The core the solver has to remember

The area a curve cannot have gave this collection a core that dissipates. It marches a superposition of play operators against a stated field, closes a loop with twelve joules per cubic metre in it, and measures a coercivity and a remanence that were never handed in. Five rungs of measurement have been taken on it since, and every one of them was taken on a bench.

That word is doing real work. A bench states the field: a drive is applied, the operators advance, and the flux is read out. It is the direction the model is written in and it is the direction nothing in a circuit ever goes.

A circuit states a voltage. The winding integrates it, so what exists at the end of a step is a flux linkage; the current is whatever the material asks for at that flux, and the field is whatever the current implies. The whole calculation runs backwards, and running it backwards is not a matter of rearranging the arithmetic.

The inverse that does not exist #

The march this site uses carries two kinds of nonlinear reactive element and both are defined by a state that is not a value. A capacitor whose capacitance moves with bias is written as a charge, q(v), so that the trapezoidal rule is applied to dq/dt = i rather than to i = C dv/dt. An inductor whose inductance moves with current is written as a flux linkage, and it needs one more thing: the inverse, i(λ), because the unknown a solver can offer an element is the voltage across it, and the voltage integrates to a flux.

That inverse is required rather than computed, and the docstring beside it says why: every saturating law worth marching has one in closed form, and a bisection buried inside a Newton loop is a way of making a convergence failure look like a physical result.

A hysteretic core has no inverse at all. Fifty milliamperes through this winding is one flux linkage on the ascending branch and a measurably different one on the descending branch, and the two are both correct. There is no function of the current that returns the flux, so there is nothing to write in the slot the element wants filled — and this is not a numerical difficulty that a better algorithm would remove. It is the physical content of the model. A part with memory is a part whose present state is not a function of its present terminal quantities, and asking for the function is asking the model to stop being what it is.

What the element carries instead #

So the element carries the core object, and the core object carries an array. Twenty-four play operators, each with its own threshold and its own remembered field, and the array is the branch: it is the answer to where has this core been, and it is the only thing that distinguishes the two fluxes at fifty milliamperes.

The march’s state vector gains it. Where a saturating inductor contributes one number to the state and a linear one contributes one number, this element contributes twenty-five — the flux linkage and the twenty-four operator fields — and a snapshot of the array is taken at the end of every accepted step, exactly as the flux linkage is.

The word accepted is the load-bearing one. Inside a step, the solver runs a Newton iteration on trial voltages, and it is emphatically forbidden to advance the operators while it does so. Every trial restores the committed snapshot before it inverts, so the whole iteration sees one branch.

Advancing them inside the loop would work, in the sense that it would produce numbers and a plot. It would also make the branch a function of how many iterations the solver happened to take, which is a march whose answer depends on its own convergence history. That failure has no symptom: the waveform looks like a waveform, the loop looks like a loop, and the only way to notice is to change the tolerance and watch the loss move. The core model already owns the discipline for this — it has a peek that reads the flux at a field without advancing anything, written for the bench and used here for the reason it was written.

Inverting against a history rather than against a curve #

What is left is a scalar solve at every trial: given the flux the trapezoidal rule proposes, find the field that produces it on the branch the operators are standing on. That is not inverting a curve, because the curve is different after every step.

Two properties make it exact rather than approximate, and both are asserted in the site’s gate rather than assumed. The flux is strictly increasing in the field — every term is non-decreasing, and the vacuum term μ₀H is strictly increasing on its own — so a root exists, is unique, and is bracketed by any pair that straddles it. And the slope is bounded below by μ₀, so a bracket of a stated width carries a residual bound without evaluating anything.

The solve is Newton on the one-sided differential permeability with a bisection safeguard, which on a monotone function cannot leave its bracket and cannot converge to the wrong root. The residual is returned rather than discarded, which is the whole difference between this and the bare bisection the flux-defined element refuses: the caller can assert that the flux on the plot is the flux the field implies, and does.

Opening the bracket around the previous step’s field rather than around zero takes the inversion from about nineteen evaluations to about five. In a march that is not an optimisation — it is the difference between a hysteretic core being affordable inside a Newton loop and not being.

Two integrals that share no arithmetic #

The verification that matters is not that the march converges. It is that the object inside the circuit is the same object the five rungs below measured on a bench.

Drive the winding from a stiff sinusoidal voltage, so the flux is a pure sinusoid of known amplitude, and integrate ∮v·i dt at the two terminals over a settled cycle. That number never touches a material variable: it is a voltage and a current at two wires, which is what an instrument sees. Then impose the same peak flux on the material directly and take ∮H dB around the loop it encloses. That number never touches a terminal.

At twenty millitesla the two agree to three parts in ten million. At fifty they agree to twelve parts in ten million, and at a hundred and twenty to thirty-one — the drift being the marching grid rather than the model, and moving the right way as the loop gets sharper.

This is the same discipline the area a curve cannot have used when it measured one loss two ways inside the model, and it is a stronger version of it: there the two integrals shared a marched trajectory, and here they share nothing but the material constants.

What the circuit sees that the bench did not #

With a series resistance in front of it — a real source, rather than a stiff one — the comparison between the two cores becomes a comparison of two circuits, and the differences are not subtle.

Twenty-two and a half volts peak at five kilohertz through four ohms into sixty turns demands a hundred and twenty millitesla and reaches a hundred and nineteen point eight. The single-valued core draws fifty point one milliamperes at the peak. The hysteretic one draws fifty-nine point five — nineteen per cent more current for the same flux, because part of the field is spent on the branch rather than on the magnetisation.

