Motor Winding AC Copper Loss Explained — Skin, Proximity & Dowell's Equation

Why a winding's AC resistance isn't its DC resistance

A winding's DC resistance is simple geometry: resistivity times length over area. But run AC current through it — and every traction motor does, at whatever frequency the drive is switching the phases — and the effective resistance climbs above that DC value. Two effects are responsible. Skin effect pushes current in a single isolated conductor toward its surface, shrinking the area it actually uses. Proximity effect is the more aggressive of the two in a real winding: the alternating field from every neighbouring turn induces eddy currents in each conductor, crowding current into thin regions and multiplying loss well beyond what skin effect alone would predict. In a slot with several layers of turns stacked on top of each other, proximity effect usually dominates.

Dowell's equation: the classical way to capture both at once

The standard tool for this is P.L. Dowell's 1966 transformer-winding analysis, which treats a stack of conductor layers as if they were sheets of foil and derives a single closed-form AC/DC resistance ratio, F_R, as a function of two things: how thick each layer is compared to the skin depth (the penetration ratio, Δ), and how many layers are stacked in the direction the field builds (m). The formula splits cleanly into a skin-effect term and a proximity-effect term that scales with m²−1 — which is why doubling the number of layers in a slot doesn't double the AC loss, it can multiply it several times over. A flat or hairpin conductor is a foil layer already, so Dowell's equation applies directly. A layer of round wires gets converted to an equivalent foil first, using a porosity factor for how tightly the round wires actually fill the layer.

Why round wire, hairpin and litz behave so differently

The three winding conductor types sit at different points on the same trade-off. Round wireis cheap and easy to wind, but a single thick strand has a lot of area exposed to the proximity field — the classic fix is to split it into several thinner strands in hand. Flat/hairpinconductors pack a slot efficiently and are the mainstream choice for modern EV traction motors, but their large, flat faces are exactly what proximity effect punishes hardest — which is why hairpin motors are usually designed with several thin conductors stacked radially (4 or 8 layers) rather than one tall one. Litz wire attacks the problem directly: many separately-insulated strands, twisted or bunched so each one only sees the average field, keep every individual strand thin enough that neither skin nor proximity effect gets much purchase — at the cost of a lower copper fill factor and a real construction limit (twist a litz bundle from too many strands in one operation and it stops behaving ideally).

Sullivan's closed form, the litz construction limit, and what the model doesn't capture

Why round wire and litz wire share the same formula, how to tell if a litz bundle is well-built, and the one real AC loss mechanism this calculator doesn't model.

Round wire and litz wire share one formula

When a conductor is thin compared to the skin depth — the usual case for both round magnet wire and individual litz strands — Dowell's equation simplifies to a single closed-form term (Sullivan & Zhang, 2014): F_R = 1 + (π·n·N)²·d⁶/(192·δ⁴·b²), where n is the number of strands per turn (just 1 for round wire), N is the number of turns stacked across the field-build direction, d is the strand diameter, δ is the skin depth, and b is the winding breadth. It's worth noticing that round wire is mathematically just litz wire with one strand — the same physics, the same formula, and that's exactly how this calculator's round-wire and litz-wire engines work under the hood.

Litz wire isn't automatically loss-free

Twisting strands together only helps if the twist is tight enough that each strand really does see the average field rather than its own local position. Push too many strands into a single twisting operation and the bundle starts behaving less like ideal litz and more like a solid conductor at high frequency — Sullivan gives a simple limit for how many strands a first twisting operation can safely combine, and this calculator checks a litz design against it. Exceed the limit and the real AC resistance will run higher than the ideal formula predicts; a well-built multi-stage ("bunch of bunches") construction is the usual fix.

What this model doesn't capture

Every formula here assumes the field builds uniformly across the slot and that every conductor in the slot carries the same current — a fair assumption for a simple series-wound coil. It breaks down for hairpin windings with multiple parallel current paths that aren't perfectly transposed: those paths can develop circulating currents between themselves, on top of the proximity loss modelled here, and that mechanism can dominate in a badly-designed parallel winding. It's a real, active research topic in EV motor design and isn't something a closed-form formula captures — it needs FEA. Treat this calculator's hairpin result as the proximity-effect floor, not the whole story, for a winding with parallel paths.

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Motor Winding AC Copper Loss: Skin Effect, Proximity Effect and Dowell's Equation