The distortion a linear model cannot have

There is a question a small-signal model cannot be asked. Not one it answers badly — one that does not exist for it.

A linear model’s output is, by its definition, a scaled copy of its input. Feed it a sinusoid and a sinusoid comes out, at the same frequency, always. It has no second harmonic, not because the second harmonic is small but because the model has no term that could produce one. Every distortion figure in every datasheet describes a quantity that lives entirely outside the model most of the analysis is done with.

Where the harmonics come from #

The mechanism is easier than it looks. Expand a device’s transfer characteristic about its bias point as a power series:

i = a₀ + a₁v + a₂v² + a₃v³ + …

The small-signal model is the first two terms and nothing else; a₁ is the transconductance. Drive that series with v = v̂ cos ωt and each power produces its own harmonics: the square of a cosine is a constant plus a cosine at twice the frequency, the cube is a cosine at the fundamental plus one at three times it, and so on.

Two consequences fall straight out, and they explain everything measured below.

The second harmonic is first order in the drive. It comes from a₂v̂²/2, so its amplitude is proportional to v̂², and the ratio of second harmonic to fundamental — which is what “distortion” means — is proportional to v̂. Double the drive, double the distortion.

The gain error is second order. The fundamental picks up a correction from the cubic term, 3a₃v̂³/4, so the gain’s fractional departure from a₁ goes as v̂². Double the drive, and the gain error quadruples from a much smaller starting point.

That difference in order is why the two boundaries in the title are seven times apart rather than close together, and it is not a property of this particular device.

The check that shares no arithmetic #

An exponential driven by a sinusoid has harmonic amplitudes that are known exactly, in closed form, and they are the modified Bessel functions of the first kind: the nth harmonic of $\exp(\hat v \cos\theta / V_T)$ is proportional to $I_n(\hat v/V_T)$. So the ratio of the second harmonic to the fundamental is $I_2(x)/I_1(x)$ with $x = \hat v/V_T$, and nothing else.

Nothing in the measurement knows that. The figure samples the curve five hundred and twelve times, transforms the samples, and reads off the ratios. The prediction sums a power series. They share the drive amplitude and the thermal voltage and no arithmetic at all.

At ten millivolts of drive the measured second harmonic is 9.6107% of the fundamental and I₂/I₁ at x = 0.3868 is 9.6107%. The worst relative disagreement over the harmonics the arithmetic can resolve is at the level of 10⁻¹⁴.

That last phrase is doing real work. At a millivolt of drive the sixth harmonic sits at 3.8×10⁻¹² of the fundamental, and a transform of samples of order one carries about fifteen digits, so comparing it is comparing rounding error. The check therefore runs only on harmonics above 10⁻⁸ of the fundamental — a floor set by measurement rather than taste: the relative error of a harmonic of size m runs at about 1.2×10⁻¹⁵/m across four decades of drive, so 10⁻⁸ buys agreement to better than 1.2×10⁻⁷, two decades inside the tolerance asserted. Harmonics below the floor are still drawn, with their measured values; they are simply not claimed about.

Why a transform rather than a series #

The measurement could have been done by differentiating the transfer characteristic three times and reading a₂ and a₃ off, and it deliberately is not. The reason is generality, and it is the same reason the rest of the site solves networks rather than manipulating expressions.

A transform takes samples. It does not care where the samples came from, so the identical routine measures the distortion of an exponential written in closed form, of a hyperbolic tangent, and of a network solved point by point with the Newton loop from a bias point is a solution. A series expansion cares a great deal: it needs symbolic derivatives, it needs them about the right point, and it stops being available the moment the characteristic is something a solver produced rather than something an author wrote down.

The cost of the transform is that it has a floor and a truncation, which the previous section had to spend two paragraphs on. That is a fair trade — a stated floor on a general method is worth more than exactness on a method that only works for the cases where the answer was already known.

