Where the mechanisms are one mechanism

How small is small signal measures one linear model failing for one reason. A tangent replaces a curve, the curve bends away from it, and the whole of the departure is the second and third terms of one series. There is nothing to combine, because there is only one thing.

Almost no real model fails that tidily. An amplifier is described as running out of three separate things — the gain–bandwidth product, the slew rate of its input pair, and the supply rails — each with its own line on a data sheet, its own units and its own boundary. Three impairments, three numbers, and a reader is left to combine them.

The arithmetic for combining them is not a matter of taste. Errors from genuinely independent mechanisms add in quadrature, which is what a noise budget correctly assumes; errors that are one mechanism seen from three sides add as magnitudes. Across thirty-five points of the plane at which this stage’s model is still within twenty-five per cent of the truth, those two answers differ from each other by as much as 38.3 per cent — and which of them to use has been decided once, at one point of one amplifier, and reported as a single number. A number has no direction, and the question is entirely about direction.

What is being combined, and what direction means #

The stage under test is the one the edge that is a region draws its plane for: a non-inverting amplifier of closed-loop gain two, built from a 10 MHz part with ±12 V rails, marched forward in time with a trapezoidal rule. Three departures live in it and they are separable by construction. Running the same march with the input pair linearised removes slewing; running it with no clamp on the compensation node removes the rails; the difference between two consecutive marches is one mechanism and nothing else.

That gives three error waveforms on one time grid rather than three error magnitudes, and the three sum to the whole exactly, because each is defined as a difference. What is not fixed by construction is the angle between them.

Treat each error as a vector whose components are its samples. Two errors of magnitude $a$ and $b$ combine to $\sqrt{a^2 + b^2 + 2ab\cos\theta}$, and $\theta$ is the angle between the vectors: a cosine of one is a straight sum, a cosine of zero is quadrature, and there is nothing else. So the whole question — which arithmetic — is a question about three cosines, and the rung below reported a single summary number in their place. One number cannot tell three slightly correlated mechanisms from two that are the same object with a third at right angles to both.

The reproduction, which comes before the departure #

A cosine between two marched waveforms is exactly the kind of quantity that returns something plausible from a mis-stated problem, so the first thing asked of it is a case whose answer is known before anything is run.

The input pair is a hyperbolic tangent. Its first departure from a straight line is cubic, so at an amplitude where that cubic term is a millionth of the linear one the slew error must be the pointwise cube of the bandwidth error. And the angle between a sinusoid and its own cube is fixed by trigonometry: $\sin^3\theta = (3\sin\theta - \sin 3\theta)/4$, the third harmonic is orthogonal to the fundamental, so the cosine between them is $3/\sqrt{10}$ — 18.4349 degrees, with no gain, no gain–bandwidth product, no tail current and no compensation capacitor anywhere in it.

Six figures of agreement, and the number agreed with is a piece of trigonometry rather than a fitted constant. That is what entitles every angle below to be quoted, in the same way that a full window reproducing a closed form entitles a field solver to disagree with it elsewhere. The habit is the one one step computed twice applies to a transient and the digits the arithmetic did not have applies to a solve whose answer looks fine: find the case where the answer is not in doubt, and check there first.

It also settles something the rung below could only observe. Two lags on one signal were said to add as magnitudes because they are both lags. They add as magnitudes because one of them is the cube of the other, which is a much stronger statement and a much narrower one — it is true of a tanh and would not be true of an independent second pole.

The departure: the third mechanism is at ninety degrees #

Drive the same stage harder at a frequency the pole can reach and the rails can too, and the three angles come apart.

Two of the three are one mechanism. The third is orthogonal to both, to two and a half parts in a thousand, while its own error runs from a third of a per cent to eighteen per cent of the signal — which is to say it is not small and not correlated. The rails add in quadrature and the two frequency mechanisms do not, and no reading of a data sheet suggests that, because the data sheet’s three numbers look alike.

The bandwidth error along that sweep is 0.615 per cent at every amplitude, unchanged to five parts in $10^{15}$. That is not a near-constancy worth rounding: a linear stage’s fractional error has no amplitude in it at all, which is what the word linear means, and it is the reason every change along the sweep can be attributed to one of the other two without subtracting anything.

There is one more thing in that figure and it is a warning about the instrument the rung below used. At 30 kHz and 14 volts of demanded output the summary correlation reads 0.0168 — a number that says these three are independent — while the cosine between the two frequency mechanisms is 0.9168 and they are as inseparable as ever. The summary is dominated by whichever mechanism is largest, and the largest one there is the rails. A single figure of merit over three vectors is not wrong so much as unable to say anything, and the pairwise angles are what it was standing in for.

