Every paper so far rode an ideal core: flux follows current instantly, forever, with no limit and no memory. Real iron breaks all three promises. It saturates — there's a ceiling on flux. It bends — the response is non-linear. And it remembers — the flux left in the core depends on where it has been. The part we called a pure change-detector turns out to hold a little state after all, and pay for it in heat.
The clean law of the toroid series, v = N·dΦ/dt with L constant, quietly assumed the flux would track the current without limit and without history. That assumption is what let every instrument stay linear and tidy. It is also false, in three specific ways that every power engineer spends their life managing:
It saturates. The magnetic domains in the iron can only line up so far. Past a certain drive, the flux stops growing — and since inductance is how much flux you get per amp, the inductance collapses. A saturated transformer is briefly almost a piece of wire.
It's non-linear. Even below saturation, flux doesn't rise in a straight line with current — the curve bends — so a clean sine of current makes a distorted flux, and the distortion shows up in the signal.
It remembers. Remove the drive entirely and the core doesn't return to zero flux. It keeps some — remanence — a frozen echo of the last thing that happened to it. To erase that echo you must actively push the field the other way. The core has a memory, and reading or overwriting it takes work that leaves as heat.
The same stubbornness that let it ignore a steady current also lets it cling to a past one. A perfect change-detector would forget instantly. Iron isn't perfect — it holds a ghost.
Drive the field H (proportional to current) and watch the flux density B respond. At first it climbs steeply — full inductance, the ideal region. Then it reaches the knee and flattens: the domains are nearly all aligned, there's no more flux to give. The slope of this curve is the inductance, and you can watch it die — past the knee, more current buys almost no more flux, and the transformer stops transforming.
This is why a transformer has a maximum volt-second product (Paper I's volt-second balance was really a saturation limit in disguise): drive it too hard or too long in one direction and the flux walks up the curve, hits the knee, and the inductance vanishes — current spikes, and the magic stops.
Now drive the field back and forth and trace the full story. The flux doesn't retrace its path — it follows a higher route coming down than going up, opening a loop. Two numbers name the memory: remanence Br, the flux still there when the drive returns to zero (the ghost), and coercivity Hc, the reverse field you must apply to force the flux back to zero (the cost of erasing). And the area enclosed by the loop is energy — the work the core eats and turns to heat, every single cycle.
The loop's area is not a diagram — it is a quantity of heat, paid once per cycle, forever. A soft core keeps the loop thin to stay cool; a hard core fattens it on purpose, to remember.
This is the honest end of the arc — the place the clean laws meet imperfect matter. And it closes a thread that ran through everything: the toroid was supposed to be a pure change-detector, deaf to anything held still. Hysteresis is the proof that it isn't, quite. The core keeps a frozen echo of its last state, and that remanence is the magnetic twin of the capacitor's own dirty secret — dielectric absorption, the "soakage" where a capacitor you've fully discharged will, minutes later, recover a ghost of its old voltage all by itself. A change-detector that quietly remembers; a charge-bucket that won't fully forget. Both ideals leak a little state.
It's a fitting place to stop. The two series showed these parts at their most elegant — carrying, crossing, rejecting, reading, all from one clean refusal to feel what holds still. This last paper shows the seam where that elegance is paid for: in a ceiling, a bend, and a ghost, settled in heat. Past here isn't more principle — it's materials science, datasheets, and thermal design, which is its own discipline and a different kind of book.
A perfect part would forget the instant the signal stopped. Real parts hold a ghost — and engineering is mostly the art of deciding how much ghost you can afford.
For the archive: this is the hinge between the idealised "communicators" work and the real-materials world — saturation and hysteresis for the magnetic side, dielectric absorption and ESR for the electric side. If the work ever continues, it continues here, in the imperfections — but the elegant arc is complete, and this is its honest last page.