The loss that depends on what it causes

There are four thermal figures in this collection before this one and every one of them is right. Two loops, and one heatsink solves a Cauer ladder of thermal resistances and capacitances with the same solver as everything else, temperature standing in for voltage and power for current. The pulse the heatsink does not feel marches a pulse train into it and finds the frequency above which the die integrates. Both answer what temperature does this power reach.

Both take the power as given. For a device whose dissipation is a property of the circuit around it that is exactly correct, and almost nothing is such a device.

A ferrite certainly is not. Its saturation flux density falls towards zero at the Curie temperature and its permeability rises towards it, and both of those are in the loop that the core the solver has to remember marches. So the loss is a function of the temperature, the temperature is a function of the loss, and the honest statement is not a product but an equation:

T = Tₐ + Rₜₕ · P(T).

Two curves, and where they cross #

The equation is easier to see as two curves than as an equation. What the thermal path can remove at a temperature is (T − Tₐ)/Rₜₕ — a straight line, because a thermal resistance is a resistance. What the part dissipates at that temperature is a curve, and the curve has to be marched: a hysteresis loop at a material whose two constants have moved, with the flux swing held rather than the field, because a winding fed from a voltage fixes the flux and lets the field be whatever the material asks for.

They cross twice.

The lower crossing is the operating point: eighty-eight point eight degrees, one point four two watts. The upper one at a hundred and ninety-one degrees is not an operating point at all. It is the boundary between a part that settles and a part that does not, and it is invisible to any method that converges — which is most of the reason the roots here are found by scanning the residual for a sign change and bisecting it, rather than by iterating the map.

The criterion is a loop gain, in kelvin #

The map is g(T) = Tₐ + Rₜₕ·P(T), and its slope is Rₜₕ·P′(T). That number is dimensionless, and a fixed point is stable exactly when it is below one.

It is the same statement what is left at crossover makes about an amplifier, arrived at from heat. Not an analogy: the same fixed-point theorem, applied to a different pair of quantities, which is why both fields need it and why a thermal engineer and a control engineer recognise each other’s diagrams.

At the operating point the loop gain here is minus nought point one nine. Not merely below one — below zero. The core’s own loss is a negative feedback: a hotter core dissipates less.

At the ignition point the same number is a hundred and twenty. Between the two the curve turns over hard, and what turns it is not the ferrite.

The wall is in the copper #

Separating the two mechanisms is the finding of this rung, and it was not the expected one.

The core’s own hysteresis loss falls the entire way — one point seven two watts at room temperature down to nought point two nine at two hundred and three degrees — because the material softens faster than anything else happens. On its own, a core cannot run away. It is a thermally stabilising element across a hundred and eighty kelvin.

The winding does the opposite, and does it violently. The saturation flux density falls as the temperature rises; the flux swing the volt-seconds demand does not; so the field needed for that swing climbs, gently at first and then without bound as the demand approaches the ceiling. The magnetising current follows the field and the copper loss follows the square of the current. From forty point six milliamperes at twenty-five degrees to twenty-three point three amperes at two hundred and three, which is nought point six milliwatts becoming three hundred and twenty-one watts.

So there is a temperature at which a wound part dissipates least, and it is not the coldest one. At this flux swing it is a hundred and eighty-seven degrees, which is far past where anybody would run the part, and the practical consequence is the shape rather than the number: over the useful range the trend is downwards and the danger is not where the intuition about heat puts it.

There is a flux swing below which there is no wall at all #

Everything above depends on the demand catching the ceiling. Below a flux swing it never does, inside the range the material has a ferromagnetic state in at all.

At fifty millitesla the headroom — the demanded swing over the saturation flux the material still has — reaches only a fraction of one before the Curie temperature arrives and the model stops having anything to say. The copper loss stays a rounding error the whole way, the total falls monotonically, and the part simply gets quieter until it stops being magnetic.

That is not an exception to the argument. It is the argument’s boundary, and it has a number, and stating it is what keeps the claim from being asserted at a slider value where it is false.

The map is not the physics #

The obvious way to solve T = Tₐ + Rₜₕ·P(T) is to iterate it: guess the ambient, ask the model what it dissipates, read off the temperature that much power reaches, repeat. Drawn as a cobweb between the map and the diagonal, that iteration converges here in thirteen steps to eighty-eight point seven six degrees — the same number the scan bisected, which is two routes to one temperature.

It converges here. The condition for the iteration to converge is |Rₜₕ·P′| < 1, which is two-sided; the condition for the device to be stable is Rₜₕ·P′ < 1, which is one-sided. Those are different questions, and the gap between them is not hypothetical: a part with a strongly negative thermal coefficient is unconditionally stable and its plain iteration diverges, oscillating between a temperature that is too hot and one that is too cold. The next rung of this ladder is exactly such a part.

There is a second and cruder way for the iteration to fail, and this core shows it. At a hundred and thirty kelvin per watt the operating point still exists and the scan still finds it — but the first iterate is the ambient plus a hundred and thirty times the cold dissipation, which is two hundred and seventy kelvin of rise, and it lands the guess past the Curie temperature, where the material has no state to be asked about. The iteration refuses on its second step while a perfectly good answer sits at a hundred and forty degrees. The iteration’s failure is about the route and not about the destination.

Where the operating point stops existing #

Raise the thermal resistance and the removal line flattens. The stable crossing walks up, the ignition crossing walks down, and at one value they touch.

At a twenty-five degree ambient that value is a hundred and eighty-two point seven kelvin per watt. At the tangency the stable root sits at a hundred and eighty-seven point nine degrees and the ignition root at a hundred and eighty-eight point seven, with loop gains of minus one point nine and plus eighteen point four — bracketing exactly one, which is what a tangency is rather than a check that it happened.