For the exponential there is a striking consequence of that last point. Every derivative of $\exp(v/V_T)$ is the function itself divided by a power of $V_T$, so the ratios $a_2/a_1$ and $a_3/a_1$ do not depend on the bias current at all. The distortion of an exponential transconductor at a given drive voltage is the same at ten microamperes and at ten milliamperes, which is why the boundary in this essay is quoted as a voltage with no current beside it, and why the site’s existing measurement found the same 28.2% of $V_T$ at every bias current tested. That is unusual. For almost any other characteristic the bias point matters enormously, and a stage biased near a point of inflection has a third-order coefficient that passes through zero — which is the basis of every “sweet spot” in a device datasheet and of a family of biasing tricks that trade one harmonic for another.

Two boundaries, seven times apart #

The site already carries one number for the edge of the small-signal model. How small is small signal measures the drive at which the linearised gain is one per cent optimistic, and gets 7.30 mV — 28.2% of the thermal voltage, a fraction that turns out to be the same at every temperature and every bias current.

That number is about the gain. Measuring the same device’s distortion gives a completely different answer.

One per cent of total harmonic distortion arrives at 1.0341 mV. One per cent of gain error arrives at 7.2999 mV. The ratio is 7.06.

Both numbers describe the same device and the same word — small — and they differ by a factor of seven, so any statement of the form “the signal is small enough” has to say small enough for what. A stage biased to run comfortably inside its gain-error budget at five millivolts of drive is producing 4.8% of second harmonic, which for an audio amplifier is a catastrophe and for a limiter is irrelevant.

The general shape holds beyond this device, because it follows from the orders. Wherever a nonlinearity is smooth, distortion grows as the first power of the drive and gain error as the second, so the distortion boundary always arrives first, and the ratio between the two boundaries grows as the tolerance tightens.

That last sentence is a prediction rather than a summary, and it can be checked. If distortion is first order and gain error second, then tightening the tolerance by a decade should move the distortion boundary down by ten and the gain boundary down by only √10 — so the ratio between them should grow by √10 = 3.162 per decade of tolerance.

Bisecting both boundaries at four tolerances gives 2.190, 7.059, 22.357 and 70.710 for ten per cent, one per cent, a tenth and a hundredth. The successive ratios are 3.223, 3.167 and 3.163. The last two are √10 to three figures, and the first is a little high because at ten per cent the drive is large enough that the higher terms of the series are no longer negligible — which is exactly the condition under which the argument was derived to fail.

A prediction that comes out right, and comes out wrong in the place its own derivation says it should, is worth more than either half alone.

What total harmonic distortion actually totals #

The figures quote a single number for distortion, and it is worth saying exactly what goes into it, because the definition contains two choices.

Total harmonic distortion here is the root-sum-square of every harmonic above the fundamental, divided by the fundamental. At ten millivolts that is 9.631%, against a second harmonic alone of 9.611% — so the second harmonic is very nearly the whole of it, and it stays that way until the drive is comparable with the thermal voltage. At fifty millivolts the second harmonic is 42.2% and the total is 44.2%, with the third contributing 12.7%.

The first choice is that the direct-current term is excluded. That is the usual convention and it is a choice: a bias shift is not distortion of the signal in the sense meant here, and including it would make a stage’s distortion figure depend on where its output happened to sit. It is worth noticing that the exponential does shift its average as the drive grows — that is what I₀(x) > 1 means — and that shift is a real effect which this measure deliberately does not report.

The second choice is the harmonic count. Truncating at twelve is fine here because the series falls away quickly, and the check that it is fine comes free: Parseval’s identity requires the mean square of the samples to equal the sum of the squares of the harmonic amplitudes, and any energy in harmonics beyond the count shows up as a discrepancy. On a square wave truncated at twenty harmonics that discrepancy is 1.01% and correctly so; on every waveform in this field it is at the level of the arithmetic.

The topology that removes half of them #

That figure is the argument of what a pair cancels and is placed here for one reason: it is the answer to the problem this essay poses, and it is a structural answer rather than a numerical one. The even harmonics do not become small; they become zero, because an odd function driven symmetrically cannot produce them. No amount of biasing, trimming or matching is involved, and no tolerance is attached.

It is the strongest kind of result available in this area, and it is worth contrasting with the alternative approach of driving the device less hard. Halving the drive halves the second harmonic. Making the characteristic odd removes it.

What the numbers mean for a stage that has to work #

It is worth turning the boundaries into the constraint they actually impose, because the arithmetic is not encouraging and the way out of it is the whole of amplifier design.

