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# Nonlinear time harmonic formulation

- To: femm mailing list <femm@yahoogroups.com>
- Subject: Nonlinear time harmonic formulation
- From: David Meeker <dcm3c@earthlink.net>
- Date: Fri, 18 Jan 2002 00:45:13 -0500

All--

A quick note--I've put up a new version of the program with some fairly minor bug-fixes (see message 1031 for a discussion of the fixes) . If you are interested, you can download it off of the femm website ( http://femm.berlios.de/ )

Anyhow, I've been thinking about implementing a nonlinear time harmonic formulation for a while, because people keep running across cases where it would be nice to have some approximation of saturation included in time harmonic problems (i.e. problems that run at a frequency other than 0 Hz). However, I've been doing a bit of head-scratching with w.r.t. a couple of aspects of this formulation, and I was wondering if anyone has thought about or wants to talk about any of these issues.

1) Hysteresis modeling. In machines that are well-designed, hysteresis losses are typically on the same order as eddy current losses. Therefore, to get a reasonable estimate of core losses, hysteresis losses really need to be addressed. In the current version of the program, a simple model of hysteresis losses is implemented in which the hysteresis is considered to be a frequency-independent phase lag between H and B. This is sort of a crude model of a complicated phenomenon, but it is pretty easy to implement and tends to give reasonable results. However, when I move to a nonlinear time-harmonic formulation, it seems like the hysteresis phase lag should no longer be constant. I'd guess that the hysteresis phase lag would converge towards zero as the material operates in an increasingly saturated regime, such that the loss per cycle converges to a constant value as H goes to infinity (because it seems like there ought to be a maximum area that could be enclosed in a major hysteresis loop). However, I don't have a good feel for a relationship between hysteresis lag and the B-H curve for the rest of the curve. Is there an intrinsic relationship that one could assume between the BH curve and a hysteresis lag parameter? I suppose that I could have the program ask for complex values of B in the BH curve definition part, but this seems like it would be a lot more desirable to have the program generate a complex-valued BH curve based on the virgin magnetization curve and a small number of other parameters.

2) Since most electric machines have a laminated structure, it would be useful to address nonlinear lamination behavior as well. The current linear time-harmonic solver uses an analytical solution eddy currents in a thin lamination to derive an apparent permeability of the lamination that accounts for the effects of hysteresis and eddy currents. For the non-linear case, an apparent BH curve that accounts for eddy currents and hysteresis needs to be generated, rather than just a single permeability. Again, the program could simply require the user to enter data points for an apparent BH curve, but that doesn't seem so desireable. It would be good, again, to infer the right behavior bulk material's BH curve (i.e. the solution to (1)) and material conductivity. Now, one could run a bunch of 1-D simulations if a bulk BH curve is known to obtain apparent permeability. However, it would be good to have a method that is more direct than running a large number of finite element runs. It seems possible to get a reasonable kludge by assuming constant permeability across the skin depth based just on the H at the surface of the lamination, but this approach is just an approximation. Any suggestions?

Dave.
--
David Meeker
http://femm.berlios.de/dmeeker

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