A boundary is a model and a tolerance

Every model has an edge opens this collection with four numbers on one frequency axis: 1.42 kHz for an ideal amplifier, 7.96 kHz for a ten-volt output, 3.97 MHz for Kirchhoff’s laws on a ten-centimetre board, 4.69 MHz for a hundred-nanofarad capacitor. The ordering is the argument, and the essay defends the fact that the four thresholds are not the same: one per cent of gain for the amplifier, ten per cent of impedance for the capacitor, one degree of phase for the board.

That defence is correct and it stops one sentence early. Having said that the four numbers are answers to four differently-worded questions, it does not ask what happens to any of them when the wording changes.

Every one of those numbers came out of a function that already takes the tolerance as an argument. Three of them take a fraction of error directly, the fourth takes a phase error in degrees, and none of the four has ever been called at anything but its default. So the collection has, in four different files, a knob it has never turned.

What is being swept, and how five different quantities share an axis #

The sweep is one argument: the fraction by which the model’s answer is allowed to differ from the better model’s. For the amplifier that is a gain ratio, for the capacitor an impedance ratio, for the small-signal model the ratio of a fundamental to the one a tangent predicts. For Kirchhoff’s laws it is a phase angle, and the conversion is the only one worth stating — a phase error of θ radians is a fractional error of θ to first order, so the tolerance goes in as degrees through $180/\pi$ rather than becoming a second knob.

The fifth boundary is an amplitude in millivolts and has haunted this ladder since its first page, because it cannot share a frequency axis with the other four. It can share a dimensionless one. Divide each boundary by its own value at one tolerance and what is left is how far it moves, which is a pure number for a frequency and for an amplitude alike. An exponent has no units, and that is the whole reason all five can be drawn together at last.

The reproduction, which comes before the departure #

Two of the five have exponents that can be known before anything is computed, and they are the calibration rather than the finding.

Kirchhoff’s boundary is $f = (\theta/360)(v_p/\ell)$ — a phase angle divided by a transit time. It is linear in its own tolerance by algebra, so the fitted exponent must come back as 1 and any departure from it is a fault in the fitting rather than a fact about circuits. It comes back as 1.000 with a worst departure from its own fitted power law of $1.7\times10^{-15}$.

A full-power bandwidth is $SR/(2\pi\hat V)$ and contains no tolerance at all. Its exponent must be zero, and it is, to $4.2\times10^{-15}$.

Those two are the instrument being made to reproduce an answer that is already known, in the sense the assumption that is a geometry insists on: a fit that has never been shown to return an exponent somebody derived on paper is a fit whose disagreements mean nothing. The other three exponents are quoted against those two.

Three exponents #

Fitted over the two decades from a tenth of a per cent to ten per cent, the five come out as 1.000, 0.513, 0.497, 0.500 and 0.000.

The amplifier is 0.513, the small-signal model 0.497, the capacitor 0.500. All three are square roots and all three are square roots for the same reason: the quantity being compared departs second order in the variable being swept. A single pole’s magnitude is $1/\sqrt{1+x^2}$, flat to first order at the origin. A capacitor’s impedance ratio is $1 - \omega^2 C(L - \tfrac{1}{2}\,\mathrm{ESR}^2 C)$, again with no linear term — which is why the capacitor that is an inductor is quoted at ten per cent rather than one. A linearised exponential’s gain ratio is $1 + x^2/8$ to leading order, the expansion behind how small is small signal. Set a second-order departure equal to a tolerance and the variable moves as the square root of it.

Kirchhoff’s boundary is 1.000 because a phase angle is first order in frequency — nothing cancels, so the error and the variable move together.

And the output’s is 0.000 because a full-power bandwidth is not an error at all. It is the frequency above which a sinusoid of that amplitude cannot be produced, which is a statement with no fraction in it and no way to accept one — the distinction the step that is too big is built on.

The residual is the part worth keeping. Two of the five are power laws and three are only asymptotically power laws, and reporting the exponents without the residuals would make those look like the same kind of statement. The amplifier’s 2.98 per cent is where its curve leaves the fit at the loose end: at thirty per cent of error a single pole is nowhere near its own expansion, and $\sqrt{2\,\mathrm{frac}}$ has stopped being the answer. The capacitor’s 0.0863 per cent is the cleanest of the three because its expansion has no term between the second and the fourth.

That is the same distinction the constant that is a window draws about a fitted ideality factor: a number extracted over a window is a statement about that window, and the residual is what says how far outside it the number survives.

