Circuit Response
every model has an edge, and the figure carries it
Where the models stop

The edges that are lengths

Almost every boundary in this collection is a frequency or an amplitude, and both of those are things a circuit designer chooses. A handful are lengths — the 0.60 millimetres a gap's field reaches into a window, the 200 microns between a track and its plane, the 10 centimetres at which Kirchhoff's laws are a degree out — and they behave differently in one way that matters: nobody chooses them at the schematic, they are set by whoever builds the thing, and they appear in no netlist at all.
15 min read 8 figures Every model has an edgeThe parasitic is the componentMeasured, not tabulated

Assumes: Every model has an edge · Kirchhoff's own frequency

Every model has an edge put four of this collection’s boundaries on one axis, and the axis was frequency. The edge that is a region added a second dimension and found boundaries that bound from both sides. The edges that move with the room added a third variable, temperature, and found that every boundary already computed had a temperature hidden in it.

All of those are quantities a circuit designer sets. A frequency is on the specification. An amplitude is a signal level. A temperature is an ambient with a number beside it.

There is a fourth kind, and it has turned up four times in this field without ever being collected: a boundary that is a distance. Not a distance in a formula — every physical constant has lengths in it — but a boundary whose independent variable is a millimetre, and which therefore cannot be read off a circuit diagram, cannot be simulated from a netlist, and does not move when the frequency does.

Where four of this site's models stop being trueIn order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 10.0 cm at 3.97 MHz, the ideal 100 nF capacitor at 4.69 MHz. 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.101001k10k100k1M10M100M1G10Gfrequency (hertz)the ideal operational amplifier1.42 kHz — a gain of 100 from a 1 MHz part is 1% low herea 10 V output at full amplitude7.96 kHz — above this the output cannot move fast enoughthe ideal 100 nF capacitor4.69 MHz — 1.2 nH of lead makes it 10% wrong hereKirchhoff's laws on 10.0 cm3.97 MHz — the board is one degree long hereeach bar is where the model may be used; the rule at its end is the numbersolved, then checked — each boundary from its own modeland one that is not a frequency: 7.3 mV
Fig. 1 The collection’s boundaries as it usually draws them: four models, four frequencies, one axis. The fifth in that figure is an amplitude and could not share the axis. Nothing on it is a length. Drag the board size and watch one of the four move — which is the first hint that a length was in there all along.

The four that are lengths

The reach of a fringing field: 0.60 millimetres. A gap in a core throws a field into the winding window, and the excess loss it causes falls off exponentially with the clearance between the gap and the copper. Fitted, the decay length is 0.60 mm — and the useful part is that it stays near 0.60 when the gap itself changes by a factor of five, so it belongs to the core’s geometry rather than to the gap.

What a millimetre of clearance from the gap is worth. computed by solving, not by drawing. The trade rule is to keep the winding two or three gap lengths from the gap, and it is quoted without a number. Here is the number, for a 1 mm gap: at 0.4 mm of clearance the winding dissipates 14.07 times its direct-current loss and its worst turn 37 times; at 3.5 mm those are 2.40 and 2.89. The excess above the far-away floor falls off as an exponential — the dashed curve, r² = 0.999 — with a reach of 0.60 millimetres, which is 0.60 gap lengths here and a different number of them at every other gap. The floor it is heading for is not one, because the winding still has its own field.
Fig. 2 The first, measured. A winding four tenths of a millimetre from a one-millimetre gap dissipates 14.07 times its direct-current loss; at three and a half millimetres it dissipates 2.40. The reach is 0.60 mm and the trade rule that says “two or three gap lengths” has the wrong variable in it.

The height of a track above its plane: 200 microns. The return current under a track spreads sideways over a distance set by that height, so the loop the pair encloses — and therefore the inductance, and therefore what it radiates — is a function of a dimension in the stack-up rather than of anything in the schematic.

The return under 10.0 cm of track, 200 µm above the plane. computed by solving, not by drawing, as an estimate from the geometry: a low-frequency path assumed to be three track-widths wide, 83.3 mΩ, and the parallel-plate inductance µ₀h/w, 125.7 nH, which is the limit for a track much wider than its height. The estimated resistance and reactance are equal at 106 kHz. Solved across the plane instead of assumed, the return does not change path at one frequency: it gathers beneath the track across a band about three decades wide, and this corner falls inside that band. Above the band the loop is the track's length times its height, 20.0 mm², and a milliamp round it at 100 MHz radiates -1.1 dBµV/m at three metres.
Fig. 3 The second, as it is estimated: a crossover at 106 kilohertz between resistance choosing the path and inductance choosing it, above which the return encloses 20 mm² rather than 50. Solved across the plane, the change is a band — half the return gathered beneath the track by 283 kHz and nine tenths by 1.42 MHz — and the top of that band is decided by a height and a sheet resistance.

