The inductance the current decides

A boundary in volt-seconds put the limit where it belongs. A core saturates at a flux density, flux is the integral of voltage over turns and area, so what a winding can be asked for is a number of volt-seconds and not a voltage or a frequency separately. The flux that walks then measured what an asymmetric drive does to that integral over many cycles.

Both treat the inductance as a constant right up to a wall. It is not a wall and it is not a constant: the material’s B–H curve bends over the whole way, so the flux per amp falls continuously, there is no current at which the part stops being an inductor, and there is no current below which it is the one the data sheet names.

The model, and the term that stops it collapsing #

B(H) here is μ₀H plus a saturating magnetisation, written as a flux linkage λ(i) with the gap’s reluctance in series. The vacuum term is not decoration. Written without it — a bare tanh — the differential permeability goes to zero at high field rather than to μ₀, the inductance collapses to nothing instead of to the air-core value, and a marched cycle’s ripple current is unbounded.

That is the same structural point the capacitance that is not one number makes about a class II ceramic, and the two models were deliberately written in the same form: a linear backbone plus a bounded term. In both cases the backbone is the vacuum, and in both cases leaving it out produces an element that behaves like nothing physical.

The one free parameter is pinned rather than fitted. dλ/di at zero current must be the reluctance answer the energy is in the gap computes from the gap and the path, and it is, to nine digits — which is the check that this rung is describing the same core as the two below it.

Three saturation currents, on one part #

The number a catalogue calls the saturation current is a point on the curve above, and which point is a choice. Ten per cent down is 1.896 A, twenty per cent is 2.395, thirty per cent is 2.694 — a spread of 42 per cent on one part, before any tolerance is considered.

All three are in use. A part specified at the 30 per cent point looks better than an identical part specified at 10, and the comparison a designer makes between two catalogue entries is a comparison between two conventions unless both are read carefully.

That is a measurement problem of the same shape as the ceramic’s test amplitude: a number that appears to be a property of the part is partly a property of how it was defined. Here it is worse in one respect, because there is no standard at all — the percentage is printed beside the current, in small type, and the two are quoted together as though the pair were one number.

The ripple a design expression cannot express #

An inductor for a switching converter is chosen from one line: $\Delta i = (V_{in} - V_{out})\,D\,T/L$. It is a rearranged v = L di/dt and it is exact while L is a number.

To find out what it is worth when L is not, the inductor is marched. The element is flux-defined — its state is λ, the trapezoidal rule is applied to dλ/dt = v, and the current is read back through the stated inverse — sitting between a switching node and the output, with the duty cycle set to $D = V_{out}/V_{in}$ so that the volt-seconds balance exactly and the direct current stays where it is put. That last detail is what makes the measurement a clean function of the operating current rather than a transient of a whole converter.

At two-thirds of an amp the three agree. At 2.56 A — ninety-five per cent of this part’s own thirty-per-cent rating — the design expression says 514 mA and the march says 1,022, twice as much, and the peak reaches 3.57 A, where the inductance is 20 per cent of nameplate. A part rated at 2.69 A is past its own rating at a direct current below it, because the ripple takes it there.

The middle curve is the interesting failure. Evaluating the expression at the inductance the part has at the operating point is the obvious repair and it is not enough: it gives 685 mA against the march’s 1,022, because the current spends part of every cycle above the operating point and the inductance up there is lower again. The correction is not a correction to a value; the value is not the right kind of object.

The quantity that does not move #

The rung below this one bounded the flux, and it is worth asking what has happened to that bound in a cycle whose current is running away.

Nothing has happened to it. Across a hundred-and-forty-fold change in the current swing, the flux swing is constant to better than two per cent, and the small excess at the top is the winding resistance taking volt-seconds of its own — an IR drop, not a magnetic effect.

That is the two rungs in one sentence. The material bounds the flux; the flux is set by the voltage and the time; both statements survive saturation exactly. What saturation breaks is the relationship between the flux and the current — and the current is the only one of the two that anybody measures, that a current-sense resistor sees, that a MOSFET has to carry and that a control loop limits.

An energy that is not half the inductance times the current squared #

The energy stored is ∫λ di, and there are two obvious ways to write it as ½Li² which give answers a factor of seventy-three apart.

At three amps the true integral is 3.66 mJ. Using the zero-current inductance gives 3.84 mJ, which is four per cent high and nearly right — because most of the current was accumulated while the part was still linear. Using the incremental inductance at three amps gives 50 µJ, which is 1.4 per cent of the answer, because the incremental value describes the last microjoule and not the first three millijoules.

Both are ½Li², both are wrong, and they are wrong by different factors in different directions. The integral is the definition and it is one line of arithmetic; the shortcut is where the trouble is.

What a control loop sees, and what it does about it #

A converter measures its inductor current — through a sense resistor, or across the switch — and a peak-current-mode loop terminates each cycle when that current reaches a threshold. Saturation therefore interacts with the control loop directly rather than as a component tolerance.

Three things follow from the numbers above and none of them is a fault condition. The peak the comparator sees is larger than the design predicts, so the loop terminates the cycle early and the output falls short. The ripple is larger, so the output capacitor’s ripple voltage is larger by the same factor. And the effective inductance is a function of load, so the loop’s own small-signal plant has a pole that moves with the operating point — which is a stability question rather than a ripple question.

The third of those is the one that has no obvious symptom. A current-mode loop’s plant has a pole at the load resistance over the output capacitance and a second one at the inductor’s own, and when the inductance halves at high load the second pole doubles. A compensation network designed at light load has more margin than intended at heavy load, which sounds harmless and is: the danger runs the other way, and a design compensated at full load has less margin as the load drops and the inductance rises back to nameplate.

