Skin Effect and Proximity Effect: How Litz Wire Solves the Loss Problem

The skin effect is the crowding of alternating current toward the surface of a conductor, which follows from Maxwell's equations: the current density decays exponentially into the depth, with a characteristic length δ = √(2/ωμσ), called the skin depth. The proximity effect is the additional eddy currents induced in a conductor by neighboring conductors carrying an alternating field; in a multilayer winding its contribution grows as the square of the number of layers. Litz wire suppresses both effects by dividing the copper cross-section into many insulated, transposed strands.

The problem of conductor losses under alternating current has an exact mathematical origin — Maxwell's equations. Not empirical lookup tables, not engineering intuition, but specific differential equations from which both the skin effect and the proximity effect are derived rigorously. In this article we work top-down: from the equations to the current density across the wire cross-section, to the Dowell equation for real windings, and to the modern numerical methods used today to compute losses in litz-wire coils.

Where the skin effect comes from: five lines from Maxwell

Inside a conductor carrying alternating current, three of Maxwell's four equations are at work — Faraday's law, Ampère's law without the displacement current (at frequencies up to hundreds of MHz it can be neglected in copper), and Ohm's law in local form:

Substitute one into another, apply the curl identity, and for the sinusoidal regime at frequency ω we obtain the Helmholtz equation for the current density:

The solution for a semi-infinite conductor decays exponentially into the depth:

And the skin depth itself is:

The current falls off exponentially and at the same time lags in phase: at each depth δ the amplitude is attenuated by a factor of e ≈ 2.72, and the phase lags by 1 radian. At a depth of 5δ from the surface the current density is already less than 1% of the surface value.

This is fundamental, and from it follows a non-obvious point: there is no "boundary" between a conducting region and a non-conducting one. Current is everywhere — it is simply negligibly small deep inside. The engineering approximation "current flows in a layer of thickness δ" is convenient but inaccurate.

The exact loss power accounts for the exponential tail and gives, per unit of surface area:

This formula is the basis for all the calculations that follow.

Round wire: the exact Bessel solution

The semi-infinite approximation works only for a flat conductor. For a round strand the problem is solved exactly through Bessel functions of the first kind with a complex argument. The current density as a function of radius r:

From this we obtain the exact ratio of AC to DC resistance:

In practice an approximation through the dimensionless parameter Q = d/δ — the ratio of strand diameter to skin depth — is used:

It is precisely from this table that the working criterion for choosing the strand in litz wire follows: Q ≤ 1, that is, d ≤ δ. Not "up to 2δ," as handbooks often state — that is the upper limiting boundary, while the practical optimum lies deeper.

Take a copper wire 1 mm in diameter at 1 MHz. The skin depth δ ≈ 66 µm, Q ≈ 15. The current density on the wire axis is roughly 0.01% of the surface value — the central part of the cross-section is electrically dead. The wire still "conducts," but only through a thin tube near the surface. Replace it with litz wire made of 100 strands of 0.1 mm (Q = 1.5, F_R ≈ 1.03), and current flows again across the entire copper area.

Proximity effect: the Dowell equation and why layers kill

The skin effect is the wire's reaction to its own field. The proximity effect is the reaction to an external field created by its neighbors. And this is where the physics turns unkind.

In 1966 P. L. Dowell published "Effects of eddy currents in transformer windings" (Proc. IEE, vol. 113), in which he derived an analytical solution for a multilayer winding made of flat foil. The idea is simple: he replaced the round wires in a layer with an equivalent foil of the same copper area, solved the one-dimensional problem, and obtained a formula for the AC resistance of an m-layer winding:

The model contains one important coefficient that Dowell introduced for the transition from foil to round wire — the winding-layer fill factor η (in the English-language literature, the porosity factor):

This is not the same as the window fill factor familiar from transformer design, k_fill = S_copper/S_window — the latter is two-dimensional (the copper fraction over the area of the whole window), whereas η is one-dimensional and describes the packing density of turns along a single coordinate (along the layer). And it has nothing to do with the geological "porosity coefficient" from building codes and standards, despite the similar English name — there it refers to the void volume in soil.

