Circuit Response
every model has an edge, and the figure carries it
Two windings, and the band between them

The flux that walks

The previous essay's saturation limit is an amplitude, and an amplitude can be respected. This one cannot. A drive whose two half-cycles differ in volt-seconds by one per cent adds the same small area to the flux every cycle, so it reaches saturation after 64 cycles — and halving the drive gives 128, and a tenth of it gives 637. Reducing the amplitude buys time in exact proportion and removes nothing. There is no amplitude at which the design is inside a limit.
16 min read 6 figures Every model has an edgeRefused, not extrapolatedMeasured, not tabulated

Assumes: A boundary in volt-seconds · The energy is in the gap

Every boundary this collection has drawn can be respected. An amplifier is one per cent low above 1.4 kHz, so stay below it. A small-signal model is one per cent optimistic above 7.3 mV, so drive it less. A core saturates above 1.10 V at 50 Hz, so apply less than that.

This essay is about the one that cannot.

A 1% imbalance, and the 64 cycles it survivescomputed by solving, not by drawing. The upper panel is the peak flux density, marched cycle by cycle, under a square drive whose positive half is 1% larger in area than its negative half. It does not settle. It walks, by the same area every cycle, and reaches 0.35 T after 64 cycles — 1280 ms at 50 Hz — against a closed form of 63.7. The lower panel is the count against the imbalance, and it rises without bound and never becomes infinite. Halving the drive gives 128 cycles, which is exactly twice: reducing the amplitude buys time and not safety, and there is no amplitude at which this design is inside a limit.00.1000.2000.3000.4000204060cyclespeak flux density (T)saturation, 0.35 Timbalance1%survives64 cycleswhich is1280 ms at 50 Hzpredicted63.7at half the drive128 cycles10%3%1%0.3%0.1%cycles survived, against the imbalancesolved, then checked — no steady state to draw64 cycles, and halving the drive gives 128
Fig. 1 The peak flux density under a drive whose positive half-cycle is one per cent larger in area than its negative half. It does not settle at a level; it walks, by the same area every cycle, and reaches saturation after 64 cycles. The lower panel is the count against the imbalance. The slider is the imbalance, and no setting of it removes the failure.

Why it walks

The flux is the integral of the applied voltage. Over one complete cycle of a drive whose two halves have equal areas, that integral returns to where it started, and the flux settles into a periodic excursion whose size the previous essay computes.

If the two halves do not have equal areas — if the positive half is a little larger, by a little longer or a little higher — then each cycle adds the difference, and the flux is a staircase rather than a periodic waveform. The amount added per cycle is

Δλ=V ϵ2f\Delta\lambda = \frac{V\,\epsilon}{2f}

for a square drive of amplitude V at frequency f with a fractional imbalance ε, and the number of cycles to saturation is the headroom divided by it:

n=λmax⁡Δλ=2f NAeBsatVϵn = \frac{\lambda_{\max}}{\Delta\lambda} = \frac{2 f\, N A_e B_{\mathrm{sat}}}{V\epsilon}

Every quantity in that expression is finite and positive, so n is finite for every imbalance. The flux always arrives.

The measurement

fluxWalk integrates the drive cycle by cycle and reports the cycle on which the flux first reaches the saturation level. The closed form above is returned beside it and is not used to produce it.

imbalance cycles at 50 Hz closed form
10% 7 140 ms 6.4
3% 22 440 ms 21.2
1% 64 1.28 s 63.7
0.3% 213 4.26 s 212.2

The agreement is within one cycle at every imbalance, which is the right way to state it: the measurement counts whole cycles, so its resolution is one, and a relative tolerance would be 14% at seven cycles and 0.5% at two hundred. Asserting a percentage failed at the largest imbalance for exactly that reason, and the check now holds the difference rather than the ratio.

The part that makes it different

Reduce the drive and the count rises in exact proportion.

amplitude cycles at 1% imbalance
as drawn 64
half 128
a tenth 637

Halving the drive doubles the time. A tenth of the drive gives ten times the time. Nothing about the sequence approaches a limit, because the mechanism does not contain an amplitude at which it stops — the flux added per cycle is proportional to the drive, and so is the time to fill a fixed headroom.

