Symmetrical Components & Fault Phasors

A balanced three-phase system is easy to analyse. Solve one phase, and the other two are the same thing shifted by 120°. Faults break that symmetry. A ground fault on phase A pulls A's voltage toward zero, leaves B and C roughly where they were, and sends current back through the earth. That is three different phasors with no shortcut.

In 1918, Charles Fortescue showed that any set of three unbalanced phasors can be written as the sum of three balanced sets:

• Positive sequence (1): equal magnitudes, 120° apart, in normal ABC rotation. This is the "healthy" part.
• Negative sequence (2): equal magnitudes, 120° apart, in reversed ACB rotation. It measures unbalance.
• Zero sequence (0): three identical phasors, all in phase. It only exists when current can return through ground or a neutral.

The payoff: in static equipment (lines, cables, transformers) the three sequences don't interact with each other. Each one has its own simple single-phase circuit, its sequence network. The fault is just a rule for wiring those three circuits together.

The operator $a = 1\angle 120^\circ$ rotates a phasor by 120°. Since $1 + a + a^2 = 0$, it is exactly what you need to pick apart the three rotations:

Read the second row: $V_1=\tfrac13(V_a+aV_b+a^2V_c)$. Multiplying Vb by a and Vc by a² rotates a positive-sequence set so that all three phasors line up with Va. They add, and anything else tends to cancel. That is the graphical construction drawn in the right-hand plot below. Numerical relays run this same calculation on filtered phasors once per processing interval (typically every ¼ to ⅛ cycle) to produce their 46, 32Q, 67Q, 59N and 51N quantities.

Each sequence network is a Thevenin equivalent seen from the fault point. Only the positive network contains a source, the pre-fault voltage E, because generators are designed to produce balanced positive-sequence voltage. The fault type sets the boundary conditions, and those decide the wiring:

For the single-line-to-ground fault, the ground (residual) current is three times the zero-sequence current:

A 74.9 MW solar plus 20 MW BESS project connects at 34.5 kV. This is a distribution-voltage POI: the plant ties directly into the utility's 34.5 kV system, so the utility supplies part of the plant's ground reference. That is not the arrangement in the larger plants of Module 0.1, where the collector sits behind the delta LV winding of the main power transformer and never sees a utility ground source. There, the plant's own grounding sources are the only zero-sequence path (see the configurations table in Module 1.2). The driving-point impedances at this POI under minimum-source conditions, before any plant grounding bank, are:

Find the bolted SLG fault current 3I₀ and compare it with the three-phase fault current. Then load the "POI as-found" preset in the calculator to check your answer.

Effective grounding: the definition and the impedance test

A system is effectively grounded when the coefficient of grounding (COG) is no more than 80 %. COG is the highest healthy-phase voltage during a ground fault, as a fraction of the normal line-to-line voltage; 80 % corresponds to an earth-fault factor of about 1.4. On a line-to-neutral base the limit is 0.8 × √3 ≈ 1.39 pu, which is where the calculator's healthy-phase readout turns red.

The impedance ratios are the conventional test that keeps COG inside 80 % for practical fault resistances (the guides' curves assume R₁ is much smaller than X₁): X₀/X₁ positive and no more than 3, and R₀/X₁ no more than 1 (the IEEE C62.92 guides). The as-found POI shows why both matter. For a bolted fault its COG is 78.8 %, just inside the limit, yet X₀/X₁ = 4.70 fails the ratio test. Add fault resistance in the calculator: between roughly 2 and 12 Ω the healthy phase rises to about 1.416 pu (COG ≈ 82 %). The ratio failure is the warning, and it is the starting point of the grounding-bank study in Module 2.2.

Sequence components aren't only for faults. Steady-state voltage unbalance is usually quoted as $|V_2|/|V_1|$. An open conductor into an ungrounded load gives equal and opposite I₁ and I₂ with no I₀. A reversed rotation shows up as pure negative sequence.

IEEE 2800-2022 uses the same language for inverter-based plants. It deals with unbalance in its continuous-operation requirements, and with negative-sequence current injection during faults in Clause 7.2.2.3.4. This course's source research also cites a negative-sequence voltage figure of about 6.7 % with a 2 s window as a ride-through condition. That figure could not be verified against the standard text, so treat it as unverified and take the actual limits from the edition and clause you are applying. The decomposer only flags unbalance above 2 % as worth a look; it deliberately doesn't draw a pass/fail line.