## Induced Eddy Currents in Simple Conductive Geometries: Mathematical Formalism Describes the Excitation of Electrical Eddy Currents in a Time-Varying Magnetic Field

## Abstract

In this paper, a complete mathematical formalism is introduced to describe the excitation of electrical eddy currents due to a time-varying magnetic field. The process works by applying a quasistatic approximation to Ampere's law and then segregating the magnetic field into impressed and induced terms. The result is a nonhomogeneous vector Helmholtz equation that can be analytically solved for many practical geometries. Four demonstration cases are then solved under a constant excitation field over all space—an infinite slab in one dimension, a longitudinal cylinder in two dimensions, a transverse cylinder in two dimensions, and a sphere in three dimensions. Numerical simulations are also performed in parallel with analytic computations, all of which verify the accuracy of the derived expressions.

- Authors:

- Univ. of Utah, Salt Lake City, UT (United States). Metallurgical Engineering

- Publication Date:

- Research Org.:
- Univ. of Utah, Salt Lake City, UT (United States)

- Sponsoring Org.:
- USDOE Advanced Research Projects Agency - Energy (ARPA-E)

- OSTI Identifier:
- 1433514

- Grant/Contract Number:
- AR0000411

- Resource Type:
- Accepted Manuscript

- Journal Name:
- IEEE Antennas and Propagation Magazine

- Additional Journal Information:
- Journal Volume: 60; Journal Issue: 1; Journal ID: ISSN 1045-9243

- Publisher:
- IEEE

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATION; 36 MATERIALS SCIENCE; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; eddy currents; quasistatics; current density

### Citation Formats

```
Nagel, James R. Induced Eddy Currents in Simple Conductive Geometries: Mathematical Formalism Describes the Excitation of Electrical Eddy Currents in a Time-Varying Magnetic Field. United States: N. p., 2017.
Web. doi:10.1109/MAP.2017.2774206.
```

```
Nagel, James R. Induced Eddy Currents in Simple Conductive Geometries: Mathematical Formalism Describes the Excitation of Electrical Eddy Currents in a Time-Varying Magnetic Field. United States. doi:10.1109/MAP.2017.2774206.
```

```
Nagel, James R. Fri .
"Induced Eddy Currents in Simple Conductive Geometries: Mathematical Formalism Describes the Excitation of Electrical Eddy Currents in a Time-Varying Magnetic Field". United States. doi:10.1109/MAP.2017.2774206. https://www.osti.gov/servlets/purl/1433514.
```

```
@article{osti_1433514,
```

title = {Induced Eddy Currents in Simple Conductive Geometries: Mathematical Formalism Describes the Excitation of Electrical Eddy Currents in a Time-Varying Magnetic Field},

author = {Nagel, James R.},

abstractNote = {In this paper, a complete mathematical formalism is introduced to describe the excitation of electrical eddy currents due to a time-varying magnetic field. The process works by applying a quasistatic approximation to Ampere's law and then segregating the magnetic field into impressed and induced terms. The result is a nonhomogeneous vector Helmholtz equation that can be analytically solved for many practical geometries. Four demonstration cases are then solved under a constant excitation field over all space—an infinite slab in one dimension, a longitudinal cylinder in two dimensions, a transverse cylinder in two dimensions, and a sphere in three dimensions. Numerical simulations are also performed in parallel with analytic computations, all of which verify the accuracy of the derived expressions.},

doi = {10.1109/MAP.2017.2774206},

journal = {IEEE Antennas and Propagation Magazine},

number = 1,

volume = 60,

place = {United States},

year = {2017},

month = {12}

}

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