Polder tensor

The Polder tensor is a tensor introduced by Dirk Polder in 1949¹ for the description of magnetic permeability of ferrites.² The tensor notation needs to be used because ferrimagnetic material becomes anisotropic in the presence of a magnetizing field.

The tensor is described mathematically as:³

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

Neglecting the effects of damping, the components of the tensor are given by

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

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

where

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

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

${\displaystyle \omega =2\pi f}$

${\displaystyle \gamma =1.11\times 10^{5}\cdot g\,\,}$ (rad /s) /(A/m) is the effective gyromagnetic ratio and ${\displaystyle g}$, the so-called effective g-factor, is a ferrite material constant typically in the range of 1.5 - 2.6, depending on the particular ferrite material. ${\displaystyle f}$ is the frequency of the RF/microwave signal propagating through the ferrite, ${\displaystyle H_{0}}$ is the internal magnetic bias field, ${\displaystyle M}$ is the magnetization of the ferrite material and ${\displaystyle \mu _{0}}$ is the magnetic permeability of free space.

To simplify computations, the radian frequencies of ${\displaystyle \omega _{0},\,\omega _{m},\,}$ and ${\displaystyle \omega }$ can be replaced with frequencies (Hz) in the equations for ${\displaystyle \mu }$ and ${\displaystyle \kappa }$ because the ${\displaystyle 2\pi }$ factor cancels. In this case, ${\displaystyle \gamma =1.76\times 10^{4}\cdot g\,\,}$ Hz/ (A/m) ${\displaystyle =1.40\cdot g\,\,}$ MHz/Oe. If CGS units are used, computations can be further simplified because the ${\displaystyle \mu _{0}}$ factor can be dropped.