The physical meaning of η is simple: η → 1 for tight packing of turns butted together, η → 0 for a sparse winding with large gaps. In the Dowell formula η enters as a factor √η inside Δ, and for a loose winding the losses decrease — the magnetic field is "smeared" over a larger volume and links with less copper. This explains the old engineering trick with a spacer cord in high-Q resonant-circuit coils: an artificially sparse layout reduces the proximity effect.

Don't be intimidated by the hyperbolic-trigonometric expression itself. What matters is this: the second bracket contains the factor (m² − 1). Proximity-effect losses grow as the square of the number of winding layers. One layer — no proximity effect at all, only the skin effect works. Two layers — already three times more. Five layers — 24 times more than a single layer.

To see numbers rather than formulas, take a typical example from the Ridley Engineering handbook (2019): a transformer winding of 5 layers of 0.3 mm copper wire at 100 kHz. The Dowell calculation gives an average resistance increase of 11.6 times. For the innermost layer it is 27 times. Increase the diameter to 0.6 mm in an attempt to "add copper and cut the losses," and the average jumps to 51 times, while the inner layer reaches 123. More copper, eight times more losses. This is a counterintuitive but rigorous result of one-dimensional electrodynamics.

Litz wire breaks this catastrophe at the root: by replacing one thick strand with a hundred thin transposed ones, it moves the winding into a regime where d ≪ δ, and the first term in the Dowell formula (the skin effect proper) becomes negligible, while the second (proximity, ∝ m²) remains but with a small Δ, because the strand is thin.

Internal and external proximity: why they must be separated

In the works of Ferreira (1994) and Sullivan–Zhang (COMPEL 2014), the proximity effect in litz wire is split into two components, and this split is fundamental for understanding the physics.

• External proximity — the field created by neighboring winding turns acts on the strand from "outside the bundle." It is precisely this that produces the (m² − 1) factor in the Dowell formula. It is fought with transposition + a sparse winding + interleaving of the primary and secondary windings in a transformer.
• Internal proximity — the field created by other strands within the same bundle acts on a given strand. It leads to that counterintuitive effect: naively splitting a thick wire into many thin strands can increase the losses if d has not dropped noticeably below δ. It is fought by choosing a sufficiently thin strand and the right packing.

Quantitatively: the integral P_prox in a litz-wire winding is the sum of two contributions — the external grows as m², the internal as n (the number of strands in the bundle). With proper design both terms are roughly equal — and that is the optimum. A strong imbalance in either direction means that there are either too few strands (the external contribution is overloaded) or too many (the internal is overloaded, and the enamel has eaten up the copper cross-section).

How the Q factor sees it: Q(f) and its peak

Everything said above concerns copper losses. But for an engineer designing a high-Q resonant circuit or a resonant converter, an integral parameter matters more: the coil Q factor Q = ωL/R. This is the very same letter Q as in the dimensionless parameter Q = d/δ, but it means something entirely different — the ratio of stored energy to energy lost per cycle. We will call it Q_coil to avoid confusion.

At low frequencies:

Q_coil grows linearly with frequency — the inductive reactance rises while the resistive part stays at the DC level. But as we approach the region where Q = d/δ becomes of order one, the denominator begins to swell because of F_R(f):

Because F_R grows faster than linearly under Dowell, beyond a certain frequency Q_coil reaches a maximum and begins to decline. This maximum is the optimal operating point of the coil, and it is what engineers look for in design. For litz-wire coils in induction heaters the Q_coil peak typically falls at 30–100 kHz; for Qi wireless-charging coils, at 100–200 kHz; for MRI coils, at tens of MHz (using special constructions).

Elektrisola publishes direct measurements of Q(f) for its standard litz wires. A typical coil of 120 strands of 0.1 mm on an air core shows a maximum Q_coil ≈ 500 at 200 kHz — 2–3 times higher than a comparable coil of single-strand copper wire of the same DC-equivalent area. This is exactly the engineering gain for which litz wire is made.

