a switch. That dual character is not a defect — the same circuit is the input stage of every operational amplifier and the core of every emitter-coupled logic gate, and which of the two it is depends only on how hard it is driven.

The saturation also explains why the pair’s gain boundary arrives so early relative to its distortion. A bounded odd function must flatten, and flattening is gain compression; the compression begins as soon as the curvature does, while the third harmonic it produces stays small because the curve is symmetric about the origin. Hence 10.4 mV for one per cent of gain error and 18.2 mV for one per cent of distortion.

The fifth harmonic, and how far down the series goes #

The pair’s odd harmonics fall away quickly, and how quickly is a useful measure of how well behaved the characteristic is.

At five millivolts of drive the third harmonic is 7.775×10⁻⁴ of the fundamental and the fifth is 7.257×10⁻⁷ — three decades further down. At twenty millivolts the third is 1.202×10⁻² and the fifth 1.745×10⁻⁴, now only two decades down. The gap narrows as the drive grows, which is the series becoming a worse and worse approximation of itself, and it narrows fast enough that by a hundred millivolts the higher harmonics are no longer negligible in any sense.

The useful consequence is that a total-harmonic-distortion figure for this device is very nearly its third-harmonic figure over the whole range where the device is being used as an amplifier: 1.2022×10⁻² of total against 1.20204×10⁻² of third harmonic at twenty millivolts. Anything that reduces the third harmonic reduces the total, and anything that claims to have reduced the total without touching the third has measured something else.

The boundaries change places #

The previous essay measured two amplitude boundaries for a single exponential and found them seven times apart, with distortion arriving first: one per cent of harmonic content at 1.03 mV and one per cent of gain error at 7.30 mV.

Run the same measurement on the pair and the order reverses.

One per cent of total harmonic distortion at 18.18 mV. One per cent of gain error at 10.41 mV. The ratio is 0.572, where the single stage’s was 7.06.

The reason is the missing second harmonic. For the single stage, the distortion at small drive is almost entirely second harmonic, which is first order in the drive and therefore arrives very early. Remove it and the leading distortion term is the third harmonic, which is second order in the drive — the same order as the gain error. Two quantities of the same order arrive at comparable amplitudes, and which of them arrives first is then a matter of the coefficients rather than of the orders.

That is a satisfying place for the argument to land. The factor of seven in the previous essay was presented as a consequence of one quantity being first order and the other second; here is a device where the first-order quantity is structurally absent, and the factor of seven duly disappears.

What breaks the symmetry, and how gently #

The cancellation is exact for an odd characteristic driven symmetrically, and both halves of that sentence are conditions that a real circuit meets approximately.

The characteristic must be odd about the drive’s zero. Two devices with different saturation currents have a characteristic that is odd about some other point, so a drive centred on zero sees a curve with a quadratic term. The second harmonic that reappears is proportional to the offset for small offsets, which is the gentlest possible failure: a one-millivolt input offset on a stage driven at twenty millivolts restores roughly a twentieth of the second harmonic the single stage would have had, not all of it.

The drive must be symmetric. A pair driven on one input with the other held fixed is a single-ended drive, and the tail node then moves — which is a common-mode excursion, and the characteristic seen from one input alone is not odd. In practice the tail’s own impedance decides how much it moves, and a good current source keeps it small enough that the pair behaves as though both inputs were driven. That is another reason the tail source’s output resistance appears in the list of costs below.

Neither failure is catastrophic and both are proportional, which is a large part of why the topology is robust enough to be universal. The symmetry does not have to be perfect to be worth most of what it promises, and the amount it delivers is proportional to how well it is kept.

Two devices, and what the second one costs #

Nothing in this collection is free, and the pair’s price is worth stating.

Half the transconductance. The tail current splits between two devices, so each carries half of it, and the differential transconductance of the pair is g_m/2 where g_m is what one device carrying the whole tail current would have had. A pair therefore has half the gain of a single stage at the same total current, and the eighteen-fold improvement in linearity is bought partly with that.

A tail current source that has to be good. The cancellation assumes the tail is constant. A tail that moves with the common-mode input turns common-mode into differential and destroys the symmetry, which is why the output resistance of the tail source is one of the numbers that decides a real pair’s common-mode rejection.

Twice the input-referred noise power. Two devices contribute noise and only their difference is signal, so the noise voltage referred to the input rises by √2. This is the boundary the noise field is about, and it is worth flagging here because it is the one cost that gets worse rather than better as everything else is improved: every technique on this page raises the amplitude at which the stage stops being linear, and none of them lowers the amplitude at which it stops being able to see anything.

Those three together are the reason a differential pair is a choice rather than an obvious improvement. Half the gain, a component that has to be good, and √2 more noise, in exchange for the exact removal of one harmonic and the halving of another. For an amplifier that will be wrapped in feedback, the trade is overwhelmingly worth it — the loop gain lost is recovered by adding a stage, and the linearity gained is not recoverable any other way. For a single stage that has to do everything itself, it is a genuine decision.

The general shape #

The pattern this essay is an instance of is worth naming, because it recurs well beyond electronics.

A symmetry in a system removes a whole class of terms exactly, rather than making them small. That is qualitatively different from a numerical improvement and it behaves differently: it does not degrade with amplitude, it does not need re-checking when a parameter moves, and it fails only when the symmetry itself is broken — at which point it fails in proportion to the breaking rather than gradually.

The measured consequence here is a second harmonic of 10⁻¹⁶ where there was one of 10⁻¹, achieved by adding a device rather than by reducing a drive. What it does not achieve is anything at all for the odd harmonics beyond a factor of two, and a design that treats “differential” as a synonym for “linear” has assumed the second half of a result whose first half is exact.

The discipline it asks for is specific and is worth carrying into any claim of this shape. Establish what the symmetry is, in the form of a property of a function rather than a description of a circuit. Then check that what it removes is removed to the arithmetic’s floor rather than merely made small, because a symmetry argument that produces 10⁻³ has an error in it somewhere and a numerical argument that produces 10⁻¹⁶ is being flattered by a coincidence. And then say what the symmetry does not remove, which in this case is everything of odd order — half of it, and only while the drive is small.

There is a version of this essay that stops after the first of those three, and it is the version most often written. It is not wrong about anything it says.