Circuit Response
every model has an edge, and the figure carries it
Power, and the part that does no work

The loop gain one temperature understates

Every thermal loop gain this collection has computed was computed at a single temperature, because a lumped fixed point has only one — and the essay that measured the gradient inside a core recorded, without measuring it, that this makes each of those numbers a lower bound. It is a lower bound by three per cent where a ferrite usually sits and by thirty-eight per cent at the well-cooled end, always in the direction that makes the part safer than the calculation said. The obvious candidate for what decides it is refused: three geometries at one Biot number are 3.3 times apart.
16 min read 7 figures Every model has an edgeTwo routes to a numberRefused, not extrapolated

Assumes: The loss that depends on what it causes · The area a curve cannot have · What a network answers, and how the answer is checked

The loss that depends on what it causes put a number on why a ferrite core does not run away: its loop gain is −0.192, negative, because this material dissipates less as it warms. Every thermal loop gain in this collection is a number of that kind — one temperature, one dissipation, one derivative — and there are several of them.

The degrees a thermocouple cannot see then solved a core as a body rather than a point, found 2.59 kelvin between its middle and its surface, and recorded something it did not measure: that a part with a gradient in it has no single temperature to take a derivative at, so every loop gain here is a lower bound rather than an estimate.

That sentence has been sitting unquantified. It is worth what a Biot number says it is worth.

What one temperature costs the loop gain of a part that has a gradientcomputed by solving, not by drawing. The thermal loop gain of a 30 mm core, solved as a body with its own internal temperature profile and again as a single lump at that profile's mean, against the Biot number. Both are negative, so the core is a stabilising feedback either way — but the body's loop is the more negative of the two at every point, by 3.0 per cent at a Biot number of 0.108 and 38 per cent at 10.8. A lumped calculation therefore understates how stable a wound part is, and the amount it understates by is not a property of the material but of how well the surface is cooled relative to how well the inside conducts. Below a Biot number of about a tenth it is worth under two per cent and the lump is the right model; at the cooled end the part has 7 kelvin inside it and half the feedback is invisible to a single temperature.100m100m110Biot number, h·d/2kmagnitude of the thermal loop gainsolved as a bodyas a single lumpat Bi0.108 → 10.8solved as a body-0.4135 → -0.0291…as one lump-0.4016 → -0.0212understated by3.0% → 38%gradient there7.5 Ksolved, then checked — one part, two thermal modelsa lump is worth its name below Bi ≈ 0.1
Fig. 1 The thermal loop gain of a thirty-millimetre core, solved twice: once as a body with its own internal temperature profile, and once as a single lump held at that profile’s own mean. Both are negative, so the core is a stabilising feedback either way — but the body’s loop is the more negative of the two at every point, by 3.0 per cent at a Biot number of 0.108 and 38 per cent at 10.8.

How a loop gain is read off a body

A lumped fixed point makes the measurement trivial: the loop gain is R·dP/dT, one thermal resistance times one slope, and both are available by definition. A continuum has no single T to differentiate at and no single R to multiply by, so the same quantity has to be got at from outside.

The way in is that a loop gain is not really about temperature at all. It is about how much a disturbance is multiplied on its way round, and for any single-pole feedback — however the pole is distributed inside the object — moving the ambient by one kelvin moves the settled temperature by 1/(1 − L). So the body’s loop gain is read by solving it twice at ambients one kelvin apart and taking the ratio, which requires no assumption about where the heat is made or how it gets out.

That is a genuinely independent route rather than a rearrangement, and it is why the two numbers can be compared at all. The lumped estimate is built from a thermal resistance and a slope; the body’s is built from two solved profiles and a subtraction. They share the material’s loss function and nothing else.

The temperature through a 20 mm core that makes its own heat. computed by solving, not by drawing. Conduction with volumetric generation, solved on forty-one cells with a surface film at each face and the loss density evaluated at each cell's own temperature — so the middle of the core makes less heat than its faces do, which a closed form for uniform generation cannot express. The peak is 115.05 °C and the surface 112.45: a gradient of 2.59 kelvin, which is 2.9 per cent of the 90.0 kelvin rise. That share is Bi/(Bi + 2) — 3.0 per cent at a Biot number of 0.063 — so it is decided by how well the surface is cooled and not by how much heat is made.
Fig. 2 The object the second route is reading, from the rung below: conduction with generation, forty-one cells, and the loss density evaluated at each cell’s own temperature — so the middle of the core makes less heat than its faces do. The peak is 115.05 °C and the surface 112.45, a gradient of 2.59 kelvin against a rise of ninety.

