The walk that stops

The flux that walks is the most uncomfortable result in this collection. A drive whose positive half-cycle carries one per cent more volt-seconds than its negative one adds the same small area to the flux every cycle, so the flux does not settle into an excursion — it walks, and reaches saturation after a count of cycles inversely proportional to the imbalance. Halving the drive doubles the count and removes nothing. The essay’s own summary is that there is no amplitude at which the design is inside a limit.

That is a correct statement about the model it was measured on, and the model is a stiff voltage source feeding a pure integrator. The flux is $\int v\,dt$ and nothing in the expression depends on the current at all. Which means the winding has no resistance in it, and the source has no impedance, and neither of those is true of anything that has ever been built.

Putting the resistance back changes the kind of answer, not its size.

What is being solved #

A square drive of amplitude $a$, whose positive half is $(1+\varepsilon)$ times as large in area as its negative one, feeding a winding of resistance $R$ on a saturating core:

$$\frac{d\lambda}{dt} = v(t) - R\,i(\lambda)$$

with $i(\lambda)$ the winding current the material demands for that flux linkage — the same single-valued $B(H) = \mu_0 H + B_{sat}\tanh(H/H_k)$ the field has used since the inductance the current decides, inverted through Ampère’s law around the magnetic path.

The mechanism is one sentence. A direct flux offset demands a direct magnetising current; that current develops a direct voltage across the winding resistance; and that voltage subtracts from the imbalance which created the offset. So the offset is not free, and the walk is not a walk but the early part of an approach.

The reason it is an approach rather than an oscillation is that $i(\lambda)$ is monotone and unbounded. Flux density saturates and field strength does not — $B \to B_{sat} + \mu_0 H$ rather than to a ceiling — so however large the imbalance, some current exists that opposes it, and the question is never whether a fixed point exists but only whether the flux reaches the material’s own limit on the way to one. That distinction is the whole of the difference between the earlier rung’s answer and this one: there, a time; here, a race between two things that both take time.

The numbers throughout are one core and one drive, stated once. A hundred turns on a core of a hundred square millimetres and a sixty-millimetre path, saturating at 0.35 tesla, holds 3.500 milliweber-turns; at fifty hertz that is 1.100 volts of peak drive, and every figure below runs at half of it — 0.5498 volts, an alternating excursion of ±277.6 millitesla, and 72.4 millitesla of headroom left over for anything else. The imbalance is what wants that headroom.

The march starts at the balanced steady state — half the flux swing below zero — rather than at zero. Starting at zero puts the whole peak-to-peak excursion above the origin in the first half cycle, which is the inrush that rung already describes and is a statement about closing a switch rather than about an imbalance. It is worth noticing, in passing, that the earlier figure’s count of 64 cycles is measured on the flux at the end of each cycle, and that its own peak flux has reached 0.9016 tesla by then, against a saturation of 0.35. Both numbers are correct measurements; they are measurements of different instants.

The reproduction, before the departure #

A march that adds an element to a model can return a plausible number for the wrong reason, so the first thing asked of it is the case whose answer is already known: the resistance set to zero, where it must become the earlier rung’s integrator exactly.

It does, and the arithmetic is short enough to write out. The core carries 3.500 milliweber-turns of volt-seconds, the drive at half its limit has an alternating excursion of ±0.27764 tesla, and each cycle adds $a\varepsilon/2f$ of flux linkage, which is $5.49779\times10^{-5}$ weber-turns. The peak of the excursion therefore reaches saturation after $1 + (\lambda_{max} - \lambda_{swing}/2)/(a\varepsilon/2f)$ cycles, which is 14.162 — so the fifteenth is the first whole cycle whose peak is outside, and the march reports the fifteenth.

The same march, asked instead for the cycle at which the flux at the end of the cycle reaches saturation, returns 115 from the balanced state and the earlier rung’s own machinery returns 64 from a demagnetised one. All of those numbers are right, and they answer different questions: which instant of the cycle, and what the flux was doing before the imbalance appeared. The one this essay uses throughout is the peak from the balanced state, because a core saturates at its worst instant and the fault being modelled is a drive that becomes unbalanced while the part is running.

There is a second reproduction hidden in the identity below, and it is the more useful of the two. At zero resistance the balance $R\langle i\rangle = a\varepsilon/2$ can only be satisfied by an infinite mean current, which no finite flux in this material produces — so the fixed point does not merely move out of reach as the resistance falls, it ceases to exist. The earlier rung’s absoluteness is that limit, stated as a property of the whole model rather than of one parameter of it.

