The neutral that carries more than a line

Three phases and the wire that carries nothing measured one of this site’s most satisfying results. Three balanced loads on a three-phase supply draw currents that sum, in the neutral, to 5.3 × 10⁻¹⁵ amperes — zero to the arithmetic, on a solve that never assumed it. The essay went on to measure what imbalance costs: eleven per cent of imbalance puts a tenth of a line current down the neutral.

That result is correct and its conditions are narrower than they look. It is a theorem about sinusoids. Three sinusoids 120° apart sum to zero because their three phasors close a triangle, and the argument uses nothing at all except that each current is a single frequency.

Almost nothing on a modern supply draws a sinusoid.

Why the third harmonic is different #

A harmonic of order h on a phase shifted by 120° is shifted by h × 120°.

For the fundamental that is 120°, and the three phasors close. For the second harmonic it is 240°, which is 120° the other way, and they close again. For the third it is 360° — which is no shift at all. The third harmonics of all three phases are in phase with one another, and phasors that are in phase do not close a triangle; they add.

So do the ninth, the fifteenth and every other multiple of three, which is why they are collectively called triplen. And they do not merely fail to cancel: they arrive at the neutral three times over, because each of the three phases contributes its own.

That is the mechanism in three sentences, and everything else in this essay is a measurement of how much it amounts to.

The exact answer, and why it is exact #

A rectifier feeding a capacitor draws current only near the peak of the supply, in a pulse. Each phase’s load conducts twice per cycle, once on each half, so there are six pulses per cycle and they sit sixty degrees apart.

While the conduction angle is narrower than sixty degrees, no two pulses overlap. The neutral current is then the union of three disjoint pulse trains, and the mean square of a union of disjoint things is the sum of their mean squares — so the neutral’s mean square is three times one phase’s, and its root mean square is √3 times a line current.

Not “more than expected”. Not “up to 1.7 times”. Exactly √3, for any pulse shape and any load, as long as the pulses do not touch.

The figure measures 1.7321 at 15°, 20°, 30°, 45° and 60° of conduction — five different waveforms, five different line currents from 0.928 A down to 0.559 A, and the same ratio to four decimals at every one. That invariance is the assertion the generator makes, and it is a far stronger claim than any single number would be.

The two routes #

The neutral is computed twice, and the two share the waveform and nothing else.

Summed in time. Three shifted copies of one phase’s current waveform, added sample by sample, and the root mean square taken over 1 080 samples of a cycle. No harmonics anywhere, no assumption about which orders survive.

Rebuilt from the spectrum. One phase’s current is decomposed into harmonics, the multiples of three are kept, each is tripled, and the root mean square is taken from the sum of their squares. Never touches the other two phases.

They agree to between 0.13% and 0.32% across the slider, and the residual is the truncation of the harmonic series rather than an error in either. That is the check that the mechanism in the section above is really the mechanism: if some non-triplen order were contributing, the second route would miss it and the two would part company.

One conduction angle is off the slider because of this comparison rather than in spite of it. At 120° the waveform is nearly sinusoidal, the triplen content has all but gone, and the rebuild is 33% away from a quantity that is only 9% of a line current — a large relative error on very nearly nothing. Widening the tolerance to admit it would have made the check useless everywhere else, so the angle was dropped and the reason recorded.

How much of a phase current is triplen #

The ratio in the neutral is the headline, and there is a second number in the figure’s panel that is worth reading beside it: the fraction of each phase current that is triplen.

At sixty degrees of conduction it is 57.7%. At forty-five degrees 57.7%. At thirty, twenty and fifteen degrees, 57.6%, 57.5% and 57.5%. It barely moves — and it barely moves for the same reason the neutral ratio does not: while the pulses are disjoint, narrowing them changes the amplitude and the harmonic amplitudes together, leaving the proportions alone.

Those two numbers are the same statement seen twice, and the relation between them is the √3. If a fraction f of each phase current is triplen and those components add coherently across three phases, the neutral carries 3f of a phase current in the triplen orders and nothing else — and 3 × 0.577 = 1.730 against a measured ratio of 1.7321. The neutral ratio is three times the triplen share to three figures, and the residue is the same truncation that separates the two routes — the harmonic list stops somewhere and the time-domain sum does not. It is a third check on the arithmetic and one the figure reports without being asked.