The exponent names the mechanism #

This is what makes the sweep a diagnostic rather than a caveat.

A boundary quoted as a single number says nothing about what produced it. Two boundaries at the same frequency, from the same part, in the same units, can be a gradient and a cliff — which is exactly what the edge that is a region found, with finite gain–bandwidth taking a full decade to go from one per cent to ten and slewing taking a fifth of one. That essay measured the difference by running each mechanism alone on a marched circuit and subtracting; it is several thousand cycles of work.

The exponent gets the same answer from arithmetic. 1 means a first-order departure, ½ means a second-order one, and 0 means a refusal. A model whose edge moves as the square root of the tolerance is a model whose error is quadratic near the boundary, and that is a statement about the mechanism rather than about the threshold anybody picked.

The two readings are the same measurement inverted. A boundary that moves as the first power of the tolerance takes a decade of frequency to go from one per cent wrong to ten; one that moves as the square root takes a factor of about three; one that does not move takes nothing at all, because the error goes from acceptable to total across a frequency that cannot be resolved.

The ordering is a function of the tolerance #

The opening figure’s argument is its ordering — amplifier, output, board, capacitor — and the ordering is not a property of the four models. It is a property of the four models and the three tolerances chosen for them, and putting all four at one tolerance moves it twice.

The board and the capacitor change places at 0.424 per cent, where both boundaries are 964 kHz. Below that tolerance the board’s edge is the lower and Kirchhoff’s laws are the first thing to fail; above it the capacitor is. Neither part has changed and neither boundary is wrong. The board’s edge moves as the tolerance and the capacitor’s as its square root, so the two curves cross, and where they cross is decided by how good an answer is being asked for.

The amplifier and the output change places at 21.7 per cent, where both are 7.96 kHz. A designer who will accept a fifth of the answer being wrong finds the ideal-amplifier model still usable above the frequency at which the output physically cannot produce the waveform — and that is not a paradox, it is the difference between an error and a refusal reasserting itself at the loose end.

So the sentence “an ordinary amplifier circuit runs out of model three thousand times sooner than a ten-centimetre board does” is true, and it is true at the tolerances the collection happens to quote. At one common tolerance the ratio is 1,600 rather than 3,000, and the two boundaries the sentence compares move by different powers, so the ratio itself is a function of the question.

One model, one mechanism, two exponents #

The sharpest case is not a comparison between two boundaries. It is one boundary, of one model, computed twice within this collection at two different exponents — and both numbers are in print.

The ideal-amplifier edge on the opening axis is a magnitude ratio: the frequency at which $|H|$ is one per cent below the ideal answer, which is 1.42 kHz for a gain of a hundred from a megahertz part. The edge that is a region computes the same boundary of the same model as a waveform difference, counting the phase lag the ideal model also claims is absent, and states plainly why: “eight degrees of lag is not a model holding to one per cent.”

Those two measures have different exponents, because a pole’s magnitude is second order in frequency and its phase is first.

A factor of 14.2 at one per cent, and 44.7 at a tenth of one, on the boundary this collection opens with. The gap widens as the tolerance tightens, because the two exponents differ: the ratio is $\sqrt{2/\mathrm{frac}}$, which is a statement about the two readings and not about the amplifier.

The marched check in that figure is worth one paragraph, because it is the essay’s only place where two genuinely independent routes meet. The algebraic edge is one line of complex arithmetic on a single pole. The dots are a bisection on a circuit integrated forward in time with a compensation capacitor, a transconductor, an output resistance and a finite open-loop gain — and they sit 5.7 per cent below the algebra at three tenths of a per cent, which is far too large to be rounding.

The cause is the stage’s finite open-loop gain, which leaves the closed-loop answer 0.100 per cent low before any frequency is applied at all. That error is in phase with the signal where the pole’s is in quadrature with it, so the two combine as $\sqrt{f^2 - e_0^2}$ rather than adding, and $\sqrt{0.003^2 - 0.001^2}/0.003 = 0.943$ against a measured 0.943. With that term accounted for the two routes agree to 0.077 per cent at the worst of four tolerances. Two routes that disagree by an amount too stubborn to be discretisation are usually two routes with one term between them, and the term here lives at zero frequency, which is the last place a frequency-domain argument looks.

One boundary, three thresholds, and 36 times between them #

The same defect has a second instance and it is the one that is entirely inside this collection’s own machinery.

