Every model has an edge
Nothing in this subject is exactly true. An ideal operational amplifier does not exist, a capacitor is not a capacitance, a small-signal model describes a tangent rather than a curve, and Kirchhoff’s laws assume that a signal crosses a circuit in no time.
None of that is a secret. What is generally missing is the number — how far each of those can be pushed before it costs something, stated as a frequency or an amplitude rather than as a caution. This essay puts four of them on one axis.
The four, and where each comes from
The ideal operational amplifier — 1.42 kHz. A part with a megahertz of gain–bandwidth product, configured for a closed-loop gain of a hundred, has ten kilohertz of bandwidth. It is already one per cent below the ideal answer at 1.42 kHz, because the loop gain that enforces the ideal answer is falling from ten hertz upward. The number comes from the gain–bandwidth product and the gain asked for, and nothing else.
A ten-volt output at full amplitude — 7.96 kHz. An amplifier’s output moves at a maximum rate, half a volt per microsecond for an ordinary part. A ten-volt sinusoid demands its highest rate at the zero crossing, and requiring that to stay within the limit gives a frequency. Above it the output is a triangle. This one is not a limit on the model but on the device, and no linear model of any order expresses it.
Kirchhoff’s laws on ten centimetres — 3.97 MHz. The current law says the current entering a node equals the current leaving at the same instant, which assumes the signal crosses the circuit in no time. On a circuit board it crosses at about 1.4 × 10⁸ metres per second, so ten centimetres is one degree of phase at 3.97 MHz. The number comes from a length and a dielectric constant.
The ideal hundred-nanofarad capacitor — 4.69 MHz. About 1.2 nanohenries of lead and via make the part’s impedance ten per cent higher than 1/(2πfC) at 4.69 MHz, and above 14.5 MHz it is an inductor. The number comes from the capacitance and the parasitic inductance.
The ordering, which is the point
Ask most readers to put those four in order and the answers cluster around the same intuition: the circuit board’s size is the exotic one, so it must be last; the capacitor is a passive component, so it must be reliable; the amplifier is the sophisticated part, so it must be capable.
The measured order is amplifier, output amplitude, circuit board, capacitor — and the first two are thousands of times lower than the last two. An ordinary amplifier circuit stops matching its model at audio frequencies. The circuit board’s physical size, which sounds like a microwave-engineering concern, is a thousand times further out.
That inversion is worth dwelling on because it explains where design effort actually goes. Nobody spends time worrying about board size in an audio circuit and everybody spends time on amplifier bandwidth, and that allocation is correct — it is just rarely justified with the numbers that justify it.
It also explains why the two limits that look most alike here are the ones least alike underneath. The amplifier’s boundary is a linear limit, curable by a better linear model: replace the nullor with a finite-gain single-pole model and the prediction is right again over a wider range. The slew boundary is a nonlinear limit, and no linear model of any order can express it.
Why the bars end where they do
Every boundary here is drawn where a stated fractional error is reached, and the fractions are not all the same: one per cent for the amplifier, ten per cent for the capacitor, one degree of phase for the board. That inconsistency is deliberate and worth defending, since a uniform threshold would be tidier.
The thresholds differ because the quantities differ in what a reader would accept. One per cent of gain is at the edge of what a well-built circuit can achieve anyway, and it is a natural place to say an amplifier has started to fail. Ten per cent of impedance is a reasonable statement of when a decoupling capacitor has stopped decoupling, and one per cent would put the boundary in a region where nobody would notice. One degree of phase across a circuit is a criterion in a different unit altogether, chosen because the conventional criterion — a tenth of a wavelength — is thirty-six degrees, which nobody would accept anywhere else in the subject.
What matters is not that the thresholds match but that each is stated, computed and movable. Every one of them appears in the figure that draws it, and a reader who wants a different threshold gets a different and equally computable number rather than a different argument.
The fifth, which is not a frequency
One boundary in this collection cannot go on this axis, and its absence is as informative as the four that are on it.
A small-signal model — a transistor’s transconductance, a diode’s dynamic resistance, a linearisation of any exponential — is wrong above an amplitude, not above a frequency. The number is about seven millivolts at room temperature, and it applies at direct current, at a kilohertz and at a gigahertz identically.
It is the boundary most likely to be crossed without noticing, because there is no frequency to avoid and no plot on which it appears. A circuit operating at a hundred hertz with a fifty-millivolt signal on a transistor’s base is outside its model, and every frequency-domain figure of it will look perfectly correct.
The shape a boundary statement has
Every number in the figure at the top of this essay has the same four parts, and separating them makes the numbers portable rather than memorisable.
A model. The thing being replaced by something simpler: the amplifier by a nullor, the capacitor by a capacitance, the circuit by a netlist.
A better model. What the first is an approximation to: a finite-gain single-pole amplifier, a three-element capacitor, a transmission line. A boundary cannot be computed without one, which is why half the figures on this site draw two curves.
A quantity and a threshold. What is being compared, and by how much it may differ: a gain, to one per cent; an impedance, to ten; a phase across a circuit, to one degree.
A variable. What is being swept to find the crossing: frequency in three cases, amplitude in the fourth.
Change any of the four and the number moves, which is why every figure here states all four rather than only the answer. It also makes the boundaries comparable: the reason the ideal amplifier and the ideal capacitor can sit on one axis is that both are frequencies found by sweeping frequency, and the reason the small-signal boundary cannot is that its variable is different.
