# Eddy current losses in electric steel by harmonics of a magnetic field

I am working on an model of a permanent magnet synchronous machine. Right now I am stuck with calculating the eddy current losses caused by the harmonics of the stator magnetic field in the electrical steel of the rotor. Or to put it differently. How do I calculate the eddy current in electric steel at high frequencies and low flux density?

I would like to use something very simple like $$P_{ec} = \sum\limits_\nu \sigma_{ec} (f(\nu-1))^2 B^2_\nu m_\nu$$ were $P_{ec}$ are the eddy current losses in watts. $\sigma_{ec}$ are the specific eddy current losses for the material, but I don't know if they are any good for frequencies above 2000Hz and I would like to calculate the losses up to 100kHz if that is even possible. $f$ is the fundamental frequency an $\nu$ is the ordinal of the harmonics. $B_\nu$ is the respective flux density and $m_\nu$ is the mass.

Now there are two big questions. I know the amplitude of the flux density in the air gap (above the surface) but how do I calculate the flux density in the lamination (electric steel: M330-35A). Apparently I have to consider the skin depth, but I have no values for the permeability at such high frequencies and comparably low flux densities. And also, how do I calculate the mass? $$m_\nu = A \delta \rho$$ ($A$ - surface of the rotor , $\delta$ - skin depth, $\rho$ - density of the electric steel) If I take the same flux density as in the air gap and calculate the masses like above, I obtain losses that are so low, that I am pretty sure they can't be right.

Does anyone have an idea how to solve this problem by adjusting the described or with another approach. I don't need a 100% accurate result. If I am 50% off that is still ok. Any references to text books or papers are also very much appreciated.

I'm no expert, but ...

MIT's "Magnetic Circuits and Transformers" discusses eddy current losses in chapter V.2. An approximate formula for a magnetic sine wave at frequency $f$ and peak amplitude $B_{max}$ (in Tesla, mks units throughout) is:

$$P_e = k_e f^2 t^2 B_{max}^2 V$$

where $k_e = \pi^2/(6 \rho)$ theoretically (but in practice is often determined from measurements for greater accuracy), $t$ is the thickness of the lamination, $\rho$ is the material's resistivity and $V$ its volume.

Skin effect tends to crowd the magnetic field towards the surfaces of the lamination; keeping the laminations thin minimizes this effect (but of course at high enough frequencies it will become significant).

I believe you can indeed sum the harmonics as you intend.

The normal magnetic field is continuous from the gap into the steel.

Another reference is Watson's "Applications of Magnetism"; he mentions that the approximate formula above generally under-predicts the actual losses.