Magnetic Permeability

The permeability of magnetic materials is usually expressed as:

$$\mu ={\mu }_r{\mu }_0$$

where ${\mu }_r$ is the relative permeability (unitless) and ${\mu }_0$ the permeability of air in $\text{H}/\text{m}$.

The permeability of air (or free space) is:

$${\mu }_{\text{air}}={\mu }_0=4\pi \times 10^{-7}$$

A good magnetic material has a high permeability, and as a result, a little effort made by the excitation (the current in the coil) will result in a large flux in the core. Magnetic flux density ($B$) and magnetic field intensity ($H$) can be related through permeability ($\mu$) to address this fact.

$$B=\mu H(\text{T})$$

This is derived by:

$$\begin{array}{rl}\mathcal{F}=R\varphi & =Hl\\ \frac{1}{\mu A}\varphi & =Hl\\ \frac{1}{\mu }\frac{\varphi }{A}& =Hl\\ B& =\mu H\end{array}$$

The relation $B=\mu H$ implies that in a highly permeable magnetic material, a small $H$ caused by a small coil current $I$ can result in a large $B$.

• High permeability is a desirable property in electric machines and transformers.
• Typical values for ${\mu }_r$ in these applications are in the range 2000-6000.
• Note that since $\mathcal{R}$ is not a constant, $\mu$ is not a constant, either; thus, the $B$-$H$ relationship in magnetic materials is not linear.