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Modelling Litz Wire using Homogenization

A PQ65 reactor wound with two parallel litz wires, modelled with homogenization and checked against measured AC resistance.

  • Component: PQ65/60 core, N95 ferrite, 4 mm centre gap; 2 parallel litz wires of 10 turns, 7 strands of 0.32 mm each.
  • Simulation: Inductance, frequency sweep 0 to 120 kHz at 1 A RMS; half model (Symmetric).
  • Winding model: Homogenization.
  • You get: Rdc, Rac, L and Q per frequency, coil and core loss.
  • Files: none.
  • In TRAFOLO: Welcome screen › Examples › Homogenization is this model with its results.
  • Run time: about 9 min for 13 frequencies on 8 CPU cores.

Model description

The reactor of Y. Otomo et al., "Analysis of Litz Wire Losses Using Homogenization-Based FEM", whose AC resistance was measured from DC to 100 kHz.

Core Core material Air gap Wires in parallel Turns per wire Strands per wire Strand diameter
PQ65/60 N95 (TDK) 4 mm, centre limb 2, not twisted together 10, one layer 7 0.32 mm
Half-sectional view of the reactor and 2D schematic of the two parallel wires
(a) Half section of the reactor, first two turns of the parallel wires highlighted. (b) Two parallel wires of 10 turns.

Modelling instructions

Setup

  1. Under DOMAIN TYPE, select Symmetric. TRAFOLO models the half above z = 0 and scales the results to the whole reactor.
  2. Under ELECTROMAGNETICS, select Inductance.
  3. Under HEAT TRANSFER, keep Off.
Setup tab with Symmetric, Inductance and heat transfer Off marked
Symmetric (1), Inductance (2), heat transfer Off (3).

Materials

  1. Under Global Materials, select Copper and click Copy.
  2. Copy Epoxy and TDK N95 100C the same way. All three appear under Project Materials.
  3. Under Default Materials, set the material of each component:
Core Coil Other Bobbin Gaps
TDK N95 100C Copper Not set Not set Epoxy
Materials tab with Copy, the three project materials and the default material per component
Copy (4), Project Materials (5), Default Materials (6).

Core

  1. On Core, click New Group. Group1core appears.
  2. Click Configure next to Group1core.
  3. Set Source to Template, Scale to mm and Gap Type to Physical.

  4. In Geometry Builder, set Type to PQ, enter the dimensions in mm and keep Mirror ticked:

    A B C D E F G
    65 30 45 21 55 26 40.8

  5. Click New Gap, enter gMin −2 and gMax 2 (a 4 mm gap), and set Limb to Central limb.

  6. Click Apply.
Group1core settings with Source, Scale, Gap Type Physical and the PQ dimensions
Group1core: Source, Scale and Gap Type (7), PQ dimensions (8), Apply (10).
PQ core in an angled 3D view with the dimension sketch and the gap table
The PQ core with the dimension sketch, and the 4 mm centre gap (9).

Coil

  1. On Coil, click New Group. Group1coil appears.

  2. Click Configure next to Group1coil.

    Coil tab with New Group, Configure and the group menu for Group1coil and Group2coil
    Coil tab: New Group (11), Configure (12), ⋮ › Copy (20), Configure for the copy (21).

  3. Set Source to Template, Scale to mm, Wire Type to Stranded & Litz and Turns per solid to 1.

  4. In Geometry Builder, set Type to Massive, enter the dimensions in mm and click Apply:

    A B C D R H N r
    0 0 1 1 15.5 24.75 10 0.5

  5. Under Translation and Rotation, set y to 0.625 mm and click Apply.

    Group1coil settings with Stranded and Litz wire, the Massive Geometry Builder, the y translation and Configure WireOne 10-turn wire in an angled 3D view with the dimension sketch

  6. Click Configure Wire. In the Group1coil_2D tab, select Homogenization and set Packing to Hexagonal, Strand Thickness to 0.32 mm and Max Frequency to ≤ 100kHz.

