Litz wire for LLC resonant converters: design by the 2025 method

At the resonant frequency (f_r) an LLC resonant converter runs with a nearly sinusoidal current, and a Dowell, Ferreira and Sullivan calculation gives the right answer for that operating point. In real service — an EV on-board charger (OBC), a server power supply, a telecom rectifier — the converter is off resonance almost all the time. Current harmonics rise to 15-25 %, the integrated leakage inductance creates a fringing field at the air gap, and the interwinding capacitance of a layered winding is comparable to the output capacitance C_oss of silicon-carbide (SiC) and gallium-nitride (GaN) switches and eats into the soft-switching (ZVS, zero-voltage switching) margin at light load. By 2025 the working method for sizing litz under LLC comes down to four steps: circuit simulation, a 2D finite-element model of the winding window (FEMM) with a homogenized litz bundle by Meeker, the Squared-Field-Derivative method (SFD) by Sullivan, and joint core-and-winding optimization. The hand calculation stays as a sanity check, not as the basis of the decision.

Below we look at where exactly an LLC falls out of the basic calculation method, how the 2025 working flow is built, and how it plays out on one real project — a 3.3 kW SiC charger at 200 kHz. The fundamentals (skin effect, the Dowell formula, the Sullivan optimum) are covered in detail in our article on skin and proximity effects; transposition and skin effect at the bundle level are in the article on choosing a litz-wire grade, and we will not repeat them here. This article is the LLC specifics and the tools people install in 2025.

The LLC transformer: 2018 → 2025

In 2018 a typical LLC transformer for an 800 W server power supply (PSU) was sized in about two hours. You took Dowell, estimated the layers, picked the strand by Sullivan's rule d_s ≈ δ/2, and put it into production. If it overheated, you redid it with a thinner strand, and that was usually enough. Over seven years the picture changed radically. LLC resonant frequencies moved from 100-150 kHz to 250-500 (up to a megahertz on GaN). On-board chargers brought a requirement for a 1:2.5 output-voltage range. The resonant inductors moved inside the transformer, and gaps of a millimeter and more appeared in the cores. The switches became SiC and GaN, and their output capacitance C_oss dropped from nanofarads to hundreds of picofarads. The transformer's interwinding capacitance (C_ps, primary-to-secondary) turned out to be in the same weight class as the C_oss of the switches themselves. Each of these changes on its own has little effect on the Dowell formula, but together they break it badly enough that the hand calculation stops being a first estimate and starts pointing in the wrong direction.

Harmonics that were not there at resonance

Everyone knows: at resonance the primary current of an LLC is a sinusoid with a total harmonic distortion (THD) below 3 %. The resonant tank filters out everything above the fundamental. If the converter always ran at the resonant frequency f_r, there would be nothing to discuss.

The problem is that output-voltage regulation in an LLC is done by varying the switching frequency. The operating frequency f_sw swings around f_r from a low f_min to a high f_max. For a charger with an output-voltage range V_out from 250 to 450 V this frequency range is designed at 1:2.2 (for example, 130 to 290 kHz at f_r = 200 kHz). At the edges of the range the current waveform changes beyond recognition.

In above-resonance operation (boost mode — when V_out is low and the gain has to be "clipped") the current becomes almost triangular. In below-resonance operation (buck mode) a characteristic "sinusoid with a hump" appears. In both cases the third harmonic jumps to 12-18 % of the fundamental, the fifth to 3-5 %. Light-load burst mode produces bursts of short pulses with an arbitrary spectrum.

The third harmonic lands at 3·f_r, that is at 600 kHz for f_r = 200. The skin depth (δ) there is half as deep. For an AWG 36 strand (d_s = 0.127 mm) at 200 kHz the ratio d_s/δ ≈ 0.65; at 600 kHz it is already 1.13. This is the rising-skin-effect regime, and the resistance factor F_R = R_AC/R_DC at the third harmonic is 5-6 times higher than at the fundamental. Computing the third-harmonic contribution to total loss:

In other words, in boost mode the third harmonic adds 12 % to copper loss. On an inverter with silicon-carbide switches and edges of a few nanoseconds (Wolfspeed C3M, Infineon CoolSiC G4) the contribution of higher harmonics is more noticeable still. Sizing an LLC only at the resonant frequency f_r in 2025 means designing in 15-20 °C of overheating above the calculated value.

