Sequence networks
The key idea
Symmetrical components give you three networks instead of one. A fault type is simply an instruction for wiring them together: one network alone, two in parallel, or all three in series. Change the connection and you change which reactances the fault current must pass through.
The idea
Symmetrical components turn one unbalanced problem into three balanced ones, so you work with three separate networks rather than one. Every generator, transformer, line and cable carries a positive-sequence, a negative-sequence and a zero-sequence reactance, and each set forms its own complete copy of the system.
Reduce each network to the point where the fault occurs and it collapses to a simple box with two terminals: the fault point F, where the short circuit is, and the reference N, the neutral or earth reference for that network. Only the positive-sequence network contains a source, because a healthy system is pure positive sequence. The negative and zero networks carry no current until a fault drives them.
From there, the central rule: a fault type is a connection. It does not change what is inside the networks — it only changes the wires between them.
- Three-phase. The fault is balanced, so only positive sequence exists. The positive network stands alone, its fault point wired straight to its reference. One network, one division.
- Line-to-line. Two phases touch and earth plays no part. The positive and negative networks go in parallel: F to F, N to N. The zero network stays disconnected, so the earthing has no effect on the result.
- Line-to-earth. One phase to earth puts all three networks in series, in a single loop. The zero-sequence reactance now sits directly in the current's path, alongside X₁ and X₂.
- Line-to-line-to-earth. Two phases to earth: the positive network in series with the negative and zero networks in parallel.
Notice the pattern: the zero-sequence network appears only in the two connections that involve earth. That is why an earthing decision can change one fault current by a factor of ten while leaving another untouched, and why a model with a guessed X₀ gives an earth-fault current that looks reasonable and is far from correct.
Try it
Switch between the fault types and watch the wires change. Then move the zero-sequence slider. Some bars move, and some stay exactly where they are.
three-phase fault current: 10.00 pu · 100% of three-phase
- Three-phase · selected10.00 pu
- Line-to-line8.66 pu
- Line-to-earth11.54 pu
- Line-to-line-to-earth (earth return)13.64 pu
A three-phase fault stays balanced, so only positive sequence exists. One network is shorted at the fault point, and the current is E ÷ X₁. Move the zero-sequence slider and nothing changes, because earth is not part of this circuit.
The model behind the numbers uses a fixed E = 1 pu source behind the positive network. The reactances are pure, with no resistance. The fault is bolted, with no arc. The slider ties X₂ to X₁. A real study keeps the resistances, works in complex numbers and scales the source by the IEC 60909 voltage factor. The connections stay exactly as drawn, and only the arithmetic becomes more difficult.
Why it matters
- The earthing enters the arithmetic in exactly one place. Solid, resistance or impedance earthing: all of it reaches the fault calculation through the single number X₀, and X₀ appears only in the two connections that involve earth.
- A line-to-earth fault can exceed the three-phase fault. In the series connection the numerator is 3, not 1. And close to a solidly earthed transformer, X₀ is often smaller than X₁. The earth fault is then the larger of the two, and switchgear rated only against the three-phase value is under-rated.
- The connection tells you which data you need. A phase-fault study runs on X₁ and X₂. An earth-fault study needs X₀ for every element in the loop, plus the winding connection of each transformer and the earthing of each star point, because those two items decide whether a zero-sequence path exists at all.
- Protection uses these currents separately. Earth-fault elements respond to the earth return, 3·I₀, so you can set them far below load current; phase elements never see it. To grade the two against each other, you have to calculate both connections.
The math, if you want itOptional — the page reads completely without it
Take the source as E = 1 pu and treat the networks as pure reactances. Every connection is then a single-loop circuit, and one division gives each fault current.
The positive network is shorted on itself, so the loop holds X₁ and nothing else:
three-phase — positive network alone
I3ph = EX₁
Two phases without earth put the positive and negative networks in parallel across the fault, so the loop carries X₁ + X₂. The faulted phase current is √3 times the loop current:
line-to-line — positive parallel negative
ILL = √3 · EX₁ + X₂
One phase to earth connects all three networks into one loop. The three sequence currents are then equal, and the faulted phase carries their sum. That is where the factor of 3 comes from:
line-to-earth — all three in series
ILE = 3 · EX₁ + X₂ + X₀
Two phases to earth put X₂ and X₀ in parallel below the positive network, and the zero network carries only its share. The current returning through the earth is 3·I0:
line-to-line-to-earth — current into earth
IE = 3 · X₂ · EX₁X₂ + X₂X₀ + X₀X₁
Four approximations sit behind these results: the source stays at exactly 1 pu with the pre-fault load ignored; the impedances are pure reactance (accurate where X is much larger than R); the fault is bolted, with no arc or earth resistance in the loop; and X₂ equals X₁.
That last one is safe for transformers, lines and cables, which offer the same impedance to a field rotating in either direction. Rotating machines do not, so a study close to a large generator or motor uses a separate X₂. A full calculation keeps the resistances, works in complex arithmetic, and replaces the 1 pu source with the IEC 60909 voltage factor c — and none of that changes a single wire in the connections above.
See it in Phasor
Phasor assembles the three sequence networks from the data you enter for each element: X₁, X₂ and X₀, plus the winding connection and star-point earthing of each transformer. It then applies the correct connection for each fault type at every bus. When an earth-fault result looks surprising, check the zero-sequence network first.