The order that stops helping

A degenerated transistor stage has an amplitude at which its second harmonic reaches any stated distortion, and this field has now derived three expressions for that amplitude. Each is one term of a reverted power series longer than the one before it. Each is nearer the measurement than the one before it. The obvious thing to do next is to derive a fourth.

This essay is about why that is the wrong instinct, and about the quantity that says so.

The three expressions come out of one construction. In normalised units — collector current in units of the bias current, drive in thermal voltages — the device relation of a stage degenerated by a factor $D$ is $u = \ln(1+x) + (D-1)x$, whose coefficients are exact and known. Reverting that series gives the transfer $x(u)$, and collecting a cosine drive through it gives the fundamental and each harmonic as a polynomial in the drive amplitude. Truncated at second order the pair gives the leading expression this field started with; truncated at fourth it gives the correction; truncated at sixth it gives the one this essay is about.

The error of a truncation is its own order #

The first measurement is the one that makes the rest predictable, and it is a slope rather than a number.

An expression correct through order $m$ is wrong by the term in $u^{m+1}$ it dropped, and for a ratio whose numerator and denominator each carry only alternate powers that arrives as $u^m$ in the ratio itself. So the leading expression’s error should grow as the square of the drive, the four-term one’s as the fourth power, and the six-term one’s as the sixth.

Fitted on the measurement rather than taken from that argument, over the decade below the point where each first reaches one per cent, the three exponents are 2.00, 4.00 and 5.88. At a degeneration of eleven they are 2.00, 4.00 and 6.01.

The fit has one piece of care in it that is worth stating, because the first version of it returned 0.84 for an exponent that should have been 2. Each truncation’s error changes sign somewhere — the term that was dropped happens to cancel the accumulated error of the terms that were kept — and the logarithm of an error dives to minus infinity there. A window that straddles such a point fits nothing at all. The fit therefore stops at the first change of sign rather than running through it, which is what turns a meaningless 0.84 into a 2.00.

What each order buys, and how fast that shrinks #

An exponent is a rate of getting worse. Turned round, it is a range: if the error of the truncation at order $m$ is about $c_m u^m$, the drive at which it reaches a tolerance $\varepsilon$ is $(\varepsilon/c_m)^{1/m}$, and the useful thing to compare between two truncations is the ratio of those two drives.

Measured at one per cent of error and a degeneration of six, the three expressions are good to 1.12, 3.88 and 7.03 thermal voltages — factors of 3.46 and then 1.81. At a degeneration of eleven they are 1.79, 5.61 and 8.28, factors of 3.14 and 1.48.

The pattern is the same at every degeneration tried and it is the whole argument of this essay. The first correction buys a factor of about three and a half in usable drive. The second buys under two. There is no reason to expect the third to buy more than about a third, and the algebra for it is already four lines of collecting terms that nobody would check by hand.

That is what a series of this kind does. It is not converging on the answer at a fixed rate; each term is smaller than the last only while the drive is small, and how small “small” has to be is what tightens as the series lengthens.

The drive at which more terms make it worse #

The ranges above end in a tolerance, which is a choice. The next number does not.

Somewhere above them there is a drive at which the six-term expression is exactly as wrong as the four-term one, and above which it is worse. At a degeneration of six that drive is 8.88 thermal voltages, where the device’s second harmonic is 11.4 per cent. At a degeneration of eleven it is 21.2 thermal voltages and 16.7 per cent.

It is found by bisecting the difference of the two magnitudes of error, on the first crossing rather than on any crossing — both errors eventually blow up, and two quantities that both blow up cross each other more than once. A plain bisection over the whole range lands on a later crossing where neither expression means anything, which is a wrong answer that looks like a right one.

Past that drive, adding terms is not merely poor value; it is negative value. The longer expression returns a number further from the device than the shorter one does, and nothing about the derivation warns of it, because every step of it is correct. The series is asymptotic: its terms shrink to a minimum and then grow, and the best answer it can give is the partial sum truncated at that minimum.

