The edge that is a region

The first essay in this field draws four boundaries on one axis and puts a rule at the end of each: the frequency above which an ideal amplifier is not one, the frequency above which a capacitor is an inductor, the frequency above which Kirchhoff’s laws are a degree out on a circuit board. It is the picture the whole collection is organised around. It also carries, in its own caption, a sentence that admits the picture is not big enough.

The fifth boundary is an amplitude rather than a frequency and cannot share this axis: a small-signal model is 1% wrong above 7.3 mV, at every frequency there is.

The note is a confession dressed as a footnote. A model does not stop being true at a point, because a model is a claim about more than one variable at a time, and the set of places where it holds is therefore a region. Every model has an edge computes each edge by holding everything else fixed and moving one thing — which is the right way to get a number and the wrong way to describe a boundary, because it gives one point on a curve and no indication of which way the curve goes.

This essay draws the curve.

One model, two axes #

The model under test is the one the field opened with: the ideal amplifier, which says that the output of a feedback stage is the closed-loop gain times the input, now, with no frequency in the statement and no amplitude either. The circuit is a non-inverting stage of gain two, built from a 10 MHz part with ±12 V rails, marched in the time domain with the same trapezoidal rule the rest of the site uses.

Three things in that circuit make the ideal answer wrong, and they are deliberately separable:

• the transconductor into the compensation capacitor, which gives the stage a pole and makes its gain fall with frequency;
• the input pair’s own tanh, which runs out of tail current and gives the stage a slew rate of 2·Vₜ·ωₜ — 3.25 volts per microsecond here, and containing no design choice, as the step that is too big established;
• the rails, which are a limit on the output and on nothing else.

Removing the first is impossible; removing the second and third is one flag each, so the same march can be run with any subset and the difference between two marches is exactly one mechanism. That is the only honest way to say which mechanism a departure belongs to, and it matters here because the whole question is whether they take turns.

The error is one number: the root-mean-square difference between the marched output and gain times the input, divided by the root-mean-square of that ideal answer, over a whole number of cycles after the start-up transient. It counts the gain that is low, the phase lag, the peak that has gone triangular and the top that has gone flat, all at once, which is what a reader who has been told “the model holds to one per cent” is entitled to assume it counts.

Counting the phase lag is a decision rather than an oversight. Every other error on this site is a ratio of magnitudes, and each of those is a statement about a spectrum; this one is a statement about a waveform, and the ideal-amplifier model’s claim is about a waveform. Eight degrees of lag is not a model holding to one per cent.

The region #

At small amplitudes the boundary is 48.8 kHz and does not care what the amplitude is: the stage has a pole at gain–bandwidth over gain, and one per cent of error arrives at a hundredth of it. That is the number the ideal amplifier, and where it stops being one computes, and it is a vertical line on this plane.

At large amplitudes the boundary bends left. By 6 V of output it is 45.0 kHz; by 11 V it is 37.9 kHz; by 11.8 V it is 36.7. Nothing about the pole has changed. What has changed is that the input pair is being asked to charge the compensation capacitor faster than its tail current allows, and that mechanism has no frequency of its own — it has a frequency per volt.

And along the top there is a third side. Above 12.19 V of output the model is more than one per cent wrong at every frequency there is, including direct current, because the rails do not move for frequency. That side is a horizontal line, and it is the only one of the three that a frequency-domain description can never produce.

Three sides, two corners, and none of it visible on a single axis.

The two numbers a data sheet gives, and where they cross #

The two dashed lines on that figure are not fits. They are the specifications, written as the boundaries they claim to be.

The first is the small-signal edge: gain–bandwidth over closed-loop gain over a hundred, which has no amplitude in it. The second is the full-power bandwidth, slew rate over 2πV̂, which has no gain–bandwidth in it. Each is quoted on its own line of a data sheet, in its own units, and a reader is left to combine them. The natural combination is the obvious one: stay left of the first and below the second and the model holds.

They cross at 10.60 V of output and 48.8 kHz. That is where a reader would put the corner of the safe region, and it is the point the two specifications were written to define between them.

