What the fourth order says about the third

The argument this essay continues produced two closed forms and one loose end. A degenerated exponential’s second harmonic is $\hat u/4D^2$ and its gain error is $|3-2D|\hat u^2/8D^4$, where $\hat u$ is the drive in thermal voltages and $D = 1 + g_mR_E$ is the factor the resistor divides the gain by. Both came from reverting the device relation as a power series and keeping three terms, both were checked against a discrete transform of a Newton-solved curve, and both were recorded with a number beside them: at a degeneration of eleven the distortion edge the first of them predicts is about seven per cent optimistic.

Seven per cent is a small number and it was left as an observation. It should not have been, because there are two quite different things it could be. It could be the series being truncated — the fourth-order term, which was thrown away, showing up at the amplitude the expression is being evaluated at. Or it could be a mechanism the series does not contain, in which case adding terms would not help and something else is going on.

The two are distinguished by one calculation: carry the reversion one order further and see how much of the seven per cent it accounts for. It accounts for 94.7 per cent of it.

The reversion, written once rather than copied #

The device relation in normalised units — collector current in units of the bias current, drive in thermal voltages — is

$$u = \ln(1 + x) + (D - 1)x$$

with $x = i - 1$. That is exact and it is the wrong way round: the drive is a function of the current and what a transfer curve needs is the current as a function of the drive. So the series has to be reverted, and the coefficients of the reverted series are what every expression here is built out of.

Reversion coefficients are published to fifth order in reference books and copying them is the obvious thing to do. It is also the thing that cannot be checked. A transcribed coefficient is trusted; a computed one can be asked a question, and the question available here is a good one. Feed the same routine the exponential’s own series — which is what $D = 1$ gives, since the logarithm’s inverse is $e^u - 1$ — and every coefficient must come back as $1/i!$. It does, to sixteen digits at the fourth order and to thirteen at the sixth, which is the arithmetic’s floor rather than the method’s.

So the routine is generic, it costs the cube of the order, and the order is six. What it returns for this relation is

$$a_1 = \frac{1}{D},\quad a_2 = \frac{1}{2D^3},\quad a_3 = \frac{3 - 2D}{6D^5},$$

$$a_4 = \frac{6D^2 - 20D + 15}{24D^7},\quad a_5 = \frac{105 - 210D + 130D^2 - 24D^3}{120D^9}$$

of which the first three are the ones the rung below this one used and the last two are new.

What a fourth coefficient is a statement about #

Driving the reverted series with $\hat u\cos\theta$ and collecting the terms at each multiple of $\theta$ gives the fundamental, the second harmonic and the third:

$$A_1 = a_1\hat u + \tfrac34 a_3\hat u^3 + \tfrac58 a_5\hat u^5,\qquad A_2 = \tfrac12 a_2\hat u^2 + \tfrac12 a_4\hat u^4$$

so the second-harmonic ratio, to one order beyond the previous rung, is

$$\mathrm{HD}_2 = \frac{\hat u}{4D^2}\left[1 + \frac{12D^2 - 34D + 21}{24D^4}\hat u^2\right]$$

The bracket is the whole of this essay. It is a correction whose size is a prediction about the uncorrected expression’s error, it is second order in the drive, and it can be evaluated at the drive the uncorrected expression is actually used at.

Do that at a degeneration of eleven and a target of one per cent. The leading expression puts the distortion edge at $4tD^2 = 4.84$ thermal voltages. The bracket evaluated there is 1.0733, so the leading expression should be about 7.3 per cent optimistic. It was measured — when the rung below this one was written, by bisection on a transform, with nothing of this expression available — to be 6.89 per cent optimistic.

That is the answer to the question this essay was written for. The seven per cent is the truncation, and the fourth-order coefficient predicts its size from a formula that knows nothing about the measurement.

The edge, at three orders #

Solving each expression for the amplitude at which the second harmonic reaches a stated target gives three numbers, and the third is a measurement.

At $D = 11$ and one per cent: the leading form gives 4.8400 thermal voltages, the next order gives 4.5445, and the bisection on the sampled curve gives 4.5281. The leading form is 6.888 per cent out and the next-order form is 0.362 per cent out, so the added term removes 94.7 per cent of the discrepancy.

At $D = 6$: 1.4400, 1.4171, 1.4169. The leading form is 1.629 per cent out and the next one 0.016 per cent — ninety-nine per cent of it removed.