That is what makes this a different kind of boundary from everything else on this site. Every other one names a value with a safe side. This one names a count, and the count is finite everywhere. There is no amplitude, no frequency, and no core size at which a volt-second-unbalanced drive is inside a limit; there is only a length of time before the failure, and design choices move it rather than remove it.

A gapped core: where the inductance goes, and where the energy is. computed by solving, not by drawing. Reluctance in series — the gap's lg/µ₀Ae and the core's le/µ₀µᵣAe — with the inductance N²/ℛ and the share of the stored energy in each proportional to its share of the reluctance. At two hundred microns on a µᵣ = 2000 core the inductance has fallen from 41.89 mH to 5.366, and 87.2% of the energy is in the gap — which is air. The share is lg/(lg + le/µᵣ), so it contains neither the turns nor the area and what decides it is µᵣ·lg against the path length; the slider shows the same gap holding 40% at µᵣ = 200 and 98% at 15,000. That is the reason a gap is a design parameter: it is the part of the magnetic circuit whose properties do not drift, do not saturate and do not depend on temperature.
Fig. 2 The headroom the walk is consuming. A gap raises the current a core can carry at a given flux and does not raise the flux the core can hold — so it buys nothing at all against this failure, which is worth knowing because it is the first remedy most people reach for.

What a design does about it instead

Since the boundary cannot be respected by staying below something, it has to be handled by a mechanism, and there are three in ordinary use. Each is a way of removing the imbalance rather than of tolerating it.

Block the direct component. A capacitor in series with the winding cannot pass a net volt-second, so the flux cannot walk. This is exact and it is why audio and communications transformers driven from anything that might have an offset have one. The cost is another corner in the low-frequency response, in series with the one the band essays measure, and a capacitor that has to be large enough not to move it.

Measure the flux and correct it. Sense the current — which above saturation is the only visible symptom, and below it is proportional to the flux — and adjust the drive to null the walk. This is what a switching converter’s flux-balancing loop does, and it is a feedback problem of the kind the feedback field owns: a slow integrator around a fast one, with the usual questions about margin.

Reset the core deliberately every cycle. Arrange the circuit so that the flux is driven back to a known point once per period rather than left to accumulate. A forward converter’s reset winding is this; so is the demagnetising interval of a flyback.

All three are structural. None of them is “use a bigger core”, and the arithmetic says why: a core twice the size doubles λmax\lambda_\mathrm{max} and doubles the count, which buys a factor of two in time against a mechanism that has all the time there is.

A 10% imbalance, and the 7 cycles it survives. computed by solving, not by drawing. The upper panel is the peak flux density, marched cycle by cycle, under a square drive whose positive half is 10% larger in area than its negative half. It does not settle. It walks, by the same area every cycle, and reaches 0.35 T after 7 cycles — 140 ms at 50 Hz — against a closed form of 6.4. The lower panel is the count against the imbalance, and it rises without bound and never becomes infinite. Halving the drive gives 13 cycles, which is exactly twice: reducing the amplitude buys time and not safety, and there is no amplitude at which this design is inside a limit.
Fig. 3 A ten per cent volt-second imbalance. The core walks into saturation in seven cycles — 140 ms — and in thirteen at half the drive. What a design does about it instead of correcting the imbalance is to leave enough flux headroom for the walk to be caught, and seven cycles is not enough time for anything slow to catch it.

Where the imbalance comes from

It is worth being concrete, because “a one per cent asymmetry” sounds like a manufacturing defect and is usually a property of the circuit working correctly.

Unequal switch drops. A bridge whose two halves conduct through devices with different on-resistances applies slightly different voltages in the two directions. A tenth of a volt out of ten is one per cent.

Unequal switching times. A gate driver whose rise and fall differ by fifty nanoseconds in a ten microsecond half-period gives half a per cent, from timing alone.

A load that is not symmetric. Anything drawing a direct current through a winding — a rectifier, a half-wave load, a fault — puts a steady magnetomotive force on the core that the drive has to work against.