Why the lump is wrong in the direction it is

The correction has a sign and the sign is not an accident.

This material’s dissipation falls as it warms, which is what makes the loop negative in the first place. Inside a body with a gradient, the hottest part is therefore the part making the least heat, and the coldest part is making the most — so a disturbance in ambient is resisted more effectively than a single-temperature calculation can express, because the parts of the object that respond hardest are the parts that are least warm.

A lumped model puts the whole object at one temperature and gets one slope. The body gets a distribution of slopes, weighted towards the steeper end, and the weighted average of a convex function’s derivative is not the derivative at the average. The gap between those two is the whole correction, and it grows with the width of the distribution — which is the gradient, which is the Biot number.

So the lumped answer is conservative, and it is worth saying that plainly before the size of it. A designer who computed −0.192 and concluded the part was stable was right, and was more right than they knew. The correction does not threaten any conclusion this collection has drawn; it says those conclusions had margin nobody had counted.

That also fixes what to look for. A correction with a known sign and an unknown size is a bound, and a bound is worth having even before the size is: whatever the number turns out to be, no thermal conclusion in this collection moves in the dangerous direction because of it.

What one temperature costs the loop gain of a part that has a gradient. computed by solving, not by drawing. The thermal loop gain of a 30 mm core, solved as a body with its own internal temperature profile and again as a single lump at that profile's mean, against the Biot number. Both are negative, so the core is a stabilising feedback either way — but the body's loop is the more negative of the two at every point, by 9.6 per cent at a Biot number of 0.431 and 47 per cent at 43.1. A lumped calculation therefore understates how stable a wound part is, and the amount it understates by is not a property of the material but of how well the surface is cooled relative to how well the inside conducts. Below a Biot number of about a tenth it is worth under two per cent and the lump is the right model; at the cooled end the part has 28 kelvin inside it and half the feedback is invisible to a single temperature.
Fig. 3 The same comparison for a material conducting a quarter as well. Every Biot number is four times larger, and the two curves separate correspondingly sooner: 9.6 per cent apart at a Biot number of 0.431 and 47 per cent at 43. The conductivity is one of the three numbers in a Biot number and, until this figure, the only one this collection had never swept.

What the correction is a correction to

Before the size of it, the thing it corrects is worth looking at directly, because the rung below measured it and the shape of it is what makes the rest of this essay’s arithmetic go the way it does.

When the inside of a core is worth solving for. computed by solving, not by drawing. The gradient between the middle of a core and its surface, as a share of the whole rise above ambient, against the Biot number. The dashed line is Bi/(Bi + 2), which is what the two closed forms give when they are divided — qd²/8k over qd²/8k + qd/2h — and the measurement follows it across two decades even though the generation is not uniform. At the poorly cooled end a ferrite core is at 2.08 per cent and the isothermal assumption is worth its name; at the well cooled end it is 78 per cent — and the gradient in kelvin has grown as well, from 2.38 to 3.37, because this material's loss falls with temperature and a cooler core dissipates more. Better cooling does not make a part more nearly isothermal; it makes the part's own gradient the thing that is left.
Fig. 4 The internal gradient as a share of the whole rise, against the Biot number, from the rung below. It follows Bi/(Bi + 2) across two decades and sits a few per cent under it at every point, because the hot middle makes less heat than the cool faces. At the poorly cooled end the share is two per cent and the isothermal assumption is worth its name; at the well cooled end it is seventy-eight.

The sentence that matters is the one about direction. Better cooling does not make a part more nearly isothermal — it removes the surface’s share of the rise and leaves the body’s own, so the fraction that is internal goes up as the cooling improves. A part on a heatsink is further from isothermal than the same part in still air, which is exactly backwards from the intuition that a well-cooled thing is a uniform thing, and it is why the correction here is largest in the arrangement a designer reaches for when they are worried about heat.

There is a second reason the share sits below Bi/(Bi + 2) rather than on it, and it is the same mechanism this whole essay is about arriving one level down. The closed form is derived for uniform generation. Here the generation is not uniform, because the middle is hotter and this material makes less heat when it is hotter — so the profile is flatter than a uniform-generation solve would give, by a few per cent, every time. The loss function’s temperature dependence is already correcting the gradient before anybody asks it to correct the loop gain.