It also gives the number a designer needs for a partial case. A load that is 20% triplen — a power-factor-corrected supply rather than a bare rectifier — puts 0.6 of a phase current down the neutral rather than 1.73. That is still more than an imbalance allowance, and it is the reason corrected supplies are specified by their harmonic content rather than by their power factor.

What it costs, in a building #

The consequence is not academic and it is the reason this essay exists as its own rung.

A three-phase distribution is conventionally wired with a neutral smaller than the phase conductors, and the justification is the theorem in the essay below this one: balanced loads put nothing in the neutral, so the neutral only ever carries the imbalance. Half the cross-section is the usual allowance and it has been standard practice for a century.

A floor of switched-mode power supplies — computers, lighting ballasts, chargers, anything with a rectifier and a reservoir capacitor — is a balanced load in every conventional sense. Each phase draws the same current. The imbalance is negligible. And the neutral carries 1.73 times a phase current, into a conductor sized for half of one.

That is a factor of three and a half over its rating, and the failure mode is thermal: the neutral is not fused, because a fuse in a neutral is a hazard, so nothing interrupts the current. It heats until the insulation fails.

The mechanism was not widely understood until the load population changed, which is exactly the shape of failure this site is built around — a model whose conditions were true for long enough that they stopped being stated.

What “balanced” turns out to mean #

The word is doing more work in the older essay than it appears to, and this is the place to be precise about it.

Balanced in magnitude and phase at the fundamental is what the theorem needs and what the three-phase essay measured. Three equal currents 120° apart.

Balanced as loads is what an installation designer means: the same equipment, in the same quantity, on each phase.

The two are the same thing only when the loads are linear. A rectifier is balanced in the second sense and grossly unbalanced in the first — its third harmonic is balanced in magnitude across the three phases and has zero phase difference between them, which is the worst possible arrangement rather than a benign one.

This is a place where the vocabulary actively misleads, and it is worth naming as such. In the language of symmetrical components, which the earlier essay already uses, the triplen harmonics are zero-sequence: identical in all three phases. Zero-sequence current has nowhere to go except the neutral, and a balanced three-wire system has no path for it at all — which is why a delta connection traps triplen harmonics circulating inside the winding rather than exporting them, and why a delta-wye transformer is the standard place to stop them.

The measurement is on a waveform, not on a phasor #

The power field has worked in phasors since its first essay, and this one does not, which is a change worth being explicit about.

A phasor is a complex number standing for a sinusoid of known frequency. It is exactly the right object for everything the field has done so far — complex power, the correction capacitor, the three balanced phases — and it is the wrong object here for a reason that is not a matter of convenience: a rectifier’s current is not a sinusoid, so there is no single complex number that stands for it.

What replaces it is the waveform itself, sampled 1 080 times a cycle, and a harmonic decomposition of that waveform when a decomposition is wanted. The field already built both — lib/spectrum.js exists because a power factor is only cos φ while the current is a sinusoid, and the distortion a linear model cannot have needed harmonic content in the semiconductor field for the same reason.

So the machinery here is shared with two other essays and the framing is not. That is the useful observation: the phasor is a compression of the waveform that is exact for one frequency and silent about the rest, and every result in this collection that was computed with phasors carries the same unstated condition the older three-phase essay did. Most of them are fine, because most of the circuits in question are linear and driven by a sinusoid. The ones that are not fine are the ones with a rectifier in them, and the power field has two.

What is not in this model #

Three things bound what has been measured, and all three make the situation worse rather than better.

The supply is stiff. The source has no impedance here, so the voltage stays sinusoidal however the current is shaped. A real supply has impedance, the pulsed current flattens the voltage peak, and a flattened peak changes the conduction angle — which moves the answer along the slider rather than off it.

The three loads are identical. Real ones are not, and the ordinary imbalance the earlier essay measures adds to the triplen current rather than replacing it. The two contributions are at different frequencies, so they add in power and the total is the quadrature sum.

Only the neutral conductor is considered. The same zero-sequence current flows in transformer windings and in any earth path that exists, and a neutral-to-earth bond gives it a second route whose share is decided by impedances nobody designed.

None of these changes the mechanism, and all of them add current to the same wire.

What is done about it #

Three responses exist and they attack different points in the chain, which makes them worth distinguishing.