The measured contour passes 38.5 kHz at that amplitude. The corner is 79.0 per cent of where the specifications put it — a fifth inside — and the missing fifth is not an error in either specification. Both are correct about their own mechanism. The region is smaller than their intersection because at the crossing both mechanisms are contributing, and a design at the corner is paying for both.

Why they do not take turns #

The temptation is to assume that two independent impairments add in quadrature, which is what independent things do and what a noise budget correctly assumes. They do not, and the reason is visible once the error is drawn as a waveform rather than as a number.

At the corner the linear stage’s error is 0.790 per cent of the signal. What the input pair’s tanh adds on top of it is 0.223 per cent. The whole is 0.999 per cent. Were the two independent the whole would be 0.821; were they the same disturbance twice it would be 1.013. It is 0.93 of the way to the second.

That is not a coincidence and it is not a subtlety. Both mechanisms are lags on the same waveform. A pole delays the output; a tail current that cannot charge the compensation capacitor fast enough delays it further. Two delays on one signal produce errors that are in phase with each other, so their magnitudes add. Quadrature is the wrong arithmetic for anything that is a lag, and it is the arithmetic a reader combining two bandwidth specifications would reach for.

The same structure appears in what the second path costs at the floor, where two noise sources genuinely are independent and genuinely do add in quadrature. The difference between the two situations is not the size of the contributions. It is whether the mechanisms share a phase angle.

Two sides, and only one of them can be crossed #

The region’s sides are drawn with the same line, at the same level, and they are not the same kind of object at all.

Finite gain–bandwidth reaches one per cent at 48.8 kHz and ten per cent at 491 kHz — a factor of 10.05, which is what an error proportional to frequency must give and is therefore a check on the measurement rather than a finding. Slewing reaches one per cent at 47.9 kHz and ten per cent at 57.0 kHz: a factor of 1.19.

So a design that crosses the left-hand side of the region by a factor of three is three per cent wrong, which may well be acceptable and is at any rate predictable. A design that crosses the diagonal side by a factor of three is not three times worse; it is off the end of the picture, with a triangular output and a gain that no longer has a value. One side is a gradient and the other is a cliff, and the region as drawn says nothing about which is which.

That distinction is the same one a band rather than an edge found in an analogue switch, where the upper and lower bounds on load resistance are set by different parts of the same component and behave differently as they are approached. It is worth stating as a general rule about this site’s own figures: an edge drawn at a stated error says where, not how quickly.

The corner moves with the question, and sometimes it is not there #

A contour needs a level, and the level is a choice. One per cent is this collection’s habit; a converter designer might want a tenth of that and a loudspeaker crossover a hundred times more.

At one per cent the corner is 0.790 of the specifications’ corner. At three per cent it is 0.791. At ten per cent it is 0.794. The pull-in is not a number about this amplifier’s parameters; it is a number about the pair of mechanisms, and it survives changing the level because near the corner the whole picture is self-similar in frequency-over-the-pole and frequency-over-the-slew-limit. Halving the level halves the small-signal edge and doubles the amplitude at which slewing catches it, and the shape between them is unchanged.

At three tenths of a per cent there is no corner. The amplitude at which the two mechanisms would become comparable is 35.3 V of output, and the rails are at 12, so the region for that reader has only two sides — a vertical one and a horizontal one — and slewing never enters the argument. The model’s validity has a different shape depending on how well it is required to hold, which is not something a boundary quoted as a number can express.

What this says about the rest of the collection #

Every boundary in this collection is a section through a region, taken at whatever value the other variables happened to have. That is not a criticism of the numbers; it is a statement about what they are, and it changes how two of them may be combined.

How small is small signal computes an amplitude at which an exponential’s linearisation is one per cent wrong, and computes it at direct current. The same transistor at a hundred megahertz has a base resistance and a junction capacitance in the way, and the amplitude at which its small-signal model fails is not the amplitude at which its exponential fails. The point the device is never at is the same observation from the other side: a specification measured at one operating point is a section through a surface.

