Symmetrical Components

Any unbalanced set of three-phase phasors decomposes into three balanced sets: positive-, negative-, and zero-sequence (Fortescue, 1918). Enter a phase set to extract its sequence components, or enter sequence components to reconstruct the phases — voltages or currents, either direction. The transformation is exact, and every step is shown with your numbers substituted.

Conversion

Phasors are RMS; angles in degrees; ABC rotation assumed. Switching V↔kV (A↔kA) converts the entered magnitudes. pu has no base defined on this page — values are used exactly as entered, and switching to or from pu does not rescale them.

Input Phasors
Phase A
V
°
Phase B
V
°
Phase C
V
°
Phasor Diagrams
Phase quantities
Sequence components (phase A reference)

Both diagrams share one magnitude scale (outer circle = larger of the two sets), so the sets are directly comparable.

Results
Residual 3·V₀
—
Unbalance |X₂| / |X₁|
—
Negative-sequence unbalance factor
Method · your values substituted

With the operator , so that and :

Fortescue Transformation

Analysis (phase → sequence)

Synthesis (sequence → phase)

Neutral / residual relations

Ground-fault (51N/51G) elements measure 3I₀. A broken-delta VT measures 3V₀. Delta windings and ungrounded wyes provide no path for zero-sequence line current. Line-to-line voltages contain no zero-sequence component.

Sequence Network Connections by Fault Type
Three-phase N₁ F N ZF Positive-sequence network only I₂ = I₀ = 0
Single line-to-ground N₁ N₂ N₀ I₁ 3ZF All three networks in series: I₁ = I₂ = I₀
Line-to-line N₁ N₂ ZF I₂ = −I₁, I₀ = 0
Double line-to-ground N₁ N₂ N₀ Bolted fault shown: N₁ in series with N₂ ∥ N₀

Each box is the complete sequence network (sources and impedances) viewed from the fault point F to the zero-potential bus N. With fault impedance, double line-to-ground places 3ZF in the N₀ branch. Fault-current magnitudes require the sequence impedances — see the scope note below.

Validation · re-computed live at load

Hand-verifiable cases. "Computed" cells are produced by the same routine that powers this page; PASS requires agreement with the expected value to 4 significant figures.

CaseInputs (phase set)Expected (analytic)ComputedResult
Scope & References
Model scope. This page transforms phasors between the phase and sequence domains — the arithmetic is exact, with no estimated values. It does not compute fault currents: fault magnitudes require the positive-, negative-, and zero-sequence impedances of the network (connections shown above) — that is a short-circuit study (SKM / ETAP / EasyPower or equivalent). ABC (positive) rotation is assumed; for ACB systems the positive- and negative-sequence results swap roles. Phasors are RMS at a single frequency: voltages line-to-neutral, currents line currents. Harmonic-domain sequence behavior (e.g., triplen harmonics acting as zero-sequence) is outside this page's scope.
  1. C. L. Fortescue, "Method of Symmetrical Co-ordinates Applied to the Solution of Polyphase Networks," AIEE Transactions, vol. 37, 1918, pp. 1027–1140.
  2. J. L. Blackburn and T. J. Domin, Protective Relaying: Principles and Applications, 4th ed., CRC Press, 2014, ch. 4.
  3. J. D. Glover, T. J. Overbye, and M. S. Sarma, Power System Analysis & Design, 6th ed., Cengage, 2017, ch. 8.