Oversize the neutral. The direct answer, and the one modern wiring practice adopted: the neutral is specified at the full phase cross-section or larger where harmonic loads are expected, rather than at half. It fixes the symptom completely and costs copper.

Trap the harmonics. A delta winding presents a circulating path for zero-sequence current inside itself and does not pass it through. A delta-wye transformer between the supply and a harmonic load therefore keeps the triplen currents on the load side, where the wiring can be sized for them, and keeps them out of the distribution behind it. This is why the transformer configuration in a building is a harmonics decision as much as a voltage one.

Stop making them. A power-factor-corrected supply draws a current shaped to follow the voltage rather than a pulse at the peak, and a current that is nearly a sinusoid has nearly no triplen content at all. This is the response that fixes the cause, it is now mandated for equipment above a modest power in most jurisdictions, and it is the reason the problem is less acute than it was — the regulation exists because of exactly the failure this essay measures.

The three compose in the obvious way, and the ordering of preference is the reverse of the ordering of cost at the point of installation, which is the usual shape of this kind of engineering decision.

Where the older essay’s imbalance number now sits #

The rung below measures what imbalance does: eleven per cent of imbalance puts a tenth of a line current down the neutral. That number is still correct and it is now the smaller of two contributions, which changes how it should be read.

The two are at different frequencies — imbalance appears at the fundamental, triplen content at the third harmonic and above — so they do not interfere and they add in power. A floor of rectifier loads with eleven per cent of imbalance on top puts

√(1.732² + 0.1²) = 1.735

line currents down the neutral, and the imbalance has contributed 0.17% of it. An installation designed with care about imbalance and no thought about harmonics has optimised the term that contributes a sixth of one per cent.

That is the sort of comparison this collection exists to make, and it is only available because both numbers were measured on solved networks in the same units. Neither essay could have made it alone.

Why the site’s usual mark is a conduction angle #

Every figure on this site carries a rule at the point its model stops being true, and almost all of them are frequencies. A handful are amplitudes and one is a size. This one is an angle, which is worth a sentence because the site’s premise says “frequency, amplitude or size” and this is a fourth kind of thing.

It is a frequency in disguise, in the sense that a conduction angle is set by how long the rectifier’s diode is forward-biased, which is set by the reservoir capacitor and the load — so it is a statement about a time compared with the supply period. But the quantity a reader can act on is the angle, and the boundary is sharp in the angle rather than in anything else: at sixty degrees the pulses meet, and one degree either side is the difference between an exact identity and a mess.

The premise’s list is a description of what has been measured rather than a constraint on what may be, and this essay adds a fourth item to it. That is worth recording rather than quietly fitting into one of the three.

The rule this model stops being true under #

Sixty degrees.

Below it the pulse trains are disjoint and the neutral carries exactly √3 line currents. Above it they overlap, the cancellation partially returns, and by ninety degrees the ratio is 1.0074.

Three phases, and the wire that carries nothing’s $5.3\times10^{-15}$ amperes is still true, and its condition — which was never stated, because for a century it did not need to be — is that every current in the system is a sinusoid. This essay is that condition, drawn.

The load that violates it is not exotic and is measured two fields away. The direct voltage that is a sawtooth marches the rectifier and reservoir capacitor behind almost every mains-powered appliance and finds it drawing its average current as a pulse over 28.8° of each half cycle, with a crest factor of 13.4 — and finds that crest factor getting worse as the reservoir capacitor is made larger, which is the direction every other consideration pushes it. So the harmonic content this essay’s neutral is carrying grows as the equipment on the phases is improved.

And the current that does no work states why the usual instrument does not report it: a phasor calculation on that waveform’s fundamental returns a displacement factor of exactly 1.000000 while the true power factor is 0.780. A three-phase installation of such loads is balanced, correctly wired, reading a perfect power factor on a displacement meter, and putting √3 times a line current down a conductor sized for none.

What else the third harmonic reaches #

A neutral current that exceeds a line current is one consequence of a load drawing current in pulses, and this field measures four others. The distortion a linear model cannot have is where the harmonics come from. The first cycle, which no steady state contains and The direct voltage that is a sawtooth are the rectifier that draws them, measured as an inrush and as a conduction angle. The capacitor that was right once is the correction that assumes a sinusoid and is defeated by exactly this current, and The far end that rises is the voltage a distorted current leaves at the end of a line. None of the five is visible in a power factor quoted as a displacement angle.