Trefoil versus flat bundling of three-phase cables — magnetic field, sheath currents, and ampacity

The guide on skin effect and proximity effect at large cable cross-sections covers why a conductor's AC resistance is higher than its DC resistance, and why that factor becomes stronger as conductors lie closer together. This article covers a related but separate practical choice for three single-core cables of the same phase group: do you lay them in trefoil formation (three cables in an equilateral triangle against each other) or in flat formation (three cables side by side in a single row)?

What trefoil formation gives you: symmetry

In trefoil formation, the three phase conductors lie in an equilateral triangle, each equidistant from the other two. This geometric symmetry has two direct consequences:

Magnetic field outside the bundle: with a balanced three-phase load, the three phase currents are equal in magnitude and shifted 120° in phase, so their vector sum is zero. In a symmetric, compact trefoil arrangement, the magnetic fields of the three phases largely cancel each other outside the bundle, so the residual magnetic field at some distance from the cable bundle is significantly lower than with a less compact or asymmetric arrangement.
Equal current distribution between phases: because each phase conductor in trefoil has the same geometric relationship to the other two (each lies at equal distance from the other two), the mutual inductive coupling is equal for all three phases. This keeps the effective AC impedance of the three phases balanced, so that at a balanced load the current also actually distributes equally across the three phase conductors.

What flat formation gives you: an asymmetric disadvantage for the outer cables

In flat formation, the three phase conductors lie side by side in a single row. As a result, the middle cable is surrounded by the two outer cables, while each outer cable has a neighbouring conductor on only one side and is free on the other side. This geometric distinction between the middle and outer positions has two consequences that do not occur in trefoil:

Unequal mutual inductance: the middle cable experiences a different (typically higher) mutual inductive coupling than the outer cables, simply because the middle cable lies closer to more current-carrying conductors. The result is that the effective AC impedance of the three phases is no longer equal.
Unequal current distribution at equal load: as a result of that unequal impedance, even with a load that is in principle balanced on the load side, the actual current can still distribute noticeably unequally across the three phase conductors — the outer cables can carry a different current than the middle one, even though the load itself is perfectly symmetric. In a design that assumes equal ampacity per cable without accounting for this transposition-like imbalance, one of the three cables can therefore end up thermally loaded more heavily than calculated.
Larger magnetic field outside the bundle: the less compact, linear arrangement cancels the magnetic fields of the three phases less effectively than the symmetric trefoil arrangement, so the residual magnetic field at a distance from the bundle in flat formation is typically higher than in trefoil, at otherwise equal current and cable spacing.

Note: in flat formation, the unequal current distribution between the outer and middle cable can be limited by periodically swapping the cables' positions along the length of the route (a form of transposition) — a measure applied especially on long, heavily loaded routes, but one that makes the installation more complex than simply applying trefoil.

When which formation is preferable

Trefoil is generally the preferred choice when the magnetic field outside the cable bundle is relevant — for example, on a route that passes a space with sensitive electronic equipment, or where exposure of people to the magnetic field must be limited — and when an equal, predictable current distribution between the three phases is desired without additional transposition measures.
Flat formation is more often applied where the physical cable tray or ladder width is limited in the other direction, or where the cables must be laid on already existing, flat support structures, and where the magnetic field outside the bundle or the mutual current balance is not a critical factor.

Practical relevance

When designing the cable route for a substantial three-phase supply — especially for long, parallel runs of cable close to sensitive equipment, measurement instrumentation, or spaces where people spend extended time — the choice between trefoil and flat formation is not an aesthetic or purely spatial choice, but has direct consequences for both the magnetic field outside the bundle and the actual current distribution, and therefore the thermal loading of each individual cable. With flat formation on a long, heavily loaded route, it must be assessed whether transposition is needed to limit the unequal current distribution between the outer and middle cable.

Common mistakes

Applying flat formation without accounting for the unequal current distribution between the outer and middle cable, causing the actually most heavily loaded cable to be thermally loaded more than the design assumes.
Not taking the magnetic field outside the bundle into account on a route past sensitive electronic equipment or an occupied space, while trefoil could have significantly limited that field.
Applying trefoil without checking the short-circuit withstand requirements for that formation — see the guide on electrodynamic forces during busbar short-circuits for the related principle that mechanical forces between conductors during a short circuit strongly depend on spacing and geometry — trefoil and flat formation produce different short-circuit forces and each requires its own cleat mounting.
Not applying transposition on a long, heavily loaded flat route while the unequal current distribution is indeed relevant at that specific length and load.