The instant a transformer is worst at

The first cycle, which no steady state contains measured the first conduction of a rectifier into an empty reservoir capacitor and found a peak of 32.4 amperes against a repetitive peak of 1.23 — and found that its size depends on when the switch was closed. The worst instant was the peak of the mains, where the full supply appears at once across an empty capacitor. The best was near the voltage zero, where the capacitor and the supply rise together. The inductance that limits, and lifts then put the transformer’s leakage inductance into that first charge and watched it divide the peak by seven.

Both treated the transformer as a source with some impedance in series. It is also an inductor in its own right — a winding on an iron core, whose magnetising current is a few per cent of the rated current in normal running and is drawn through the primary whether anything is connected to the secondary or not. That branch has an inrush of its own, and it is governed by a different quantity from the capacitor’s. The capacitor cares about voltage, and a voltage can jump. The core cares about flux, and a flux cannot.

This essay measures that inrush with the secondary open, so that nothing but the magnetising branch is present, and finds that the instant which is kindest to a reservoir capacitor is the cruellest one for the iron behind it.

The core, and how it is marched #

The transformer is a 230-volt, 50-hertz primary on a core of grain-oriented silicon steel: ten square centimetres of iron, a twenty-centimetre magnetic path, saturation at 1.8 tesla, and twenty microns of effective gap where the laminations butt together. The turns are chosen so that the working peak flux is 1.4 tesla, a common design point — 739 turns — and the primary winding has two ohms of resistance.

The iron is described by its B–H law, $B = \mu_0H + B_{\text{sat}}\tanh(H/H_k)$, the same saturating form the inductance the current decides uses for a gapped inductor: an initial permeability of five thousand, falling through a knee to the permeability of free space once the material can magnetise no further. Ampère’s law around the path, iron and gap together, turns that into a flux linkage as a function of the winding current, and its inverse is what the march needs.

The circuit is one equation: the flux linkage changes at the rate the supply voltage, less the winding’s drop, allows,

$$\frac{d\lambda}{dt} = v(t) - R\,i(\lambda),$$

and it is marched with the trapezoidal rule, solving for the new flux at every step by Newton’s method on the core’s own law. Two checks hold it honest. Every run is repeated with half as many steps and must give the same largest current to a part in a thousand, which is the guard against a step too coarse for the knee. And at the voltage-zero switch-on the same branch is also built as a netlist, with the core as a flux-defined element, and solved by the general-purpose transient solver used elsewhere in this subject; the two give 87.6 amperes to a part in ten thousand.

The second check also found something worth recording. Switched on mid-cycle, the netlist solver’s answer converged only in proportion to its step — 32.0 amperes at two thousand steps a cycle, 35.3 at eight thousand, 36.1 at thirty-two thousand — while the scalar march gave 36.37 at five hundred. The general solver starts a step from a source that has just jumped and averages across the jump. At the voltage zero there is nothing to jump, which is why the two routes agree exactly there.

Flux has to start from zero #

The reason the voltage zero is the bad instant is a single line of calculus. The flux linkage is the integral of the voltage applied, so a supply $V\sin(\omega t + \theta)$ switched on at angle θ produces

$$\lambda(t) = \frac{V}{\omega}\left[\cos\theta - \cos(\omega t + \theta)\right].$$

The second term is the settled flux, a sinusoid swinging by $\pm V/\omega$. The first is a constant offset fixed by the instant of switching — and it is there because the flux at the moment of switching is zero, while the settled flux at that instant is generally not.

Switched on at the voltage peak, θ is ninety degrees, the offset is zero, and the flux starts exactly on the settled waveform: in steady state the flux crosses zero at the voltage peak, because it lags the voltage by a quarter cycle. Nothing is out of place and the first cycle is the hundredth cycle.

Switched on at the voltage zero, the settled flux should be at its negative extreme, but the real flux is at zero. The offset is the whole of $V/\omega$, and the flux swings from zero to twice its working peak in the first half cycle. A core designed to work at 1.4 tesla is asked for 2.8 — and it cannot provide it. Past 1.8 tesla the iron is saturated, its permeability has collapsed to that of air, and the winding behaves like an air-cored coil whose inductance is a few millihenries rather than eleven henries. The current needed to push the flux further rises by orders of magnitude.

