Symmetrical Components Calculator with Phasor Diagrams

Symmetrical components are the standard way to analyze unbalanced conditions in three-phase power systems. Fortescue’s theorem states that any set of three phasors, however unbalanced, can be decomposed into three balanced sets: a positive-sequence set rotating in normal phase order, a negative-sequence set rotating in reverse order, and a zero-sequence set of three identical phasors. Fault analysis, protection settings, and power quality limits are all written in this language.

Enter the three phase voltages (or currents) below as magnitude and angle, or simply drag the phasor tips in the diagram. The calculator decomposes the set in real time, draws the sequence components, and evaluates the voltage unbalance factor against the IEC and NEMA limits.

Symmetrical Components Calculator

Decompose unbalanced three-phase phasors into positive, negative, and zero sequences

Phase A
|V|
°
Phase B
|V|
°
Phase C
|V|
°
Magnitude in any consistent unit (V, kV, pu, A). Angles in degrees, counter-clockwise positive.
Voltage unbalance factor |V₂|/|V₁|
— %
Awaiting inputs
Phase quantities · drag the dots
Sequence components
Positive V₁
—
Negative V₂
—
Zero V₀
—
|V₀|/|V₁|
—
[V0 V1 V2]^T = (1/3) [1 1 1; 1 a a^2; 1 a^2 a] [Va Vb Vc]^T a = 1∠120°
Fortescue decomposition of an unbalanced three-phase set. Unbalance limits: IEC 61000-2-2 and EN 50160 allow up to 2% voltage unbalance (NEMA MG-1 requires derating motors above 1%). Works for voltages or currents.
Unbalance factor
— · —
Negative- to positive-sequence ratio, the quantity power quality standards limit. Updates live as you type or drag.
Negative sequence V₂
—
What heats motor rotors and drives unbalance protection. A balanced set has exactly zero.
Zero sequence V₀
—
Run the calculator to see the zero-sequence component here.

How the decomposition works

With the operator \(a = 1\angle 120^\circ\), the sequence components of the set \((V_a, V_b, V_c)\) are

\[ \begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix} = \frac{1}{3} \begin{bmatrix} 1 & 1 & 1 \\ 1 & a & a^2 \\ 1 & a^2 & a \end{bmatrix} \begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} \]

The transformation is exact and reversible: each phase quantity is the sum of its three sequence contributions, for example \(V_a = V_0 + V_1 + V_2\). The same equations apply to currents. Because each balanced sequence set sees its own independent impedance network in a symmetrical system, the method turns one hard unbalanced problem into three easy balanced ones.

Worked example

Load the SLG fault preset: phase A collapses to 0.15 pu while phases B and C stay near 1 pu. The decomposition gives roughly V1 = 0.73, V2 = 0.25, and V0 = 0.32 pu. That signature, a large zero-sequence component together with a large negative-sequence component, is exactly how protection relays recognize a single-line-to-ground fault: ground (zero-sequence) elements pick up on V0 and I0, and the fault type is classified before the breaker ever opens.

Interpreting the unbalance factor

  • VUF < 1% (well balanced): normal healthy-system territory. NEMA MG-1 allows full motor loading up to 1% voltage unbalance.
  • VUF 1 to 2% (within IEC limit): permitted by IEC 61000-2-2 and EN 50160 for steady state, but NEMA already requires motor derating, and sustained values here are worth investigating (blown capacitor fuse, unbalanced single-phase loads, open delta).
  • VUF > 2% (excessive): outside the standard limits. Expect extra rotor heating in machines, possible negative-sequence protection operation, and current-limit issues in inverters. Large values together with V₀ usually mean a fault or an open conductor, not just load unbalance.

A note on zero sequence: it only flows where a return path exists. Delta windings trap it, which is why the winding connection of transformers decides how ground-fault current distributes through a network, and why V0 at a bus is a reliable indicator of ground involvement on that side of the transformer.

Working in per-unit?

Grab the free Per-Unit System Cheat Sheet: base conversions, impedance change of base, and sequence network basics on one printable page.

Get the cheat sheet →

Frequently asked questions

Does this work for currents as well as voltages?

Yes. The Fortescue transformation is the same for any set of three phasors. Enter current magnitudes and angles and the outputs are I0, I1, I2. For currents, 3I0 is the residual (neutral or ground) current that ground relays measure.

Which definition of unbalance does the calculator use?

The true (IEC) definition: VUF = |V₂|/|V₁| × 100%. The simpler NEMA definition based on line-voltage deviations from the average approximates this but can differ noticeably for the same set of voltages; the sequence-based definition is the one used in modern standards and studies.

Why is the zero-sequence set drawn as a single phasor?

The zero-sequence set consists of three identical phasors, equal in magnitude and angle, so on a phasor diagram they lie exactly on top of each other. The calculator draws one phasor V0 representing all three.

Can the angles be entered in any reference?

Yes. Sequence magnitudes are unaffected by a common rotation of all three phasors; only the sequence angles shift with the reference. Most people set phase A near 0° by convention, which the presets do.

References

  • C. L. Fortescue, “Method of Symmetrical Co-ordinates Applied to the Solution of Polyphase Networks,” Transactions of the AIEE, vol. 37, 1918.
  • IEC 61000-2-2, Electromagnetic compatibility (EMC): Compatibility levels for low-frequency conducted disturbances.
  • NEMA MG-1, Motors and Generators, section on operation under unbalanced voltage.
  • J. L. Blackburn, Symmetrical Components for Power Systems Engineering, Marcel Dekker, 1993.