The obvious law, tested

A Biot number is hd/2k — a surface coefficient, a thickness and a conductivity — and it is the natural candidate for what decides the correction. It is the dimensionless group that says how much of a body’s rise is internal, the gradient is exactly what a lumped model discards, and the correction plainly grows with it. It is the right shape of answer.

A Biot number decides the gradient and not what the gradient costs. computed by solving, not by drawing. The correction to the loop gain against the Biot number, for three families that reach each Biot number a different way — by cooling harder, by getting thicker, and by conducting worse. If the dimensionless group were the whole story the three would lie on one curve. They do not: geometries within a third of one Biot number spread by 3.3 times, because they sit at 69, 37, 165 degrees and this material's loss curve is not equally bent at all of them. A second-order expansion in the profile's variance does not repair it either — measured against that predictor the correction comes out with the wrong sign. What survives is weaker than a law and is what the measurements support: the correction is always in the safe direction, it grows with the Biot number, and below a tenth of one every geometry drawn here is inside 2.1 per cent whichever route it took to get there.
Fig. 5 The correction against the Biot number, for three families that reach each Biot number a different way: by cooling harder, by getting thicker, and by conducting worse. If the group were the whole story the three would lie on one curve. They do not — geometries within a third of one Biot number are 3.3 times apart, because they sit at 69, 37 and 165 degrees and this material’s loss curve is not equally bent at all of them.

A Biot number knows the shape of the profile and nothing about the loss function it is a profile of. Two cores with the same fraction of their rise inside them can have gradients of two kelvin and of ten, at forty degrees and at a hundred and sixty, and the loss curve’s bend over those two intervals is not the same bend. The group decides the gradient; it does not decide what the gradient costs.

The obvious repair fails too, and it fails informatively. A second-order expansion says the correction should go as the curvature of P(T) times the variance of the profile — a two-parameter law, one from the geometry and one from the material. Measured against that predictor it comes out with the wrong sign and between 1.2 and 16 times the wrong size, which says the correction is not a small perturbation about the mean at all: the profile is wide enough, relative to the curvature of the loss, that the leading term is not leading.

So there is no law here to carry away, and the honest result is weaker and still useful: the correction is always in the safe direction, it grows with the Biot number, and below a tenth of one every geometry tried is inside 2.1 per cent however it got there. That is a bound and a threshold rather than a formula, and it is what the measurements support.

The third factor

A Biot number is hd/2k — a surface coefficient, a thickness and a conductivity — and it decides everything above. The rung below swept the first of those across two decades and drew the result that better cooling makes a part less nearly isothermal, not more.

It did not sweep the third. A ferrite’s thermal conductivity is between about 3 and 5 watts per metre per kelvin, an iron powder’s is lower, and a bonded composite’s can be lower again — and unlike the surface coefficient it is not a design choice made at the end but a property of the material chosen at the beginning, for magnetic reasons, by somebody who was not thinking about heat.

Sweeping it moves the Biot number as hard as anything else does. At four watts per metre per kelvin the correction to the loop gain runs from 3.0 per cent to 38 across the cooling sweep; at one, from 9.6 to 47; at eight, from 1.6 per cent to 32. The material’s magnetic properties and its thermal ones are chosen together and priced separately, which is the ordinary shape of the trouble in this field — the current inside the iron is about the same material’s electrical conductivity mattering for a reason nobody selected it for.

Where a lump is worth its name

The practical form of all this is a threshold, and it is a clean one.

Below a Biot number of about a tenth the correction is under 2.1 per cent in every geometry tried — by all three routes to it — and the lumped calculation is the right model — not an approximation to be apologised for, but the correct answer to three figures for a fraction of the work. That covers a small ferrite in still air, which is most of what this collection has actually modelled, and it is why the earlier numbers stand.

Above about one, the correction is tens of per cent and the lumped calculation has stopped being a model of the part. In between is a band where it is worth knowing but not worth solving for.

The threshold is useful precisely because a Biot number is cheap. It needs no solve, no loss function and no iteration — a thickness, a conductivity and a surface coefficient, multiplied and divided — so the decision about whether the expensive model is needed can be made before the expensive model is built. That is the same shape of result as the edges that are lengths: a boundary computed from quantities a designer already has, deciding which of two models to reach for.

The cooling below which there is no answer at all

There is a second boundary in the same sweep and it is not a modelling threshold. It is the material declining.

