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# [femm] Re: Problems with axisymmetric problems

- To: femm@egroups.com
- Subject: [femm] Re: Problems with axisymmetric problems
- From: Dcm3c@aol.com
- Date: Tue, 5 Oct 1999 01:02:53 EDT
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    > At the Vienna University of Technology we are
    > making experiments with magnetic pulse
    > fields. Essentially, the pulse has the shape
    > of a damped cosine and we wanted to compare
    > the experimental results with time harmonic
    > solutions of Maxwell's equations as a first
    > approximation. I have used FEMM to calculate
    > the eddy currents in copper and iron samples.
    > For cylindrical samples the results are
    > comparable, but for spherical samples we have
    > found rather large differences.
    >
    > After some searching in the library I found
    > an analytical solution for the eddy current
    > density in a sphere in a homogeneous
    > sinusoidally varying magnetic field. The
    > results for "large" spheres (around 5 cm) are
    > in excellent agreement. However, small
    > spheres of a few millimeters in diameter,
    > which have been used in our experiments, give
    > unreliable results.

    [...]

    > I welcome any comments!

    Thanks for the note--your writeup is pretty interesting. However, I think
    that femm is actually doing the correct thing in the case of the small
    spheres; something seems to be wrong with the analytical solution presented
    for this case.

    To see the problem, the easiest case to consider is the one corresponding to
    Figure 31, located at

    http://atp6000.tuwien.ac.at/MAGNET/WS/projects/proj3/node10.html

    In this case, a copper sphere with a radius of 3.65 mm and a conductivity of
    sigma=56.82 MS/m is exposed to a source field of a Bsrc=1 Tesla amplitude
    varying at 109.89 Hz (omega=690.46 rad/sec). This apparently corresponds to
    the miniturnsphere.fem example problem.

    This is a good case to consider because the radius and frequency are small
    enough that the reaction field from the eddy currents can be neglected (The
    skin depth at this frequency and conductivity is 6.4 mm. Since the skin
    depth is substantially greater than the radius of the sphere, neglecting the
    reaction currents for the purpose of estimating the induced current density
    is reasonable). When you can ignore the reaction currents, you can
    substitute directly into Faraday's law to get an expression for the eddy
    current density:

    J=-j*omega*sigma*r*Bsrc/2

    (see http://members.aol.com/gmagnetics/scholz.pdf for details of how I got
    this...)

    This formula predicts an eddy current density of 71.6 MA/m^2 at the farthest
    radius of the sphere. Looking at the finite element solution, the amplitude
    of the induced current density at the point (r=3.649,z=0) is 71.63 MA/m^2,
    showing a good agreement.

    Now, evaluating the sphere.nb Mathematica notebook under the above conditions
    yields a current 106.8 MA/m^2, which is substantially larger than one might
    expect.

    So, what is the difference? I loaded sphere.nb into Mathematica and took the
    power series about omega=0 using the Series[] function and subsitituted in
    mu0 for mu, since we are considering the copper sphere. The result is:

    J=-j*(3/4)*omega*sigma*r*Brc

    Now, this doesn't match the low-frequency limiting case that that one can
    obtain from Faraday's law. There is an extra factor of 3/2 in there for this
    limiting case.

    As a sanity check, I also ran this problem on the student version 3.4a of
    Quickfield. The resulting amplitude of the flux density for the same point
    was 71.5 MA/m^2 from this solution, which closely matches both the femm and
    simple Faraday solutions. (Check out
    http://members.aol.com/gmagnetics/msphere.zip for the qfield version of this
    problem. If you check out this model, make sure that you set the results to
    complex or peak to get the `right' values for comparison to femm--Quickfield
    defaults to RMS values, whereas femm only displays the currents as complex
    amplitudes)

    > In addition, the new release 2.1a of femmview
    > seems to display wrong eddy current densities.
    > For example, the eddy current density does not
    > vanish at r=0. However, if the results of fkern
    > 2.1a are displayed with femmview 2.1,
    > everything seems to be alright. But I have not
    > tested the new version more thoroughly, yet.

    Well, it looks like I really "fixed" this one good.... Anyhow, you are
    correct--I did screw up the eddy current densities in release 2.1a. Thanks
    for pointing this out to me. I have updated the version of the program
    available on the femm homepage. (http://members.aol.com/gmagnetics/setup.exe)
    Even though it still says 2.1a, it now has the fixed version of femmview in
    it. Rather than download the entire distribution again, you can download the
    file:

    http://members.aol.com/gmagnetics/femmview.zip

    This has the fixed version of femmview in it. Just replace the femmview.exe
    in you \program files\femm\bin directory with the new version in this zip
    file.

    Dave.
    --
    http://members.aol.com/dcm3c

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