Kirchhoff’s own frequency reports a ten-centimetre board as one degree long at 3.97 MHz, and beside that number the same computation returns a tenth-of-a-wavelength figure of 143 MHz. Both are printed. A tenth of a wavelength is thirty-six degrees, so the two rules of thumb for one boundary are exactly 36 times apart — a ratio with no material, no frequency and no geometry in it, because it is the ratio of two chosen angles.

And there is a third. The refusal this collection raises when a lumped model is applied to an electrically large circuit fires at ten degrees, not at one. So the site reports its edge at 3.97 MHz, declines to answer above 39.7 MHz, and sits inside a convention that says 143 MHz — three thresholds for one boundary, spanning a factor of thirty-six, with the middle one being the site’s own.

The measured departures settle it. At one degree the two models disagree by three parts in ten thousand; at the tenth-of-a-wavelength rule they disagree by thirty per cent, which is not a boundary of a model but the end of it. The conventional rule is not conservative and it is not a rounding of the strict one — it accepts an answer that is wrong by a third.

And the exponent is 2 rather than 1, which is the same trap once more. A model with one inductance and one capacitance already carries the first-order term of the line’s expansion, so what is left is second order — the phase length is proportional to the tolerance and the impedance error to its square, and the two exponents belong to the same boundary. Which of them a reader gets depends entirely on what the lumped model is being compared against: a wire, or two elements.

What this does not say #

It does not say the collection’s numbers are wrong. Every boundary on the opening axis is right at the tolerance it was computed at, and the essay that draws them defends those tolerances well. What is missing is one number beside each, which costs nothing to compute and which turns a threshold into a curve.

It does not say a common tolerance is better. One per cent of gain is a reasonable place to say an amplifier has failed, and one per cent of a decoupling capacitor’s impedance is a place nobody would notice; putting both at one number would make the picture tidier and the boundaries less useful. The argument is that the choice is a variable and should be reported as one.

And it does not give the exponent to a boundary that has no expression. The three square roots above came out of expansions that can be done on paper, and the fit merely confirmed them. A boundary found by bisecting a marched circuit — the kind the edge that is a region and a band rather than an edge produce — has an exponent that is a measurement like any other, and getting it costs a sweep rather than a line of algebra.

Two limits that arrive together #

There is one number in this collection that shows what the sweep is for outside its own field.

The inductor one mode cannot see computes a common-mode choke’s differential corner at 7.86 MHz, from a leakage inductance and the resistances either side of it. On the five-centimetre board that essay is about, Kirchhoff’s laws are a degree out at 7.94 MHz. The two agree to one per cent, and they have nothing whatever to do with each other: the first is set by a coupling coefficient and a source impedance, the second by a length and a dielectric constant.

That coincidence is worth exactly as much as the tolerances behind the two numbers. Move the phase criterion from one degree to three and the board’s edge goes to 23.8 MHz and the coincidence evaporates; the choke’s corner does not move, because it is a first-order response and its corner is defined by a half-power point rather than by an accepted error. So the honest statement is that two limits of unrelated kinds land within a per cent of each other on one board at the strictness this collection quotes, and that neither of them appears on a schematic — which is the point the edges that are lengths makes about the second of the two.

The number worth carrying #

Three exponents: 1, ½ and 0.

A boundary whose edge moves in proportion to the tolerance is a first-order departure, and tightening the requirement by ten costs a decade of the variable. A boundary whose edge moves as the square root is second order, and the same tightening costs a factor of three. A boundary whose edge does not move is not an error boundary at all; it is a refusal, and asking for a looser one buys nothing.

The habit that goes with it is one line long. Ask what a quoted boundary’s tolerance was, and then ask what it does when that tolerance moves — because the first question has an answer that is often missing and the second has an answer that is nearly always cheap. This collection’s own boundaries took a sweep of forty-one points and no new machinery to place in three classes, and two of the three classes were things somebody already knew and had never written beside the number.

What the sweep bought here is a re-ordering the opening figure could not show, a factor of 14.2 between two numbers for one amplifier, and a factor of 36 between two rules of thumb for one board. None of those is a correction. All three are the second half of a statement that had only ever been made in halves, which is the same shape as the edges that move with the room: a number, the condition it was computed under, and the multiplier that carries it to a different condition. There the condition was a temperature. Here it is how wrong the answer is allowed to be, and unlike a temperature it is not in the room — it is in whoever asked.