The flux the reset leaves behind

A boundary in volt-seconds put a core’s limit in the units a drive is specified in: the flux swings by the integral of the voltage, the core saturates when the swing reaches $N A_e B_{sat}$, and a symmetric drive — a square wave or a sine with no direct component — swings the flux from $-B_{max}$ to $+B_{max}$ and back. Every essay that followed kept the drive symmetric, or nearly: the flux that walks and the walk that stops studied what a small imbalance between the half cycles does, and found it to be a walk the winding’s resistance eventually holds.

Many drives are not symmetric at all. A forward converter’s transformer is driven positive for the switch’s on-time and then left to reset: a diode and a reset winding let the magnetising current fall back to zero, and nothing drives it negative. A single-ended pulse transformer does the same. The volt-seconds of each pulse are spent pushing the flux up from wherever the reset left it, and the assumption every textbook makes for such a drive — that the flux starts each pulse at zero, so a one-way drive has half the swing of a symmetric one — is an assumption about the material that only a material with no memory satisfies.

A real core remembers. At zero current it sits at its remanence, and the remanence is flux the next pulse cannot use.

The core, and how a one-way cycle is marched #

The core is the one the area a curve cannot have built to give a material a loop: the site’s saturating tanh curve split into a reversible part and a superposition of play operators, each holding its output until the field moves more than its threshold away. The model’s two material parameters are a coercive spread and a reversible fraction; here they are set to make a power ferrite with a squareness of a quarter — 90.6 mT of remanence against a saturation of 0.35 T, and a coercive field of 38 A/m — beside a soft ferrite with a remanence of 22 mT and a square-loop core, ten times more permeable, whose loop is nearly rectangular. The geometry is the first essay’s: 100 turns on 100 mm² of core with 60 mm of magnetic path, so a tesla of swing is ten milliweber-turns.

A cycle is two moves. The pulse: the flux is raised from wherever it is to the peak the design allows, 0.3 T here, the core inverted at each step to find the field it needs. The reset: the field is lowered until the winding current is zero — which on an ungapped core is $H = 0$, and on a gapped one is where $N i = l_e H + l_g B/\mu_0$ vanishes — and the core is left there. The march repeats until the flux at the start of a pulse stops moving, which takes one cycle.

Ungapped, the reset ends at $H = 0$ on the descending branch of the minor loop the pulse traced: 90.8 mT, a hair above the major loop’s remanence. The next pulse starts there and can add only 209 mT before it reaches 0.3 T. The symmetric drive’s 0.6 T of swing has become 209 mT, not the 300 the textbook half promises.

How much of the swing a one-way drive keeps #

The loss depends on how square the material’s loop is, and on how hard the design drives it.

At a peak of 0.3 T the soft ferrite keeps 46.3 per cent of the symmetric swing — its 22 mT of remanence costs little — and the power ferrite keeps 34.9. The square-loop core keeps 14.7: it starts every pulse at 212 mT, and of the 0.3 T it is allowed it can use 88 mT. A square-loop core is the material of choice for a magnetic amplifier or a saturable reactor, where remembering is the point; driven one way as a transformer it is nearly useless without help.

Lowering the peak does not rescue the share. The remanence of the minor loop falls as the peak does, but the swing falls faster: the power ferrite keeps 26.4 per cent at 0.1 T against 34.9 at 0.3. A designer who backs off the flux density to gain margin in a one-way drive gains less than the symmetric arithmetic suggests, because at low flux the loop’s remanence is a larger fraction of its excursion.

In volt-seconds the consequence is direct. The first essay’s core holds 3.5 milliweber-turns from zero to saturation; a symmetric drive to 0.3 T uses six; a one-way drive on the power ferrite, 2.09, and on the square-loop core 0.88. A forward converter designed on the textbook’s three milliweber-turns per pulse saturates its core on the second pulse.

Why a symmetric drive never noticed #

The remanence was always in the core, and a symmetric drive pays nothing for it. Driven from $-B_{max}$ to $+B_{max}$, the flux passes through the remanent point on the way down and its mirror image on the way up, and both are simply points on a loop the drive is traversing anyway; the volt-seconds of each half cycle carry the flux the full $2B_{max}$ whatever the loop’s shape. The remanence appears as loss — the loop’s area, which the area a curve cannot have computed — but not as missing swing.