The cooling below which a 30 mm core has no steady state at all. computed by solving, not by drawing. The hottest point in the core against how well its surface is cooled. Above 12.8 watts per square metre per kelvin there is a steady state and the curve gives it. Below that the solve does not return a large number — it refuses, by name, because the material has no ferromagnetic state above its Curie temperature of 210 °C and a loss cannot be computed in a state that does not exist. The boundary is not a modelling choice: it is the surface rise q·d/2h reaching the Curie point, so it moves with the thickness and with the flux and not at all with the conduction. Still air is about five to ten in these units, which puts an unfanned core of this size on the wrong side of it.
Fig. 6 The hottest point in a thirty-millimetre core against how well its surface is cooled. Above 12.8 watts per square metre per kelvin there is a steady state and the curve gives it. Below that the solve does not return a large number — it refuses, because the material has no ferromagnetic state above its Curie temperature of 210 °C and a loss cannot be computed in a state that does not exist.

The refusal is not a numerical guard bolted on. The surface rise of a slab is qd/2h whatever its conduction does, so below some surface coefficient the whole body is past the Curie point and there is nothing for the material model to describe. That is a boundary the material declares about itself, and it is exactly the kind the answer that is perfect and absurd is about — a solver that returned a number here would be lying about a state that does not exist.

And it moves in proportion to the thickness, which the closed form says it must. Measured at 10, 20, 30 and 40 millimetres the boundary sits at 4.3, 8.5, 12.8 and 17.1 watts per square metre per kelvin — 0.43 per millimetre, to two figures, at every one. That is qd/2h solved for h, and it does not contain the conductivity at all, because a surface rise is a surface rise however well the inside conducts.

Still air is worth about five to ten in these units. A thirty-millimetre core of this material at this flux, unfanned, is on the wrong side of that line — which is a statement about a part rather than about a model, and it is the reason the earlier essay’s own figure had to choose its lowest surface coefficient by arithmetic rather than typing one in.

Where the boundary goes when the part gets smaller

The refusal above is for a thirty-millimetre core, and it moves — which is what says it is physics rather than a guard.

The cooling below which a 10 mm core has no steady state at all. computed by solving, not by drawing. The hottest point in the core against how well its surface is cooled. Above 4.3 watts per square metre per kelvin there is a steady state and the curve gives it. Below that the solve does not return a large number — it refuses, by name, because the material has no ferromagnetic state above its Curie temperature of 210 °C and a loss cannot be computed in a state that does not exist. The boundary is not a modelling choice: it is the surface rise q·d/2h reaching the Curie point, so it moves with the thickness and with the flux and not at all with the conduction. Still air is about five to ten in these units, which puts an unfanned core of this size on the wrong side of it.
Fig. 7 The same sweep for a ten-millimetre core. The boundary has moved from 12.8 watts per square metre per kelvin to 4.3, and the curve above it is the same shape. Measured at 10, 20, 30 and 40 millimetres it sits at 4.3, 8.5, 12.8 and 17.1 — 0.43 per millimetre, to two figures, at every one.

Proportional to the thickness and independent of the conductivity, which is what qd/2h solved for h says it should be: a surface rise is a surface rise however well the inside conducts, so a material that spreads heat perfectly still cannot get it out through a film that is not there. That is a useful thing to be able to say without a solve — the boundary is one multiplication — and it is the honest counterpart to the Biot threshold above. One of the two numbers a designer needs here comes from a closed form and the other does not, and knowing which is which is most of the value.

And the two boundaries are about different failures. The Biot threshold says when a lumped model stops describing a part that is working. The Curie boundary says when there is no working part to describe. A design can be well inside the first and outside the second, which is a thirty-millimetre core in still air: perfectly well modelled as a lump, and with no steady state to model.

Why the second route was necessary

One more thing about the method, because the measurement would not exist without it and the trick generalises past this problem.

A lumped loop gain is a product of two things a lumped model hands over for free — a thermal resistance and a slope. A continuum hands over neither. There is no single temperature to take the slope at, and the thermal resistance is not one number either: the path from the middle of the body to the ambient is not the path from its face to the ambient.

What made it measurable is that a loop gain is defined by what it does rather than by what it is made of. Displace the ambient by a kelvin, let the object settle, and the settled temperature moves by 1/(1 − L) — for any single-pole feedback, whatever the internals. So the body’s loop gain comes out of two solves and a subtraction, with no assumption at all about where heat is made or how it leaves. That is the same move one number from two measurements makes for a coupling coefficient: measure the loop by perturbing it rather than by modelling it.

