A boundary in volt-seconds

Every boundary in this field so far has been a frequency, and the previous essay’s model — reluctance in series, inductance from a geometry — has no boundary at all: it is linear, so it holds at any amplitude and any frequency, forever.

Real cores do not. Past some flux density the material’s permeability collapses, the reluctance rises by orders of magnitude, and every number in the previous essay becomes wrong at once. This essay measures where, and the answer is in a unit this collection has not used before.

Why it is not a current #

The natural guess is that a core saturates at a current, since the current is what a winding carries and what a designer measures. That is true for a fixed inductance and it is the wrong way round.

The flux linkage in a winding is λ = ∫v dt, and the flux density is that divided by N·Aₑ. Nothing in either expression is a current. The current is related to the flux through the inductance — λ = L i — but the inductance is exactly what stops being constant at saturation, so quoting the boundary as a current is quoting it in terms of a quantity that is only defined below it.

Quoting it as a volt-second product avoids that entirely:

$$\lambda_{\max} = N A_e B_{\mathrm{sat}}$$

which for a hundred turns on a 100 mm² core saturating at 0.35 T is 3.500 mWb-turn. Three numbers, none of them a current, none of them a frequency, and none of them dependent on the waveform.

Where the frequency comes from #

A sinusoid of peak amplitude V at frequency f has a flux amplitude of V/2πf, so the voltage a winding may carry is

$$V_{\max} = 2\pi f\,N A_e B_{\mathrm{sat}}$$

and the entire frequency dependence of a transformer’s rating is that one division. A transformer “rated for 50 Hz” is a transformer whose volt-second product was divided by 2π × 50 once, by somebody who then printed the answer.

Read the other way it is more useful. The seconds a stated voltage may be applied for is $\lambda_\mathrm{max}$/V — 3.5 ms at one volt, 350 µs at ten, 35 µs at a hundred — which is exactly the number a switching design needs and is not a frequency at all. A pulse of the wrong width saturates a core that a sinusoid of the same amplitude and the same fundamental frequency would not.

Measuring it rather than inverting the formula #

The closed form above could simply be printed. It is instead used as the thing being checked, and the measurement is a march.

fluxMarch integrates the applied voltage sample by sample with the trapezoidal rule and reports the largest excursion of the flux from its own mean. saturationSweep bisects on the amplitude until that excursion reaches $B_\mathrm{sat}$. The two share the core’s dimensions and nothing else — one is an integral of a sampled waveform, the other is a product of three constants and a frequency.

Fitted across three decades, the exponent is 1.000000.

The point of doing it this way is not the agreement. It is that the same integrator, having been checked against a case with a closed form, can then be pointed at cases that have none — an asymmetric drive, a switched waveform, a transformer energised at the wrong instant — where there is no formula to invert and the flux has to be followed.

That is the next essay’s subject, and the reason this one exists first.

Two details that decide the answer #

The excursion is measured from the mean, not from zero. A flux waveform sitting on an offset saturates on its peak, and the offset is set by the drive’s history rather than by its amplitude. Reporting the absolute peak would make a symmetric drive look worse than it is — the integrator’s constant of integration is arbitrary — and an offset one look better. Taking the excursion from the waveform’s own mean removes the arbitrary constant and keeps the real asymmetry.

The integration is trapezoidal rather than a held sum. The noise field has already recorded what the difference costs: its two routes to a noise voltage disagreed by two to seven per cent until two discretisation errors were found, one of them a trapezoidal rule interpolating between held samples. Here the integration is over a smooth waveform sampled two thousand times a cycle, so the trapezoidal error is a part in 10⁶ and the 0.001% agreement above is what remains of it.

What the turns slider shows, and what it does not #

The figure’s slider is the number of turns, and it moves the whole line up in proportion — which is right, since $\lambda_\mathrm{max}$ contains N linearly — and it hides a trade that is worth making explicit.

Doubling the turns doubles the voltage the winding can carry and quadruples the inductance, because inductance goes as N². So more turns is unambiguously better for saturation and for the lower band edge at once, which is the only place in this field where two levers point the same way.