Kirchhoff’s own frequency computes a frequency from a size, which is already a two-variable statement collapsed to one by fixing the board. Its region is a curve in (frequency, size) and the essay draws exactly that curve, which is why it is the one edge in the opening figure that carries a slider.

And which picture sets the upper edge is this essay’s argument in a different field, arrived at before the general case was stated: two candidate mechanisms for one transformer’s upper corner, which one binds depending on the load, and a crossover at 1500 Ω where they change places. That is a corner on a region, measured, without the plane being drawn.

The one-variable numbers are still the right ones to quote #

None of this is an argument for putting a two-dimensional contour on a data sheet. The contour above took several thousand marched cycles to bisect and it is specific to one closed-loop gain; the specifications are specific to the part and can be measured on a bench in an afternoon. They are the right numbers.

What is wrong is the arithmetic a reader is left to do with them. Three rules come out of the measurement and all three are cheap:

Lags add, they do not combine in quadrature. Two bandwidth-like impairments at comparable size give an error close to their sum. If a budget is being built from a gain–bandwidth and a slew rate, adding the fractional errors is a good estimate and adding them in quadrature is optimistic by the factor measured above.

The corner is about a fifth inside where the two lines cross. For this pair of mechanisms, at any level the rails allow. That is a design margin, not an error bar.

And a full-power bandwidth is a different kind of number from a small-signal bandwidth. The first is the frequency above which the waveform cannot be produced, and it is approached in a fifth of a decade. The second is the frequency at which the answer is one per cent low, and it is approached over a decade. Putting them on one plane forces them to be commensurable and they are not; what the plane does is make the incommensurability visible, which is the honest outcome.

What is not in this model #

No temperature. The gain–bandwidth product of a real part moves by tens of per cent between −40 and +125 °C, and the slew rate moves with it because both are the same tail current and the same capacitor. Every number above is at 300 K, so the region drawn here is one slice of a three-variable object, which is exactly the criticism this essay makes of a one-variable boundary.

No load. The stage above drives nothing, so the output stage never runs out of current. That mechanism has its own boundary and its own region, and it is the subject of the current above which there is no impedance: above the output stage’s rating the quantity being measured stops existing rather than becoming inaccurate, which is a fourth side of a different shape entirely.

And only one waveform. The region is drawn for a sine, because a sine has an amplitude and a frequency and therefore a place on the plane. A step does not; nor does a data pattern, nor music. The error a slew rate does to a square wave is not read off a sine’s contour at the square wave’s fundamental, and the step too large to have an impedance is what happens when the drive is a step instead.

The habit this belongs to #

The site’s rule has always been that no model is drawn without the frequency, amplitude or size at which it stops being true. That sentence has three nouns in it and every figure so far has picked one.

Picking one is not a simplification of the boundary. It is a section through it, and a section through a corner is a point that says nothing about the two curves meeting there. The measurement in this essay is what the sentence has meant all along, drawn once at the cost of a few thousand marched cycles, so that the numbers on the other pages can be read for what they are: correct, cheap, and each of them one point on something larger.

Which suggests where the next region would be drawn rather than leaving it as a principle. The candidates are the boundaries whose two mechanisms are already measured separately and never together: the current above which there is no impedance has an input pair’s limit and an output stage’s rating binding at different loads — 19.4 per cent against 4.2 at 0.47 nF, 0.9 against 2.3 at 22 nF — which is a region in the load-capacitance and load-current plane with a corner somewhere between them. Two requirements pulling one capacitor has a stability floor and a droop optimum crossing at about five microfarads, which is a region in the capacitance and series-resistance plane. And a band rather than an edge already has both of its edges in one variable and a third quantity closing them, which is this essay’s shape one dimension further on.

In each of those the second axis exists, the machinery to sweep it exists, and the reason it has not been swept is that a curve is cheaper than a surface. This essay is the argument that the surface is sometimes worth the difference — and the corner at 38.5 kilohertz, against two quoted numbers crossing at 48.8, is what the difference bought here.