The size of the circuit: ten centimetres. Kirchhoff’s laws are a statement about a circuit small compared with a wavelength, and the frequency at which they fail is proportional to one over the size.

Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, the ideal 100 nF capacitor at 4.69 MHz, Kirchhoff's laws on 1.00 cm at 39.7 MHz. 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.
Fig. 4 A one-centimetre circuit. Kirchhoff’s laws are a degree out at 39.7 MHz, and the other three boundaries have not moved at all — they belong to parts rather than to distances.

And the packing of a winding: a ratio of lengths. The porosity factor and the window fill are both dimensionless, which makes them look like different objects from the three above; they are not, since both are one length divided by another and both are set on the winding machine.

Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 30.0 cm at 1.32 MHz, the ideal 100 nF capacitor at 4.69 MHz. 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.
Fig. 5 Thirty centimetres: 1.32 MHz. A factor of thirty in length for a factor of thirty in frequency, which is the defining property of an edge that is a length.

A fifth, which is a length that is a time

There is one more in the collection and it belongs in the list with a qualification, because it is a length that has been converted into something else before anybody sees it.

A transmission line’s delay is its length divided by a velocity, and every reflection argument in the lines field is written in nanoseconds rather than in centimetres — the staircase in time counts round trips, and a round trip is twice a length. The conversion is exact and it hides the provenance: a delay is a number a designer can put on a timing budget, and the centimetres it came from are on a drawing somebody else owns.

The same conversion happens to every one of the four above, sooner or later. A clearance becomes a watt. A window fill becomes an alternating-current resistance. A track height becomes an inductance. And once converted, the number joins the electrical model and stops being visibly a length at all — which is exactly how a boundary set by a dimension ends up being verified by a simulation that has no dimensions in it.

What they have in common

Three things, and the third is the one worth the essay.

They do not move with frequency. A reach of 0.60 mm is 0.60 mm at ten kilohertz and at one megahertz, unlike every boundary how small is small signal and its neighbours compute. The consequence of standing inside it moves with frequency — the excess loss goes as the square of a field that goes with the current — but the boundary itself is a property of a geometry and a permeability, both of which are constants. That is unusual in this collection: almost every other edge here is a frequency, and a frequency moves when anything moves.

They are set by manufacture rather than by design. A designer chooses a switching frequency, a turns count, a wire gauge and a core. A designer does not choose how much former is under the first layer, or whether the winder pulled the tape tight, or which of two identical-looking bobbins the line was using that week. Every one of those is a length, and the four boundaries above are functions of exactly that kind of length. The nearest thing in the rest of the collection is the tolerance that is not on any part, where a response’s spread comes from an interaction no single component’s specification contains.

And they are absent from every electrical representation of the circuit. This is the important one. A schematic has no lengths in it. A netlist has no lengths in it. A simulation of a converter has an inductance, a resistance and a core loss model, and every one of those is a number that was computed somewhere else from a geometry the simulation cannot see. So a length-shaped boundary survives every electrical review that a design gets, and shows up in the first thermal measurement of the first build — which is exactly where it is most expensive to find.

Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, the ideal 100 nF capacitor at 4.69 MHz, Kirchhoff's laws on 3.00 mm at 132 MHz. 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.
Fig. 6 Three millimetres — the smallest the slider offers, and the inside of a package: 132 MHz. What the four lengths on this page have in common is that each one turns into a frequency by dividing a velocity by it, and nothing about the components enters.

The one that is a length and a frequency at once

Two of the four are not purely lengths, and the way they are not is instructive.

Kirchhoff’s boundary is a length and a frequency: a ten-centimetre circuit is lumped below 3.97 megahertz and a one-centimetre circuit below 39.7. So the boundary is really a product — a size times a frequency — and the size is one factor of it.

The skin depth is the same shape. A conductor’s resistance departs from its direct-current value where the skin depth is comparable with the radius, and the skin depth is √(2ρ/ωµ): another length, this time one the material sets rather than the manufacturer.

Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 1.00 m at 397 kHz, the ideal 100 nF capacitor at 4.69 MHz. 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.
Fig. 7 A metre, where the boundary is 397 kHz and has fallen below the capacitor’s own self-resonance. This is the one that is a length and a frequency at once: it is computed from a size, and it appears on the same axis as three boundaries that are computed from parts.

That distinction — a length somebody chose against a length physics computed — is what separates the two halves of this list. A skin depth and a wavelength are properties of a material and a frequency, and they can be worked out at the desk. A gap clearance and a window fill are properties of an object that has been built, and they cannot.

Why the collection kept finding them here

Three of the four are in the magnetics field and that is not an accident.

A magnetic component is the one part in an ordinary circuit whose internal geometry is the design. A resistor is a value; a capacitor is a value and two parasitics; a transistor is a model card. A transformer is a window with copper arranged in it, and every parameter it has — the magnetising inductance, the leakage inductance, the alternating-current resistance, the interwinding capacitance, the fringing loss — is a statement about where things are relative to each other.

So a magnetic part is where a length-shaped boundary is most likely to be found, and it is also where it is hardest to see, because the part arrives with a data sheet that has numbers on it and no drawing — the same gap between a value and an object that the assumption that is a geometry measures inside a winding.

Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 3.00 m at 132 kHz, the ideal 100 nF capacitor at 4.69 MHz. 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.
Fig. 8 Three metres, the end of the slider: 132 kHz. Across the six sizes this page has drawn — 3 mm to 3 m — the boundary runs 132 MHz, 39.7 MHz, 13.2 MHz, 3.97 MHz, 1.32 MHz, 397 kHz and 132 kHz, and the product of length and frequency is 397 kHz·m at every one. The collection kept finding these here because a field solver is the only thing that produces them, and a netlist has no lengths in it at all.

Why the collection did not notice sooner

The honest answer is that it had not built anything whose geometry it could solve.

Every boundary in the first three rungs of this ladder came out of a lumped model: a netlist, a transfer function, a marched circuit. A lumped model is one in which the geometry has already been reduced to element values, and a boundary computed from element values is necessarily a frequency, an amplitude or a temperature — because those are the only variables left.

Adding a field changed what could be asked. The three magnetic boundaries above all came from solving a cross-section rather than a netlist, and the first thing a cross-section has that a netlist does not is a coordinate. So this rung is less an observation about electronics than about what a model is capable of noticing: a boundary can only be expressed in the variables the model has, and a lumped model has no lengths.

That is also the warning to attach to it. There are certainly length-shaped boundaries in the fields this collection has not solved a field for, and they will stay invisible for the same reason — not because nobody looked, but because the representation being looked at cannot hold one.

What to do about a boundary nobody can see

The three responses are not equivalent and they are worth ranking.

Turn it into an electrical number and put it on the drawing. A clearance from the gap becomes a minimum first-layer former thickness with a tolerance; a window fill becomes a stated margin; a track height becomes a stack-up requirement. This works and it is what a mature design does, and its weakness is that it moves the problem into a document rather than removing it.

Design so the boundary does not bind. Split the gap so there is no concentrated fringing field; interleave so the leakage does not depend on a single insulation thickness; use a plane so the return path has no choice about where to go. This is better and it costs area, turns or layers.

Or measure the built object. A length-shaped boundary is invisible to simulation and obvious in a thermal image, which is why the first article of a magnetic part gets a thermal survey and the first article of an amplifier does not — and why the degrees a thermocouple cannot see is worth knowing before reading one.

The three questions to ask of a length-shaped limit

A frequency limit gets three standard questions: where is it, how sharp is it, and what happens past it. A length limit needs different ones, and the collection’s four suggest what they are.

Who controls it? A skin depth is controlled by physics and a frequency; a gap clearance is controlled by a winder. The first can be designed against and the second can only be specified and inspected. This question is the one that decides whether the limit belongs in an analysis or in a drawing note.

What does it scale with? The reach of a fringing field turned out to scale with the leg rather than with the gap, which is why the trade rule stated in gap lengths is wrong in both directions. Getting the scaling right is usually more valuable than getting the coefficient right, because a coefficient is one design and a scaling is all of them.

And how much variation is there? A reach of 0.60 millimetres means half a millimetre of winding variation is a factor of two in the excess loss. A boundary whose variable has a manufacturing tolerance comparable with the boundary itself is not a limit, it is a distribution — which is the same shape of statement the tolerance that is not on any part makes about component spreads, arriving from a dimension rather than from a value.