Energising at the wrong instant. Closing a switch onto a mains transformer at a voltage zero crossing rather than at a peak asks the flux to start half a cycle’s worth away from where the steady state wants it, which is the inrush current every transformer’s specification mentions. It is the same arithmetic with one large step instead of many small ones.

The first two are of the order of a per cent in a well-built circuit, which is why the figure’s default is one per cent rather than something dramatic. At that value and this core the answer is 1.28 seconds.

A 5% imbalance, and the 13 cycles it survives. computed by solving, not by drawing. The upper panel is the peak flux density, marched cycle by cycle, under a square drive whose positive half is 5% larger in area than its negative half. It does not settle. It walks, by the same area every cycle, and reaches 0.35 T after 13 cycles — 260 ms at 50 Hz — against a closed form of 12.7. The lower panel is the count against the imbalance, and it rises without bound and never becomes infinite. Halving the drive gives 26 cycles, which is exactly twice: reducing the amplitude buys time and not safety, and there is no amplitude at which this design is inside a limit.
Fig. 4 Five per cent: thirteen cycles, 260 ms. Halving the imbalance has doubled the time, exactly, because the walk is a sum and the flux limit is fixed — so the count of cycles is the inverse of the imbalance and nothing else enters.

The check that this is a mechanism rather than a fit

The gate holds four things about the walk and it is worth listing them, because between them they distinguish a mechanism from a curve that happens to fit.

Every imbalance saturates. Four values across two decades, all of them reaching the threshold inside the run. A mechanism with a safe side would have produced a run that did not.

The count matches the closed form to within one cycle. Four values, each within the measurement’s own resolution — which is what agreement means for an integer.

The count rises monotonically as the imbalance falls. Trivially expected and worth holding, because a march with an accumulation error would eventually stop being monotone and the failure would look like scatter.

Halving the drive doubles the count and a tenth of it multiplies by ten. This is the one that carries the essay’s conclusion, and it is checked against the closed form rather than against the base measurement, so that the integer resolution enters once rather than twice.

That fourth check is the interesting one to have written, because it is a claim about a scaling rather than about a value — the same shape as the tolerance exponents in the ladder essay, where what is asserted is that a cascade’s error grows as the 0.99 power and a ladder’s as the 2.00, rather than that either is a particular size. A scaling is much harder to satisfy by accident than a number is.

Why the count and not a time

The measurement reports a number of cycles and converts it to seconds only in the caption, and the choice is worth defending because seconds is what a designer wants.

The count is the quantity the mechanism produces. Each cycle adds a fixed area, so what the flux does is count, and the number of steps to fill the headroom is independent of how fast the steps are taken. Doubling the frequency halves the time and leaves the count where it is — because doubling the frequency also halves the area each half-cycle contributes.

That has a consequence a time would obscure. The count depends on the imbalance and the amplitude and not on the frequency, so a switching converter running at 100 kHz and one running at 50 Hz with the same fractional imbalance and the same fraction of full flux survive the same number of cycles — about a second and a half in one case and 640 microseconds in the other.

Which reframes the design problem usefully. The correcting mechanism does not have to act within a fixed time; it has to act within a fixed number of cycles, which is the natural unit for a control loop that runs once per cycle anyway. A flux-balancing loop with a bandwidth of a hundredth of the switching frequency has a hundred cycles to work in, and this essay’s figure says how many it has before the core is full.

A 2% imbalance, and the 32 cycles it survives. computed by solving, not by drawing. The upper panel is the peak flux density, marched cycle by cycle, under a square drive whose positive half is 2% larger in area than its negative half. It does not settle. It walks, by the same area every cycle, and reaches 0.35 T after 32 cycles — 640 ms at 50 Hz — against a closed form of 31.8. The lower panel is the count against the imbalance, and it rises without bound and never becomes infinite. Halving the drive gives 64 cycles, which is exactly twice: reducing the amplitude buys time and not safety, and there is no amplitude at which this design is inside a limit.
Fig. 5 Two per cent: 32 cycles, 640 ms, and 64 at half the drive. The count and not a time is what this figure reports, because the time depends on the switching frequency and the count does not — a converter run at twice the frequency saturates after the same number of cycles and half the wall-clock time.