  7. Next to Fill Factor, click Calculate, set Strand Type to Round, Strand thickness to 0.32 and N strands to 7, and click Accept. Fill Factor becomes 0.7168. Then click Generate Mesh: the 2D model of the strand bundle appears on the right. Without it the run stops.

    Group1coil_2D settings with Homogenization, Hexagonal packing, 0.32 mm strands, Max Frequency 100 kHz, Fill Factor and Generate Mesh2D mesh of the hexagonal strand model

  8. Back on Coil, click ⋮ next to Group1coil and choose Copy. Click Configure next to the new Group2coil, set y to −0.625 mm and click Apply. In its Configure Wire tab, click Generate Mesh.

    Two parallel 10-turn wires in an angled 3D view
    Both 10-turn wires in the Coil tab 3D view.

Max Frequency sizes the mesh of the 2D strand model for the skin depth at that frequency; the model is still solved at every frequency of the sweep. The example keeps ≤ 100kHz for its 110 and 120 kHz points, where the skin depth (0.19 mm at 120 kHz) is only a little finer than at 100 kHz: Rac Loss still rises with the square of the frequency, 17.3 mΩ + 41.0 mΩ × (120/100)² = 76 mΩ against the simulated 75.4 mΩ. For a sweep well beyond 100 kHz, choose ≤ 1MHz.

Assembly

  1. On Assembly, click Assemble Geometry.
Assembly table with air, both wires, core and gap and their materials
Assembly: the bodies and their materials.
Assembled half model in an angled 3D view
The assembled half model: core, gap and both wires.

ElMag

  1. Under Primary Winding, set Excitation to Irms [A] and Coil group connection type to Parallel.

    Primary Winding with Irms excitation and Parallel connection
    Excitation Irms [A] and Coil group connection type Parallel (22).

  2. In the coil group table, set both groups:

Coil Group Winding Connection
Group1coil Primary Series
Group2coil Primary Series
Coil group table with Group1coil and Group2coil both Primary and Series
Coil groups (23): both wires Primary, Series.
Front view of the half model with both wires coloured as primary
ElMag 3D view: both wires are coloured as Primary.

Circuits

Nothing to change: TRAFOLO builds the circuit from the ElMag settings.

Primary circuit table with ground, FEM coil and current source
The automatic primary circuit: ground, FEM coil and current source.
Circuit schematic with ground, FEM coil and current source
Schematic of the primary circuit.

Waveform

  1. Under SWEEP BY, select Frequency and set Fixed Irms to 1 A.
  2. Click Generator next to Primary Frequency [Hz] List, set Step to 10000 and Max to 120000, and click Accept.
  3. Add 0.0 at the start of the list for the DC point.
Waveform tab with Frequency sweep, Fixed Irms 1 A, Generator and the frequency list
Frequency (24), Fixed Irms 1 A (25), Generator (26), the 13 frequencies (27).

Heat

  1. Under GLOBAL SETTINGS, set Steady-State Temperature to 20 °C. It sets the copper conductivity.

    Heat tab global settings with Steady-State Temperature 20 degrees
    Steady-State Temperature 20 °C (28).

Mesh

  1. Under GLOBAL SETTINGS, set Mesh Algorithm to Gmsh and Mesh Refinement to Coarse.

    Mesh global settings with Gmsh and Coarse
    Mesh Algorithm Gmsh and Mesh Refinement Coarse (29).

  2. Click Compute Mesh.

    Compute Mesh button
    Compute Mesh (30).

With homogenization the mesh need not resolve strands or the skin depth: TRAFOLO's automatic sizes (0.4 mm in the wires) give about 240,000 elements.

Mesh of the half model with the mesh information
The mesh of the half model: 240,063 elements.

Solve

  1. Set Parallel cores to the number of performance cores of your CPU and keep Solver Preset Fast.

    Solve tab with Parallel cores and Solver Preset Fast
    Parallel cores (31) and Solver Preset Fast (32).

  2. Click Simulation Setup, then Run Simulation.

    Simulation Setup and Run Simulation buttons
    Simulation Setup, then Run Simulation (33).

