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Re: Transient Magnetic Field Analysis

mjcraft78 wrote:

I am attempting to model the magnetic field of an electromagnet with a
step input of current to the coil (this is an axisymmetric problem).
I understand that FEMM can model the effect of eddy currents &
hysteretic losses on a "magnetic circuit" for a given frequency, but
currently does not have the capability of modeling a DC bias. Is
there some way that I could correlate the AC response at several
different frequencies to predict the behavior for a step input?

This is sort of an interesting question. As long as linear materials are an OK assumption, you certainly could infer the transient response from a series of steady-state AC simulations. You'd want to perform a number of simulations at different frequencies, probably chosen at even increments on a log scale. At each frequency, you'd evaluate your desired output--say flux linkage of a coil or something divided by the input current--being careful to record your result as a complex-valued number. Taken as a whole, you could view these data points as points along a transfer function between current and your desired output.

To use this information to infer transient response, you'd first fit a transfer function to your data points. You'd probably want to pick a form like:
G(s) = (n0 + n1*s + n2*s^2 + ...)/(1 + d1*s + d2*s^2 + d3*s^3 + ...)
It is possible to arrange things so the the coefficients can be determined by the least-squares solution to an overdetermined system of linear equations, or you could attack it directly as a nonlinear least-squares fit. You'll probably have to experiment with different orders of transfer function to find a good fit over your frequency range of interest.

(Alternatively, you might use a transfer function with fractional order poles--eddy currents often produce a response that can be compactly represented as a fractional order transfer function like k/(1+tau*s^(1/2)), where k and tau are parameters that you'd choose to fit your data. However, you have to eyeball the data to pick the "right" form of transfer function. One oddball case that I ran across was well approximated by a form like k/(1+tau*s^(1/4). Although fractional order transfer functions are really good at representing the dynamics of systems with eddy currents, they are not as useful as integral order systems, because the Laplace transforms are hard to invert and fractional order simulations are hard to simulate in the time domain.)

Once you have some transfer function G(s) that is a "good enough" fit to your finite element runs, you can use it to compute the step response. The Laplace transform of a unit step is (1/s). To get the step response, you need to compute the inverse Laplace transform of [1/s * G(s)]. Since I'm a Mathematica junkie, I'd use the Mathematica command:
InverseLaplaceTransform[G/s, s, t]
to analytically compute the Laplace transform to get the time response. Alternatively, you could convert G(s) into a discrete time representation (e.g. via Tustin transformation or Euler approximation, see http://lorien.ncl.ac.uk/ming/digicont/digimath/sampled2.htm or http://alpha400.ee.unsw.edu.au/elec2042/e2042num/e2042num.html for some nice explanations) and directly simulate the response to a step input.

It is also not impossible to derive analytical models for these sorts of actuators that include the eddy current effects. A nice paper about modeling eddy currents in magnetic thrust bearings (perhaps similar to your topology) is at http://www.ifr.mavt.ethz.ch/publications/kucera95a.pdf Although Kucera doesn't present it, you can use the continued fraction expansion of tanh (like I describe in http://femm.berlios.de/dmeeker/pdf/chop2.pdf -- another paper about modeling eddy currents in magnetic bearings) to infer a finite order approximation of the transfer function with eddy currents.


Additionally, I've used FEMM to calculate the inductance of the
circuit, but I'm guessing that this does not take into account the
generation of eddy currents due the rate of change of the field?
Please let me know if I'm way off here or what I can do to model this
problem accurately.

If you do an AC simulation and calculate the inductance via (integral of A.J)/i^2, you get a complex-valued inductance that /does/ take into account the effects of eddy currents. The reason that it is complex-valued is that the eddy currents introduce a phase shift between the coil current and the flux that links the coil. You can also use the imaginary part of this inductance to compute the losses due to the eddy currents.

Dave.

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