Two boundaries removed, and one moved

The essay before this one found three amplitudes where one was expected. Driving an exponential with a sinusoid produces a mean that has moved, a fundamental that is not the tangent’s prediction, and harmonics that are not zero, and each of those is one per cent wrong at a different drive: 1.03, 5.17 and 7.30 millivolts at room temperature.

A differential pair is the standard answer to all of that. Its transfer is a hyperbolic tangent rather than an exponential and everybody knows what a tangent does: it is odd, so the even-order products cancel, and the stage is more linear.

More linear by how much, in each of the three?

The answer is not “by a factor” in any of the three cases. Two of the boundaries do not move at all — they cease to exist — and the third moves by a factor that is $\sqrt2$ in the limit and is measurably not $\sqrt2$ at the criterion anybody uses.

Two quantities that are zero rather than small #

A pair’s differential output is $I\tanh(v/2V_T)$, and $\tanh$ is an odd function. Driven by a sinusoid — itself odd about the origin over a period — the output has no mean and no even harmonics.

The figure measures that rather than claiming it. At a drive of two thermal voltages, which is eighty times the amplitude where a single transistor’s second harmonic reaches one per cent, the pair’s mean comes out at $-3.2\times10^{-17}$ and its second harmonic at $9.1\times10^{-17}$. Those are the floating-point residue of summing four thousand samples of an odd function, and the same transform on a single transistor at the same drive gives a mean of 2.280 times quiescent and a second harmonic of 0.689.

That is a change of kind rather than of degree and the distinction matters. A quantity that is small gets larger when conditions change. A quantity that is zero by symmetry stays zero as long as the symmetry holds, whatever the drive, and the drive can be made arbitrarily large without producing any of it.

So two of the three boundaries from the essay before it are gone. There is no amplitude at which a pair’s operating point has shifted by one per cent and no amplitude at which its second harmonic reaches one per cent, because neither quantity is ever anything but zero.

The third moves by forty per cent #

The gain error survives, because it is even in the amplitude and $\tanh$ has odd terms that produce it.

Expanding, $\tanh u = u - u^3/3 + \dots$, and the fundamental of $\tanh(a\sin\theta)$ is $a(1 - a^2/4 + \dots)$. Against the exponential’s $a(1 + a^2/8 + \dots)$ that is a coefficient twice as large and of the opposite sign — a pair’s gain compresses where a single transistor’s expands — and the amplitude at which each is one per cent from its tangent differs by the square root of the ratio of the coefficients.

There is a factor of two in the argument as well, since the pair sees $v/2V_T$ rather than $v/V_T$, and the two factors combine:

$$\frac{a_{\text{pair}}}{a_{\text{single}}} = \frac{2\cdot\sqrt{4\varepsilon}}{\sqrt{8\varepsilon}} = \sqrt2$$

So a pair is $\sqrt2$ more linear in the only sense that survives, which is forty-one per cent and not an order of magnitude.

The figure bisects both on the measured waveform rather than substituting. At a criterion of one part in ten thousand the ratio is 1.41433 against $\sqrt2 = 1.41421$. At one per cent it is 1.42610, which is 0.84 per cent above $\sqrt2$ — measurably not the limit, because the two devices’ next terms differ as well as their leading ones, and one per cent is not a small enough criterion for the leading terms to be the whole answer.

In millivolts at room temperature: 7.304 for a single transistor and 10.42 for a pair.

What a pair was actually bought for #

Put the two results together and the honest account of a differential pair is a strange one.

As a linearity device it is transformative. The second harmonic goes from 25 per cent at a drive of one thermal voltage to zero, exactly, at every drive. There is no amount of signal that produces even-order distortion in a balanced pair. That is not a forty per cent improvement or a twenty-decibel one — it is the removal of a whole family of products, and it is why every low-distortion input stage, every mixer that needs port isolation and every modulator is built this way.

As a headroom device it is nearly useless. The amplitude at which the small-signal gain calculation stops being right moves from 7.3 mV to 10.4. A designer who reaches for a pair because the input swing is too large for one transistor has bought forty per cent of amplitude, and if the swing was twice too large it is still too large.

The confusion between those two is common and it comes from “more linear” being used for both. The useful repair is to name the quantity: a pair is infinitely more linear in even orders and $\sqrt2$ more linear in gain compression, and those are different requirements belonging to different circuits, exactly as that essay’s three boundaries are.