boost · f_sw 130 kHz · V_out 450 V

I₁ (fundamental) 6.5 A

I₃ (3rd) 14 %

I₅ (5th) 3 %

resonance · f_sw 200 kHz · V_out 350 V

I₁ (fundamental) 8.4 A

I₃ (3rd) 3 %

I₅ (5th) ≈ 0

buck · f_sw 280 kHz · V_out 250 V

I₁ (fundamental) 7.8 A

I₃ (3rd) 15 %

I₅ (5th) 4 %

First harmonic (f_sw) Third harmonic (3·f_sw) Fifth harmonic (5·f_sw)

Data captured by a PLECS simulation of a real 3.3 kW charger. At resonance the spectrum is clean and a thin litz strand handles it perfectly. At the edges of the regulation range (boost and buck) the third harmonic jumps to 14-15 %, and its R_AC at the triple frequency is 5-6 times higher than the fundamental's. Sizing litz only at the resonant frequency f_r in this case means missing the loss by 12-20 %.

The field at the air gap

The basic Dowell model assumes that the field in the winding window is plane-parallel and runs along the axis. This works in a gapless transformer, where the whole magnetic flux is closed through the iron.

In an LLC it does not work out that way. The resonant tank needs a leakage inductance L_r: it is either built on a separate inductor (the historical option) or integrated into the transformer through the magnetic geometry and a gap. In 2025 manufacturers do only the second option: fewer parts, cheaper assembly, more compact.

The gap in an LLC works out to 0.3-1.5 mm, sometimes more. The field above it does not go straight into the iron — it spreads "fan-like" into the winding window. Strands that end up within the first one or two millimeters of the gap fall into a local field 5-10 times the average design field strength H.

The exact coefficients were published by Mühlethaler in 2012 (PhD thesis at ETH Zürich, the Swiss Federal Institute of Technology, open PDF). The local resistance factor F_R as a function of distance from the gap:

What these numbers mean in hardware: "on average over the winding" the transformer is built right, the loss is acceptable. But the first two or three turns at the gap heat to 180-200 °C, the enamel degrades within a month, and after six months of service you get breakdown, arcing, replacement. This is probably the most common warranty-return cause for mid-quality on-board chargers.

It is fixed with a distributed gap (two or three sections), a manufacturing clearance of 3-5 mm between the winding and the core face, and, in high-power designs, copper Faraday shunts that divert the fringing field. None of these options can be sized on paper. They are sized only by 2D finite-element modeling of the winding window.

A schematic map based on Mühlethaler's data (ETH 2012), verified in 2D FEMM models. The inner two turns of any winding that end up within 1-2 mm of the gap fall into a local field 5-10 times the average. Local loss grows quadratically, and the strand temperature in this zone reaches 180-200 °C while the winding average is 110 °C. It is fixed three ways at once: a distributed gap (two or three sections instead of one), a manufacturing clearance of at least 3 mm from the core face, and, in high-power designs, copper Faraday shunts.

When soft switching (ZVS) is lost at light load

The third point is the least obvious. And the one that most often catches out designers who have moved from silicon insulated-gate bipolar transistors (Si IGBTs) to silicon-carbide field-effect switches.

An LLC delivers 96-98 % efficiency through soft switching of the primary switches (ZVS, zero-voltage switching — switching at zero drain-source voltage). For ZVS to work, the magnetizing current I_m must recharge the output capacitances of both switches within the dead time t_d (the pause between turning one half-bridge switch off and the other on). The condition:

In this sum C_total the engineer usually plugs in C_oss(M1) + C_oss(M2) and forgets the third term — C_ps, the transformer interwinding capacitance.

When the switches were Si IGBTs with a C_oss of a few nanofarads, a C_ps of 50-100 pF was lost in the noise. In 2025 on-board chargers and server power supplies use SiC or GaN: the Wolfspeed C3M0021120K has C_oss ≈ 80 pF, the GaN GS66516B about 60. A transformer's C_ps with two P/S junctions (a P-S-P layering, primary-secondary-primary) is typically 30-60 pF, with five junctions (P-S-P-S-P-S) already 90-250.

At full load it all works: I_m is large, ZVS holds. At 10 % load I_m drops tenfold, the ZVS boundary moves above the operating point, and the converter starts switching hard. On the scope this shows as 30-80 MHz ringing on the drains, plus +5…10 W of extra loss and +20 °C on the switch package.

