The harmonic that names an inrush

The instant a transformer is worst at switched a 230-volt primary onto the mains with its secondary open and measured the first peak of its magnetising current: 87.6 amperes at the voltage zero, 791 times the settled magnetising peak, because the core’s flux has to start from zero and the voltage-zero instant leaves it offset by a whole working peak. It ended with two questions it had not asked. How long does the inrush last? And what does it look like to the equipment whose job is to notice a transformer in trouble?

The second question is the more pointed one. A large transformer is protected by a differential relay, which measures the current flowing into the primary and the current flowing out of the secondary, scaled by the turns ratio, and trips if they disagree. A fault inside the transformer — a winding shorted to the core, two turns shorted together — makes them disagree. So does an inrush. The magnetising current flows into the primary and nowhere else, and to a relay that compares the two sides it is indistinguishable in kind from a fault, and often much larger in size than the threshold.

The relay cannot simply wait it out, because a real internal fault must be cleared in a few cycles. It has to recognise the inrush by its shape, and the feature it uses is the second harmonic. This essay measures that feature on the marched current, cycle by cycle, against fault currents with the largest offset they can have, and finds both why it works and where its margin lies.

The same core, marched for longer #

The apparatus is the one the instant a transformer is worst at built: a 739-turn primary on ten square centimetres of silicon steel worked at 1.4 tesla, saturating at 1.8, with two ohms of winding resistance and the secondary open. The core’s law is a saturating B–H curve without hysteresis, and the circuit is the single equation $d\lambda/dt = v(t) - R\,i(\lambda)$, marched by the trapezoidal rule with Newton’s method on the flux at each step and checked by repeating every run at half the step count.

What is new is the length of the march and what is read from it. Forty cycles are marched, and each cycle of the current is resolved into its harmonics separately — a Fourier series over exactly one mains period, the same calculation a numerical relay performs on a sliding one-cycle window. The second harmonic is then quoted as a share of the fundamental in that cycle, which is the ratio relays act on.

How the inrush decays #

The flux offset that causes the inrush is not permanent, and the reason is the winding resistance. A current made of one-sided pulses has a non-zero mean, so the voltage it drops in the winding has a non-zero mean too, and that average voltage is subtracted from what the core receives. Cycle by cycle the net volt-seconds walk the flux back toward the centre, the excursions into saturation shrink, and the pulses shrink with them.

The resistance does two things, both visible in the figure. It lowers the first peak, as the earlier measurement found, by stealing volt-seconds during the first pulse. And it speeds the decay, because a larger resistance drops a larger mean voltage for the same one-sided current. A winding of one ohm takes fourteen cycles to bring the peak down a hundredfold; one of four ohms takes four. This is the well-known observation that large, efficient transformers have long inrushes: their winding resistance is small in proportion, and their inrush can persist for seconds.

The figure also shows something less obvious. After the first few cycles the decay slows sharply and the curves flatten well above the settled 111 milliamperes. At cycle forty the three cores still draw between 163 and 227 milliamperes. The explanation is that the decay has two regimes. While the core is being driven into saturation every cycle, the pulses are amperes, the winding drops volts of mean, and the offset unwinds fast. Once the excursions are back under the knee, the core’s inductance has returned to eleven henries, the current is a few hundred milliamperes, the mean drop is a fraction of a volt, and what remains of the offset unwinds with an L/R time constant of several seconds. The marched mean flux says so directly: with two ohms it is 65 per cent of the working peak in the first cycle, 30 per cent by the fifth, and still 24 per cent at the fortieth.

The flux that walks measured the same physics from the other side: a small asymmetry in a drive waveform accumulating flux in a core until saturation stops it. Here the walk runs the other way, from saturation back toward the centre, and it is fast where the resistance has something to work with and slow where it does not.

A one-sided pulse is rich in second harmonic #

The reason the second harmonic identifies an inrush is visible in the first cycle’s spectrum, set beside the spectrum of the current it has to be distinguished from.

The inrush in its first cycle is one narrow pulse per period, above half its peak for under four milliseconds of the twenty. A periodic train of narrow one-sided pulses has a spectrum that falls slowly, with every harmonic present: the narrower the pulse, the flatter the spectrum, and in the limit of a spike every harmonic is as large as the fundamental. The second harmonic is 79 per cent of the fundamental and even the third is half of it.