What a pair cancels, and what it only halves makes the same distinction for a different set of quantities — the ones a pair removes outright and the ones it merely shares — and the two essays are the same observation about symmetry applied to distortion and to offset.

The sign is opposite, and that is worth something #

Buried in the expansion is a fact the $\sqrt2$ hides: the two devices’ gain errors have opposite signs.

A single transistor’s fundamental is $a(1 + a^2/8)$ — the gain expands with amplitude, because an exponential curves upward and a sinusoid spends more of its excursion where the curve is steep. A pair’s is $a(1 - a^2/4)$ — the gain compresses, because a tangent flattens.

So the two are not two grades of the same defect. They are defects in opposite directions, and two stages in cascade, one of each, have gain errors that partly cancel. A single-ended stage driven at amplitude $a$ followed by a pair driven at amplitude $b$ has a combined third-order coefficient of $a^2/8 - b^2/4$, which is zero when $b = a/\sqrt2$ — the same $\sqrt2$, arriving as a design condition rather than as a ratio of boundaries.

Whether that is buildable is a separate question and the answer is usually no: the two amplitudes are set by the gains between the stages and the cancellation is first order in everything, so it survives a few per cent of mismatch and not more. What it does explain is a measurement that otherwise looks like luck — a two-stage amplifier whose third-order intercept is much better than either stage’s, which happens when the stages’ curvatures oppose and is exactly what the pair that is worse than either finds going the other way when they agree.

The general point is that a sign is information and a magnitude is not. Two errors of 3 per cent that add give 6; two that oppose give zero at one amplitude and a residual with a different exponent everywhere else. A budget built from magnitudes cannot tell those apart, which is the same complaint where the mechanisms are one mechanism makes about adding errors in quadrature when they are not independent.

The cancellation belongs to the symmetry, not to the topology #

The even orders are zero because the transfer is odd, and the transfer is odd because the two halves are identical. Neither of those is a property of the circuit diagram; they are properties of the parts being matched.

The figure measures what happens when they are not. Twenty millivolts of offset between the two inputs — which is a poor but entirely ordinary mismatch for an untrimmed pair of discrete transistors — displaces the operating point from the tangent’s symmetry point, and the even orders come back: a mean of $1.0\times10^{-2}$ and a second harmonic of the same order, where both were $10^{-17}$.

Fifteen orders of magnitude, from twenty millivolts of imbalance. That is the refusal in this essay and it is what keeps the zeros honest. A cancellation this complete is a cancellation that depends entirely on something, and the something is not the pair — it is the matching.

The practical consequence follows directly. A monolithic pair on one die, at one temperature, with a few hundred microvolts of offset, has second-order products fifty decibels below what an untrimmed discrete pair produces, and both are called differential pairs. The topology is worth nothing on its own; what is being bought is the matching, and the rejection the parts have is the same observation made about a different figure of merit.

It also means the even-order performance is a drift specification rather than a static one. An offset that is trimmed to zero at twenty-five degrees is tens of microvolts at eighty, so the even-order distortion of a trimmed pair has a temperature coefficient, and it is the offset drift’s.

The tail, which is the symmetry nobody draws #

The odd transfer requires the two collector currents to sum to a constant, and what makes them sum to a constant is the tail current source. It is drawn as an arrow in a circle and it is the part of the pair the cancellation actually depends on.

A tail source with finite output resistance is a resistor, and a resistor does not hold the sum constant: the common-mode voltage at the two emitters moves with the drive — at twice the signal frequency, because it follows the magnitude of the excursion rather than its sign — and the sum of the two currents moves with it. That is an even-order product, arriving at the output through whatever common-mode gain the following stage has, and it is proportional to the tail resistance rather than to any mismatch.

So there are two independent routes for even orders into a pair’s output and they need different cures. Mismatch between the halves is cured by matching, and is what the refusal above measures. A finite tail impedance is cured by making the tail source stiffer — a cascode, a current mirror with degeneration — and no amount of matching touches it.

The distinction matters because the two have different signatures. Mismatch produces even orders that are proportional to the offset and roughly independent of frequency; a finite tail impedance produces even orders that follow the tail source’s own output impedance, which falls with frequency as its parasitic capacitance takes over. A pair whose second-harmonic distortion is flat with frequency has a matching problem; one whose second harmonic rises with frequency has a tail problem, and the two are distinguishable on a bench in one sweep.