A one-way drive uses only the upper half of the loop, and only the part of it above where the reset leaves the core. The remanence is therefore not a cost the material imposes on every drive; it is a cost the reset imposes, by stopping at zero current rather than driving on into the negative branch. That is why the cures are all about the reset: a gap moves where zero current is, a resonant reset overshoots it, and a second, opposite pulse — which makes the drive symmetric again — removes the question entirely.

The same core also behaves differently in the one-way cycle in a way a designer will meet at the peak. The magnetising current at 0.3 T is the current on the ascending branch of the minor loop the pulse traces, and the inductance the current decides found how steeply that current rises as the flux approaches the knee. A one-way drive that starts at 90.8 mT instead of zero reaches the same peak flux with the same peak current — the core does not care where the pulse began — but it reaches it with 91 mT less of the pulse’s volt-seconds spent, so a pulse sized for the full swing overshoots the peak by exactly the remanence and lands on the steep part of the curve.

The gap, and the line it draws #

The standard cure is an air gap, and the figure that explains it is the shear line. With a gap the winding current is $N i = l_e H + l_g B/\mu_0$, so zero current is not $H = 0$ but

$$H = -\frac{l_g}{\mu_0 l_e}\,B,$$

a straight line through the origin with a negative slope, drawn dotted in the hero figure. The reset now ends where that line meets the descending branch — at a negative field, below the remanence. And since the descending branch reaches $B = 0$ only at $H = -H_c$, the meeting point can be no further left than that, which bounds the start:

$$B_0 \le \frac{\mu_0 H_c\, l_e}{l_g}.$$

The marched starting points obey it. For the power ferrite, a 0.02 mm gap brings the start from 90.8 to 56.0 mT; 0.1 mm to 22.0, against a bound of 30.2; 0.5 mm to 5.5, against 5.7. The square-loop core, whose coercivity is half the power ferrite’s, comes down twice as fast: 12.5 mT at 0.1 mm. At small gaps the shear line is steep and meets the descending branch high up, above the knee, where the field has not yet reached $-H_c$, so the start sits well under the bound. As the gap grows the line flattens, meets the branch near the coercive field, and the start settles onto the bound, falling as one over the gap.

That gives the gap a design rule in one line: to start each pulse within $B_0$ of zero, the gap must be at least $\mu_0 H_c l_e/B_0$. The coercivity is the material property that sets it, not the remanence — a material with a large remanence and a small coercivity is cheap to correct, because its loop is tall but narrow and the shear line crosses it near the axis.

What the recovered swing costs #

A gap is not free. It is a reluctance in series with the core, and the winding has to drive flux through it.

The peak magnetising current is the ungapped core’s 132 mA plus $l_g B/(\mu_0 N)$, to a per cent at every gap, because at the peak the core sits on the same branch however much gap is in series with it. So the current grows linearly with the gap while the swing approaches its 0.3 T ceiling only as one over it, and the exchange gets steadily worse. The first 0.05 mm recovers 55 mT of swing for 119 mA of extra current. Going from 0.2 to 0.5 mm recovers 7 mT for 716 mA.

Magnetising current in a forward converter is not wasted outright — the reset winding returns most of the stored energy to the supply — but it flows in the switch and the primary at the worst moment, the end of the on-time, and it sets the reset circuit’s size. The energy is in the gap found a gapped inductor storing nearly all of its energy in the gap’s volume; here that stored energy is what the reset has to dispose of every cycle, and it is proportional to the gap. The sensible gap is the one at the knee of this figure — a few hundredths of a millimetre for a power ferrite — where the swing has come most of the way back and the current has not yet started to dominate.

The gap a core already has #

A gap of a few hundredths of a millimetre is not exotic. Two halves of a ferrite core clamped together meet on ground faces, and the faces are flat to some microns rather than to nothing, so every assembled core has a residual gap of perhaps five to twenty micrometres before anyone has asked for one. On this essay’s core that is between a tenth and a third of the 0.05 mm that recovers most of the swing, and it explains why a forward converter on a two-piece core often works better than the ungapped arithmetic says it should: the assembly has already sheared the loop a little. It also explains why the same design on a toroid — which has no mating faces and no residual gap — can saturate where the two-piece version did not. A specified gap is the way to make the shear deliberate rather than a property of how tightly the clip was fitted. The gap that is bigger than it is adds one correction: the field fringing round a gap enlarges its effective area, so a gap has less reluctance than its length suggests and shears the loop a little less — the bound $\mu_0 H_c l_e/l_g$ should be read with a slightly shorter effective gap than the one that was ground.