What this rung does not claim

It is not a new mechanism. Every boundary listed was measured in its own essay with its own machinery; this rung is a statement about what they have in common, and the only thing it adds is the observation that four of them share a kind of variable.

And “length” is doing some work. A window fill is dimensionless, a porosity is dimensionless, and a clearance is in millimetres. What they share is not the unit but the provenance: each is a measurement of the built object rather than a value assigned to a component, and each is therefore absent from the representation the design was verified in.

That is the honest form of the claim, and it is narrower than “there are boundaries that are lengths”. There are boundaries whose variable is a fact about the physical arrangement, and no schematic contains one.

The number worth carrying

There is no single number here, which is itself the point. What is worth carrying is the question: which of this design’s limits are set by things the drawing does not contain?

For a magnetic component the answer is most of them. For a circuit board it is the stack-up and the return path. For a discrete amplifier it is very nearly none, which is why the habit of asking does not develop until somebody has built a transformer.

Two more that were not on the list

Two boundaries elsewhere in the collection belong on this axis and are easy to miss, because in both cases the length is inside a package rather than on a board.

The current that does not reach the input is one: a teraohm across a board from a fifteen-volt rail is fourteen millivolts of error through a gigohm source, a humid morning takes that resistance down two decades, and the whole of the effect is in a resistance nobody drew, between two features whose spacing was chosen by a layout tool. A guard ring takes the same error to a nanovolt, and the design parameter is a clearance.

The resistance that is below zero is the other and it is the sharpest, because the length is not even a design feature. An emitter follower’s output resistance is 5.5 Ω at direct current and −21.9 Ω at 257 MHz, and the sign does not belong to the transistor: with an ideal source at the base there is no negative band at all, and a hundred nanohenries of wire between the source and the base produces one from 110 to 301 MHz. Seven centimetres of lead, which is a length nobody specified, decides whether a circuit oscillates — and the input that pushes back finds the threshold at 74.9 nanohenries with ten ohms of source, rising to 913 with a hundred, so the repair is the opposite of the instinct.

Which extends this essay’s question by one clause. It is not only which of a design’s limits are set by things the drawing does not contain, but also which of them are set by things that vary between one build and the next without anybody having decided to change them. A gap length is chosen once by whoever makes the core; a lead length is chosen implicitly, differently, by whoever assembles each unit.

That second class is the one with no design remedy, and the honest response to it is the one the input that pushes back arrives at: make the circuit insensitive to the length rather than trying to control the length. Ten ohms of source resistance puts the oscillation threshold at 74.9 nanohenries, which is seven centimetres of wire and is inside what an assembly will produce; a hundred ohms puts it at 913, which is most of a metre and is not. The fix for a follower that oscillates is to make the thing driving it worse, and that is a schematic change against a boundary that is not on the schematic.

The general form of that remedy is worth stating because it is the only one available for this whole class. A length cannot be specified into a netlist, and a design that depends on one is a design whose behaviour is decided by whoever builds it. What can be done is to move the threshold far enough that every length an assembly plausibly produces is on the safe side of it — which is not a smaller sensitivity but a larger margin, and which is why the numbers in this essay are worth having as numbers rather than as cautions. A margin cannot be sized against a warning.

Part 4 on model edges

One argument about Model edges, and one of 5 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

Linearising an exponential at 125 °C, and what it costs. The linear model understates the gain by 1% at 9.69 mV and by 10% at 30.2 mV. The thermal voltage at this temperature is 34.3 mV, so "small compared with Vₜ" is not the criterion — 28% of Vₜ is already 1% wrong. The edges that move with the room Part 3 — Every boundary in this collection is quoted at one temperature and most of them are functions of it. The small-signal edge is proportional to the thermal voltage, so it runs from 5.67 millivolts at −40 degrees to 9.69 at +125 — a factor of 1.71, the ratio of the absolute temperatures exactly. A realised Q is 1.54 per cent high at one end of that range and 2.58 at the other. The numbers are right; the condition attached to them was left off, and it is the same condition every time. Five boundaries, one tolerance, and three exponents. computed by solving, not by drawing. Each of five model boundaries re-solved at forty-one tolerances from 0.1% to 30%, divided by its own value at 0.1% so that an amplitude in millivolts and four frequencies can share one axis — an exponent has no units. Fitted over the two decades to 10%: Kirchhoff's laws 1.000, the ideal amplifier 0.513, the small-signal model 0.497, the ideal capacitor 0.500, and the full-power bandwidth 0.000. A boundary set by a first-order departure moves in proportion to the tolerance, one set by a second-order departure moves as its square root, and a refusal does not move at all — so relaxing the tolerance from 0.1% to 10% buys a factor of 100 on the board and 10.0 on the capacitor. A boundary is a model and a tolerance Part 5 — Every edge in this collection is computed from a fraction of error nobody states, and the four on its opening axis use three different ones. Swept over two decades, each boundary moves as a power of that fraction — Kirchhoff's laws exactly as the first power, the amplifier and the capacitor and the small-signal model as its square root to within three per cent, and a full-power bandwidth not at all. The exponent identifies the mechanism, and it re-orders the axis twice: the board fails before the capacitor below 0.424 per cent, and the output before the amplifier above 21.7.