Three frequency bands — and where litz wire actually works

Modern models (Wojda & Kazimierczuk, 2012; Geng et al., IET Power Electronics, 2021) divide the operation of litz wire into three regimes:

The boundary between the mid and high bands is a separate, interesting piece of physics.

Why litz wire dies above 2 MHz: inter-strand capacitance

Every pair of adjacent strands in litz wire is a microcapacitor. Enamel 5–15 µm thick with a relative permittivity ε_r ≈ 3.5–4 (polyurethane, polyesterimide) forms a flat gap. The inter-strand capacitance per unit length for a typical 0.1 mm litz wire comes out on the order of 1–10 pF/m per pair of adjacent strands, and accounting for the many hundreds of pairs — tens of nanofarads per meter of total capacitance.

At frequency f the reactance of this capacitance is:

At 100 kHz for 10 nF we get 160 Ω — a thousand times larger than the ohmic resistance of the strands, so the capacitance does not interfere. At 5 MHz for the same capacitance it is only 3.2 Ω, already comparable to the AC resistance. Instead of flowing through the enamel (which is formally an insulator), the current begins to "spill over" between strands through the capacitive coupling. The transposition loses its meaning, and electrically the litz wire turns into a single thick strand.

Two further troubles add to this:

• The self-resonant frequency (SRF) of the coil. The winding's parasitic capacitance resonates with the inductance at the SRF, above which the component stops being an inductance and becomes a capacitance. For litz-wire coils the SRF is usually 2–10 MHz.
• Capacitive switching losses:
  P_C = C · V² · f

  P_C

  capacitive losses, W

  V

  voltage amplitude across the capacitance, V

  C

  parasitic capacitance of the winding, F

  f

  switching frequency, Hz

  At frequencies above 1 MHz and voltages of 400–800 V these losses can exceed the gain from the reduced ohmic resistance. Paradoxically, in some flyback converters switching from single-strand wire to litz wire worsens the overall efficiency precisely because of this.

Practical takeaway: the working range of litz wire is 10 kHz to 1 MHz. With reservations, up to 2–3 MHz. Above 5 MHz, you are already into special constructions with increased enamel thickness or flat film conductors.

What is used today: from Dowell to 3D FEA

The Dowell equation is a one-dimensional approximation. For most transformers its accuracy is sufficient, but the deviation from reality can reach 60% with an unfortunate winding geometry (Nan & Sullivan, IEEE Trans. Power Electronics, 2009). This pushed the industry toward numerical methods.

Today's toolkit looks like this:

• The Ferreira method (1994): an exact Bessel solution for each strand, supplemented by superposition of the magnetic fields of its neighbors. More accurate than Dowell for sparse windings, less accurate for dense ones.
• Sullivan–Zhang (COMPEL 2014): an analytical model that accounts for the twist pitch, directions, and multistage construction. Implemented in MATLAB and released as open source on the Thayer School (Dartmouth) website.
• 2D FEA with discrete transposition (Roy, Sakly, IEEE 2014): the key idea is to solve a 2D FEA for parallel strands and then mathematically "shuffle" them in the impedance matrix according to a given twisting protocol. Accuracy within 10% for the R_AC/R_DC ratio.
• 3D FEA with a litz model in Ansys Maxwell: the litz wire is modeled as a single cylinder with anisotropic conductivity. This makes it possible to compute coils with thousands of turns without drawing each strand.

In 2022 Lin et al. proposed an effective homogenization method for transient FEA that compresses the computation of a 10,000-strand winding to a few minutes of CPU time. By comparison, directly modeling each strand in the same problem would take several days.

What this gives the practicing engineer: modern litz wire is not selected "from a lookup table." For a specific product an FEA model is run with the real geometry of the window, core, and gap — and the optimum number and diameter of strands is found, minimizing the sum of copper and core losses. Dowell has remained in the engineering textbooks, but in serious developments of EV onboard chargers (OBCs), medical MRI coils, and wireless charging, numerical models are at work.