It is fixed in one of two ways. First: increase I_m, that is reduce the magnetizing inductance and recompute the gap. Second: reduce C_ps, that is simplify the layered layout. And here a trade-off appears. P-S-P-S-P-S in theory gives lower proximity-effect loss than P-S-P. But it is also the one that breaks ZVS at light load. On real 1-3 kW designs the optimum is exactly P-S-P: one junction, minimum C_ps, an effective number of winding layers m_eff = 1 for each winding. P-S-P-S-P-S is academically more correct, in practice worse. This is counterintuitive, and little is written about it.

Minimum load at which ZVS holds

P-S-P 8 %

P-S-P-S 22 %

P-S-P-S-P-S 38 %

C_ps ≈ 50 pF — optimum for SiC/GaN C_ps ≈ 120 pF — a compromise C_ps ≈ 220 pF — breaks ZVS at light load

Computed from the condition I_m·t_d ≥ 2·V_in·(C_oss + C_ps) for a typical 3.3 kW charger on Wolfspeed C3M0021120K switches (C_oss ≈ 80 pF, t_d = 200 ns, V_in = 400 V). The shorter the green bar, the longer ZVS holds as the load falls. A P-S-P layering keeps ZVS down to 8 % — ideal behavior at no load. P-S-P-S-P-S forces the converter to switch hard already at 38 % load, where the switches develop 30-80 MHz ringing and +5…10 W of extra loss.

The 2025 calculation workflow

By 2025 a six-step working method has settled in the industry.

Step 1

Circuit simulation

LTspice, PLECS

→ i(t) at 4 points

→

Step 2

Spectral analysis

FFT or full i(t)

→ I_k, k = 1, 3, 5…

→

Step 3

2D window model

FEMM 4.2 / ANSYS Maxwell, Meeker homogenization

→ B(x, y, t)

→

Step 4

SFD integration

Python + scipy

→ P_prox for the whole winding

→

Step 5

Joint optimization

GeckoMAGNETICS, NSGA-II, differential evolution

→ Pareto front

→

Step 6

Neural-network model

MagNet, Tang et al. 2024 graph neural networks

→ ×1000 speed-up

The minimum-configuration stack (LTspice, FEMM, Python) is fully free and covers steps 1-4 for projects up to 5 kW. Steps 5-6 make sense on volume projects from 5000 units/year: GeckoMAGNETICS gives a Pareto front over {P_loss, mass, price} in an evening, MagNet returns a loss estimate in milliseconds — and that is what lets the optimization loop sweep 10⁵-10⁶ designs instead of a few hundred.

You start with circuit simulation. LTspice (free) or PLECS (paid, cheaper than ANSYS, more convenient for power electronics). Capture the primary i_pri(t) and secondary i_sec(t) winding currents not at resonance but at four corner points: low V_in × full load, high V_in × full load, any V_in × 10 % load, and burst mode (for no load). The key thing at this step is for the switch model to account for the real C_oss(V) dependence and the core model for the real B-H hysteresis loop, not idealized ones. Without that the simulation lies about the harmonics.

Decompose the currents into a spectrum. Either a fast Fourier transform — FFT — for the subsequent summation of individual harmonic contributions, or take i(t) whole (for the SFD method, see below). You get which frequencies and which amplitudes the current flows at in each conductor.

Build a 2D model of the winding window in FEMM or ANSYS Maxwell. FEMM is an open 2D finite-element analysis package (FEA). The geometry here is real: a real core, a real gap, a real layout of the primary and secondary windings with interwinding insulation. The most important thing at this step — the litz wire is not modeled strand-by-strand. For a typical 500×0.07 mm wire that would mean a million mesh nodes and a week of computation per solution. Instead Meeker's homogenized model (D. Meeker, 2012) does the work: the bundle is replaced with an anisotropic material with an effective frequency-dependent conductivity σ_⊥(f) that analytically accounts for the proximity-effect loss between strands "inside" the homogenized fill. Accuracy versus a full FEA that draws every strand stays within 5-10 % for bundles up to 1000 strands. The FEMM analysis of an LLC rests on this simplification.