The site’s shortest list of boundary kinds

The collection has been keeping an informal count of what a boundary can be measured in, and this essay adds the strangest entry so far.

A frequency, an amplitude, a size, a duration, a level, a volt-second product, a resistance — and now a count of cycles, which is unlike all of them in one respect: it is not a property of a design that can be compared with a requirement. A specification cannot say “the imbalance shall be less than X” and be safe, because every X gives a finite count.

What it can say is “the flux shall be balanced by a mechanism”, which is a statement about the structure of the circuit rather than about a value in it. That is a genuinely different kind of requirement and it is the reason this failure mode has a name of its own in the trade — staircase saturation, flux walking, flux imbalance — where most boundaries do not.

A 0.5% imbalance, and the 128 cycles it survives. computed by solving, not by drawing. The upper panel is the peak flux density, marched cycle by cycle, under a square drive whose positive half is 0.5% larger in area than its negative half. It does not settle. It walks, by the same area every cycle, and reaches 0.35 T after 128 cycles — 2560 ms at 50 Hz — against a closed form of 127.3. The lower panel is the count against the imbalance, and it rises without bound and never becomes infinite. Halving the drive gives 255 cycles, which is exactly twice: reducing the amplitude buys time and not safety, and there is no amplitude at which this design is inside a limit.
Fig. 6 Half a per cent, the smallest imbalance drawn: 128 cycles, 2560 ms, 255 at half the drive. Across the five imbalances the count runs 7, 13, 32, 64 and 128 against 10, 5, 2, 1 and 0.5 per cent — the product is 64 to 70 at every one of them. The shortest list of boundary kinds this site keeps has three entries: a frequency, an amplitude and a count, and this is the only boundary in the collection that is a count.

The one-large-step version

Inrush is the same arithmetic with one step instead of many, and it is worth doing because it is the case a reader is most likely to have met.

A transformer in the steady state has a flux waveform that lags the applied voltage by a quarter cycle — the flux is the voltage’s integral — so at the instant the voltage passes through zero going positive, the flux is at its negative peak. Close a switch at that instant and the flux starts from zero instead, so the first half-cycle takes it to twice the normal peak.

Twice the normal peak is far into saturation for any core designed with sensible headroom, which is why the inrush current of a mains transformer is tens of times its rated current, and why it is worst when the supply is connected at a voltage zero and absent when it is connected at a voltage peak.

The arithmetic is this essay’s with ε = 1 and one cycle: the whole imbalance arrives at once. And the remedies are the same three — a series impedance to limit it, a controlled switch that closes at the right instant, or a deliberate reset — which is a reassuring sign that the framing is the right one: a mechanism that explains both the slow walk and the single step, with the same expression and a different value of one variable.

There is a wrinkle worth keeping. Real transformers show inrush on some switch-ons and not others, and the reason is that the core retains a remanent flux from however it was last switched off. Starting from a remanence of the wrong sign makes the first half-cycle worse than the factor of two above; starting from the right sign makes it better. That is a property of the material’s hysteresis loop, which this essay’s linear integrator does not have, and it is the second thing the field’s non-linear extension would add.

Two things this measurement does not include

The core’s own non-linearity is not modelled. The march is of a linear flux integral against a fixed threshold, so what it reports is the cycle on which the flux would reach the saturation level. A real core’s permeability begins falling before the nominal saturation flux, so the current starts rising a little earlier and the departure is less abrupt than the model implies — in the direction of arriving sooner rather than later, so the count here is an optimistic bound rather than a pessimistic one.

Any resistance in the winding limits the walk. A real winding has resistance, and a direct component of flux implies a direct component of current, which develops a voltage across that resistance opposing the imbalance. So a real core with a resistive winding and a voltage drive reaches an equilibrium offset rather than walking without bound — at an offset where the resistive drop cancels the imbalance.