A stage whose distortion must stay below a tenth of a per cent may be driven to 103 µV. A single bipolar transconductor with a load resistance chosen to give a gain of a hundred therefore has an output swing of about ten millivolts, which is not a useful amplifier. Ask for a hundredth of a per cent and the drive falls to 10 µV and the swing to a millivolt.

Three things are done about that, and each is a different part of this collection.

Make the characteristic odd, which removes the second harmonic entirely and buys a factor of about eighteen in allowable drive at one per cent — the differential pair reaches one per cent of distortion at 18.2 mV against the single stage’s 1.03 mV. That is the largest single improvement available and it costs a second device.

Degenerate the stage, which is to say put a resistance in series with the emitter so that most of the input voltage falls across something linear. The device sees a fraction of the drive, its distortion falls with that fraction, and the gain falls by the same factor. This is the trade nobody escapes: linearity bought at exactly the price of gain.

Apply feedback, which is the feedback field and reduces distortion by the loop gain — the same loop gain that sets the bandwidth, the margin and the ringing. A stage with forty decibels of loop gain has a hundredth of the distortion, and has spent its forty decibels.

All three are the same transaction seen from different sides: gain is the currency and linearity is what it buys. The reason the boundaries in this essay are worth measuring rather than assuming is that they set the price.

What the field’s rule looks like here #

Every figure on this site carries the frequency, amplitude or size at which the model in it stops being true, and this page is one of the few where the answer is squarely an amplitude and where there are two of them.

The honest statement of the small-signal model’s domain therefore has two numbers in it: excursions below about a millivolt if harmonic content matters, and below about seven millivolts if only the gain does. Quoting one without saying which is being measured is how a specification comes to mean two different things to the person who wrote it and the person who reads it.

What moves an amplitude edge, given that frequency does not #

Three things do, and each has an essay measuring by how much.

Symmetry, which removes a whole harmonic rather than reducing it. What a pair cancels, and what it only halves measures both halves of what that is worth: a differential pair’s second harmonic comes out at $10^{-16}$ of the fundamental — the arithmetic’s own floor, not a small physical residue — while its third comes out at exactly half the single stage’s, which is a reduction and not a cancellation. So the odd-order edge moves by a factor of two and the even-order one moves to infinity.

Local feedback, which linearises the curve. What a resistor in the emitter buys measures the exchange rate and finds the two sides of the trade different sizes: dividing the gain by six moves the distortion edge by thirty-five times — the square of the factor — and the gain edge by only twelve, so the two close up and a well-degenerated stage stops being limited by its linearity.

And temperature, which moves the edge without anybody touching the circuit. The edges that move with the room finds the small-signal edge proportional to the thermal voltage, running from 5.67 millivolts at −40 °C to 9.69 at +125 — a factor of 1.71, the ratio of the absolute temperatures exactly.

Which is the useful summary. An amplitude edge is moved by the arrangement rather than by the bandwidth, it is moved a long way by symmetry and a modest way by feedback, and it moves on its own with the room by rather more than most designs allow for.

There is a fourth thing that does not move it, and naming it is the point of this essay’s title. Global feedback around a stage divides the distortion by $1+T$ and moves nothing about the stage itself — and how much of the amplifier gets through shows that division running out exactly as fast as the loop gain does, rising above one near crossover so that the loop makes the distortion worse there than no loop at all. So a distortion figure quoted at a kilohertz for a circuit crossing over at six megahertz is not a property of that circuit at twenty, and the amplitude at which the underlying curve bends has not moved by anything.

Which is the distinction the title is about, stated at the end rather than the beginning. A small-signal model has no second harmonic because it is linear by construction, and a feedback loop has less second harmonic because it divides one that exists. The first is a property of a description and the second is a property of a circuit, and only the second has a number, an amplitude and a frequency above which it stops.

What the harmonics are used for #

Harmonic content is the measurement four later results in this field are fitted to. What a pair cancels, and what it only halves removes the even ones exactly. What a resistor in the emitter buys moves the amplitude at which they arrive. Where the two exponents come from derives the exponents this measurement fits, and The point the device is never at extrapolates them to a figure of merit the device never reaches. How small is small signal is the boundary all four are distances from.