Positive sequence

What it is (precise)

Take any three phasors — currents or voltages, however unbalanced — and they can be rewritten, exactly and uniquely, as the sum of three balanced sets. That is Fortescue's symmetrical-component decomposition, and its bookkeeping runs on one tool: the operator a, a unit phasor that rotates whatever it multiplies by +120°. Applying it twice rotates by 240°, applying it three times returns the start, and the three positions sum to zero — 1 + a + a² = 0.

The positive-sequence component is extracted as X₁ = (Xa + a·Xb + a²·Xc)/3, and for a perfectly balanced set in the normal a-b-c order, where phase b lags phase a by 120°, the transform hands back X₁ = Xa with the other two components exactly zero. Balanced systems contain positive sequence alone; the decomposition adds nothing until something pushes the phases apart.

What makes the positive sequence more than the first row of a matrix is what it does in iron and copper. Fed into a three-phase winding, a positive-sequence current set produces a magnetic field rotating at synchronous speed in the machine's normal, forward direction — the field every rotating machine on the plant is built around, and the component that carries the useful power. Because 1 + a + a² = 0, a positive-sequence set also sums to zero at every instant, which is why a balanced plant needs no neutral conductor and why balanced load current leaves no residual for a ground relay to read.

The other two components each have their own entry. The negative sequence is the mirror set, rotating in the reversed a-c-b order — the component unbalance is measured by and machines pay for, covered in the negative-sequence entry.

The zero sequence is three co-phasal phasors that need a neutral or the earth to flow at all — ground faults and triplen harmonics, covered in the zero-sequence entry. And the rotation order itself, the wiring-level question of which sequence a set of conductors delivers, belongs to phase sequence: swapping two conductors reverses it without changing a single magnitude.

Why every single-line study is a positive-sequence model

The one-line diagram draws one conductor for three because it assumes balance, and every study built on it inherits the assumption. A load-flow model, a balanced short-circuit calculation and an RMS stability model are all positive-sequence networks: one phase's worth of impedances, with the other two phases implied by symmetry.

The physics cooperates because sources live only in the positive-sequence network — a machine's EMFs are a balanced set, so the negative- and zero-sequence networks are passive, carrying current only when an asymmetry couples them in. A bolted three-phase fault, the classic balanced disturbance, draws positive-sequence current alone: I₁ = E/(Z₁ + Z_f), with nothing at all flowing in the other two networks.

Unbalanced disturbances break the symmetry and connect the passive networks at the point of asymmetry — in series for a single line-to-ground fault, in other arrangements for other fault types — which is where the decomposition stops being invisible and becomes the working method of the fault study. The ground-fault arithmetic and the paths zero-sequence current can take are the zero-sequence entry's subject; what belongs here is the label on the model itself.

When an interconnection study calls for a positive-sequence RMS model, it means exactly this balanced representation — and transmission providers in inverter-heavy regions increasingly demand EMT models alongside it, because ride-through and weak-grid behavior do not appear correctly in a positive-sequence RMS frame.

The impedance on the nameplate

When a transformer nameplate or a cable datasheet quotes a single impedance, that figure is the positive-sequence impedance — the impedance a balanced current set sees. For stationary, symmetric equipment the positive- and negative-sequence impedances are equal, because reversing the rotation of a balanced set changes nothing a static element can detect: Z₁ = Z₂ for transformers, cables and transposed lines. Rotating machines are the exception — their negative-sequence impedance is a separate, smaller number, taken up in that entry.

The zero-sequence impedance is the outlier in the other direction, because co-phasal currents share a common return and the phase-to-phase coupling adds instead of cancelling; for overhead lines Z₀ typically lands around 2 to 3.5 times Z₁. One nameplate number therefore covers two of the three sequence networks, and the third has to be assembled separately — from winding connections and neutral treatments rather than from ratings, which is the zero-sequence entry's territory.

The percent impedance in every short-circuit calculation is this same positive-sequence figure — each transformer's contribution to bus fault level scales with 1/z — and a transformer adds an angle as well as a magnitude: a vector group with clock number n shifts positive-sequence quantities by n × 30° and negative-sequence quantities by exactly −n × 30°, the arithmetic differential protection has to compensate for.

The nameplate figure also sets only part of the fault picture. Where the system's zero-sequence impedance comes out lower than its positive-sequence impedance, the single line-to-ground fault exceeds the three-phase level, so a study quoting only the balanced figure has not established the worst case.

What the PCS locks onto

A power conversion system meets the positive sequence as a control input. A grid-following PCS runs a phase-locked loop on its terminal voltage, and what the loop synchronizes to is the positive-sequence component of that voltage — the balanced rotation whose angle and magnitude the rest of the control chain consumes.

The current references the converter builds from that angle, the P-f and Q-V droop laws the plant controller runs, and the real and reactive power metered at the point of interconnection are positive-sequence quantities in the same sense: the plant's entire steady-state control hierarchy operates on the component this entry names.

Under fault, the same frame decides what the plant contributes. A synchronous machine feeds a bolted fault with a subtransient current of roughly 3 to 8 per unit, sustained by trapped flux — a physical response. An inverter is limited by its semiconductors to roughly 1.0 to 1.3 per unit, and the magnitude, phase and sequence content of that current are a control decision.

A legacy grid-following converter controlled in a balanced positive-sequence frame simply regulates its negative-sequence current toward zero, even during an unbalanced fault — what that does to protection, and what modern interconnection standards require instead, is the negative-sequence entry's subject.