Compute the loss with the Squared-Field-Derivative method (SFD). The SFD method (Squared-Field-Derivative, Sullivan, IEEE Transactions on Power Electronics, 2001) gives the proximity-effect loss in litz wire for an arbitrary current waveform through the integral of the squared time derivative of the magnetic flux density:

You take the magnetic flux density B(x, y, t) from FEMM (one solution per harmonic with subsequent summation, or a time-domain solution for an arbitrary i(t) waveform), integrate over the winding volume, and get P_prox for the whole winding at once. There is no need to add the contributions of individual harmonics — it is a single operation. On systems with gallium-nitride switches and nanosecond edges, SFD is effectively the only workable method: the harmonic series there does not converge in a reasonable number of terms.

Run it through joint core-and-winding optimization. The historical scheme (choose the core first, then the winding) gives a local minimum, not a global one. Kolar's group at ETH (Mühlethaler, Krismer, Papamanolis) showed in works from 2012-2020 that joint optimization over all parameters at once — core, gap, litz wire, interleaving, insulation — gives entirely different designs. The objective functions: total loss P_total, mass, price, efficiency at partial load. The solvers are the differential evolution algorithm or the multi-objective genetic algorithm NSGA-II to build the Pareto front; Bayesian optimization for cases where each FEA solution is expensive in time. Open source: GeckoMAGNETICS (a spin-off of the ETH project, the academic version is free). The key paper: Papamanolis et al., IEEE Open Journal of Power Electronics, vol. 1, 2020, open access, required reading.

Optionally plug in machine-learning models. If you have to sweep a million design variants, FEMM on each would take months. Here MagNet does the work — an open database from Princeton University (Sullivan, Wang) with ten million measured waveforms on various ferrites, plus a neural network that predicts core loss from the B(t) waveform with an error below 3 %. Prediction time is milliseconds. The MagNet Challenge 2024 produced several more free models. For winding loss similar approaches develop more slowly, but Tang and co-authors from the University of California, Davis showed a graph neural network for litz wire in 2024 with an 8 % error versus a full FEA at the same prediction speed.

In optimization this works as a two-level loop. The outer loop is a genetic algorithm or differential evolution. The inner loop is a neural-network model instead of a full FEA. The result is a sweep of 10⁵-10⁶ designs in hours instead of months. For volumes of tens of thousands of units per year this pays off within a quarter.

Where the basic theory breaks

The formulas of Dowell, Ferreira and Sullivan, from their 1966-1999 years, have not gone anywhere and are not obsolete; they remain the best way to understand the physics of winding loss. What the skin effect is, why proximity grows as m², where the strand-diameter optimum lies — without that basis you cannot meaningfully apply either SFD or FEMM. What changed is the domain of validity. The basic formulas work correctly at a single frequency, for a strictly plane-parallel field, with no gaps, with no account of C_ps — that is, on a lab bench or in a textbook example. On a real LLC with a gap, GaN switches and wide-range regulation, the result diverges from the measurement by 30-50%, and you cannot predict in advance which way the error will go.

On production projects the hand calculation is still done in 2025, but as a sanity check for the FEMM numbers, not as the result. A three-fold discrepancy — look for an error in the model. A 20-30% discrepancy — that is exactly the correction for harmonics, the gap and edge effects that the basic theory does not account for.

A 3.3 kW SiC charger at 200 kHz — how it is done in practice

To make this less abstract, let us work through a typical single-phase on-board charger.

What the textbook calculation gives. At f_r = 200 kHz and 100 °C the skin depth is δ = 0.196 mm. The Sullivan optimum for m_eff = 1 (a P-S-P layered layout) gives a strand diameter d_s ≈ 0.15 mm. For an RMS current I_rms ≈ 9 A and a current density of 4 A/mm² you need ≈ 2.3 mm² of copper, the nearest catalog step is 140×0.15 (2.5 mm²). The resistance factor F_R at 200 kHz is taken as 1.4. Add up the loss, you get 4.2 W. Off to build it.

After assembly you get: at the high V_out = 450 V (boost mode, f_sw = 130 kHz) nothing overheats, all nominal. At the low V_out = 250 V (buck mode, f_sw = 280 kHz) the winding in the gap area is at 168 °C, the enamel at its limit. At 8 % load ZVS is lost, the switches ring, efficiency falls from 96 to 92 %. Back to the rework bench.

What should have been done from the start. Run the PLECS simulation at three corner points:

The third harmonics at 840 kHz and 390 kHz are significant. Computing the AC resistance R_AC at these frequencies: for a 0.15 mm strand at 840 kHz the ratio d_s/δ ≈ 1.6 — already an intermediate regime, F_R near 3, and locally in the hot zone at the gap near 8. It turns out that 0.15 mm is already too thick for the upper part of the range, and the optimum shifts to the 0.1 mm series (315×0.1 at 2.5 mm²).