It is a different origin of harmonics from the one the distortion a linear model cannot have measures in a transistor, where a smooth curvature generates a second harmonic in proportion to the drive. Here the curvature is a cliff, and the harmonics come from the pulse shape it cuts out.

A settled magnetising current, by contrast, has no even harmonics at all. Its waveform is symmetric — each negative half-cycle is the positive one inverted — and a waveform with that half-wave symmetry contains only odd harmonics, the cancellation what a pair cancels, and what it only halves measures in a differential stage. The second harmonic in an inrush is therefore a direct measure of its asymmetry, which is to say of the flux offset. It is not an incidental feature of the waveform; it is the offset, read in the frequency domain.

The fault current carries a large mean as well — 0.75 of its fundamental in the first cycle, because a fault through an inductive line switched at the worst instant starts with a full decaying offset. But that offset is a smooth exponential added to a sine. Over one cycle, an exponential decaying with a time constant of ten radians of the mains has a little curvature, and the second harmonic it produces is 7.5 per cent of the fundamental. The two currents both have offsets; only one has its offset delivered in pulses.

Where the restraint’s margin lies #

A relay that restrains when the second harmonic exceeds a threshold — 15 per cent is a common setting — works if every inrush stays above it and every fault stays below. The question is which side is closer to the line.

For the faults, the second harmonic falls with each cycle as the offset decays, and it is largest when the offset decays fastest within a cycle — that is, at low X/R. A fault through a line with X/R of thirty has a slowly decaying offset that is nearly constant over a cycle, and so nearly no second harmonic: three per cent in the first cycle. At X/R of ten it is 7.5 per cent. At X/R of three the offset falls by most of its size within the first cycle, the exponential’s curvature is large, and the second harmonic reaches 14.6 per cent, a fraction of a per cent under the threshold.

So the margin on the fault side is set by the most resistive fault, not by the largest one. That is a useful thing to know about a protection setting, because it is not where intuition places the danger: a heavy, highly inductive fault is the safe case for restraint, and a fault through an arc or a long resistive cable is the one that approaches the line. A setting of 15 per cent is, on these numbers, chosen just above the worst plausible fault.

On the inrush side the margin is wide in the early cycles and narrows slowly. The inrush is 20 per cent or more for twelve cycles and does not cross 15 per cent until cycle thirty, by which time its current is 205 milliamperes. A differential relay typically ignores anything under a fifth or so of the transformer’s rated current, and a transformer of this core size carries a load current of amperes — the winding current the current that sizes the transformer measured for a rectifier load is several amperes root-mean-square — so by the time the restraint would fail, the current is too small to operate the relay anyway. The second harmonic outlasts the inrush it identifies, because the harmonic tracks the flux offset and the offset decays more slowly than the current.

Deeper saturation, less second harmonic #

That account has one more variable in it, and it is the one that has caused relays to misoperate in practice. The first cycle’s second harmonic is not a fixed property of inrush; it depends on how deeply the core is driven into saturation, and that depends on the working flux.

A core worked harder is pushed further past the knee by the same doubled swing, so it spends longer in saturation each cycle and its current pulse is wider. A wider one-sided pulse is closer to a half-wave rectified sine, which has a second harmonic of 42 per cent of its fundamental; a narrower one is closer to a spike, whose harmonics are all as large as the fundamental. So the deeper the saturation, the less second harmonic. Over the range drawn, the first cycle falls from 90 per cent at 1.0 tesla to 76 per cent at 1.7.

At these numbers the effect does not threaten the 15 per cent setting, and that is the honest reading of the figure. But it shows the mechanism by which modern transformers came to threaten it. Cores of high-permeability, low-loss steel are worked closer to saturation and have sharper knees, and when a core also starts with a residual flux in the unlucky direction, the pulse can be wide enough to bring the second harmonic of an inrush down toward the teens. Relay manufacturers responded with lower thresholds, cross-phase blocking — restraining all three phases if any one shows enough second harmonic — and waveform methods that look for the flat dead band between pulses rather than for a harmonic at all.

