HOMOGENIZATION. Hands-on modeling of litz in 3D FEM #9

Litz wire can be considered a composite material consisting of 2 parts - conductor and dielectric.

Think of an analogy - a wall of bricks with glue between them. When analyzing the wall's overall strength and stress distribution, would the engineer include every part and detail in the calculation model?

Likely not.

Instead, it will represent a wall with simple geometry while defining "effective" material properties that capture the combined behavior of the bricks and glue. The magic lies in how we estimate those anisotropic properties.

Periodicity reduces complexity

All geometrical or time-dependent periodic characteristics can be represented in "periodic space." For example, instead of solving a transient sine wave with a dozen timesteps, we can define it as a harmonic problem describing frequency and amplitude thanks to the discrete Fourier transform.

In the context of litz, the Dowell equation is based on the assumption of periodicity in one dimension.

Can we use Dowell for litz?

Yes, because it has 1D periodicity. However, we must remember that it will work only for magnetic fluxes parallel to winding layers. This is a fundamental assumption. If this is not the case, we should prepare for a significant error.

Homogenization of 2D periodic structures

If we neglect twisting and uneven spaces between strands, the litz/stranded has a periodic structure in 2 dimensions.

The first fundamental condition to remember when using the model for modeling litz/stranded is that:

Wire diameter << Twist length

This means the ratio between the strand length Ls and wire length Lw is close to 1. Based on "Stranded Wire With Uninsulated Strands as a Low-Cost Alternative to Litz Wire" by Xu Tang et al., the Twisted Length Factor can be defined as

where n is the number of strands, ds is the strand diameter, Ka is the packing factor, and p is the pitch (distance for one full twist). We need to use this scaling factor to account for larger DC resistance. The closer the value to 1, the better. For litz, usually, this factor is around 1.01, meaning that the factor accounts for roughly 1% of error.

Another condition to ensure all strands have the same impedance is

Twist length << Wire length

If this is not true, strands of the same litz wire will have different currents, losses, etc.

Packing

Another factor to consider is the structure of the strands' packing. Square and rectangular shapes are mostly used for stranded wires, while hexagonal shapes better represent the random structure of litz.

Unit cell

Strand geometrical model that represents periodic structure is called a cell. For hexagonal packing, the smallest cell consists of 7 strands.

The model implemented in ElmerFEM uses 6 homogenization parameters. Four are related to the cell's response to the external magnetic field (complex values for 2D), which gives us magnetic reluctance (or permeability, if you like). Another two define impedance (complex value).

To learn more about the implementation of the model, refer to the ElmerFEM webinar.

To calculate homogenization parameters, these 6 parameters are computed for each frequency. In the case of 100kHz, 0.32mm strand thickness, and 0.7168 fill factor, the magnetic flux response looks as follows:

The impedance is estimated by running a current through the central strand

Let's solve a case

The article we chose for validation this time is "Analysis of Litz Wire Losses Using Homogenization-Based FEM" by Y. Otomo et al., where authors use slightly different homogenization techniques based on pre-calculating complex permeability using Bessel functions, etc. The implementation in ElmerFEM is more generic since parameters are calculated numerically for arbitrary cell structures. Still, in the case of evenly distributed round litz strands, we expect both methods to provide similar results.

This time, we will focus on two of the most challenging cases from the article.

One layer of litz winding

PQ65 with a 4mm gap. Winding consists of litz 7 x 0.32 mm. One layer, two wires wound in parallel, 10 turns. Winding is excited with a 1V voltage for a frequency from DC to 100kHz (in TRAFOLO up to 120kHz).

We build two windings, each consisting of 10 turns, and then connect them in parallel using circuits.

The mesh is rather coarse. A mesh sensitivity study has shown that resistance errors are below 5% on average.

The homogenization model setup in TRAFOLO requires the user to specify the packing type, strand thickness, and fill factor. These two parameters calculate the distance between strands.

Simulation for 13 frequency points on 10 CPU cores lasts less than 5 minutes. As a result, we get inductance values, winding resistance, and 3D field distributions like winding loss distribution in windings. With it, we could more accurately simulate losses near gaps and predict hotspots.

The total winding resistance

Looks too good to be true?

TRY IT YOURSELF - This is a solved model for use in the TRAFOLO software. You can get a trial at no cost here.

Realistic case

PQ65 with a 4mm gap. Winding consists of litz 7 x 0.32 mm. 14 parallel wires form 2 layers, each making 10 turns. Winding is excited with a 1V voltage for a frequency from DC to 100kHz (in TRAFOLO up to 120kHz).

To be clear, we are not 100% sure this is the exact winding pattern used in the measured prototype, as the wire length would differ.

Simulation for 13 frequency points on 10 CPU cores lasts less than 45 minutes. As expected from a transition from DC to AC, wind losses in parallelly connected wires move from the inner part of the winding to the region near the gap.

Calculated DC resistance [2.9mΩ] and AC resistance at 100kHz [68.3mΩ], which is a 32 times difference. At 100kHz, the difference between TRAFOLO and measurement is about 39%, which is relatively high.

It could be a different winding pattern or a packing factor, but we are not sure yet. If you have ideas for the discrepancy, we would love to hear them.

THIS IS THE CASE - a solved model for use in the TRAFOLO software. You can get a trial at no cost here.

The final remark

This is undoubtedly a great feature. Its simplicity of use and accuracy are balanced, and with further improvements, it might become a no-brainer.

However, it is not all that rosy, and there are also some challenges. The main ones are

• The need for a transient homogenization model. For example, you cannot solve winding losses in the flyback transformer accurately with this harmonic model. Support for transient waveforms will be available next year.
• Currently, the model does not consider twisting. The first-order correction is easy to implement. A twisted Length Factor will be added in the following releases.
• You should expect slower convergence or even divergence in cases involving many litz wires connected in parallel (it is not a problem when connected in series). We think the reason is that the current depends on the impedance of each litz wire, while the latter one is computed based on the external magnetic field, which depends on the current distribution in the winding, and so on. The problem becomes strongly nonlinear.
• The model does not consider the uneven distribution of strands. Might add another 5%. We would love to hear your suggestions.