It also supplies the calibration. At a small enough Biot number the body is a lump, and the two routes have to agree — they do, to under two per cent below a Biot number of a tenth, by three different routes to that Biot number. A method that could not reproduce the lumped answer where the lump is right would have no standing to disagree with it anywhere else, which is the argument the assumption that is a geometry makes for a field solve and the reason its calibration comes before its result.

What this does not settle

It does not re-price the individual loop gains this collection has published. Each was computed for its own geometry at its own Biot number, and the correction has to be applied case by case; what is established here is the rule for applying it and the threshold below which nobody need bother.

It does not extend to the winding. The copper’s loss rises with temperature where the core’s falls, so a winding is a destabilising feedback where a core is a stabilising one, and a part whose loss is mostly copper has a loop gain of the opposite sign. That is the mechanism the loss that depends on what it causes found running away when the flux ceiling fell — and a winding is thin, well coupled to its own surface and usually at a Biot number where none of this matters.

And it is one dimension. A real core is a shape with corners, and the hottest point in a shape with corners is not on the axis of a slab. The Biot threshold would survive that; the specific percentages would not.

The number worth carrying

Three per cent at a Biot number of a tenth, thirty-eight at ten, always in the direction that makes the part safer than the calculation said — and no single number that predicts which.

The habit that goes with it is about what a recorded caveat is for. The sentence that every loop gain here is a lower bound was written down honestly, by the essay that had just discovered the reason for it, and it sat in the record as a warning for exactly as long as it took somebody to ask how large a warning it was. A caveat with no number in it cannot be acted on: it is equally consistent with “ignore this” and “the model is wrong by half”, and a reader has no way to tell which. Measuring it turned it into a threshold, and a threshold is a thing a design can be checked against.

Part 6 on thermal feedback

One argument about Thermal feedback, and one of 6 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

The temperature through a 20 mm core that makes its own heat. computed by solving, not by drawing. Conduction with volumetric generation, solved on forty-one cells with a surface film at each face and the loss density evaluated at each cell's own temperature — so the middle of the core makes less heat than its faces do, which a closed form for uniform generation cannot express. The peak is 115.05 °C and the surface 112.45: a gradient of 2.59 kelvin, which is 2.9 per cent of the 90.0 kelvin rise. That share is Bi/(Bi + 2) — 3.0 per cent at a Biot number of 0.063 — so it is decided by how well the surface is cooled and not by how much heat is made. The degrees a thermocouple cannot see Part 5 — Every thermal answer in this collection has been one temperature, and a core makes its heat in its volume and loses it from a surface, so it has two. Solved as a conduction problem, a twenty-millimetre core in still air is 2.59 kelvin hotter in the middle than on the outside — 2.9 per cent of a ninety-kelvin rise, which is why the lumped answer has been good enough. Cool the same core on a plate and the gradient does not shrink; it grows to 3.37 kelvin and becomes 78 per cent of what is left.

