Sequence networks
The key idea
Symmetrical components give you three networks instead of one. A fault type is an instruction that tells you how to wire those three networks together. You use one network alone, two networks in parallel, or all three networks 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 problems. You therefore work with three separate networks, not one. Every generator, transformer, line and cable carries a positive-sequence reactance, a negative-sequence reactance and a zero-sequence reactance. Each set of reactances forms its own complete network for the system.
Reduce each network to the point where the fault occurs. Each network then becomes a simple box with two terminals. One terminal is the fault point, written F, which is the place of the short circuit. The other terminal is the reference, written N, which is the neutral or earth reference for that network. Only the positive-sequence network holds a source, because a healthy system is pure positive sequence. The negative and zero networks carry no current until a fault drives them.
The central rule follows. A fault type is a connection. The fault type does not change the contents of 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, and a wire connects its fault point directly to its reference. One network gives one division.
- Line-to-line. Two phases touch, and earth is not part of the fault. The positive and negative networks go in parallel: F to F, and 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 is now directly in the path of the current, together with X₁ and X₂.
- Line-to-line-to-earth. Two phases go to earth. The positive network is then in series with the negative and zero networks in parallel.
The list shows one pattern. The zero-sequence network appears only in the two connections that involve earth. For this reason, an earthing decision can change one fault current by a factor of ten and leave another fault current unchanged. A model with a guessed X₀ gives an earth-fault current that looks reasonable but is far from correct.
Try it
Switch between the fault types and watch the wires change. Then move the zero-sequence slider. Some bars change, and some bars stay 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. The earthing sets X₀, and X₀ appears only in the two connections that involve earth. Solid earthing, resistance earthing and impedance earthing all reach the fault calculation through that one number.
- A line-to-earth fault can exceed the three-phase fault. In the series connection the numerator is 3, not 1. X₀ is often smaller than X₁ close to a solidly earthed transformer. 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. It also needs the winding connection of each transformer and the earthing of each star point. Those two items decide whether a zero-sequence path exists.
- Protection uses these currents separately. Earth-fault elements respond to the earth return, 3·I₀, so you can set them far below the load current. Phase elements do not respond to that current. To grade the two against each other, you must 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 that 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 sum produces the factor of 3:
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 of the current. The current that returns through the earth is 3·I0:
line-to-line-to-earth — current into earth
IE = 3 · X₂ · EX₁X₂ + X₂X₀ + X₀X₁
Four approximations support these results. The source stays at exactly 1 pu, and the model ignores the pre-fault load. The impedances are pure reactance, which is accurate where X is much larger than R. The fault is bolted, so no arc or earth resistance appears in the loop. X₂ is equal to X₁.
That last approximation is safe for transformers, lines and cables. They offer the same impedance to a field that rotates in either direction. Rotating machines do not, so a study close to a large generator or motor uses a separate value for X₂. A full calculation keeps the resistances and works in complex arithmetic. It also replaces the 1 pu source with the IEC 60909 voltage factor c. 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. That data is X₁, X₂ and X₀, plus the winding connection and the star-point earthing of each transformer. Phasor 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.