At $D = 1$, the bare device, there is nothing to remove: the leading form is out by seven parts in a hundred thousand, because at one per cent of distortion a bare exponential is being driven at forty millivolts of a thermal voltage and the series is nowhere near its own limits.

The error of a truncated series, as a power law #

There is a stronger statement available than “the correction helps”, and it is the one that identifies the mechanism rather than merely repairing the number.

If the leading expression’s error is the first neglected term, then that error must be proportional to the square of the drive it is evaluated at — because the neglected term is two orders up in a series whose alternate terms vanish by symmetry. The drive it is evaluated at is different at every degeneration factor, since the edge itself moves as $D^2$, so this is a claim that can be fitted across the whole axis rather than checked at a point.

Fitted over degeneration factors from two to eleven, at a target of one per cent, the exponent comes out 1.96. At half a per cent, 1.94; at three per cent, 2.09; at five per cent, 2.25.

The drift is itself the story. As the target rises the drives rise, the next neglected term begins to contribute, and the fitted exponent walks upward away from two. So the assertion in the figure is a bracket — the exponent is between 1.8 and 2.4 — rather than a number, because a tolerance tight enough to fail at five per cent would be a tolerance chosen to pass at one.

What the fifth coefficient buys, which is a different question #

The fourth coefficient corrects the second harmonic. The fifth corrects the gain error and the third harmonic, because those are odd quantities and the even coefficients do not touch them:

$$\frac{A_1}{a_1\hat u} - 1 = \frac{3a_3}{4a_1}\hat u^2 + \frac{5a_5}{8a_1}\hat u^4$$

and the leading part of that is the $|3-2D|\hat u^2/8D^4$ the previous rung derived.

That expression has a null. Its coefficient vanishes at $D = 3/2$ exactly, where the exponential’s own convexity and the local feedback’s compression cancel, and the essay that found the null observed that the third harmonic must vanish there too — measured at $9.5\times10^{-10}$ against $1.04\times10^{-4}$ on the bare device, five orders down, with the residue attributed to the fourth-order term.

The fifth coefficient says what that residue is. At $D = 3/2$, $a_3$ is zero and $a_5$ is $3.25\times10^{-4}$, so the gain error there is not zero but quartic in the drive, and the third harmonic is not zero but fifth order. Neither is a small non-zero coefficient of the expected order; both are the next order, which is what a cancellation leaves behind and is a different statement from a cancellation that is merely good.

The two harmonics do not converge at the same rate #

One more consequence falls out of the coefficient list and it is easy to miss, because it is a statement about two quantities rather than about either.

The second harmonic’s leading term is $a_2$ and its correction is $a_4$; the gain error’s leading term is $a_3$ and its correction is $a_5$. Those are different pairs of coefficients, so there is no reason for the two expressions to lose accuracy together, and they do not.

At a degeneration of six and a drive of two thermal voltages the second-harmonic expression carried to four terms agrees with the transform to 1.4336 against 1.4346 per cent — seven parts in ten thousand. The gain-error expression at the same drive is worse, and the reason is $|3 - 2D|$: at $D = 6$ that coefficient is nine, at $D = 2$ it is one, and at $D = 3/2$ it is zero, so the ratio of the leading term to its own correction is not a smooth function of the degeneration at all. Near the null the corrected expression is entirely the correction.

So “how accurate is the closed form” has two answers on the same device, and which one applies depends on which quantity is being asked about. That is worth stating because a reader who has checked one of them is likely to assume the other.

Where a series stops being the right object #

A reverted series is a local statement, and there is a drive above which adding terms stops being the efficient thing to do. It is worth saying where that is, because it bounds the whole approach rather than any one expression.

The reverted series for $x(u)$ has a finite radius of convergence, set by the nearest singularity of the relation being inverted. For $D = 1$ that is the logarithm’s branch point at $x = -1$ — the current going to zero — which is a drive of $-\infty$ in one direction and gives a radius of one in $x$. For larger $D$ the linear term pushes the singularity further away in $u$, which is the formal reason the expressions get better with degeneration at a fixed drive and the informal reason degeneration is used at all.

But the useful bound is not the radius. It is the drive at which the next term is a per cent, and that is what the figure above measures directly, at each degeneration, from three quantities rather than from an analysis of singularities. At $D = 11$ and a one per cent target the answer is that the third order is 6.9 per cent out and the fourth 0.36; at five per cent it is 94 and 14. A reader who wants an expression good to a per cent at five per cent of distortion needs a sixth coefficient, and by then the sensible thing is to bisect the curve.