Next — FEMM. We draw the PQ 50 core with a 0.8 mm gap (the center leg cut in half, a distributed gap), and set the winding blocks as Meeker's anisotropic material. We solve at 200, 600, 1000 kHz. Two millimeters from the gap we see a field strength H = 38 kA/m against an average of 6 kA/m over the winding. The local resistance factor in this zone is F_R = 7.2.

We do the SFD integration over the winding volume for the PLECS current waveforms. We get the primary-winding loss for the three modes:

buck · 280 kHz · V_out 250 V

Dowell at f_r 3.8 W

FEMM + SFD 5.9 W

Miss +55%

resonance · 200 kHz · V_out 350 V

Dowell at f_r 4.2 W

FEMM + SFD 4.6 W

Miss +10%

boost · 130 kHz · V_out 450 V

Dowell at f_r 2.5 W

FEMM + SFD 3.4 W

Miss +36%

At resonance the basic method is accurate to 10 %, within the engineering margin. But at the edges of the regulation range (boost and buck modes), where the charger spends most of its real operating time, the error reaches 36-55 %: the calculated 3.8 W becomes an actual 5.9, the winding heats 15-20 °C more, and the enamel at the gap degrades. The FEMM numbers are verified with a thermocouple on samples.

The basic calculation misses in buck mode by 55 %. That is no longer a "correction factor" but a rethink of the thermal management.

We iterate variants in a short loop. We pull the clearance between the winding and the gap from 1.5 to 3 mm — this removes the local peak at the gap. If a P-S-P-S-P-S layering was planned, we change it to P-S-P: the ZVS problem disappears, C_ps drops from 180 pF to 50. We change the primary from 140×0.15 to 315×0.1 — the same copper cross-section but a lower F_R at the harmonics. We change the secondary from litz wire to 0.15 mm × 5-layer copper foil; it is simpler geometrically, has a lower F_R, half the DC resistance R_DC, and litz wire for a 21 A secondary gives no such advantage anyway.

After the rework: total copper loss P_{Cu_total} = 7.8 W against 11 in the first version, ZVS works down to 11 % load, the maximum temperature in the window is T_max = 118 °C at an ambient temperature T_amb = 85 °C. Efficiency 96.7 % — within the brief. The price of this is three or four days of engineering work, a PLECS license (or free LTspice), FEMM (free), and the SFD Python script — half a day. At a volume of 5000 chargers a year it pays off in a month and a half.

What the industry works with

The minimum sufficient stack for serious LLC work up to 5 kW is LTspice, FEMM and Python. Fully free, covers 90 % of the work. For production with optimization you add PLECS, GeckoMAGNETICS and ANSYS Maxwell; it pays off at volumes from 5000 units/year.

What we do for LLC projects

For LLC we run four standard series by strand diameter: 0.071 mm (40 constructions), 0.1 mm, 0.15 mm and 0.2 mm — each from 0.5 to 20 mm² of copper. A first-pass series selection by operating frequency goes like this: 0.071 mm — for converters above 300 kHz (gallium-nitride LLCs, induction heaters); 0.1 mm — the main series for LLCs at 150-300 kHz; 0.15 mm — for 50-150 kHz and power-factor-corrector (PFC) inductors; 0.2 mm — for 25-80 kHz and traction converters. The stranding pitch is 10-40 mm, multi-bundle transposition (type 1, 2 or 3 depending on frequency), and the resistance-factor tolerance F_R is ±5 %. A rectangular profile is available on top of any series when you need to add 20 % of window fill (covered in the article on rectangular litz wire). Composite litz foil goes to high-current secondaries, and transposed 3×5 and 5×5×5 constructions to frequencies above 300 kHz. Before ordering we ask for the operating-frequency range (not just the rated point), the rectifier topology, the core geometry and the planned winding-interleaving configuration. With this data we run the FEMM model before delivery and tell you whether the wire fits. This is part of the work on an order from 10 kg, not a separate service.

A ready-made "litz for LLC 200 kHz" catalog does not exist and cannot exist. The same wire by construction (for example, 50×AWG 36) gives F_R = 1.4 in one LLC and 2.8 in another with the same nominal operating point. The difference comes from the window geometry, the gap and the winding configuration. It makes sense to select the wire for a specific core and a specific operating mode, not "in general".