Why the window is one cycle exactly #

The ratio a relay acts on is only meaningful if the window it is computed over is right, and that is a measurement question before it is a protection one. A Fourier series over exactly one mains period resolves a periodic waveform into harmonics with no leakage between them: the second harmonic it reports is the second harmonic present. Over a window that is not a whole period, energy from the fundamental leaks into every other bin, and a waveform with no second harmonic at all reads as if it had some. The window that forgives the period measured exactly this leakage for a root-mean-square converter: a window 1.3 per cent too long reads a sine 0.6 per cent wrong.

For a relay the leakage is not symmetric in its consequences. The fundamental is the large component, so even a small fraction of it leaking into the second-harmonic bin adds a few per cent to the ratio, and a fault — whose true second harmonic sits a fraction of a per cent under the threshold at low X/R — can then read as an inrush and be restrained when it should trip. This is why numerical relays track the system frequency and resample to keep their window at one true period, rather than assuming fifty hertz.

There is a second subtlety, which the per-cycle reading in the figures hides. An inrush is not periodic: each cycle’s pulse is smaller than the last. A one-cycle window over a decaying pulse train sees a slight trend inside the window, and that trend has a spectrum of its own — exactly the mechanism that gives the fault current its second harmonic. In the inrush the trend is a small correction on a large ratio. In the fault it is the whole of the ratio. The same windowing arithmetic produces the signal in one case and the confusion in the other.

The ratio is also blind to everything outside its window. It cannot know that the current it is looking at began with an 88-ampere pulse thirty cycles earlier; it sees only the shape of the present cycle. That is a deliberate design choice, since a relay must decide from what it can measure now, and it is the reason the second harmonic’s persistence matters: a feature that disappeared with the large current would leave the relay to judge the tail of an inrush on its size alone.

What the model leaves out #

The core has no hysteresis, so it has no remanence, and remanence is the single largest influence on how low an inrush’s second harmonic can go. A core that starts with residual flux in the direction the switching offset adds to saturates more deeply on the first cycle and produces a wider pulse; one that starts in the opposite direction may not inrush at all. The core the solver has to remember carries a hysteretic core law that would put a number on this, and the shape of the conclusion — deeper saturation, less second harmonic — would carry over.

It is a single phase. A three-phase transformer’s three limbs share a magnetic circuit, and the inrush in each phase depends on the switching instants of all three; a delta winding also lets a circulating current carry some of the even harmonics, so the line currents a relay sees have less of them than the winding currents do. Both effects move the numbers and are why relays restrain on all three phases at once.

The current transformers that feed a real relay are omitted. They are themselves iron-cored, and a long unidirectional inrush can push a protection current transformer toward saturation, which distorts the measured current and adds harmonics of its own — the same volt-second limit a boundary in volt-seconds draws, met by the instrument instead of the plant.

Finally, the faults are modelled as linear R–L circuits switched at their worst instant, which gives their largest offset and so their largest second harmonic. A fault involving an arc is not linear, and its own distortion adds harmonics that this comparison does not include.

The number to keep #

An inrush’s second harmonic is its flux offset read in the frequency domain, because a settled magnetising current is half-wave symmetric and has none. Marched cycle by cycle, an inrush begins at 79 per cent second harmonic and stays above the usual 15 per cent restraint for thirty cycles, by which time its current has fallen from 88 amperes to 205 milliamperes. Fault currents with the largest offset they can have peak at 14.6 per cent — at X/R of three, where the offset decays fastest — so the restraint’s margin on the fault side is set by the most resistive fault. Deeper saturation widens the pulse and lowers the inrush’s second harmonic, from 90 per cent at 1.0 tesla to 76 at 1.7.

Still open: the remanent core, the three-phase bank, and the sympathetic inrush #

The remanent core. Marching a hysteretic core law through repeated switch-off and switch-on would give the distribution of first-cycle second harmonic over random closing instants and random residual flux — the number that decides how often a given restraint setting fails.

Three limbs. A three-phase core shares its flux between limbs, so the offset in one phase is not independent of the others, and a delta winding circulates part of the even harmonics. Marching the coupled magnetic circuit would say how much second harmonic survives into the line currents.

The sympathetic inrush. Energising a transformer beside one already running can drive the running one into a long, low inrush of its own, through the voltage drop the first one’s inrush causes in the shared supply. The two currents then decay together for seconds and their second harmonics interact, a case relays have been known to misjudge.