Two things follow that a designer can use. The crossing sits well beyond the range either expression is usable in — 8.88 thermal voltages against 7.03 — so in practice the diminishing returns arrive long before the reversal, and the reversal is a fact about the mathematics rather than a trap in the workshop. And the ordering of the two errors is the only diagnostic available from inside the algebra: if a longer expression disagrees with a shorter one by more than the shorter one’s own estimated error, the drive is past where either belongs.

What to do instead, which is cheaper than the algebra #

The measurement these three expressions are chasing costs nothing to make.

A transfer curve solved by Newton’s method at a few hundred points, sampled at a cosine drive, transformed, and its second line divided by its first, gives the harmonic at that drive to a part in $10^{10}$. Bisecting the drive until the harmonic reaches a target gives the edge to twelve digits in forty iterations. On the machine that rendered this figure the whole operation is a few tens of milliseconds, and it has no range at all: it is as accurate at forty per cent of distortion as at a thousandth of one.

So the honest answer to “what does the sixth order give” is that it gives a closed form with a stated range, and that a closed form with a stated range is worth having for two reasons that are not accuracy. It shows the shape of the dependence — $4V_T t D^2$ says the edge is proportional to the target and to the square of the degeneration, which no bisection reports — and it can be inverted, so a designer choosing $D$ for a distortion budget starts from an expression rather than from a search.

Neither of those improves with a sixth term. The shape is in the leading term; the range is what the corrections buy; and past the range, the bisection is the answer.

The check that these coefficients are the right ones #

None of the above would be worth reading if the coefficients were transcribed.

The reversion is written out generically — given the coefficients of $u(x)$, return those of $x(u)$ — rather than by copying the five published values, and the check on it is the one a table cannot pass: feed it the exponential’s own series and it must return $a_i = 1/i!$. It does, to sixteen digits, at every order. Feed it the degenerated device’s series and the third coefficient comes out $(3-2D)/6D^5$, which is what the essay one rung down derived by hand and is where the $D = 3/2$ pole in that expression comes from.

The harmonic weights are the other half. A cosine raised to an even power carries a second harmonic of a specific size — a half for $\cos^2\theta$ and $\cos^4\theta$, and 15/32 for $\cos^6\theta$ — and getting the last of those wrong would move every number in this essay by six per cent without moving any of them in a way that looked wrong. The weights are checked by the only test available: the three truncations must agree with each other, and with the measurement, in the limit of small drive. They agree to a part in $10^6$ at a twentieth of a thermal voltage.

The drives this is all about are not small #

It is worth putting the numbers in volts, because “thermal voltages” makes everything sound safely theoretical and the drives in question are not.

A thermal voltage is 25.85 millivolts at room temperature. So at a degeneration of six the leading expression is trustworthy to one per cent up to 29 millivolts of drive at the base, the four-term one to 100 millivolts, and the six-term one to 182. The crossing at which the longer expression becomes the worse one is at 230 millivolts.

A hundred millivolts of base drive is not an unusual signal. It is a line-level audio input into a stage with a gain of ten, or the output of a preceding stage that was designed for exactly this one. So the range these expressions have is not comfortably outside the region anybody uses — it runs right through it, and which expression a designer should be using depends on the amplitude in a way that is easy to get wrong in either direction.

At a degeneration of eleven the numbers are 46 millivolts, 145 and 214, with the crossing at 548. More degeneration buys more range in absolute terms because it buys linearity; the shape of the trade does not move, which is what makes the exponents worth having.

Why the sign changes are not a nuisance #

The zero crossings that had to be kept out of the fit are worth a paragraph of their own, because they are the clearest evidence that the series is behaving the way this essay claims.

An error that changes sign means the truncation passed through the right answer on its way to being wrong in the other direction. That can only happen if the omitted terms alternate — the term the truncation dropped is subtracting where the terms it kept were adding — and it is exactly what the coefficients do: the reversion of this device’s series is alternating in sign from the third coefficient onward.