Why this is worth an essay rather than a footnote #

The result could be written in three lines and it has taken rather more, because what it demonstrates is a habit rather than a coefficient.

A closed form that fits a measurement is evidence of very little. A closed form that fits a measurement and predicts the size of its own error is a different kind of object: it has been asked a question it could have failed, and the question was available only because somebody had written down what the expression neglected. A discrepancy recorded as a number and never explained is a debt: it looks like a fact, it is quoted like a fact, and nothing in it says whether the model or the arithmetic is the thing that is wrong.

The second thing it demonstrates is that an error term is a measurement instrument. Nothing here measured the drive at which the series fails by studying the series. It measured a distortion edge three ways, took the difference, and fitted an exponent to it — and the exponent, 1.96, is the thing that says the difference is a truncation rather than a mechanism.

Where the series is used, and what replaces it #

A fourth-order term used to bound a third-order one is a statement about a series rather than about a device. The order that stops helping is where the same series is measured past its own radius, and adding terms makes it worse. Where the two exponents come from is the derivation the series is a truncation of. What a resistor in the emitter buys is the parameter it is expanded in. The point the device is never at is the extrapolated figure of merit the series is usually quoted to support, and A bias point is a solution, not a choice is the alternative to expanding anything — solve the device and read the answer off.

What is checked #

Four assertions, and the second is the one that would catch a wrong reversion rather than a wrong number.

That the fourth-order term accounts for more than four fifths of the gap the leading expression leaves at the largest degeneration drawn, at every distortion target the slider offers, and that the corrected expression is closer to the measurement than the uncorrected one in every case.

That the reversion returns $1/i!$ for the exponential’s own series, which is what $D = 1$ reduces to. That is the assertion a transcribed table of coefficients could not have.

That it explains less at a larger target, which is the correction’s own edge in the same quantity as everything else in this field — a correction term is a small-signal expression too.

And that the leading expression’s error grows as about the square of the drive it is evaluated at, fitted across the degeneration axis and bracketed rather than quoted, because the exponent itself drifts as the next term arrives.

Where this ladder stops, and why it stops rather than continuing #

Carrying a series one order further and measuring what the new term explains is a procedure that can be repeated, and the rung above this one is where repeating it is measured rather than assumed.

The order that stops helping takes three expressions, each one order longer than the last and each nearer the measurement, and finds the rule that governs all of them: the error of an expression truncated at order $m$ grows as the $m$-th power of the drive — measured at 2.00, 4.00 and 5.88 — so every added order buys a range that ends sooner than the last one bought. Above 8.9 thermal voltages the six-term expression is further from the device than the four-term one.

Which is the honest answer to the obvious question this essay raises. The fourth-order term corrects the third by 7.33 per cent against a measured 6.89 and takes the residue to 0.362 per cent, and a fifth would correct that in turn — over a smaller range. An asymptotic series does not converge to the answer; it approaches it and then leaves, and the optimal truncation is a property of the drive rather than of the algebra.

That is a boundary of exactly the kind this collection collects, on an unusual axis: the variable is the order of an expression, and what fails is not a circuit but a description of one. The digits the arithmetic did not have is the only other place here with a boundary of that shape, and it is the complementary case — there the procedure fails while the object it describes is perfectly well behaved.

The practical use of a correction term whose own error is measured is narrower than it sounds and is worth stating. It is not that the four-term expression should be used instead of the three-term one; it is that having both makes the residue visible, and a residue with a known growth law is a stated range rather than an unknown. Three terms with no fourth is an expression whose error nobody can bound; three terms with a fourth computed is an expression whose error is 6.89 per cent at a stated drive and grows as the square of it.

That is the same argument one step, computed twice makes about a marched response, where the difference between two routes is the trapezoidal rule’s own error and falls by a factor of four every time the step is halved. In both cases the second computation is not the answer; it is the error bar, and it is the only way to get one. And in both cases the error bar has its own edge, which is the recursion this collection keeps arriving at: a bound is a model too, and the honest thing to do with it is to say where it stops. Which is not a regress in practice, because the sequence terminates: at some order the correction is smaller than the measurement’s own noise, and the next term is unmeasurable rather than merely small. On a solved curve that floor is the arithmetic’s rather than an instrument’s, which is why this collection can carry the argument one term further than a bench could — and why the place it stops is decided by the order that stops helping’s finding about the series rather than by anything about the measurement.