The measurement a bench makes #

Two ways of measuring this are in common use and they disagree in a way the model explains.

An impedance analyser measures the small-signal inductance at whatever direct current a bias supply can push through the winding, and what it returns is the top curve of this essay — the slope of λ at that current. That is the right number for a small signal on a large bias and it is not the number a converter’s ripple obeys, because a converter’s ripple is not small.

A volt-second measurement — apply a known voltage for a known time and measure the current reached — returns the chord rather than the slope, which is a charge-average inductance analogous to the ceramic’s charge-average capacitance. It is closer to what a converter does and it is still not it, because the current in a converter starts from a direct bias rather than from zero.

The march is neither of those and needs no inductance at all: the state is the flux, the element is λ(i), and the answer comes out as a waveform. That is the same move the capacitor that remembers makes with a charge, and in both cases the reason is identical — a quantity defined as a derivative cannot be marched through a range over which it varies.

Where the part actually fails #

Above about three amps in this design the march stops looking like a triangle at all. The core saturates part way through the on-time, the inductance falls to its air-core 0.05 µH, and the current rises at 140 amps a microsecond until the switch turns off — 55 A at three amps of load, limited only by the winding resistance and the duration.

This is a stable periodic orbit rather than a divergence, and the reason is the flux result above: the volt-seconds still balance, so λ returns to where it started every cycle, so the current does too. The part is not destroyed by the mathematics. It is destroyed by the 55 amps, which the switch was not sized for and which the current limit sees a few hundred nanoseconds too late — a fault whose signature is a switch failure with no obvious cause on a supply that had been running for months.

Choosing a part, with the curve rather than the number #

The design procedure that produces the mistake above is entirely reasonable and is worth writing out, because the place it goes wrong is one step and not the whole of it.

Choose the ripple as a fraction of the load current, usually a third. Rearrange the design expression for L. Look up a part whose inductance is near that and whose saturation current is above the peak — load plus half the ripple. Check the copper loss. Done.

The step that fails is the third, and it fails twice. The saturation current is quoted at a percentage that is not stated in the same units the comparison needs, and the peak used in the comparison was computed from the nameplate inductance rather than from the inductance at the peak. At 2.56 A of load in this design, the procedure computes a peak of 2.82 A, compares it against a 2.69 A rating and flags a small margin problem. The march says the peak is 3.57 A, which is 33 per cent past the rating rather than 5 per cent past it.

The repair is not a safety factor. It is to compute the peak from the curve rather than from a value: the peak is where the flux the volt-seconds deliver puts it, and the flux is exactly predictable, so a single evaluation of the inverse λ⁻¹ at (λ_dc + ½Δλ) gives the right answer without marching anything.

That is the same relationship the copper that makes it worse found between the winding and the frequency, arriving one level up: two constraints on one part pulling opposite ways, and an optimum that exists only because they do.

What is not modelled #

No hysteresis and no core loss. λ(i) here is single-valued, so a cycle traverses the same curve both ways and dissipates nothing in the core. A real ferrite has a loop whose area is a loss per cycle, and the loss rises steeply with flux density — so the thermal consequence of running near saturation is not in these figures at all, and it is usually what fails first in continuous operation.

No temperature. Saturation flux density falls with temperature — a power ferrite loses about a third of it between 25 °C and 100 °C — so a part that is comfortable on the bench is at its knee in a hot enclosure. Every current in this essay moves down as the assembly warms, and the loss that warms it rises as it does, which is a positive feedback loop of the kind two loops, and one heatsink measures for a semiconductor.

And no distributed gap. The model has one lumped gap in a high-permeability path, which is a gapped ferrite. A powdered-iron or sendust core has its gap distributed through the material, and its inductance falls much more gradually with current — a soft saturation that is a design choice, chosen precisely so that the failure above is a gentle derating rather than a cliff.

The shape of the rung #

The two rungs below computed a boundary and this one measured what is on the far side of it, and the surprise is which quantity survived. The flux bound is exact everywhere, including deep in saturation, including in a cycle whose current has grown by a factor of a hundred and forty. It is the current — the quantity the design expression is written in, the quantity the sense resistor measures, the quantity the switch has to carry — that is not bounded at all.

An inductance is a ratio between two things, and only one of them was ever constrained.

That is the same reading the first cycle, which no steady state contains gives of an inrush: a quantity that is well behaved in the description everybody uses, and unbounded in the one the hardware lives in. Here the well-behaved description is magnetic and the hardware’s is electrical, which is why the boundary the two rungs below computed is both exactly right and no help at all.

Three saturation currents, and which one a data sheet means #

The same part having three saturation currents depending on which per cent it was quoted at is the result a designer meets first, and it belongs to a class this collection keeps finding rather than to magnetics.

The constant that is a window is the general form: a quantity defined as a derivative or by a threshold has a value at every operating point and no value anywhere, so a single printed number is a measurement condition that was not printed. There it is a diode’s ideality factor returning 1.23 to 1.98 from eight fits to one curve; here it is a saturation current returning three values from one L(i) curve and three choices of “how much inductance loss counts as saturated”.

The capacitance that is not one number is the same defect on the other reactive component, and the parallel is close enough to be worth drawing: a ten-microfarad ceramic is 2.000 µF as a slope, 5.814 µF as a charge average and 2.105 µF as a bridge reads it, three answers to three questions, all correct. An inductor’s nameplate value is the zero-current slope and a converter operates at a bias, which is exactly the substitution that essay warns about.

Which is why the volt-second boundary is the one worth designing against. It is not a threshold on a curve and not a derivative — it is $N\cdot A_e\cdot B_{sat}$, a single number with no frequency in it, and it bounds the flux exactly whichever per cent anybody else was quoting.