Permanent Magnet Knee Point Flux Density

The knee point flux density is the value of the flux density inside a permanent magnet below which the demagnetization process becomes irreversible.

When the field $\mathbf{H}$ is non-zero, the magnetic flux density $\mathbf{B}$ inside a permanent magnet changes linearly with the field as:

The relation between $\mathbf{B}$ and $\mathbf{H}$ is called the magnetic constitutive relation.

When the magnetic field $\mathbf{H}$ is opposite to the magnetization direction (the direction of remanence), the magnetic flux density magnitude decreases (known as the demagnetization process). When the magnet demagnetization is strong, the linearity of the constitutive relation no longer holds, and the magnet undergoes irreversible changes. A result of this irreversible demagnetization is that, after the removal of the external field, the magnet flux density does not return to its original remanence value. The value of the flux density inside the permanent magnet, below which the demagnetization process becomes irreversible, is known as the knee point flux density.

The knee point flux density ${\mathbf{B}}_k$ of a magnet can depend on the temperature. If either (or both) the remanence ${\mathbf{B}}_r$ or the permeability $\mu$ of the magnet depend on the temperature $T$ , then the knee point flux density can be obtained from the standard knee point flux density ${\mathbf{B}}_{sk}$ , standard temperature $T_s$ , remanence ${\mathbf{B}}_{sr}$ , and permeability ${\mu }_s$ .

where ${\mathbf{S}}_{\alpha }=\frac{{\mathbf{B}}_r}{{\mathbf{B}}_{sr}}$ is the intrinsic flux density scale, ${\mathbf{H}}_{sk}=\frac{{\mathbf{B}}_{sk}-{\mathbf{B}}_{sr}}{{\mu }_s}$ the standard temperature knee point field strength, ${\mathbf{S}}_{\beta /\alpha }=\frac{{\mathbf{S}}_{\beta }}{{\mathbf{S}}_{\alpha }}$ strength/intrinsic flux scale ratio, ${\mathbf{S}}_{\beta }=\frac{{\mathbf{H}}_k}{{\mathbf{H}}_{sk}}$ field strength ratio, ${\mathbf{H}}_k=\frac{{\mathbf{B}}_k-{\mathbf{B}}_r}{\mu }$ the knee point field strength. In practice, due to the model and measurement errors, the quantity ${\mathbf{S}}_{\beta /\alpha }$ is capped from above and below as $\frac{1}{{\mathbf{S}}_{\max }}\le \frac{{\mu }_s-{\mu }_0}{\mu -{\mu }_0}\le {\mathbf{S}}_{\max }$ with some extreme value of ${\mathbf{S}}_{\max }=5$ .

This value is finite only if $\mu >{\mu }_0$ .

When both remanence and permeability are independent of the temperature, then the knee point flux density is equal to the standard knee point flux density ${\mathbf{B}}_k{|}_{\{B_r,\mu \}}={\mathbf{B}}_{sk}$ . If only the permeability is temperature-dependent, then ${\mathbf{S}}_{\alpha }{|}_{B_r}=1$ and ${\mathbf{B}}_k{|}_{B_r}={\mathbf{B}}_{sk}+({\mathbf{S}}_{\beta /\alpha }-1){\mathbf{H}}_{sk}{\mu }_0$ . If only the remanence is temperature-dependent, then ${\mathbf{S}}_{\beta /\alpha }{|}_{\mu }=1$ and ${\mathbf{B}}_k{|}_{\mu }={\mathbf{S}}_{\alpha }{\mathbf{B}}_{sk}$ .