What holds the fixed point #

A fixed point that depends on the details of a magnetisation curve is a fixed point nobody can quote. This one does not.

At a fixed point the flux is periodic, so the voltage across the winding integrates to nothing over a cycle, so the mean current through it is the mean applied voltage divided by the resistance. The mean applied voltage is $a\varepsilon/2$ — the direct component of the imbalanced square — and that is the whole statement:

$$R \cdot \langle i \rangle = \frac{a\varepsilon}{2}$$

No permeability, no saturation flux, no turns, no frequency. The material decides what flux that current corresponds to and has nothing to do with the current itself.

It is worth being clear about what that identity is and is not. It is not a small-signal approximation and it is not a statement about this material: it is bookkeeping on a periodic waveform, and it holds for any element whose state is the integral of the voltage across it. What makes it worth a figure is that it converts an awkward question — how far does a nonlinear core drift under an unbalanced drive — into two independent ones. The first, how much direct current flows, has a one-line answer with nothing in it. The second, what flux that current corresponds to, is where all the material’s behaviour lives, and it turns out to have an answer nobody would guess.

Two routes are kept here for a specific reason rather than out of habit. A fixed point is exactly what an explicit integrator gets wrong when its step is too long, and the cheap route above is a forward integration of one scalar equation — chosen because it makes a bisection over the resistance affordable, and suspect for precisely that reason. The netlist march is the same circuit written as a source, a resistor and a nonlinear inductor and solved by the machinery what a network answers describes: trapezoidal, implicit, with the flux linkage as the state. The two share the core model and nothing else.

They also disagree, mildly and instructively, until the march is run long enough. The march starts from a demagnetised core and the direct route from the balanced steady state, so the march carries an energising transient the other does not; at seventy cycles it reported the identity 4.9 parts in $10^4$ short and looked like a disagreement about physics. Three settling times of the direct route’s own measurement is what closes it, which is the pairing one step computed twice insists on, arriving as a question about how long rather than how small.

The offset is smaller than the obvious calculation #

The identity fixes the current. Reading the flux off it is the step where the obvious thing is wrong.

The obvious thing is to look up the flux at which the magnetisation curve carries that direct current — the offset that would produce it if it were the only thing flowing. It is not the only thing flowing. There is an alternating excursion of nearly three hundred millitesla either side, and the magnetisation curve is convex, so the mean current over the cycle is dominated by the part of it spent far up the curve rather than by the part spent near the offset.

That the error is in the safe direction is worth stating and worth not relying on. The stiffness holding a direct offset is the average of the incremental stiffness over the excursion the offset is riding on, so it depends on the alternating amplitude — which means the correction is not a property of the core, and a design running at a smaller flux swing gets less of it. At the limit of a vanishing swing the two routes coincide, which is the geometry in which the obvious calculation is exactly right and nobody operates.

The same shape appears wherever a small quantity rides on a large excursion through a curved law: it is why the incremental inductance of a biased core is not the inductance at the bias, which two inductances at one current measures on the same material from the other side.

The boundary a design can use #

The offset is only interesting if it arrives before saturation does. Below some resistance it does not, and that resistance is what the whole rung produces.

Proportionality is the useful half. A tenth of an ohm per one per cent of imbalance, on a hundred-turn winding driven at half its volt-second limit, is a requirement almost any real winding meets by accident — which is the practical resolution of the earlier rung’s discomfort. The walk is real, the count is right, and on nearly every part that has ever been wound the fixed point arrives first.

That the proportionality is not exact is itself informative and the figure prints the departure rather than smoothing it. The ratio is 0.1000 ohms per per-cent at the small end and 0.1105 at the large one, a rise of ten per cent over two decades, and the direction is the one the mechanism predicts: a larger imbalance demands a larger offset, a larger offset sits on a stiffer part of the magnetisation curve, and a stiffer material returns more current for the flux it is given. So the resistance needed grows very slightly less than in proportion, and the linear rule is conservative where it is wrong. A rule of thumb that errs in a stated direction is worth more than one quoted to three figures at a single point, which is the argument every model has an edge makes about ranges generally.

The other half is not proportional at all and is the more important one.