What the angle is a function of #

The plane has two axes and the amplifier has at least two design parameters, so the angle between the two frequency mechanisms could in principle be a function of four things. It is a function of one.

The variable is the slew demand: $2\pi f\hat V/\mathrm{SR}$, what the sine asks of the tail current divided by what the tail current can supply. Neither the closed-loop gain nor the gain–bandwidth product appears except through it. Below a demand of about a tenth the cosine is $3/\sqrt{10}$ and the second mechanism is simply the cube of the first; at a demand of one — the sine asking for exactly the available slew rate — it has fallen to about 0.84; by eleven times the slew rate it is 0.1055 and the two mechanisms have very nearly become independent, which is a fact about a waveform that has stopped being a sine rather than about an amplifier anybody would ship.

The demand is the quantity a full-power bandwidth is quoted at, which is worth saying because it puts the two in the right relation. A full-power bandwidth is the frequency at which the demand reaches one: the point at which a sine of that amplitude can no longer be produced at all. The angle is untouched most of the way there — a fall of 0.17 per cent at a quarter of the demand, 0.79 at a half, 2.47 at three quarters — and then moves suddenly, to 32.40 degrees at a demand of one, which is an eleven per cent fall in the cosine arriving inside the last quarter. So the collapse variable is already on the data sheet, under a different name and used for a different purpose, and the angle it governs turns at exactly the frequency the refusal does.

That collapse is why one measurement at one corner of one amplifier was worth generalising, and it is also the reason nobody could have known it was. A single point on a plane carries no information about which of its coordinates matters. This is the same move the edges that are lengths makes for a boundary set by a size and a boundary is a model and a tolerance makes for one set by how much error is allowed: find the dimensionless group and the family of curves becomes one curve.

The other axis, where the answers come out reversed #

The rung below measured how sharply each side of the region is approached, and measured it in frequency — the axis two of the three mechanisms live on. Along amplitude the answers invert.

Finite gain–bandwidth has no amplitude edge. Not a distant one; none. Its fractional error is a property of the frequency and of nothing else, so at a frequency where it is under one per cent no amount of drive will take it there, and a designer who has decided a signal is too large has said nothing about it at all.

That asymmetry decides what a reduction in signal level can buy, and it is measurable in one line. The same stage at 30 kHz asked for 14 volts is 0.6152 per cent wrong from bandwidth, 0.1928 from slewing and 9.2961 from the rails, for 9.3317 in total. Halved to seven volts it is 0.6152 per cent wrong from bandwidth, 0.0310 from slewing and nothing at all from the rails, for 0.6445 — a factor of fourteen and a half for six decibels. Halve it again and the answer barely moves, because the remaining error is the pole’s and the pole does not care. The only remedies for that term are a faster part or a lower closed-loop gain, and the second of those turns out to change the shape of the region rather than its size.

The rails and the tail current are both cliffs, and they are the same cliff to two per cent. Under a fifth more drive — 16.1 per cent for one and 18.3 for the other — takes each from one per cent wrong to ten. That matters because the rung below found slewing to be a cliff in frequency — 1.19 times from one per cent to ten, against 10.05 for the pole — and left the third mechanism unmeasured on either axis. It is a cliff too, and by almost exactly the same factor, which says that both are the same kind of failure: a quantity that has run out, rather than a term that is growing.

And the clipping edge is at 12.1855 volts at one kilohertz, 12.1855 at ten and 12.1859 at forty — the same amplitude to under three parts in a hundred thousand across a factor of forty in frequency. That side of the validity region is horizontal, and this is the measurement that says so rather than the drawing that assumes it.

The corner moves with a design choice, and then leaves #

The slew rate of this stage is $2V_T\omega_t$ and contains no closed-loop gain. The small-signal edge is $\omega_t/A$ and is inversely proportional to it. So the amplitude at which the two become comparable must move when the gain moves, and the direction is decidable before anything is marched.

Working the cancellation through gives an input amplitude with no circuit in it at all: $2V_T\sqrt{1-\varepsilon^2}/\varepsilon$, which at one per cent is 5.170 volts. The march bisects 5.299 at a gain of two — the two routes agree to 2.5 per cent, and the residual is not slack in either. It is the marched small-signal edge sitting at 48.8 kHz where a single pole puts it at 50.0, which is a separate measurement and accounts for the whole of the difference.