Results

At 1 A RMS (Results › Summary and Inductance):

Frequency (kHz) Circuit Inductance (µH) Rac Loss (mΩ) Rac Loss / Rdc Q Factor Core Loss (mW)
0 – 17.3 1.00 – 0
10 23.04 17.7 1.02 82 0.16
50 23.04 27.9 1.61 258 2.5
100 23.03 58.3 3.37 245 8.2
120 23.03 75.4 4.36 227 11.2

Inductance

23.0 µH, constant within 0.1 % from 10 to 120 kHz. Circuit Inductance (from the terminal voltage and current) and Energy Inductance (from the stored magnetic energy) agree at every frequency. The 4 mm gap holds most of the magnetic energy, so the ferrite's permeability hardly matters: with µr 3000 instead of 4000, L changes by less than 0.5 %.

Inductance plot with circuit and energy inductance from 0 to 120 kHz
Circuit and Energy Inductance per frequency: 23.0 µH.

AC resistance

The winding resistance rises from 17.3 mΩ at DC to 58.3 mΩ at 100 kHz (3.4 times) and 75.4 mΩ at 120 kHz (4.4 times). Above 40 kHz the 0.32 mm strands are thicker than the skin depth (0.21 mm at 100 kHz), and the fringing field of the 4 mm gap crosses the turns next to it, so these carry the highest loss (field picture below). For the same copper area, finer strands cut the proximity loss roughly with the square of the strand diameter; more distance between the winding and the gap lowers the fringing-field loss.

Rac Loss and Rdc curves per frequency
Rac Loss (purple) against Rdc (orange).
Inductance results table from 0 to 30 kHz
Inductance table: L, XL, Rac Circuit, Rac Loss, Rdc and Q Factor per frequency.
Coil loss density at 120 kHz, highest in the turns next to the gap
Coil loss density at 120 kHz: highest in the turns next to the gap (Coil Losses).

Q factor and core loss

Q Factor = Reactance XL / Rac Circuit. It rises while the resistance stays near Rdc, peaks at 265 around 60 to 70 kHz, and falls once the AC resistance grows faster than the frequency. Here Rac Circuit equals the winding resistance (Rac Loss) within 2 %: the core loss, computed separately from the material's core-loss model, is not part of it. The core loses 11 mW at 120 kHz and 1 A (peak flux density 0.015 T), 13 % of the total; including it, Q at 120 kHz would be 200 instead of 227.

Results Summary with coil, core and total loss per frequency
Summary: losses per frequency at 1 A RMS; scroll right for the higher frequencies.
Magnetic flux density in the core at 120 kHz
Magnetic flux density at 120 kHz, 1 A RMS: at most 0.015 T (Magnetic Flux).
Core loss density at 120 kHz
Core loss density at 120 kHz (Core Losses).

The example ships without 3D field data: run it before Open in ParaView (ParaView (post-processing).

ParaView layout with coil loss, flux density and core loss of the reactor at 120 kHz
Coil loss, flux density and core loss at 120 kHz in ParaView.

Validation

TRAFOLO's Rac Loss against the AC resistance measured by Otomo et al. (measured values read from the published plot, ±0.5 mΩ):

Frequency (kHz) Measured (mΩ) TRAFOLO (mΩ) Difference
0 15.5 17.3 +11 %
10 20.2 17.7 −12 %
50 29.4 27.9 −5 %
70 37.8 37.8 0 %
100 52.6 58.3 +11 %

TRAFOLO stays within 12 % of the measurement from DC to 100 kHz.

Winding resistance per frequency, TRAFOLO against the measurement of Otomo et al.
Winding resistance per frequency: TRAFOLO against the measurement of Otomo et al.

Background

Why homogenization

Litz wire limits the skin effect in each strand, but the proximity effect from neighbouring turns and the gap field remains. Resolving it strand by strand needs elements of about a third of the skin depth (0.06 mm at 120 kHz) in every strand. Homogenization replaces each wire with an equivalent material that TRAFOLO computes from a 2D model of the strand bundle, once per frequency, so the 3D mesh treats each wire as a solid. It assumes ideally twisted strands; the current split between the two parallel wires is computed.

Method details: Homogenization of Litz and Stranded Windings.

Further reading