Polder tensor

The Polder tensor is a tensor introduced by Dirk Polder[1] for the description of magnetic permeability of ferrites.[2] The tensor notation needs to be used because ferrimagnetic materials become anisotropic in the presence of magnetizing field.

The tensor is described mathematically as:[3]

${\displaystyle B={\begin{bmatrix}\mu &j\kappa &0\\-j\kappa &\mu &0\\0&0&\mu _{0}\end{bmatrix}}H}$

where:

${\displaystyle \mu =\mu _{0}\left(1+{\frac {\omega _{0}\omega _{m}}{\omega _{0}^{2}-\omega ^{2}}}\right)}$

${\displaystyle \kappa =\mu _{0}{\frac {\omega \omega _{m}}{{\omega _{0}}^{2}-\omega ^{2}}}}$

${\displaystyle \omega _{0}=\gamma \mu _{0}H_{0}\ }$

${\displaystyle \omega _{m}=\gamma \mu _{0}M\ }$

and ${\displaystyle \gamma =17.6\cdot g\ }$ [kHz/(A/m)] is a gyromagnetic ratio and g is a factor between 1.9-2.4 depending on ferrite material. Magnetizing frequency (f) is expressed as ${\displaystyle \omega =2\pi f}$, H₀ is a bias field, M is magnetization and ${\displaystyle \mu _{0}}$ is magnetic permeability of free space.