Where this goes next

Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 10.0 cm at 3.97 MHz, the ideal 100 nF capacitor at 4.69 MHz. 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. Every model has an edge Four assumptions this collection runs on, with the frequency at which each stops being true, on one axis. The ordering is not the one most readers would guess — an ordinary amplifier circuit runs out of model at 1.42 kHz, three thousand times sooner than a ten-centimetre circuit board does. The ideal amplifier is good to 1% over a region, and its corner is 21% inside the specifications. computed by solving, not by drawing. The 1 per cent contour of the ideal-amplifier model for a non-inverting stage of gain 2 built from a 10 MHz part, drawn over frequency and output amplitude at once. Each point is bisected on a marched circuit: the error is the root-mean-square difference between the marched output and 2 times the input, which counts the gain that is low, the phase that is late and the peak that is flat. Three mechanisms bound the region — finite gain–bandwidth on the left, the input pair's slew rate on the diagonal, and the rails at 12.19 V along the top. The two dashed lines are the numbers a data sheet gives: a small-signal edge at 48.8 kHz with no amplitude in it, and a full-power bandwidth of slew rate over 2πV̂ with no gain–bandwidth in it. They cross at 10.60 V and 48.8 kHz; the measured contour passes 38.5 kHz at that amplitude, which is 0.790 of it. The edge that is a region Every boundary this collection has drawn is a number on one axis, and the figure that gathers four of them admits in its own caption that the fifth is an amplitude and cannot go there. Drawn on both axes at once, the ideal amplifier's one per cent boundary is a region with three sides and a corner — and the corner sits at 38.5 kilohertz where the two numbers a data sheet quotes cross at 48.8, because the two mechanisms are lags on the same waveform and add as magnitudes rather than in quadrature. 10.0 cm of track, solved as a lumped circuit and as a line. The two agree to 0.030% at 3.97 MHz, where the track is one degree long, and to 30.1% at 143 MHz, where it is a tenth of a wavelength. Above that the lumped model is not approximately right; it is describing a different object. Kirchhoff's own frequency The current law says the current entering a node equals the current leaving it at the same instant, which assumes the signal crosses the circuit in no time. It crosses at about two-thirds the speed of light, so the law has a frequency of its own — set by nothing but the physical size of the board. The fringing field, and the turns standing in it. computed by solving, not by drawing. The slot on the left is the gap, cut through the centre leg to the core's own symmetry plane where the potential is zero. Flux crossing it does not stay in the slot: it bulges into the window and crosses the copper at right angles to the layers, which is the one direction Dowell's expression and every ladder in this collection assumes has no field in it. The turns are shaded by their own loss. The worst is turn 4, level with the gap, at 32.5 times its direct-current dissipation; the best is 1.39 times. Same wire, same current, same winding, and a spread of 23.4 between them. The turns nearest the gap A gapped inductor's flux does not turn a corner into the iron on its way out of the gap; it bulges into the window and crosses the copper at right angles to the layers. Four tenths of a millimetre from a one-millimetre gap, the worst turn of an eight-turn winding dissipates 37.5 times its direct-current loss and the winding as a whole 14.1 times. Move the same winding three millimetres further out and those become 2.9 and 2.4 — and the distance that governs it is 0.60 millimetres, which is not the gap length and does not scale with it.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down, the 8 sharing most with it of 21.

The objects named here

The third axis, after the field and the idea: the things themselves, and every essay that touches each one.

Design tradeoffFringing fieldLumped-elementMeasurement conditionModel rangeModel refusalParasiticsProximity effectReturn current