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

Question

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.

Answer 1

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.