That second one is important enough to state plainly rather than bury: the walk is unbounded only for an ideal voltage drive into a lossless winding. With resistance in the loop, the flux settles at a direct offset whose size is the imbalance’s voltage divided by the winding resistance, times the inductance — which for a small enough resistance is still far past saturation, and for a large enough one is not.

Which is exactly the shape of every model on this site: the failure described here is real, is the one that occurs in practice, and has its own boundary — the winding resistance at which the equilibrium offset falls below saturation. That boundary is measurable with the same machinery, it is not measured here, and it is the clearest thing this essay leaves open.

What the lower panel of the figure is for

The figure has two panels and the lower one is the argument rather than an illustration.

The upper panel shows one drive walking: a staircase rising to the saturation line, at one imbalance. It is convincing and it is a single case, and a reader would be right to ask whether the imbalance chosen is representative.

The lower panel answers that by drawing the count against the imbalance across two decades — 7, 22, 64, 213 and 637 cycles at 10%, 3%, 1%, 0.3% and 0.1% — on logarithmic axes, where it is a straight line of slope −1. That straight line is the claim: the relationship is exactly inverse, it has no knee anywhere, and extending it to the right predicts a finite count at any imbalance however small.

The site’s habit of drawing a sweep beside a case has caught things before. The ladder’s convergence to the wave limit is a power law with an exponent of −0.54 and an 8.7% residual, which is only visible as a fit across four section counts; the averaging law separates a white sequence from a pink one by two exponents rather than by two values. A single case with a number under it is an anecdote; a sweep with an exponent is a mechanism.

Here the exponent is exactly −1 and the mechanism is a division, which is about as simple as this collection’s results get. What makes it worth the space is not its complexity but its shape: it is the one result in this collection where the safe side of the boundary is empty.

What has to be added to make it safe

A boundary with no safe side is a boundary that a circuit rather than a component has to answer, and the answers are all of one kind: something that measures the walk and corrects it.

The cheapest is a series capacitor, which cannot pass a direct component and therefore cannot supply a volt-second imbalance — and which is why half-bridge converters have one where push-pull converters need current-mode control instead. What it costs is a voltage that varies with duty cycle and a part carrying the full winding current, which is the pair that is worse than either’s component in a place where its series resistance is dissipating rather than damping.

The one this collection can price is the drive’s own asymmetry, and the delay that is two delays is where it is measured: a comparator’s two propagation delays differing by ten nanoseconds saturates a hundred-turn core in 231 cycles, and more hysteresis improves the duty cycle and worsens the volt-seconds at the same time. So the imbalance this essay treats as a given is, in a real converter, a difference between two numbers on a semiconductor’s data sheet — and the 64 cycles measured here is what one per cent buys.

Which is the practical form of “no amplitude is inside the limit”. The design has to contain a mechanism that observes the flux or the current and pushes back, and every such mechanism is a feedback loop with its own stability question — so the boundary is not removed but exchanged for one of the kind what is left at crossover is about.

Part 2 on saturation

One argument about Saturation, and one of 4 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it:

