eppy - Eddy Current Simulations for Induction Welding

This module provides a basic simulation framework to calculate eddy currents in flat, non-magnetic and isotropic, plates. The framework consists of two modules:

coil_geom to generate various coil geometries.
eppy to calculate and visualize magnetic fields and eddy currents.

The eppy module itself consists of:

Functions to calculate and visualize the magnetic field generated by a coil. The field strength is calculated using Biot-Savart.
Functions to calculate and visualize the eddy currents in a flat non-magnetic plate. The eddy current distribution is calculated using the approach proposed by Nagel.
A text parser that allows running simulations via an input file.

Example

Before continuing, allow me to first to whet your appetite. Consider a non-magnetic square plate with a thickness of 1 mm, an edge length of 250 mm, and an isotropic conductivity of 25 kS/m. Eddy currents are generated in the plate by means of a square coil oriented parallel to the plate at a distance of 10 mm. The coil, which has an edge length of 50 mm and corner radii of 5 mm, is supplied with an alternating current at a frequency of 250 kHz. The animation below shows the problem outline and coil current on the left, the z-component of the magnetic field at the location of the plate in the center, and the eddy current density in the plate on the right.

The calculation takes approximately 5 seconds of which most time is spend generating the system matrix. The actual solving takes a fraction of a second. Pretty cool, no? The script used to generate the animation can be found in the folder with examples.

Installation and usage

You can simply clone the repository to your folder of choice using git:

git clone https://github.com/wjbg/eppy.git

Use of this module via an interactive computing environment (e.g. Jupyter) is recommended. All functions are reasonably well-documented and the annotated examples should be sufficient to get you started. Alternatively the module can be run from the command prompt to read an input file with the simulation details. For example, in MS Windows:

python eppy.py examples\meander_coil.txt

Background

This section provides (very) brief information on the generic approach used to calculate the magnetic field generated by the coil and the induced eddy currents in the laminate. Regarding the latter, the reader is kindly referred to the excellent paper by Nagel in which the mathematical approach clearly outlined.

Magnetic field

The amplitude of the magnetic field generated by the coil is determined using the Biot-Savart law.

With reference to the image above, the Biot-Savart law relates the magnetic field <img src="https://latex.codecogs.com/svg.latex?\mathbf{B}"/> at position <img src="https://latex.codecogs.com/svg.latex?\mathbf{r}"/> to the steady currents in for example a wire or a coil. In differential form the Biot-Savart law yields:

with <img src="https://latex.codecogs.com/svg.latex?\textrm{d}\mathbf{\ell}"/> an infinitesimal length of the conductor carrying a current <img src="https://latex.codecogs.com/svg.latex?I"/>, while <img src="https://latex.codecogs.com/svg.latex?\mu_0"/> is the magnetic constant. The displacement vector <img src="https://latex.codecogs.com/svg.latex?\mathbf{r}"/> represents the direction and distance from the conductor element to the point where the field is evaluated. The magnetic field as the result of a current through a wire can be obtained by integrating the equation above along the path of the wire:

Eddy currents

The eddy currents in a plate are determined using the method described by Nagel. The plate is assumed to be very thin, which means that the eddy currents will be constant over the plate thickness and have no out-of-plane component. In addition, the plate is non-magnetic and has an isotropic and uniform conductivity. The imposed magnetic field is assumed to be sinusoidal, which allows the use of a phasor description where all time derivatives equal <img src="https://latex.codecogs.com/svg.latex?i\omega"/>, with <img src="https://latex.codecogs.com/svg.latex?\omega"/> the angular excitation frequency.

Considering the Maxwell-Faraday equation in differential form:

where <img src="https://latex.codecogs.com/svg.latex?\mathbf{E}"/> is the electric field intensity and <img src="https://latex.codecogs.com/svg.latex?\mathbf{B}"/> is the total magnetic field intensity. Using Ohm's law:

this yields:

Nagel now introduces another vector field <img src="https://latex.codecogs.com/svg.latex?\mathbf{T}"/>, called the electric vector potential, such that:

Rearranging then gives:

which can be solved numerically for <img src="https://latex.codecogs.com/svg.latex?\mathbf{T}"/> using for example (as is done here) the finite difference method. The in-plane eddy currents can then be calculated as:

Citation

In case you make use of this work for scientific publications of any kind, please make sure to cite the author of the underlying mathematical framework used for determining the eddy currents:

@article{Nagel2019,
  author =       {Nagel, James R.},
  journal =      {IEEE Transactions on Magnetics},
  title =        {Finite-Difference Simulation of Eddy Currents in
                  Nonmagnetic Sheets via Electric Vector Potential},
  year =         {2019},
  volume =       {55},
  number =       {12},
  pages =        {1-8},
  doi =          {10.1109/TMAG.2019.2940204}}

In addition, the use of (parts of) this code can be cited as:

@misc{eppy2021,
  author =       {Grouve, Wouter J.B.},
  title =        {Eppy - Eddy Current Simulations for Induction Welding},
  year =         {2021},
  publisher =    {GitHub},
  journal =      {GitHub repository},
  howpublished = {\url{https://github.com/wjbg/eppy}}}

License

Free as defined in the MIT license.

Eppy

Install / Use

README