That divergence is the honest boundary. A resistance can hold an offset down; it cannot make room for one. As the alternating swing alone approaches the volt-second limit the headroom an offset must fit inside goes to zero, and the resistance required to keep the offset inside it grows without bound — so safety here is bought with headroom and only trimmed with resistance. A design at a fifth of its volt-second limit needs 13.4 milliohms and one at a quarter needs 19.5; the same design at 63 per cent needs 1.03 ohms, and above about 64 per cent the alternating swing alone reaches saturation with no imbalance at all, so no resistance is relevant to it.

Which is the correct reading of the earlier rung’s conclusion rather than a refutation of it. Reducing the amplitude does buy time in exact proportion when the source is stiff, and that is what was measured. What it also buys, once there is a resistance, is a boundary — and the boundary moves far faster than the amplitude does.

The percentages above are shares of that one number, which is why they transfer. A drive at 63 per cent of the limit is a drive whose alternating excursion uses 63 per cent of the flux the core has, on any core and at any frequency, and the resistance it needs is set by what is left over. What does not transfer is the resistance itself: an ohm is an ohm, and whether a winding has one depends on how many turns of what wire are on what former.

What it does not say #

It does not say the resistance is free. Every milliohm of it dissipates the load current squared, and the resistance that grows with frequency says how much more of it there is than a direct-current measurement admits. Adding winding resistance to hold off a walk is a poor trade against fixing the imbalance, and the figures above are a margin calculation rather than a design method.

It does not extend to a stiff source. A bridge driving a transformer through switches with milliohm on-resistances, or a winding fed from a low-impedance rail, is close enough to the model of the earlier rung that its conclusion stands — and the resistance the fixed point needs is the total around the loop, of which the winding’s own is only a part.

And it does not cover a core with memory. The material here is single-valued, so the offset can move back down the same curve it moved up; a hysteretic core of the kind the area a curve cannot have built has remanence, and an offset established under a fault does not simply retrace when the fault clears. What the fixed point becomes on a material with two branches is a rung this ladder has the machinery for and has not climbed.

Finally, the upper end of the resistance axis has an edge of its own and the figures respect it rather than running through it. The alternating magnetising current also flows in the winding, so a resistance comparable with the winding’s own magnetising impedance drops the alternating swing as well as the direct offset. Every quantity plotted above is measured only where the swing is intact to two per cent, and the resistance at which that stops being true is found by measurement rather than assumed — the same discipline every model has an edge asks for, applied to the axis of a figure instead of to a frequency.

What it opens #

The nearest question is the one that made the imbalance real in the first place. The delay that is two delays measures a relaxation oscillator whose comparator has different rise and fall delays and finds a duty cycle of 50.28 per cent — a volt-second imbalance of 0.276 per cent, which walks a hundred-turn core in 231 cycles. That essay’s transformer has no winding resistance either. At 0.276 per cent the critical resistance here is about twenty-eight milliohms, which the transformer in that circuit certainly exceeds, so the walk it describes very probably stops — and where it stops, and whether the resulting offset costs the design anything, is a calculation neither essay has made.

The other question is about the approach rather than the fixed point, and there is already a measurement of it lying about. Settling to the floor of double precision takes 246 cycles at 0.15 ohms and 11 at 6 — a factor of 22 for a factor of 40 in resistance, where an ordinary $L/R$ would give 40. Multiplying the two out, the product of resistance and settling time rises steadily from 36.9 to 66 ohm-cycles across the axis, which says the effective inductance holding the offset is not constant: it grows as the offset shrinks, by 1.8 times over that range, because a smaller offset sits on a less stiff part of the same curved law. That is the convexity of the previous section seen in time rather than in amplitude, and it is the reason a settling time cannot be read off a nameplate inductance.

The number worth carrying #

A tenth of an ohm, at a one per cent imbalance and half the volt-second limit, is the difference between a countdown and a margin. The requirement scales with the imbalance and diverges with the drive.

The habit that goes with it is about which element a conclusion belongs to. The walk’s absoluteness — no amplitude, no frequency, only a time — came from an integrator with nothing in the loop to oppose it, and the element that opposes it is the one nobody draws: a winding resistance that appears in no magnetic model, is left out of every flux calculation, and decides here whether the failure exists at all. A conclusion that survives every parameter of the model it was measured on is a conclusion about the model’s elements, and the way to test it is to add one.