Re: [femm] hist. losses in FEMM

[David Meeker <dmeeker@xxxxxxxx>]

jasiu2003 wrote:
>   Hi,
>
> I will solve the nonlinear problem with hysteressis losses.
> How can I define the PHI_{hmax} parameter for nonlinear materials?
>
> Thanks in advance.
>
> Jan
This is a good question--it's not exactly obvious how this works
internally.  FEMM models hysteresis by defining a complex-valued BH
curve.  The program starts with the "virgin" magnetization curve that
the user enters and generates an "effective" B-H curve by evaluating
the amplitude of the fundamental of B(H sin(t)) for each value of H on
the user-defined B-H curve.  There are alternative ways of getting an
"effective" B-H curve, which yield similar, but not exactly identical,
results, but this way seems to make the most sense to me. The effective
curve typically flattens out at higher flux densities than the DC curve.

Anyhow, the effective B-H curve is then used to evaluate a
non-hysteretic permeability, mu = B(effective)/H.  The maximum value of
mu over the defined range of H, which we could be denoted as mumax, is
the determined.  A complex-valued B-H curve is then constructed, where
the B corresponding to each value of H:

B= B(effective)*exp(-j*Phi_hmax*mu/mumax)

where j=sqrt(-1) and Phi_hmax is the parameter that you were talking
about above.

So, what does this all have to do with losses?  Well, we could define
mu_h, the hysteretic permeability, as:

mu_h=mu*exp(-j*Phi_hmax*mu/mumax)

The hysteresis loss per unit volume per cycle is then directly related
to the imaginary part of mu_h:

loss per cycle = - PI*Im(mu_h)*|H|^2

This is sort of a general _expression_, regardless of how one choses to
structure mu_h.  The way that FEMM models mu_h, the imaginary part is:

Im(mu_h)= - mu*sin(Phi_hmax*mu/mumax)

Since the "hysteresis angle", Phi_hmax, is usually a fairly small angle
("usually less than 20 degrees" per Stoll's "Analysis of Eddy
Currents"), the imaginary part of mu_h is closly approximated by:

Im(mu_h)= - Phi_hmax*(mu^2)/mumax

If we jam this into the loss _expression_, we get:

Power loss = PI*(Phi_hmax*(mu^2)/mumax)*|H|^2 
              = PI*(Phi_hmax/mumax)*|B|^2

What the power loss form assumed by FEMM is actually doing is just
assuming that hysteresis losses vary as |B|^2, and you can use the
above _expression_ to pick a best-fit value of Phi_hmax based on
experimental data for hysteresis power loss versus flux density.  At
any rate, this hysteresis angle=Phi_hmax*mu/mumax is suggested in an
obscure paper by O'Kelly, "Hysteresis and eddy-current losses in steel
plates with nonlinear magnetization characteristics," Proc. IEEE,
119(11):1675-1676, November 1972, in which OK results were reported for
this method.

It is interesting to note, however, that this dependence is different
from Steinmetz's Equation (e.g.
http://scienceworld.wolfram.com/physics/SteinmetzsEquation.html), a
widely used way of guesstimating hysteresis losses, which suggests a
dependence that's more like B^(1.6) at the flux density levels where
you'd actually design an electric machine to operate.  I didn't pick
this form, however, because B^(1.6) would imply a hysteresis lag angle
ill-behaved at low levels of flux density unless the B-H curve has a
very specific form as B goes to zero.

Dave.


-- 
David Meeker
dmeeker@xxxxxxxx
http://femm.berlios.de/dmeeker