Where this goes next

The temperature through a 20 mm core that makes its own heat. computed by solving, not by drawing. Conduction with volumetric generation, solved on forty-one cells with a surface film at each face and the loss density evaluated at each cell's own temperature — so the middle of the core makes less heat than its faces do, which a closed form for uniform generation cannot express. The peak is 115.05 °C and the surface 112.45: a gradient of 2.59 kelvin, which is 2.9 per cent of the 90.0 kelvin rise. That share is Bi/(Bi + 2) — 3.0 per cent at a Biot number of 0.063 — so it is decided by how well the surface is cooled and not by how much heat is made. The degrees a thermocouple cannot see Every thermal answer in this collection has been one temperature, and a core makes its heat in its volume and loses it from a surface, so it has two. Solved as a conduction problem, a twenty-millimetre core in still air is 2.59 kelvin hotter in the middle than on the outside — 2.9 per cent of a ninety-kelvin rise, which is why the lumped answer has been good enough. Cool the same core on a plate and the gradient does not shrink; it grows to 3.37 kelvin and becomes 78 per cent of what is left. The heat a core makes against the heat its path removes, and the two temperatures where they are equal. computed by solving, not by drawing. The rising straight line is what the thermal path can carry away at a temperature — (T − 25)/45 watts, a line because a thermal resistance is a resistance. The curve is what the wound part actually dissipates at that temperature, marched from a hysteresis loop at a material whose saturation flux and permeability both move with temperature. They cross twice. The lower crossing at 88.8 degrees is the operating point and its loop gain is -0.192 — negative, so the core is a stabilising feedback and not a destabilising one. The upper crossing at 191.1 degrees is an ignition temperature: above it the part cannot get rid of what it makes. The slider moves the thermal resistance. The loss that depends on what it causes Every thermal figure in this collection has had the power handed in. A ferrite's has no business being: its saturation flux falls with temperature, its permeability rises, and both move the loss. Closing that loop makes the temperature a fixed point rather than a product — and the fixed point has a stable root at 89 degrees whose loop gain is negative, an ignition root at 191 whose loop gain is 120, and a thermal resistance of 183 kelvin per watt at which the two touch and neither exists. The temperature a part cannot come back from, and how long it takes to leave. computed by solving, not by drawing. The same fixed-point equation as the rung below, marched in time with a thermal capacitance rather than solved for its steady states: C dT/dt = P(T) − (T − T_a)/R_th, stepped adaptively on the temperature change because dT/dt goes through zero at each fixed point. Every trajectory starting below 191.1 °C returns to 88.8, however far above the operating point it began; every one starting above it leaves the material's range entirely, the closest in 0.2 minutes. The two nearest starts are 3.0 kelvin apart. The ignition temperature is a boundary in the STARTING CONDITION, and no steady-state analysis contains one. The boundary that is a starting point A wound part with a stable operating point at 88.8 degrees and an ignition temperature at 191.1 will never reach the second, because nothing takes it there. Marched in time rather than solved for its steady states, the same equation says what does: a trajectory starting at 189.6 degrees settles back and one starting at 192.6 leaves the material's range in twelve seconds — two starts three kelvin apart. And an overload of four times the normal loss is survivable for ever, while seven times is survivable for seventeen minutes. The eddy term is f² below the skin-depth frequency and f^1.5 above it, both exactly. computed by solving, not by drawing. Eddy-current loss in a 0.35 millimetre lamination held at a mean flux of 1 tesla, against frequency, with the classical uniform-flux expression drawn beside it and the local exponent across the top. Below the frequency at which the sheet is two skin depths thick — 465 Hz here — the two agree and the exponent is 2.0000. Above it the flux is confined to a layer whose thickness falls as one over the square root of the frequency, and the exponent is 1.5000: three halves, exactly. It does not arrive there monotonically — it undershoots to 1.4847 at ξ = 3.28 and comes back up, which is the bounded cosine term the asymptotic statement drops. At 1.00 kHz the classical term is already 1.74 times the truth. The solve and the closed form agree to 7.5e-3 per cent across six decades. The current inside the iron Every core-loss law has an eddy term of the form d²f²B²/6ρ, and it is derived by assuming the flux is uniform across the lamination — an assumption that is a frequency and that the expression does not carry. Solved instead as a diffusion, the exponent is exactly 2 below the frequency at which the sheet is two skin depths thick, exactly 1.5 above it, and it undershoots to 1.485 on the way. For a 0.35 mm sheet the crossing is 465 hertz, so at a kilohertz the classical term is 1.74 times the truth and at ten kilohertz it is forty-seven times. The best foil thickness for 4 layers, for three currents with the same fundamental. computed by solving, not by drawing. The loss of a portion of 4 layers against foil thickness, with the loss weighted by the current in each harmonic rather than computed for one frequency. A sinusoid wants 0.6631 skin depths and lands at 1.3368 times the direct-current resistance — four thirds, the rung below's constant, reproduced. A triangular ripple wants 0.6432, which is the same answer to within 3.0 per cent, so a winding carrying one needs none of this. A square current of the same fundamental wants 0.3838 — thinner by a factor of 1.728 — and lands at 1.8313, which is not four thirds and is not any constant the geometry knows. Building to the sinusoid's answer costs 16.8 per cent more loss. The optimum a spectrum moves The best foil thickness for a winding is derived for one sinusoid and quoted as a property of the geometry: a minimum at four thirds of the direct-current resistance, whatever the layer count. Weight the loss by the current in each harmonic instead and a square current of the same fundamental wants foil 1.728 times thinner and lands at 1.83, and a narrow pulse wants it 3.68 times thinner. Four thirds is a property of the current. The constant that replaces it for an ideal square edge is exactly two, and a real winding sits between them at a place its edge rate decides.

The objects named here

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

CoreDesign tradeoffGrid convergenceLoop gainMeasurement conditionModel rangeModel refusalTemperature coefficientThermal runawayVerification