Polder tensor

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:


 * $$B = \begin{bmatrix} \mu & j \kappa & 0 \\ -j \kappa & \mu & 0 \\ 0 & 0 & \mu_0 \end{bmatrix} H$$

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


 * $$\mu = \mu_0 \left( 1+ \frac{\omega_0 \omega_m}{\omega_0^2 - \omega^2} \right) $$
 * $$\kappa = \mu_0 \frac{\omega \omega_m}{{\omega_0}^2 - \omega^2}$$

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.