The voltage a winding may carry, which is a volt-second limit read at a frequency. computed by solving, not by drawing. The dots are bisections on a marched flux — the voltage integrated sample by sample until the peak excursion reaches 0.35 T — and the line is N·Ae·Bsat·2πf. They agree to 0.001% over three decades, and the fitted slope is 1.000000: exactly proportional, because flux is the integral of voltage and nothing else. The quantity that belongs to the core is the 3.500 mWb-turn, which has no frequency in it. A transformer "rated for 50 Hz" is a transformer whose volt-second product was divided by 2π × 50 once. A boundary in volt-seconds Part 1 — A core saturates on the integral of the voltage applied to it, not on the current through it and not at a frequency. The quantity that belongs to the core is N·Ae·Bsat — 3.500 mWb-turn here — and it has no frequency in it at all. Everything a data sheet says about a transformer's frequency rating is that one number divided by 2πf once: measured on a marched flux, the voltage a winding may carry is proportional to frequency to a fitted exponent of 1.000000. One part, three saturation currents: 1.90 A, 2.40 A, 2.69 A. computed by solving, not by drawing. B(H) is μ₀H plus a saturating magnetisation, written as a flux linkage, and the inductance drawn here is dλ/di — the slope of that flux, which is what a small signal on a direct current actually meets. It is 18.92 µH at no current, 16.73 µH at two amps and 10.43 µH at three, and it never reaches zero: the vacuum is still there, so the part falls to its air-core 0.05 µH and stays. The three marks are the ten, twenty and thirty per cent drops different manufacturers print as the saturation current — 1.896, 2.395, 2.694 amps, a spread of 42 per cent on one part. The inductance the current decides Part 3 — The rung below bounded the flux and the boundary is exact: the volt-seconds decide the flux swing whatever the material does, and a cycle whose current ripple runs from 519 mA to 75 A has the same flux excursion to better than two per cent. What saturation breaks is the relationship between that flux and the current — so a ripple the design expression puts at 514 mA is 1,022 mA at 2.56 A of load, the peak reaches 3.57 A where the part is at a fifth of its nameplate inductance, and the same part has three saturation currents depending on which per cent it was quoted at.

Where this goes next

The voltage a winding may carry, which is a volt-second limit read at a frequency. computed by solving, not by drawing. The dots are bisections on a marched flux — the voltage integrated sample by sample until the peak excursion reaches 0.35 T — and the line is N·Ae·Bsat·2πf. They agree to 0.001% over three decades, and the fitted slope is 1.000000: exactly proportional, because flux is the integral of voltage and nothing else. The quantity that belongs to the core is the 3.500 mWb-turn, which has no frequency in it. A transformer "rated for 50 Hz" is a transformer whose volt-second product was divided by 2π × 50 once. A boundary in volt-seconds A core saturates on the integral of the voltage applied to it, not on the current through it and not at a frequency. The quantity that belongs to the core is N·Ae·Bsat — 3.500 mWb-turn here — and it has no frequency in it at all. Everything a data sheet says about a transformer's frequency rating is that one number divided by 2πf once: measured on a marched flux, the voltage a winding may carry is proportional to frequency to a fitted exponent of 1.000000. A gapped core: where the inductance goes, and where the energy is. computed by solving, not by drawing. Reluctance in series — the gap's lg/µ₀Ae and the core's le/µ₀µᵣAe — with the inductance N²/ℛ and the share of the stored energy in each proportional to its share of the reluctance. At two hundred microns on a µᵣ = 2000 core the inductance has fallen from 41.89 mH to 5.366, and 87.2% of the energy is in the gap — which is air. The share is lg/(lg + le/µᵣ), so it contains neither the turns nor the area and what decides it is µᵣ·lg against the path length; the slider shows the same gap holding 40% at µᵣ = 200 and 98% at 15,000. That is the reason a gap is a design parameter: it is the part of the magnetic circuit whose properties do not drift, do not saturate and do not depend on temperature. The energy is in the gap A ferrite core is chosen for its permeability and then deliberately cut, and the cut is not a compromise. Reluctance adds in series, so a two-hundred-micron gap in a µᵣ = 2000 core holds 87.0% of the stored energy while the ferrite holds thirteen — and the share is lg/(lg + le/µᵣ), a ratio of two lengths, containing neither the turns nor the area. The material chosen for its permeability holds almost none of what the component stores. Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 10.0 cm at 3.97 MHz, the ideal 100 nF capacitor at 4.69 MHz. The fifth boundary is an amplitude rather than a frequency and cannot share this axis: a small-signal model is 1% wrong above 7.3 mV, at every frequency there is. Every model has an edge Four assumptions this collection runs on, with the frequency at which each stops being true, on one axis. The ordering is not the one most readers would guess — an ordinary amplifier circuit runs out of model at 1.42 kHz, three thousand times sooner than a ten-centimetre circuit board does.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down, the 8 sharing most with it of 12.

The objects named here

The third axis, after the field and the idea: the things themselves, and every essay that touches each one.

Flux densityMagnetic pathModel rangeSaturationVolt-seconds