Symmetrical components — asymmetric fault analysis per Fortescue

The guide on short-circuit current calculation per IEC 60909-0 covers the equivalent-voltage-source method for calculating Ik'' — but at its core, that method in its basic form applies directly to one specific case: the symmetrical three-phase fault (all three phases short-circuited simultaneously), where the network remains fully balanced after the fault occurs. The vast majority of real faults on a network are, however, asymmetric: a single line-to-ground fault, a fault between two phases, or a fault between two phases and earth. For those cases a simple single-phase equivalent circuit is not enough — this requires the method of symmetrical components, developed by Charles Fortescue (1918).

Fortescue's theorem

Fortescue showed that any arbitrary unbalanced (asymmetric) three-phase phasor system can be mathematically decomposed into the sum of three symmetrical sub-systems:

Positive-sequence component (1) — three phasors of equal magnitude, 120° apart, in the same phase sequence as the original, healthy network.
Negative-sequence component (2) — three phasors of equal magnitude, also 120° apart, but in the reverse phase sequence.
Zero-sequence component (0) — three phasors of equal magnitude, in phase with one another (no mutual shift).

Any real, potentially strongly asymmetric voltage or current condition during a fault can be written exactly as the sum of these three components. The network is then modelled as three separate networks, each symmetrical in itself — and therefore each solvable with a single-phase equivalent circuit — each with its own impedance Z1, Z2 and Z0.

Why three networks instead of one

Positive-sequence network (Z1): this is the same network that applies under normal, symmetrical operation — the same impedances as in the ordinary short-circuit current calculation of IEC 60909-0.
Negative-sequence network (Z2): for static equipment (cables, lines, transformers), Z2 is virtually equal to Z1. For rotating machines (generators, large motors), Z2 differs noticeably from Z1, because the rotating field of the negative-sequence current turns in the opposite direction relative to the rotor — see also the guide on negative-sequence protection (46), which is based on this same phenomenon.
Zero-sequence network (Z0): this network deviates the most, and only exists once an actual return path for the (in-phase) zero-sequence current is present. A delta-connected winding blocks the zero-sequence path to the outside world entirely (the zero-sequence current circulates internally within the delta); an ungrounded star point likewise offers no external zero-sequence path. Only with a grounded star point — directly or via an impedance, see the guide on NGR resistance earthing and the guide on the zigzag earthing transformer, which exists specifically to create a zero-sequence path where none naturally exists — does a path for Z0 exist.

The four fault cases

Each fault type corresponds to a specific way in which the three networks are interconnected:

Symmetrical three-phase fault (3φ): only the positive-sequence network is involved: Ia1 = Ea / (Z1 + Zf), Ia2 = Ia0 = 0. This is exactly the case that the equivalent-voltage-source method of IEC 60909-0 calculates in its basic form.
Single line-to-ground fault (1φ-earth, SLG): the three networks are connected in series: Ia1 = Ia2 = Ia0 = Ea / (Z1 + Z2 + Z0 + 3·Zf). The total fault current to earth is then 3×Ia0.
Phase-to-phase fault (2φ, LL): the positive- and negative-sequence networks are connected in parallel, in opposition, with the zero-sequence network not involved: Ia1 = −Ia2 = Ea / (Z1 + Z2 + Zf).
Phase-to-phase-to-ground fault (2φ-earth, LLG): all three networks are interconnected in parallel — the most complex combination, with an outcome that, depending on the situation, can come out higher or lower than for the other fault types.

A counter-intuitive consequence: the ground fault can give the highest current

Note: in a solidly or low-resistance grounded network, it is not a given that the symmetrical three-phase fault always produces the highest short-circuit current. When the zero-sequence impedance Z0 is smaller than the positive-sequence impedance Z1 — which can occur close to a solidly grounded star-delta transformer with a relatively low zero-sequence reactance — the calculated current of a single line-to-ground fault can actually exceed that of the symmetrical three-phase fault. Whether this applies to a specific installation depends entirely on the actual Z0/Z1 ratio of that network and that transformer, and must be calculated case by case — this is not a universal rule.

Practical relevance

Symmetrical components are not just a calculation method, but also the basis for several practical protection functions already covered elsewhere on this site: negative-sequence protection (46) detects I2, zero-sequence voltage protection (59N) detects V0, and earth-fault overcurrent protection (50N/51N) detects I0. Understanding where these quantities come from — namely as one of the three components of Fortescue's decomposition — also clarifies why each of these protections is specifically sensitive to the fault type it was designed for, and why, for instance, 50N/51N does not operate on a phase-to-phase fault without earth involvement (where I0 remains zero).

Common mistakes

Assuming the three-phase fault is always the governing (highest) fault when rating switchgear — at a low Z0/Z1 ratio, a single line-to-ground fault can give the highest current.
Setting Z2 equal to Z1 for rotating machines — for generators and large motors this is a noticeable simplification that can lead to an incorrect fault current calculation.
Assuming a zero-sequence path exists where it does not — for example with a delta-connected winding or an ungrounded star point, where Z0 is, in practice, infinite (no path).
Mixing up the three asymmetric fault types (SLG, LL, LLG) by applying the same network coupling — each fault type has its own specific combination of series/parallel connection of the three networks.