Above it there is no solution. Not a large temperature — no solution, which is what a thermal runaway actually is, stated as the disappearance of a root rather than as a number that got big. The boundary is bisected on whether a root exists, because that is a question with a yes and a no in it; watching two curves approach and calling it when they are close enough needs a tolerance nothing justifies.

The same boundary read from the other end is the number that appears on a specification. Nobody sells a converter with a thermal resistance on its label; they sell it with a maximum ambient, and at sixty kelvin per watt this part’s is a hundred and thirty-five degrees.

What is still handed in #

Three things on this page are stated rather than solved for, and it is worth being explicit about which.

The thermal resistance is one number. A real path has a junction, a case and an ambient with their own capacitances, and the ladder that models it is already in this collection; a fixed point on a single resistance is the steady state that ladder settles to, which is right for a part running continuously and wrong for one that is pulsed. The pulse the heatsink does not feel is the boundary between those two, measured, and it applies here unchanged.

The temperature is uniform. A core has a hot spot in its centre and a cooler surface, and the mechanism that matters most — the flux ceiling — is decided by the hottest part rather than by the average. That makes this a lower bound on the feedback and not an estimate of it.

And the material model is two coefficients. A saturation flux that falls as a mean-field power of the reduced temperature and a permeability that rises linearly: the two forms a catalogue’s curves are fitted to, with the coercivity tied to the same reduced temperature as the flux. Every conclusion here is downstream of that choice, and the one that is least sensitive to it is the shape — a negative loop gain over the useful range and a wall in the winding — because the wall is a consequence of the flux ceiling closing at all rather than of how fast it closes.

A hundred evaluations, and why the memo is not an optimisation #

Locating the boundary is a bisection over the whole fixed-point calculation, and the fixed-point calculation scans a grid of temperatures. Every one of the twenty-odd bisection steps therefore asks for the dissipation at the same hundred-odd temperatures, and each of those marches a hysteresis loop with two settling cycles at every one of three hundred and sixty points on it.

Without a memo on the temperature that boundary costs a hundred thousand marched cycles. With one it costs a hundred, and the memo is keyed on the temperature alone because the closure has already fixed everything else — which is why the model is handed to the solver as a factory rather than as a two-argument function.

The same reasoning forced a smaller decision one level down. The bisection on the ambient moved the scan grid with each trial, so no two trials asked about the same temperature and the memo hit nothing. Pinning the grid to the lowest ambient the bisection will try fixes it, and it fixes a second thing that was not the motivation: a boundary located on a moving grid has a resolution that depends on where the boundary is, which is not a property anybody wants in an answer.

What the loop gain buys #

The number worth carrying out of this rung is minus nought point one nine, and the reason is that it is negative.

A wound component’s own core loss stabilises it. Every decibel of the intuition that a hot magnetic part is on its way to being a hotter one comes from somewhere else — from the winding, whose resistance rises with temperature and whose current rises faster once the flux ceiling starts closing, and from a switch or a diode sharing the same heatsink. The ferrite is the well-behaved part of the assembly.

The number worth carrying out of the machinery is that a temperature can be a fixed point with more than one solution, and that the interesting one is the one nothing converges to. Scanning for sign changes costs a hundred evaluations and finds both; iterating costs thirteen and finds one, and gives no indication that the other is there.

That is the same asymmetry every derivative, and the one that is zero reports about a stationary point, and stable, and unstable with less gain reports about a loop that is unstable at a gain below the one it is stable at. A converging method answers the question it was asked. What it cannot do is report that a second answer exists, and on this page the second answer is the one a design has to stay away from.

Three roots, and what each of them is worth #

A stable root at 89 degrees, an ignition root at 191 and a thermal resistance at which the two touch is the whole of what a steady-state analysis can say, and the three rungs above this one are what each of the three turns into once time is allowed in.

The boundary that is a starting point is the ignition root made usable. A part with a stable point at 88.8 degrees and an ignition temperature at 191.1 will never reach the second, because nothing takes it there — and marched rather than solved, a trajectory starting at 189.6 degrees settles back while 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 survivable for ever while seven times is survivable for seventeen minutes.

The degrees a thermocouple cannot see is the stable root made honest. It is a single temperature, and a core makes its heat in its volume and loses it from a surface — 2.59 kelvin between centre and surface in still air, growing to 3.37 on a cold plate, which is the direction nobody re-checks an assumption in.

And the thermal resistance at which the roots touch is the one quantity of the three that a designer sets rather than discovers, which is why it is worth quoting as 183 kelvin per watt rather than as a criterion. It is a mounting, and it is the only term in this essay that a bill of materials contains.

Where the loop comes from, and what feeds it #

Closing the loop needs a loss that depends on temperature, and this field has measured which parts of the loss do.

The exponent nobody put in is the one that matters most, because it says the flux exponent is a local slope running from 2.94 at half a millitesla to 1.46 near saturation. A falling saturation flux therefore moves the material along that curve as it warms, and the loop gain depends on where on it the design sits — steeply at low flux and hardly at all near saturation. Two designs with the same loss at room temperature can have loop gains a factor of two apart.

The current inside the iron adds a second temperature path that acts in the other direction: the eddy term goes as $1/\rho$, and a metal’s resistivity rises with temperature, so that part of the loss falls as the core warms. Above the frequency at which the sheet is two skin depths thick — 465 hertz for a 0.35 mm sheet — that term dominates, which makes a laminated core at high frequency a negatively fed-back system where a ferrite at low frequency is a positively fed-back one.

Which is why the roots on this page are quoted for a stated material at a stated flux rather than as a property of magnetics. The sign of the loop is a property of which loss mechanism dominates, and that is decided by frequency and lamination thickness rather than by anything thermal.