The consequence is a prediction rather than an observation, and it is the sharper half of the page. The output corner is that amplitude times the gain, the rails are fixed, so past a stated gain the corner is above the rails and there is no corner: the region has a vertical side and a horizontal one, and slewing never enters the argument at all. The crossing is measured at a gain of 2.270 and the closed form $\varepsilon V_{sat}/2V_T$ puts it at 2.321. At two per cent tolerance it is 4.59 and at three per cent 6.91.

So the shape of an amplifier’s validity is decided by the closed-loop gain, and a stage of gain ten does not have a smaller version of the region a stage of gain two has — it has a region of a different shape. That is the same discovery a band rather than an edge makes about an analogue switch, where the two bounds on load resistance belong to different parts of one component, and it is the reason a boundary quoted as a number cannot carry it.

One thing does not move. The corner sits at 0.7842, 0.7875, 0.7897 and 0.7908 of where the two specifications cross, for gains of 1.2, 1.5, 2 and 2.2 — a fifth inside, constant to eight parts in a thousand. The rung below found that fraction unchanged when the tolerance moved and could not say whether it was a property of the pair of mechanisms or of the numbers. It survives the gain moving too, which is a second axis and a stronger claim.

The arithmetic that is actually right #

Three cosines give a budget rule directly, and it is neither of the two anybody reaches for. Add the bandwidth and slew errors at $3/\sqrt{10}$, then add the clipping error to that in quadrature:

$\varepsilon^2 = \varepsilon_{bw}^2 + \varepsilon_{sl}^2 + \tfrac{6}{\sqrt{10}}\varepsilon_{bw}\varepsilon_{sl} + \varepsilon_{cl}^2$

Over thirty-five points spanning two closed-loop gains, half a volt to seven of drive and 10 to 50 kHz — every point at which the model is within twenty-five per cent of the truth — that rule is worst by 2.93 per cent. Pure quadrature is worst by 17.08 per cent low. A straight sum of all three is worst by 36.68 per cent high. The cheap rule is the right one and it costs one extra multiplication.

The optimistic answer is the dangerous one and it is the one a budget written by habit produces, because quadrature is what a noise budget does and a noise budget is the budget most engineers have written most often. The difference between the two situations is not the size of the contributions; it is whether the mechanisms share a phase angle, exactly as what the second path costs at the floor finds for two noise sources that genuinely do not.

What it does not say #

It does not hold once slewing is large. The orthogonality of the rails against the other two is measured under 0.0025 at 30 kHz, where the slew mechanism reaches 0.37 per cent. At 45 kHz, where it reaches 26.5 per cent, the worst clip cosine is 0.1442; at 60 kHz, where it reaches 59.7, it is 0.6846. A flat-topped output and a triangular one are not independent shapes, and past that point the decomposition is describing a waveform nobody would call an amplified sine. That is the edge of this page’s own model, and it is a frequency: at a gain of two, about 30 kHz for a signal at the rails.

It is one waveform. Every angle here is measured on a sine, because a sine has an amplitude and a frequency and therefore a place on the plane. A step has neither, which is why the step that is too big and the step too large to have an impedance measure the same slew rate against a different question and get different numbers from it.

And the third mechanism is a soft clamp. The rails here are two exponentials with a fifty millivolt knee, which is what an output stage does and what a Newton loop can be marched through. A harder limit would sharpen the cliff and would not move the angle, since orthogonality is about shape rather than about magnitude — but that is an argument rather than a measurement, and it is not made here.

The number worth carrying #

Two of an amplifier’s three departures are one mechanism at 18.4349 degrees, exactly the angle between a sinusoid and its own cube; the third is at ninety, to under three parts in a thousand. Errors that share a phase add as magnitudes and errors at right angles add in quadrature, so the right budget does both in one line and is within three per cent where the two familiar rules are seventeen per cent low and thirty-seven per cent high.

The habit that goes with it is shorter, and it generalises past amplifiers. Every model has an edge collects boundaries as numbers, which is what a boundary has to be to go on a page. A number has no direction, and the moment two of them are combined the direction is the whole of the answer — so a model with several failure mechanisms is not described by their sizes, and the measurement that decides how to add them has to be made on waveforms rather than on magnitudes. Where that measurement has never been made, the safe assumption is a straight sum and the common assumption is quadrature, and they are different by a factor that is large enough to design wrongly on.