## Complex permeabili

A useful tool for dealing with high frequency magnetic effects is the complex permeability. While at low frequencies in a linear material the magnetic field and the auxiliary magnetic field are simply proportional to each other through some scalar permeability, at high frequencies these quantities will react to each other with some lag time. These fields can be written as phasors, such that $H=H_0 e^{j omega t} qquad B=B_0 e^{jleft(omega t - delta ight)}$

where $delta$ is the phase delay of $B$ from $H$. Understanding permeability as the ratio of the magnetic field to the auxiliary magnetic field, the ratio of the phasors can be written and simplified as $mu = frac{B}{H} = frac{ B_0 e^{jleft(omega t - delta ight) }}{H_0 e^{j omega t}} = frac{B_0}{H_0}e^{-jdelta}$,

so that the permeability becomes a complex number. By Euler's formula, the complex permeability can be translated from polar to rectangular form, $mu = frac{B_0}{H_0}cos delta - j frac{B_0}{H_0}sindelta = mu^prime - j mu ^{primeprime}$.

The ratio of the imaginary to the real part of the complex permeability is called the loss tangent, $andelta = frac{mu^{primeprime}}{mu^prime}$,

which provides a measure of how much power is lost in a material versus how much is stored.

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