The figure shows the flux reaching 2.21 tesla rather than 2.8, and the difference is the winding resistance doing the only thing it can. While the saturated core draws tens of amperes, the two-ohm winding drops a substantial fraction of the supply, and that drop is volt-seconds the core does not receive. The resistance is the reason the inrush ends at all, and why it ends at a finite current.

This is the volt-second boundary a boundary in volt-seconds drew for a single pulse, met by the mains on the first half-cycle. The core can absorb a fixed number of volt-seconds in one direction before it saturates, and a supply switched on at the voltage zero spends twice its settled allowance in one go.

The opposite of the capacitor #

The two inrushes, the capacitor’s and the core’s, now look like mirror images, and they are.

A capacitor’s current is $C\,dv/dt$, so its inrush comes from a voltage that jumps: switched on at the peak, the full 17 volts of the rectifier’s secondary appears across an empty capacitor in an instant. An inductor’s voltage is $d\lambda/dt$, and its trouble comes from a flux that cannot jump: switched on at the voltage zero, the flux begins where the settled flux would not be. One element objects to a step in voltage and the other to a step in its time integral, and a sine has its steps in the two quantities a quarter cycle apart.

The consequence is awkward for anything that has both. A transformer feeding a capacitor-input rectifier — the arrangement in nearly every linear power supply ever built — has a best instant for its reservoir near the voltage zero and a best instant for its core near the voltage peak, and no switch-on angle is kind to both. A power supply switched on at random takes the worst of whichever it lands on, which is why both inrushes appear in a fuse’s surge rating, and why the protection that is gone by the second time finds the thermistor limiters built for the one also absorbing the other.

The working flux decides how large #

The size of the inrush is governed by how far the doubled flux overshoots saturation, and that depends on the one design choice that sets everything else about a transformer’s core: the working flux density.

A core worked below half its saturation flux never inrushes badly, because twice its working flux still fits inside the material: at 0.8 tesla the doubled swing reaches 1.6, below the knee, and the worst first peak is 82 milliamperes, only 2.7 times the settled magnetising current. The knee in the core law is soft, so there is no exact threshold, but the curve turns sharply upward between 0.9 and 1.0 tesla, exactly where twice the working flux crosses the saturation flux. By 1.0 tesla the first peak is 22 amperes, and by 1.4 tesla it is 87.6.

Nobody designs a mains transformer at 0.8 tesla, and the reason is copper and iron. For a given voltage the turns are inversely proportional to the working flux, so halving the flux doubles the turns, which doubles the winding resistance in a given window or doubles the window for the same resistance; and the core loss and size scale with it too. The current that sizes the transformer measured the winding heating a rectifier imposes; the working flux is the other end of the same trade, and inrush is part of its price. Toroidal transformers, worked close to saturation because their cores have no gaps to waste, are notorious for exactly this, which is why they are the usual home of soft-start circuits.

The settled magnetising current itself rises sevenfold between 0.8 and 1.7 tesla, as the working point climbs toward the knee. That is the slow part of the cost and the inrush is the fast part. Below half of saturation the inrush is a small multiple of a small current; above it, it is a large multiple of a larger one.

How precisely the peak must be found #

If the voltage peak is the safe instant, a switch that closes there removes the inrush entirely. This is point-on-wave switching, and it is used on large power transformers, where the inrush can trip protection and sag the supply for a whole substation. The question it raises is how precise the timing has to be.

The tolerance is tighter than it looks. Within fifteen degrees of the peak — 0.83 milliseconds either side — the offset is $V\cos\theta/\omega$, at most 26 per cent of the working peak, and the flux reaches 1.76 tesla, just under the knee. Thirty degrees off, the offset is half the working peak, the flux reaches 2.1 tesla, and the current is 328 times the settled value. The curve is steep because the offset grows linearly away from the peak while the knee is a cliff.

A relay’s closing time varies by a few milliseconds, which is most of the window, so a mechanical contactor cannot do this reliably. Controlled switching uses a breaker with a measured, repeatable closing time and a controller that schedules the command against the voltage it observes. Even then, the calculation here assumes that the core starts with zero flux. A real core does not: when it was last switched off, its flux stopped wherever the supply happened to leave it, and the iron’s hysteresis keeps some of it. That residual flux moves the safe instant away from the peak by an amount the controller has to know.

What a fuse integrates #

A fuse does not respond to a peak. It melts when the heat delivered to its element, $\int i^2\,dt$, reaches the element’s own rating, so the quantity that matters to it is the energy of the pulse rather than its height. The same march gives that number directly.