Why the better model is not simply used instead
An obvious objection to this whole apparatus: if a better model exists for every boundary drawn, why not use the better model everywhere and stop discussing the worse one?
Three reasons, and they are the reasons idealisations survive.
The simpler model is comprehensible. “The gain is the ratio of two resistors” is a sentence a designer can hold, combine with other sentences, and use to make decisions. “The gain is the solution of a five-element network including a controlled source and a compensation capacitor” is not.
The simpler model is invertible. Design proceeds backwards from a requirement, and the ideal relations can be solved for the components. The better models generally cannot; they can only be evaluated, which makes them tools for checking rather than for choosing.
The better model has a boundary too. The finite-gain single-pole amplifier is itself an idealisation — it has no slew limit, no offset, no noise, no supply rails. Replacing an idealisation with a slightly less idealised one does not escape the problem; it moves it.
Which is the actual argument for the rule this site runs on. The answer to a model with a boundary is not a model without one, because there is no such thing. It is a model whose boundary has been computed, so that a reader knows which side of it they are working on.
What a model’s range is, and is not
There is a distinction running under all of this that is worth making explicit, because it separates this collection’s rule from ordinary caution.
A model’s range is not derivable from the model. Nothing inside the ideal-amplifier idealisation knows about gain–bandwidth product; nothing inside 1/(2πfC) knows about lead inductance; nothing inside Kirchhoff’s laws knows about the speed of light. Each boundary comes from the thing the model was an abstraction of, and it has to be brought in from outside.
That is why a presentation which gives only the models has, without saying so, discarded the information needed to know when to trust them. The models are correct; the omission is what makes them dangerous. And the omission is invisible, because a model with no stated range looks exactly like a model whose range is infinite.
The rule this site runs on is a response to that: no model is drawn without the frequency, amplitude or size at which it stops being true. The number goes in the caption strip, in the same place in every figure, computed rather than quoted.
What the rule costs and what it buys
Enforcing it is not free. Every figure needs a second computation — the model’s boundary as well as its prediction — and several of them need a second network, because the honest comparison is between the idealisation and a better model rather than between the idealisation and an assertion.
What it buys is that the figures stop being illustrations. A curve drawn with a rule on it at 1.42 kHz is making a claim that could be wrong, and that has been checked; a curve drawn without one is making no claim about its own validity at all, and so cannot be checked.
It also changes what a slider is for. Every draggable figure here moves the parameter that sets a boundary — the lead inductance, the closed-loop gain, the slew rate, the length of track, the temperature — so moving it is watching the boundary move rather than watching a shape change. And since every frame was generated at build time by the same code with the same assertions, a reader turning the handle is testing the claim’s generality on frames the build has already tested.
Boundaries that are not on this site
Four models have boundaries drawn here and the subject has many more. Naming a few of the absent ones makes clearer what kind of statement the rule is asking for.
Temperature. Every semiconductor parameter moves with it, most of them substantially: a transistor’s transconductance is inversely proportional to absolute temperature, a diode’s forward voltage falls about two millivolts per degree, and a resistor’s value moves by a stated number of parts per million. Those are not boundaries of models so much as a variable the models do not carry — though the small-signal amplitude limit, which is proportional to kT/q, is a genuine example of one moving with temperature, and its figure has a temperature slider for that reason.
Supply voltage. An amplifier’s output cannot exceed its rails, which is a boundary in volts with no frequency attached. It is omitted here because it is obvious and because nothing interesting is computed by stating it.
Noise. Every model in this collection is deterministic, and a real measurement has a floor below which nothing is visible. That bounds the useful range from below rather than above, which is a different shape of statement and would need its own machinery.
Manufacturing spread. Every number quoted here is a nominal value, and real parts vary. A gain–bandwidth product specified as one megahertz typically means “at least one, and possibly three”, which moves the boundary by the same factor. That is worth remembering whenever a figure here quotes a boundary to three digits: the method is exact and the input is not.
The rule as an editorial constraint
A last observation about what enforcing this does to a collection, since it is the decision the whole site is built on.
A rule that governs every figure is a strong constraint, and the useful ones are those that occasionally make a figure impossible. This one does. A figure whose model has no computable boundary either has to acquire one — usually by drawing the better model beside it, which is more work and a better figure — or it does not get made.
That is the test a governing rule has to pass. A rule that never prevents anything is a slogan, and a rule that prevents everything is a different subject. This one has rejected several figures during this build and improved more: the comparison of an ideal source with a real one, the nullor drawn against a three-element amplifier, the lumped model drawn against a transmission line, and the capacitor drawn against its own symbol are all figures that exist in the shape they do because a number had to go in the caption strip.
The refusals, which are the other half
A boundary that is only reported is a comment. The models in this collection decline outside their range, and the declining is exercised rather than described.
The ideal amplifier asked for its prediction above the closed-loop bandwidth raises an error naming the frequency, the gain and the bandwidth available. The small-signal model asked for a large-signal answer raises one naming the amplitude at which it is already wrong by the stated fraction. The lumped model applied to an electrically large circuit raises one naming the size and the phase length. Each of those refusals is triggered inside a figure, and the resulting message is checked, so an assertion that quietly stopped rejecting would fail the build.
That last point is the one worth carrying away, and it applies well beyond this subject. Half the value of a check is what it rejects. A test suite in which every assertion passes and none has ever been shown to fail is a test suite that might be testing nothing, and the only way to find out is to feed it something it must refuse.