Polder tensor

Template:Use dmy dates The Polder tensor is a tensor introduced by Dirk Polder for the description of magnetic permeability of ferrites. 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:

$B=\left[\begin{array}{ccc}\mu & j\kappa & 0\\ -j\kappa & \mu & 0\\ 0& 0& {\mu }_0\end{array}\right]H$

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

$\mu ={\mu }_0(1+\frac{{\omega }_0{\omega }_m}{{\omega }_0^2-{\omega }^2})$

$\kappa ={\mu }_0\frac{\omega {\omega }_m}{{{\omega }_0}^2-{\omega }^2}$

where

${\omega }_0=\gamma {\mu }_0{H}_0$

${\omega }_m=\gamma {\mu }_{0}M$

$\omega =2\pi f$

$\gamma =1.11\times 10^5\cdot g$ (rad / s) / (A / m) is the effective gyromagnetic ratio and $g$, the so-called effective g-factor (physics), is a ferrite material constant typically in the range of 1.5 - 2.6, depending on the particular ferrite material. $f$ is the frequency of the RF/microwave signal propagating through the ferrite, $H_0$ is the internal magnetic bias field, $M$ is the magnetization of the ferrite material and ${\mu }_0$ is the magnetic permeability of free space.

To simplify computations, the radian frequencies of ${\omega }_0,{\omega }_m,$ and $\omega$ can be replaced with frequencies (Hz) in the equations for $\mu$ and $\kappa$ because the $2\pi$ factor cancels. In this case, $\gamma =1.76\times 10^4\cdot g$ Hz / (A / m) $=1.40\cdot g$ MHz / Oe. If CGS units are used, computations can be further simplified because the ${\mu }_0$ factor can be dropped.