Which is also why each truncation is so accurate for so long and then fails so abruptly. An alternating series with terms that shrink has an error bounded by its first omitted term, which is tiny while the drive is small; when the terms stop shrinking, the bound stops holding and there is nothing gradual about it. The curves in the first figure are flat for two decades of drive and then turn nearly vertical, and that is the reason.

The practical consequence is a warning rather than a method. A truncation evaluated near one of its own sign changes agrees with the measurement beautifully, at a drive where the next order disagrees with it badly — so a spot check at one amplitude can certify an expression that is about to fail. The site’s own habit covers this: an assertion is made across a range, not at a point, and the range is stated.

What this rung says about the three below it #

The ladder this essay ends has an unusual shape, and it is worth naming.

The first rung measured what a resistor in the emitter buys — gain traded for linearity, at an exchange rate. The second derived the two exponents behind that trade. The third asked whether a seven per cent discrepancy in the second’s expression was the series being truncated or a mechanism the series does not contain, and found it was the truncation, by carrying one more term and watching ninety-five per cent of the gap close.

The fourth cannot ask the same question again. Carrying one more term does close more of the gap, at small drives, by an amount that is now predictable from the exponent rather than interesting in itself — and at large drives it opens the gap instead. So the ladder does not end because the subject ran out. It ends because the tool did: the series has a range, the range has been measured, and past it the question has to be asked of the device rather than of the expansion.

That is the site’s own premise pointed at one of its own methods. Every model here is drawn with the frequency, amplitude or size at which it stops being true; a truncated series is a model of a device, and this is the amplitude at which it stops being one.

What the series is a series in #

A truncated series with a radius nobody quotes is the shape of three results in this collection. The straight lines, and where they are not the curve is the same defect on an asymptotic construction rather than a power series. The distortion a linear model cannot have is the first term of this series taken alone, and it is exactly the term this page says stops being enough. What a resistor in the emitter buys is the parameter the series is expanded in. The exponent that is a square is the same accounting on a device whose law is a polynomial to begin with, and A bias point is a solution, not a choice is the alternative to expanding at all — one Newton iteration against eleven terms.

What is checked #

The three exponents are asserted against the truncation orders they should equal, to within a fifth, with the fit refusing any window that crosses a sign change. The ranges are asserted to increase with order, and the factor each buys to shrink. The crossing is asserted to exist, to be the first one, and to lie beyond the range either expression is usable in. And the reversion is fed the exponential’s own series and must return the reciprocal factorials, which is the check that separates a derivation from a transcription.

What is not checked here, and is stated rather than measured: whether an eighth-order truncation buys the third of a factor the pattern predicts. Writing it is four lines of collecting terms and checking it is the same measurement again, and the reason it is not here is the reason this essay exists — the answer is known in advance and is not worth the algebra.

The three rungs this ends, and the one boundary of the same kind #

The ladder below this one is three essays long and each of them was an improvement that this one bounds.

What a resistor in the emitter buys measured two exponents and could explain neither. Where the two exponents come from reverted the degenerated transfer curve and explained all three of its results exactly, including a singularity at a degeneration factor of three halves. And what the fourth order says about the third carried the reversion one term further and found the leading expression 6.89 per cent optimistic where the correction predicts 7.33, taking the residue to 0.362 per cent.

Read as a sequence, the obvious continuation is a fifth term, and the finding here is that there is no point in one. What is unusual is that the boundary is not a value of a physical quantity but a property of a procedure: an asymptotic series approaches its answer and then leaves, and the optimal truncation depends on the drive.

The only other boundary of that shape here is the digits the arithmetic did not have, where a ladder synthesis stalls at order nine and doubling the precision makes it worse — the object being described is fine and the procedure describing it is not. Both belong on an axis the edge map cannot carry, and both are found the same way: by doing the next step and measuring whether it helped.

Which is the practical instruction this ladder ends on. An expression carried one order further is cheap; deciding whether the extra order is worth having is not a matter of judgement but a measurement, and the measurement is the same one every time — evaluate both truncations against the device across the range of interest and see where the longer one stops being nearer.