Circuit Response
every model has an edge, and the figure carries it
Concept

Symmetrical components — where it appears

Three unbalanced phasors rewritten as three balanced sets: one turning in the supply's own order, one turning against it, and one with all three in phase. The decomposition is a change of basis and exact for any three phasors, and a neutral conductor carries three times the third set and is blind to the second.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The neutral of a three-phase supply with one phase 30% off. computed by solving, not by drawing. Balanced, the three line currents sum to 4.6e-16 of one of them and the neutral carries nothing. With one phase 30% heavier the neutral carries 2.65 A against a line current of 11.50 A. The neutral reaches a tenth of a line current at 11.1% imbalance.

Three phases, and the wire that carries nothing

Three sources a third of a cycle apart, feeding three equal loads, return a current of 5×10⁻¹⁵ amperes down the wire between the star points. That is zero, and the whole of three-phase distribution rests on it. What is worth measuring is how fast it stops being zero, and the answer is that an eleven per cent imbalance in one phase puts a tenth of a line current down a conductor often sized on the assumption that it carries none.

power · Three-phase The two sequences a neutral current says nothing about. computed by solving, not by drawing at 61 imbalances. Three 20 Ω loads on a 230 V, 50 Hz star supply, one of them raised by a fraction of itself, with the neutral in place. The zero-sequence current is the one the neutral carries three times and is the only one a neutral ammeter reads; the negative sequence is a balanced set of three phasors rotating the other way. At 30.0 per cent imbalance it is 0.8846 A against 10.6154 A of positive sequence, 8.333 per cent, against 8.333 per cent from x/(3 + 2x). Two per cent arrives at 6.250 per cent imbalance, bisected on the network.

The half the neutral does not carry

A star load unbalanced in one phase produces two things, not one. The neutral carries three times the zero-sequence current, which is the half a neutral ammeter reads; the other half is a negative-sequence set of exactly the same size, rotating backwards, that the neutral never sees. With 0.5 Ω of line in front of 20 Ω loads, losing a phase entirely puts 50.00 per cent negative sequence in the current and 0.8265 per cent in the voltage a switchboard meter reads.

power · Three-phase The harmonics of one phase of three balanced rectifier loads conducting 60°, each sorted into the sequence its three phasors form. computed by solving, not by drawing: each harmonic of the three phases decomposed into zero, positive and negative sequence. Every one is a single sequence to the arithmetic's floor: h1 positive, h3 zero, h5 negative, h7 positive, h9 zero, h11 negative, h13 positive, h15 zero. A balanced set shifts harmonic h by h × 120° between phases, so the third and its multiples are not shifted at all — zero sequence — the seventh is shifted as the fundamental is, and the fifth the other way round. Relative to the fundamental, the zero-sequence harmonics together are 72.5%, the negative 24.1% and the positive 16.7%.

The sequence a harmonic belongs to

Symmetrical components are defined on phasors, so they describe one frequency, and a rectifier load's current has many. Taken harmonic by harmonic, three balanced loads give a single sequence at every harmonic, to the arithmetic's floor: the third and its multiples zero sequence, the seventh, thirteenth and nineteenth positive, the fifth, eleventh and seventeenth negative — because harmonic h is shifted h × 120° between phases. The sequence decides where each harmonic can go. Connected line to line, the loads put no triplen in a line at all: 3.32 A of them circulates round the delta and never leaves it. Fed through a delta–wye transformer beside a second group, the fifth and seventh cancel in the supply and the eleventh and thirteenth remain, because the transformer turns the two sequences opposite ways.

power · Three-phase

Named alongside it

The objects these essays reach for when they reach for this one.

Neutral currentThree-phaseImbalanceTriplen harmonicsDesign tradeoffLine impedanceLine lossMeasurement conditionPower factorSuperpositionTotal harmonic distortionVerification

All concepts