That is the same diagnostic logic the rejection four resistors decide applies to a difference amplifier’s common-mode rejection: two mechanisms, two frequency signatures, and the shape of the curve says which one is binding.

What the odd transfer does not buy #

That $\tanh$ is a differential pair’s exact transfer. It is, for two ideal exponential devices with a perfect tail current and no emitter resistance. Series resistance linearises it further — which is what a resistor in the emitter buys and is a much larger effect than the $\sqrt2$ — and a finite tail-current source does the opposite. The model here is the pair alone.

That the third harmonic is negligible. It is smaller than a single transistor’s second at every amplitude on the axis and it is the dominant distortion of a pair, so it is what a pair’s specification is about. What this essay establishes is that it is not worse than the single device’s odd orders, so the even-order cancellation is not paid for.

That “odd function, no even harmonics” needs a measurement. It does not need one to be believed and it does need one to be checked, which is a different thing: the transform used here is the same one that produced the single transistor’s numbers, and a systematic error in it would show up as a non-zero even order in the pair. The $10^{-17}$ is a check on the instrument as much as on the device.

That twenty millivolts is a typical mismatch. It is a poor one, chosen to make the return of the even orders unmistakable. A monolithic pair is at a few hundred microvolts and its even-order products are correspondingly smaller — by the ratio of the offsets, since the leakage of even orders is first order in the imbalance for small imbalances.

Zeros measured rather than argued, and the imbalance that ends them #

Every harmonic is measured on the waveform rather than taken from a series, which is what makes the vanishing ones a measurement instead of an assumption.

The mean and the even orders are checked to be below $10^{-12}$ at a drive of two thermal voltages — eighty times the amplitude at which a single device’s second harmonic reaches one per cent — and the single device is checked at the same drive to have a mean above twice quiescent, so the comparison is between two measurements rather than between a measurement and an expectation.

The $\sqrt2$ is checked as a limit, at a criterion of a part in ten thousand, to two parts in a thousand — and the value at one per cent is checked to be measurably above it, so the figure states where its own tidy ratio stops being tidy.

The third harmonic is checked to be smaller than the single device’s, which is the claim that the cancellation is not bought from the odd orders.

And the cancellation is refused with twenty millivolts of imbalance, where the mean and the second harmonic return to $10^{-2}$ — fifteen orders above where they were.

A symmetry is a stronger thing to have than a factor #

The structure this essay adds is a distinction between two ways a circuit can be better.

A factor is an improvement in a quantity. It holds at one amplitude and a different amplitude gives a different factor; it degrades as conditions move; and it can be traded against something. The $\sqrt2$ in gain compression is a factor, and it is a small one.

A symmetry is a statement that a quantity is identically zero. It holds at every amplitude, does not degrade with the drive, and cannot be traded because there is nothing to trade. The even-order cancellation is a symmetry, and it is what a pair is actually for.

The price of a symmetry is that it depends on a condition rather than on a margin, and conditions fail abruptly. A factor of $\sqrt2$ becomes a factor of 1.3 when something drifts; a symmetry becomes nothing at all, and the fifteen orders of magnitude in the refusal above are what “nothing at all” looks like when it happens. That is the trade, and it is why matched pairs are made on one die and trimmed, and why the specification that matters for a pair’s distortion is its offset rather than any of its distortion numbers.

The same pattern turns up wherever a cancellation is doing the work. The mismatch that cancels itself and the far end that cancels are two more, and in both the useful question is not how large the cancellation is but what it is conditional on.

Still open: the degenerated pair, the leakage as a function of offset, and the third harmonic’s own boundary #

Where an emitter resistor and a pair meet. Degeneration linearises a single device by feedback and a pair linearises by symmetry, and the two are usually applied together. Whether the $\sqrt2$ survives degeneration, is multiplied by it, or is swamped by it, is a computation on one netlist, and what a resistor in the emitter buys has the apparatus for one half of it.

The even orders as a function of the imbalance. Twenty millivolts brings them back to $10^{-2}$ and a few hundred microvolts should bring them back proportionally less. Whether the leakage is first order in the offset over the useful range — and where it stops being — would turn the matching specification into a distortion specification directly, which is the number a designer actually wants.

And the third harmonic’s own one-per-cent boundary. The essay before it found three boundaries for a single device by asking three questions of one function. A pair has only two of those questions available, and the second of them — the harmonic boundary — belongs to the third harmonic rather than the second. Locating it would complete the comparison: two boundaries for a pair against three for a single device, with a ratio for each.