Switched on at the voltage zero, the first cycle of the magnetising current delivers 22.0 ampere-squared seconds. Over the first ten cycles the total is 22.6, so 97 per cent of the heating happens in the first cycle alone: the pulse is tall and narrow — above half its peak for only 3.8 milliseconds — and the cycles after it, though still large by the standard of the settled current, contribute little. Expressed as a root-mean-square over that first cycle, the inrush is 33.2 amperes.

For a transformer of this size, whose rated primary current would be a couple of amperes, that is the whole problem of primary fusing in one number. A fuse fast enough to protect the winding against a sustained overload of a few times its rating has a melting integral of a few ampere-squared seconds, and a single unlucky switch-on delivers several times that. This is why transformer primaries are fused with time-lag fuses, whose elements carry a thermal mass that absorbs a short pulse of high current, and why the fuse’s catalogue lists an inrush rating separate from its continuous one. It is also why a fuse that has survived a hundred switch-ons can fail on the hundred-and-first: the angle is random, and only a fraction of closings land near the voltage zero.

That randomness is worth putting a number on, because it explains a familiar complaint. With the switch closing at a uniformly random angle, the flux offset is uniformly distributed in its phase, and the measured tolerance curve says the first peak exceeds ten times the settled current everywhere outside 75 to 105 degrees — five-sixths of all closing angles. Most switch-ons inrush, and a minority inrush badly, so the worst case is met often enough to be designed for and rarely enough to be missed in a short test.

What the model leaves out #

The core law is anhysteretic: it has saturation but no loop, so it has no remanence. That is the largest omission. A real core switched off at an arbitrary instant keeps a residual flux, typically a few tenths of a tesla in silicon steel, and the next switch-on starts from it rather than from zero. If the residual has the same sign as the offset the switching creates, the two add, and the worst case becomes twice the working flux plus the remanence. The core the solver has to remember carries a hysteretic model that could be marched here instead, and the number it would change is the worst case rather than the shape.

The winding is a single resistance in series with the core. A real primary’s resistance is distributed through the winding, and its leakage inductance — the part of its field that does not link the core — adds to the air-core inductance during saturation and lowers the peak further. A transformer’s saturated inductance is also larger than the model’s, because the flux that leaves the saturated iron still links the winding through the space around it, which a one-dimensional path model does not hold; the band a turns ratio holds over and the energy is in the gap are the two places this subject treats those parts of the field. Both effects lower the absolute peaks drawn here and neither moves the angle at which they occur.

The supply is a perfect source. A real mains has impedance, and a large transformer’s inrush is large enough to sag the voltage it is drawn from, which reduces the volt-seconds and the inrush together.

Finally, only the first few cycles are drawn. The inrush does not end after one cycle: the offset decays as the winding resistance, carrying a one-sided current, slowly walks the flux back to the centre, and the rate at which it does so — and what a protection relay sees while it does — is a separate measurement.

The number to keep #

A transformer’s magnetising inrush is decided by flux, not voltage: a supply switched on at angle θ leaves the flux offset by $V\cos\theta/\omega$, so the voltage zero is the worst instant and the voltage peak the best — the reverse of a capacitor-input rectifier. A 230-volt primary worked at 1.4 tesla draws 87.6 amperes at the voltage zero, 791 times its settled magnetising peak. Below half of saturation the doubled swing fits in the iron and the inrush is a few times the settled current; above it the inrush grows by orders of magnitude. Closing within fifteen degrees of the peak keeps it under ten times the settled current.

Still open: the decay, the harmonic it carries, and the remanence #

How long the inrush lasts. The offset is unwound by the winding resistance, which sees a one-sided current and so drops a voltage with a non-zero mean. How many cycles that takes, and how it depends on the resistance, decides how long the inrush is visible to anything that watches the current — a fuse, a relay, an upstream transformer.

What a protection relay sees. A transformer’s differential protection compares the current into it with the current out of it, and an inrush is current in with nothing out: it looks exactly like an internal fault. Relays tell the two apart by the second harmonic, which a one-sided pulse train is rich in and a fault current is not. Whether that holds over the whole decay is a measurement on this same marched current.

The remanence. A hysteretic core switched off and on again starts from its residual flux. The best closing angle then moves to where the prospective flux equals the residual, and the worst case grows by the remanence. Marching a core with a loop would put numbers on both.