The price is elsewhere and it is real. More turns is more wire in the same window, so the wire is thinner and the winding resistance rises as roughly N² as well — which eats the midband gain, and does it worse at high frequency where the resistance is already rising with the skin depth. And more turns is more turn-to-turn capacitance, which moves the self-resonance down.

So the honest form of “more turns is better” is: better for the two low-frequency boundaries, worse for the two high-frequency ones, and the design sits wherever those four meet. Every quantity in that sentence has a figure in this field.

What the boundary looks like on the site’s own axis #

This is the sixth kind of quantity a boundary on this site has been measured in, and it is worth putting them together because the list has grown enough to be a statement about the collection.

The volt-second is the first that is a product of two units, and it behaves like one: it can be spent as a large voltage for a short time or a small one for a long time, and a design has to respect the product rather than either factor. That is why a transformer’s specification is genuinely two numbers — a voltage and a frequency — rather than one, and why quoting either alone is quoting half of a product.

Two waveforms with the same amplitude and different limits #

The volt-second framing makes a comparison possible that a voltage rating alone cannot express, and it is the practical reason for insisting on it.

Take three waveforms of the same peak amplitude V at the same fundamental frequency f, and ask how much flux each puts into a core:

A sinusoid integrates to V/2πf, which is V/(6.28f).

A square wave holds V for half a period, so it integrates to V/(4f) — 57% more flux than the sinusoid at the same peak voltage and the same frequency.

A triangle integrates to V/(8f), which is 21% less than the sinusoid.

So a transformer rated at a hundred volts on a sinusoid saturates at sixty-four volts on a square wave of the same frequency, and would take a hundred and twenty-six on a triangle. None of those three numbers is available from a rating in volts at a frequency; all three are one division of the volt-second product.

This is the same observation the digital field made about the quantisation formula’s 1.76 dB — that it is a statement about a sinusoid’s crest factor and changes when the waveform does — and the same one the power field made about a rectifier’s power factor, where a phasor calculation returns exactly 1.000000 about a current that is not a sinusoid. Three fields, three figures of merit defined on a waveform and quoted about a system.

The pattern is worth naming as an instruction: when a specification is a single number about an amplitude, find out which waveform it was defined on. In this field the answer is almost always a sinusoid and the application is almost always a square wave, which is a factor of 1.57 in the direction that saturates the core.

What saturation does when it arrives #

The measurement above finds the boundary and stops there, which is deliberate, and it is worth saying what is on the other side and why this essay does not go.

Past $B_\mathrm{sat}$ the material’s incremental permeability falls towards that of air, so the inductance collapses — by a factor of the relative permeability, which for a µᵣ = 2000 ungapped core is a factor of two thousand. The current, which was the flux divided by the inductance, therefore rises by that factor, essentially instantaneously, and what limits it is whatever resistance is in the loop.

That is not a gradual departure and it is not a distortion. A transformer driven into saturation presents a near short circuit to its source for the remainder of the half-cycle, and the failure is usually of the switch rather than of the magnetics.

Modelling it needs a non-linear reluctance, which needs Newton’s method on the magnetic circuit — the machinery the semiconductor field already has for an exponential device, pointed at a B–H curve instead. That is a real and reachable piece of work and it is not in this phase. What is here is the boundary, measured, in the unit that belongs to it.

Why a gap helps, and by exactly how much #

The previous essay’s arithmetic gives the answer directly and it is worth connecting the two.

Saturation is a limit on B, and B is the flux over the area. A gap does not change the area and does not change the flux for a given applied volt-second, so a gap does not change the saturation voltage at all. $\lambda_\mathrm{max}$ is N·Aₑ·$B_\mathrm{sat}$ with no reluctance in it.

What a gap changes is the current at which that flux is reached. The gap raises the reluctance, so the same flux needs more ampere-turns, so the winding can carry more current before the core is full. A gapped core stores more energy — the ½L i² with a smaller L and a much larger i — which is what the previous essay’s 87% is about.

