Symmetrical components — Fortescue decomposition

Reference notes

Use Next → on the narrator above to step through Fortescue's decomposition: the unbalanced raw set, the three balanced sequence sets (positive, negative, zero), the decomposition formulae, and where the technique shows up in real power-system analysis.

Why we need this

Per-phase analysis assumes balanced three-phase: three equal magnitudes, exactly 120° apart. Most steady-state load-flow studies fit that assumption. But unbalanced conditions — single line-to-ground faults, open conductors, unbalanced loads, non-linear-load harmonics — destroy the symmetry, and a per-phase model can no longer give the right answer. Symmetrical components rescue us: any three phasors can be uniquely decomposed into three balanced sets that can be analyzed per-phase, then re-superposed.

The three sequence sets

Positive sequence (+)

Three equal-magnitude phasors, 120° apart, rotating in the same sense as the original system (A → B → C). With a = 1∠120° as the rotation operator:

This is the "normal" component — it carries real power, produces torque in motors, and is what you'd see in a perfectly balanced system.

Negative sequence (−)

Three equal-magnitude phasors, 120° apart, but rotating in the opposite sense (A → C → B). The a and a² roles swap relative to positive sequence:

Negative sequence shows up only under unbalance. In a motor it produces a braking torque, induces rotor losses, and is what relay 46 (negative-sequence overcurrent) trips on.

Zero sequence (0)

Three equal-magnitude phasors, all in phase — no rotation at all:

Their sum is 3 V₀, not zero. That means zero-sequence currents need a return path (neutral wire or earth) to flow. With no neutral, zero-sequence current is forced to zero by KCL. Triplen harmonics (3rd, 9th, 15th, …) are zero-sequence — they're in phase across all three phases, so a delta winding captures them as circulating currents and a tertiary delta protects the rest of the system from harmonic distortion.

The Fortescue decomposition formulae

Given any three phase phasors V_a, V_b, V_c:

Notice the elegant symmetry: in V₊ we multiply V_b by a and V_c by a²; in V₋ we swap them. And reconstruction is just addition:

Where this matters in practice

Single line-to-ground fault

SLG faults are the most common power-system fault (about 70–80% of all faults). For a fault on phase a with the others healthy, the boundary conditions are I_b = I_c = 0 and V_a = Z_f · I_a. Substituting into the Fortescue matrix yields:

So in the sequence-network model, the three networks (Z₊, Z₋, Z₀) are connected in series. The fault current is:

where Z_f is the fault impedance (often taken as zero for bolted faults). Other fault types reduce to other interconnections of the same three sequence networks.

Transformer zero-sequence behaviour

• Yy0 with both neutrals grounded: zero-sequence sees the same path as positive.
• Yy0 with isolated neutrals: zero-sequence is blocked on both sides — open circuit in the Z₀ network.
• Yd with grounded Y: Z₀ provides a path on the Y side that ends in the delta (which presents short to neutral from the rest of the system's view). On the delta side: open.
• Dd: zero-sequence cannot enter or leave on either side.

The structural rule: a delta winding always blocks zero-sequence from crossing it; a grounded wye provides a path.

Motor protection

Continuously running an induction motor on unbalanced supply produces both negative-sequence current (which heats the rotor disproportionately) and a torque component opposing rotation. The standard rule of thumb is that a 5% voltage unbalance causes ~30% increase in motor losses — which is why ANSI relay 46 (negative-sequence overcurrent) is a core induction-motor protection element.

Keyboard shortcuts

• → next step  ·  ← previous step
• R replay narration  ·  M mute / unmute  ·  F fullscreen