Over a complete cycle the source delivers seventeen point two microjoules to the hysteretic winding and two and a half hundredths of a nanojoule to the single-valued one, which is zero to the resolution of the march. That is the whole of the difference between a core that can get warm and a core that cannot, priced in the only currency a power supply has.

The gap, and how much of the current stops being history #

Ampère around the path is N·i = le·H + lg·B/μ₀, and only the first term carries the material’s history. The second is a length and a flux density: single-valued, exactly linear, and completely indifferent to where the core has been.

So opening a gap moves a growing share of the winding current into a term with no memory in it. At half a millimetre most of the ampere-turns are the gap’s; at one millimetre it is ninety-six per cent, and the part has become an air-cored inductor with some iron in it to keep the flux together.

That last clause is the part worth carrying. A gap does not reduce core loss — the fourth rung of this ladder established that at fourteen figures on a bench — and now the circuit says it too, from a different direction: the loss density moves by seven per cent while the peak current moves by a factor of twenty-seven. The energy is in the gap priced the storage this buys; what the march adds is that the price is paid in current and the loss is not part of the transaction.

The defect that was in the solver and not in the element #

The first run of this circuit reported a worst node-sum residual of two hundred and thirty-five milliamperes on a sixty-milliampere circuit. That is not a small error; it is the answer being wrong.

It was not the new element. The identical run with the single-valued core reported the identical residual, which is what said so — and is why both runs exist rather than only the interesting one.

The cause is the Newton damping. The march limits how far a device’s voltage may move in one iteration, and the default limit is a fifth of a volt, which was chosen for semiconductors — where a fifth of a volt is an enormous excursion, as a bias point is a solution spends its length establishing. The voltage across this winding has to travel twenty-two volts in the first step of the march, and at a fifth of a volt an iteration the sixty-iteration budget covers twelve of them. Sizing the limit from the drive takes the residual to nine parts in ten thousand billion and the worst iteration count to six.

What this does not yet let the field do #

The element exists and it is verified, and there are three things it is one step away from rather than capable of.

It has no temperature. The loss is computed and nothing is done with it, which is the omission the next rung is about: a core’s dissipation raises its own temperature, its permeability and its saturation flux both move with temperature, and the loop that closes is a fixed point of the same shape the diode that conducts backwards solves for a junction.

It carries no eddy currents. The current inside the iron solves the diffusion inside a lamination and gets two exact exponents out of it; that solve is a frequency-domain statement about a sinusoid, and putting it into a march means a distributed state inside the sheet rather than one operator array at its surface.

And its operators are advanced once per step. They are rate-independent, so their state depends on the path the field took and not on how fast, and a step that jumps the field past a reversal lands them where a continuous traverse would only if the traverse is monotone inside the step. That holds for every waveform here at a sane step count, and the sub-stepping factor is exposed rather than assumed so the assumption can be tested.

Where this leaves the field #

Five rungs of this ladder measured a material. This one puts the material in a circuit, and the two statements it makes are worth separating.

The modelling statement is that a state which is not a number can be threaded through an ordinary trapezoidal march without any new solver: one Newton loop per step, one snapshot per accepted step, one monotone scalar inversion per trial, and Kirchhoff’s current law rebuilt from the element laws and checked at every step of every run on this page.

The physical statement is smaller and sharper. A hysteretic core asks its source for nineteen per cent more current than the single-valued model predicts at the same flux, delivers seventeen microjoules a cycle to the material instead of nothing, and stops caring about either as soon as a gap takes the ampere-turns away from the iron. None of that is available from an inductance, because an inductance is a derivative at a point and this part does not have one — the same objection the capacitance that is not one number raises against a class II ceramic’s printed value, arrived at from the magnetic side.

Two inductances at one current made that last point on a bench, with two branch slopes at one bias that never converge. The circuit makes it again in the only place it costs anything: the current a supply has to find.

Twenty-four numbers the Newton loop must not touch #

The construction that makes this work — state variables carried alongside the solve and forbidden to the iteration — is worth naming as a construction, because it is what separates a model with memory from a model with a complicated law.

Every other nonlinear element in this collection is handled by a bias point is a solution, not a choice’s recipe: replace the device by a conductance and a current source that agree with its own law at the present guess, solve, repeat. That works because the law is a function — one voltage, one current — so a guess can be improved. A play operator is not a function of the present field, so there is no law to linearise and nothing for Newton to converge on; what there is instead is a rule for updating the operator once the step’s field is known.

The diode that conducts backwards is the same construction in the transients field, arrived at independently: a junction that holds charge has one extra state variable, updated after the step rather than solved within it, and it produces a behaviour — 3.83 amperes backwards for 48 nanoseconds — that no refinement of an instantaneous law approaches.

The two together say what a state variable is for. It is not a more accurate parameter; it is the thing that makes the model’s answer depend on how it got here, and both essays measure a quantity that is identically zero without one.

Three parts in ten million, between two integrals #

The circuit and the bench agreeing on the loss to three parts in ten million is the check this whole construction rests on, and its value is entirely in the two routes sharing nothing.

One integral is taken at the two wires — voltage times current, integrated over a cycle, which is what an instrument connected to the winding would report and which knows nothing about play operators. The other is taken inside the material — $\oint H\,dB$ over the locus, which is what the model is made of and which knows nothing about the circuit it is embedded in. A state variable updated in the wrong place, a locus traced in the wrong direction, or an operator updated before the step rather than after would satisfy one of those and not the other.

That is the same construction what a network answers sets up for a linear solve — a current law rebuilt from the element laws and an energy balance computed by a route that reads no resistance — and it is the answer to that essay’s own limitation, which is that neither of its two checks can see a coupling. A hysteretic core cannot be checked by either, and this pair is what replaced them.