So the two essays’ boundaries are independent, which is the useful part. The volt-second limit is a property of the turns, the area and the material; the current limit is a property of those plus the gap; and a designer picks the area and turns for the volt-seconds and the gap for the current, with neither choice disturbing the other.

That independence is unusual on this site, where almost every pair of design levers interacts. It is worth naming because it makes the design procedure short: size the core for the volt-seconds, then gap it for the current.

What a bench measurement of this looks like #

The volt-second product is unusually easy to measure and unusually rarely quoted, which is worth one closing section because the procedure follows directly from the definition.

Apply a constant voltage to the winding and watch the current. While the core is linear the current rises as a ramp — a constant voltage across an inductance gives di/dt = V/L — and at saturation it leaves the ramp abruptly and heads for whatever the loop resistance allows. The volt- seconds are the applied voltage times the time to that departure.

That is the whole method. It needs a voltage source, a current probe and an oscilloscope, it produces the quantity that belongs to the core rather than a quantity about somebody’s sinusoid, and it works identically on a gapped core, an ungapped one, a transformer with one winding open or a winding on a core nobody has documented.

Two cautions, both of them consequences of things already in this field.

It has to be a short pulse. The measurement ends in saturation by construction, so the current at the end is large and the energy in it has to go somewhere. Doing it with a switch and a single pulse rather than repetitively is the ordinary practice.

The departure is sharp and the eye is a good detector of it. This is the rare case on this site where a boundary is genuinely abrupt rather than a per cent of something — the permeability falls by a factor of the material’s µᵣ, so the current’s slope changes by that factor — and there is no tolerance to argue about. Compare that with the small-signal boundary, which is a per cent chosen by the person quoting it, or with Kirchhoff’s own frequency, which is one degree of phase because somebody picked one degree.

Saturation is the first boundary in this collection that is not a matter of how much error is acceptable. It is a place where a material stops doing the thing the model was about, and the model does not degrade towards it — it holds, and then it does not.

An amplitude that can be respected, and one that cannot #

The two rungs above this one take that boundary and do the two things left to do with it, and their answers are opposite.

The flux that walks is the one that removes the comfort. This essay’s limit is an amplitude and an amplitude can be respected; that one measures a drive whose two half-cycles differ in volt-seconds by one per cent, which adds the same small area to the flux every cycle and reaches saturation after 64 of them — with halving the drive giving 128 and a tenth of it 637. Reducing the amplitude buys time in exact proportion and removes nothing, so there is no amplitude at which the design is inside a limit.

The inductance the current decides is the one that bounds what saturating actually costs, and its answer is narrower than the alarm suggests. The volt-seconds decide the flux swing whatever the material does — a cycle whose current ripple runs from 519 mA to 75 A has the same flux excursion to better than two per cent — so what saturation breaks is the relationship between that flux and the current. 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.

Read together, the three say what N·Ae·Bsat is for. It bounds a flux exactly and permanently; it says nothing about a current, which is a material property with three published values; and it can be walked to by a drive that never exceeds it.

A frequency rating that is one number divided once #

The measured exponent of 1.000000 for the voltage a winding may carry against frequency is the result that makes a data sheet’s frequency rating readable, and the reading is unflattering: every such rating is $N\cdot A_e\cdot B_{sat}$ divided by $2\pi f$, so the entire frequency dependence a transformer is sold with contains no information beyond one number in webers.

Which means the rating transfers exactly to a waveform it was not measured on, provided the volt-seconds are computed rather than assumed. A sinusoid and a square wave of the same peak voltage and the same frequency do not have the same volt-second area — the square is $\pi/2$ larger — so a part rated at fifty hertz sinusoidal is rated for a lower square-wave voltage at the same frequency, by exactly that factor and by nothing else.

The one thing the rating cannot carry is the direction the flux that walks is about, since a walk is a property of the difference between two half-cycles rather than of either one’s area. A drive inside the rating on every cycle and asymmetric by one per cent reaches saturation after 64 of them, and no volt-second number printed on a part addresses it.