The reset’s own volt-seconds #

What the reset has to do follows from the same picture. During the on-time the flux rises from its start to the peak; during the reset it must come back down by the same amount, so the reset winding’s volt-seconds must equal the switch’s. Ungapped, the power ferrite’s reset has 209 mT to undo; with 0.1 mm of gap, 278 mT, because the reset now carries the flux below the old remanence to the shear line. A gap therefore asks more of the reset as well as of the switch: the reset voltage times the reset time must grow with the swing it recovered.

That couples the gap to the converter’s maximum duty cycle. A reset winding with as many turns as the primary resets at the input voltage, so the off-time must be at least as long as the on-time, and the duty cycle cannot exceed a half whatever the gap. The gap buys swing per pulse, not a longer pulse: it lets a design run a higher flux density at the same duty cycle, or the same flux density with fewer turns. A boundary in volt-seconds wrote the core’s limit as a number in weber-turns with no frequency in it; for a one-way drive the number the designer can spend per pulse is that limit less the remanence’s share, and the gap is the knob that moves the share.

The first pulse is the exception #

The march exposes a practical trap that a single measurement would miss.

A core that has been demagnetised — by a symmetric drive that decayed, by heating above its Curie point, or by never having been used — starts its first pulse at zero, and that pulse has the whole 0.3 T of headroom. The second pulse starts from whatever the first left behind, and from then on every pulse starts from the same point. A bench test that fires one pulse into a fresh core and measures the current sees a margin the converter will never have in operation. The square- loop core shows it most starkly: 0.3 T available on the first pulse and 88 mT on every one after.

Two inductances at one current found the same history-dependence in a core’s small-signal inductance, which depends on which branch the core last came down. A unipolar drive is the large-signal version: the history is one pulse long, and it is always the same pulse.

What the march assumes #

The reset here is ideal: the winding current is let fall to exactly zero and the core relaxes along its descending branch with nothing else acting. A real forward converter’s reset winding clamps the voltage during reset, so the flux comes down at a rate the clamp sets, and the reset must be long enough for the current to reach zero — if the next pulse arrives first, the core starts higher still, and the walk of the flux that walks begins. A resonant reset drives the flux below the remanence on purpose, into the negative branch, and recovers some of the lost swing without a gap; that is a symmetric drive in disguise, and this essay’s limit does not apply to it.

The material is a model with two parameters. The play-operator core reproduces remanence, coercivity and the wiping-out of minor loops, and it was calibrated to make the power ferrite’s squareness about a quarter, typical of a manganese–zinc power material; it does not model temperature, which moves a ferrite’s remanence and coercivity substantially, nor the rate-dependence of a real loop at the frequencies where eddy currents matter, which the current inside the iron measured.

The numbers #

Driven one way to 0.3 T and reset to zero current, a power ferrite with 90.6 mT of remanence starts every pulse after the first at 90.8 mT and can add 209 mT — 34.9 per cent of a symmetric drive’s swing, against the textbook half. A soft ferrite keeps 46.3 per cent; a square-loop core 14.7. A gap moves the reset onto the shear line $H = -l_g B/(\mu_0 l_e)$ and bounds the start by $\mu_0 H_c l_e/l_g$, which the march approaches as the gap grows: 22.0 mT at 0.1 mm against a bound of 30.2, 5.5 mT at 0.5 mm against 5.7. The recovered swing costs magnetising current linearly — the ungapped 132 mA plus $l_g B/(\mu_0 N)$ — while the swing returns only as one over the gap.

Still open: a clamped reset, a temperature, and the resonant reset priced #

A reset with a clamp and a deadline. The march lets the current fall all the way to zero. A clamped reset brings the flux down at a fixed rate, and if the off-time is shorter than the time that takes, the core starts the next pulse above its remanence. The off-time below which a one-way drive walks to saturation, against the clamp voltage and the material, is the boundary a forward converter’s maximum duty cycle is set from.

The remanence at temperature. A ferrite’s remanence and coercivity both fall as it warms towards its Curie point, so a one-way drive gains swing as it heats — and a gap sized at room temperature is oversized at full load. Whether the gap can be chosen so that the swing is flat with temperature is a calculation with the material’s temperature model in it.

A resonant reset, priced. Letting the magnetising inductance ring with the switch’s capacitance during the off-time drives the flux below the remanence and recovers swing without a gap. What it costs — the voltage the switch sees during the ring, against the swing recovered — is the comparison that would say when a gap is the better cure.