Symmetrical components
The key idea
Any unbalanced set of three phasors can be rewritten as the sum of three perfectly balanced sets: positive sequence, negative sequence and zero sequence. Balanced problems are easy — so solve three easy problems and add the answers.
The idea
Balanced three-phase circuits are easy because every phase is a copy of its neighbors. One phase tells you everything — that is why a single-line diagram works.
A fault on one phase destroys that symmetry, and suddenly all three phases must be tracked at once. In 1918, Charles Fortescue found the escape: any unbalanced set of three phasors is exactly equal to the sum of three balanced sets —
- Positive sequence. Three equal phasors rotating A→B→C — the normal, healthy pattern. In a balanced system, this is all there is.
- Negative sequence. Three equal phasors rotating the wrong way, A→C→B. It appears only when phases become unequal — and it makes healthy rotating machines heat up.
- Zero sequence. Three equal phasors all pointing the same way. They cannot cancel in the phase conductors, so this current must return through the earth or neutral — which is why it exists only for faults involving earth.
Each set, being balanced, solves like the easy case — one phase each. Three easy solves, one addition, and the unbalanced answer falls out.
Try it
Sag phase A and watch the negative and zero sequence appear from nothing.
positive |V₁|
0.800 pu
negative |V₂|
0.200 pu
zero |V₀|
0.200 pu
Unbalanced: negative and zero sequence have appeared. Each set on its own is perfectly balanced — that is the trick. Solve three easy balanced problems, add the results, and you have solved the hard unbalanced one.
Why it matters
- Every unbalanced fault calculation runs on this. Fault types map to combinations of the three sequence networks: line-to-line uses positive and negative; faults to earth bring in zero sequence.
- Equipment data comes in sequence terms. The Z₁, Z₂, Z₀ columns in a cable datasheet and a transformer's zero-sequence behavior are exactly these components. A model missing Z₀ cannot compute any earth fault.
- Relays measure it directly. Modern relays compute negative and zero sequence current in real time — a clean "something is unbalanced" signal that ignores balanced load entirely.
The math, if you want it
With the rotation operator a = 1∠120°, the components of phasors VA, VB, VC are: V₁ = ⅓(VA + a·VB + a²·VC), V₂ = ⅓(VA + a²·VB + a·VC), V₀ = ⅓(VA + VB + VC). The transformation is exact and reversible: VA = V₁ + V₂ + V₀, with B and C recovered via a. The widget above computes exactly these sums.
See it in Phasor
Every unbalanced fault Phasor calculates is a sequence-network solution: you enter Z₁, Z₂ and Z₀ for each element, and the results report the sequence currents alongside the per-phase values.